Nearly mode I fracture toughness and fatigue delamination propagation in a multidirectional laminate fabricated by a wet-layup Текст научной статьи по специальности «Физика»

УДК 539.421.5

Nearly mode I fracture toughness and fatigue delamination propagation in a multidirectional laminate fabricated by a wet-layup

T. Chocron and L. Banks-Sills

The Dreszer Fracture Mechanics Laboratory, School of Mechanical Engineering, Tel Aviv University, Ramat Aviv, 6997801, Israel

Five double cantilever beam specimens were tested quasi-statically to obtain a GIR resistance curve. In addition, nine double cantilever beam specimens were tested in fatigue to obtain a Paris-type relation to describe the delamination propagation rate da/dN where a is delamination length and N is the cycle number. Displacement ratios of Rd = 0.10 and 0.48 were used for five and four specimens, respectively. The specimens were fabricated by means of a wet-layup process from carbon fiber reinforced polymer plies. The interface containing the delamination was between a unidirectional fabric and a woven ply. The fracture toughness and fatigue delamination propagation protocols are outlined. The mechanical and thermal residual stress intensity factors were obtained by means of finite element analyses and the conservative M-integral along the delamination front. They were superposed to determine the total stress intensity factors. It was found that the total mode I stress intensity factor dominates the other two stress intensity factors. Thus, nearly mode I deformation was achieved. Interpolation expressions for the mechanical and thermal residual stress intensity factors were determined using three and two-dimensional fittings, respectively. Results are presented with an expression for GIR determined. Moreover, the fatigue data is described including threshold values and master-curves. These results shed light on the behavior of delamination propagation in multidirectional laminate composites.

Keywords: constant amplitude, fatigue delamination growth rate, fiber reinforced composites, fracture toughness, resistance curve, R-ratio

DOI 10.24411/1683-805X-2018-16014

Вязкость разрушения типа I и усталостное расслоение слоистого пластика с различной ориентацией волокон, полученного методом мокрого формования

T. Chocron, L. Banks-Sills

Тель-Авивский университет, Рамат-Авив, 6997801, Израиль

В ходе квазистатических испытаний пяти двухконсольных образцов получена кривая сопротивления GIR. Также проведены испытания девяти двухконсольных образцов на усталость и получено уравнение типа Пэриса для описания скорости распространения расслоения da/dN, где a — длина расслоения, N — число циклов. При этом использовали коэффициенты смещения Rd = = 0.10 и 0.48 для пяти и четырех образцов соответственно. Образцы изготавливали методом мокрого формования из полимерных слоев, армированных углеродным волокном. На границе раздела между однонаправленной тканью и плетеным полотном задавали расслоение. Представлены данные по распространению усталостного расслоения и вязкости разрушения. С использованием анализа методом конечных элементов и М-интеграла найдены коэффициенты интенсивности механических и термических остаточных напряжений вдоль фронта расслоения. Путем их наложения определены суммарные коэффициенты интенсивности напряжений. Показано, что коэффициент интенсивности напряжений нормального отрыва доминирует над двумя другими коэффициентами интенсивности напряжений, за счет чего деформация протекает нормальным отрывом. С помощью трехмерного и двумерного сглаживания получены интерполированные выражения для коэффициентов интенсивности механических и термических остаточных напряжений соответственно. Результаты представлены с использованием кривой GIR. Описаны усталостные характеристики, включая пороговые значения и обобщенные кривые. Полученные результаты дают представление о поведении расслоения в слоистых композитах с различной ориентацией волокон.

Ключевые слова: постоянная амплитуда, скорость распространения усталостного расслоения, волокнистые композиты, вязкость разрушения, кривая сопротивления

© Chocron T., Banks-Sills L., 2018

1. Introduction

Composite materials, in general, and carbon fiber reinforced polymers (CFRP), in particular, have evolved greatly since the 1960s. Currently, composite materials are used in industries, such as automotive manufacturing, air, marine and space structures, civil construction, medical equipment and more. These significant developments, especially in the aerospace industries, stemmed from the fact that polymer composites have low specific weight, with respect to metals. These composites are characterized by high ratios of toughness to weight and strength to weight, which allow them to tolerate defects. These features enable such structures to have longer life cycles and lower fuel consumption, which reduces economic costs [1].

Composite laminates may develop delaminations between plies, as a result of stress raisers, such as corners, free surfaces, an interface between different materials, and more [2]. Quasi-static and cyclic nearly mode I deformation that may cause structural failure are the main focus of this study. For the first, a GIR resistance curve is determined to characterize the mode I energy release rate GI which leads to catastrophic delamination propagation for a given delamination length a. For the second, a Paris relation between delamination propagation length per cycle da/dN and a function of GI, f (GI), is obtained as

=D[f (cor. (i)

dN

In Eq. (1), D and m are power law fitting constants. A number of investigations [3-10] made use of Eq. (1) with various functions for f (GI).

In Sect. 2, the material and specimen used in this study are presented. In addition, the test protocol for each test type is discussed, including a description of the test setup. Obtaining the integrated averaged mode I energy release rate is discussed in Sect. 3 for determining GIR, GImax and GImin. For fatigue, the method of calculating a and da/dN for each cycle is described in Sect. 3, as well. In Sect. 4, the results from fracture toughness and fatigue delamina-tion propagation tests are presented. In addition, a technique for determining GI threshold values and achieving master curves will be described. Finally, a discussion and

conclusions of this investigation is presented in Sect. 5. It may be noted that there is a separate section entitled: Supplementary material, where all sections, figures and tables are enumerated with a prefix S.

2. Material and methods

2.1. Specimen and material

The specimen used in this study is the double cantilever beam (DCB) as presented in Fig. 1. Load blocks were used to apply opening displacements. The specimen length, width and thickness are l, b and h, respectively. The initial delamination length a0 was measured from the center of a load block hole to the delamination front. All specimens were fabricated by a water jet process from the same plate. The width and thickness ranges, for all specimens, were between 19.74 and 20.08 mm and between 4.75 and 5.01 mm, respectively. The length of all specimens was about 200 mm and a0 was about 54 mm. The measurements were carried out guided by the standards in [11, 12] with some differences as described in Sect. S1 of the Supplementary material. The specimen measurements for the fracture toughness test specimens are presented in Table S1; those for fatigue delamination propagation test specimens may be found in Tables S2 to S4. It should be noted that the standards are for unidirectional (UD) material; whereas, the specimens considered here consist of a multidirectional (MD) laminate.

The composite laminate was fabricated from carbon fibers in an epoxy matrix with a small amount of glass fibers in the UD ply. The mechanical properties and coefficients of thermal expansion (CTEs) of those three constituents are presented in Table S5. The laminate consisted of 19 plies of three different types as illustrated in Fig. 2. The first type is a UD-fabric with carbon fibers in the 0°-direction (along the xx - axis) which included a small percentage of glass fibers transverse to the carbon fibers. The two other ply types are plain, balanced weaves with tows in the 0°/ 90° and the +45°/-45°-directions, respectively. The upper ply of the interface is the UD-fabric and the lower ply is the weave with tows in the +45°/-45°-directions as shown in Fig. 2. The laminate was manufactured by means of a wet-layup and cured in an oven at 85°C.

The weight fraction of each component (carbon and glass fibers, epoxy and voids) in each ply were found by dissolving the matrix in an acid bath. For the UD fabric, the carbon was burned off to measure the weight fraction of the glass fibers. The equivalent volume fractions are presented in Table S6. The material properties of the constituents, as well as their volume fractions were used as input to the high-fidelity generalized method of cells (HFGMC) micromechanical model [13] to obtain the effective mechanical and thermal properties of the UD-fabric and the weave. These properties are presented in Tables S7 and S9, respectively. The UD-fabric is modeled to be transversely

x2

x1

+45°/-45

0° UD

+45°/-45°

x2 0° (UD) Xe xi

+45°/-45° (weave)

Fig. 2. Laminate plies and interface (color online)

isotropic [14] with five independent effective mechanical properties and two CTEs. The glass fibers were accounted for by increasing the Young's modulus of the matrix by means of the rule of mixtures. The weave is tetragonal [14] and described by six independent effective mechanical properties and two CTEs.

2.2. Fracture toughness test protocol

The fracture toughness standards [11, 12] make use of a DCB specimen to measure the mode I delamination toughness of UD laminates. These standards were used as guidance in testing the MD laminate composed of a UD-fabric and woven plies. It is emphasized that the delamination is between two plies with fibers in different directions making it an interface crack between two effectively anisotro-pic (homogeneous) materials. Moveover, the delamination plane is not a symmetry plane. Thus, there are three deformation modes: 1, 2 and III. The contribution of the shear modes will be discussed in the sequel.

The test protocol used here followed the two standards and that developed in [10]. Fracture toughness tests were performed on five DCB specimens denoted as FT-wet-1-0/ (j = 4, ..., 8); FT represents fracture toughness; wet indicates that the specimens were fabricated by means of a wet-layup; the first numeral represents the batch number; the second numeral represents the specimen location in the plate, with respect to the other specimens.

The system used in the test is described in Sect. S3 and presented in Fig. S5 in the Supplementary material. As prescribed in the standards, the tests were conducted under displacement control. Automatic test instructions were written using Instron WaveMatrix computer software [15] which interacts with the Instron servohydraulic loading machine (model number 8872; High Wycombe, England) used in the tests. An Instron 250 N Dynacell load cell was used with an accuracy of about ±0.25% of the measured load greater than 2.5 N. In the first test stage, the displacement rate was set to increase at 1 mm/min, until a load drop of at least 2 N occurred or the delamination propagated between

3 and 5 mm, as prescribed in the standards. Then, the specimen was unloaded with a displacement rate of 4 mm/min until the value of the load acting on the specimen was 3 N. A small positive load was chosen so as not to induce compression in the specimen. The standards recommend a displacement rate of between 0.5 and 5 mm/min for loading; for unloading, the displacement rate should be less than 25 mm/min. The rates used in this protocol were in keeping with the standards. In the next step, the displacement rate was again set to 1 mm/min. This displacement rate was used until the delamination propagated by at least 60 mm. Finally, in the last step of the test, unloading was performed to 3 N with a displacement rate of 8 mm/min. Every 0.5 s during a test, an image was taken using a LaVision digital camera (model Imager Pro SX, Gottingen, Germany). From the images and after the test, it was possible to measure the delamination length a as it propagated. Concurrently, values of the load and displacement were also recorded. It may be noted that there was full synchronization between the load, displacement and images. Each test took approximately 60 min. The specimens were stored in an environmental conditioning chamber (M.R.C. BTH80/-20, Holon, Israel) at least one week before a test was performed with a temperature of 23 ± 1°C and a relative humidity (RH) of 50 ± 3%. The ASTM standard requires a temperature of 23 ± 3°C and relative humidity of 50 ± 10%. During each test, the room temperature and RH were measured every five minutes.

2.3. Fatigue test protocol

Tests were performed on five specimens, FTG-wet-1-0/ (j = 9, ..., 13) with a cyclic displacement ratio R = 0.10, and on four, FTG-wet-1-0j (j = 14, ..., 17) with Rd = 0.48; FTG represents fatigue and Rd is given by

Rd = d min/d max- (2)

In Eq. (2), dmin and dmax are the minimum and maximum actuator displacements, respectively, during a constant amplitude fatigue test. In theory, Rd should be equal to the cyclic load ratio Rp which is given by

R Pmm/ ^max' (3)

where Pmn and Pmax are the minimum and maximum loads, respectively, acting during a fatigue cycle. In practice, Rd and Rp are not equal. The only standard dealing with mode I fatigue deformation is Ref. [16]. This standard focuses on determining the number of cycles N leading to delamina-tion propagation initiation as a function of the mode I maximum energy release rate GImax in a cycle for a DCB UD laminate specimen. Currently, there is no standard for dela-mination propagation of MD laminates, and in particular, for woven plies. Since the present study focuses on delami-nation propagation, the test protocol is different from that described in the standard; rather, it follows that presented in [10]. However, this study is consistent with the spirit of the standard in preparation of the specimens, use of suitable equipment, maintenance of environmental conditions, etc.

The system used for the fatigue tests is the same as that used for the fracture toughness tests. Again, displacement control was applied. Every 10 or 15 min during the tests, measurements of temperature and RH were recorded. These tests lasted up to 9 days running continuously.

Correctly determining the actuator position is an important aspect in the test procedure. The position of the actuator, once connected to the specimen, was recorded as d0 as shown in Fig. 3. First, a quasi-static opening displacement at a rate of 1 mm/min was applied in order to propagate the delamination between 3 and 5 mm, as indicated in the standard. The procedure was carried out as

described in Sect. 2.2 for the fracture toughness tests. The

*

maximum actuator displacement dmax was recorded as shown in Fig. 3, a. It is used as the maximum displacement in the fatigue tests. Next, unloading was performed at a rate of 8 mm/min until the load P = 3 N was reached. There are two difficulties associated with the test system. The first problem relates to the uncertainty in a load less than 2.5 N as explained in Sect. 2.2. This is especially problematic for Rd = 0.10. The second problem results from some freedom in the test system which may be a result of the connections between the specimen and the test machine. It may be observed in Fig. 3, b in the load-displacement (P-d) curve

that the minimum displacement does not return to its initial position d0. The unloading curve was extrapolated linearly as shown by the dotted orange line in Fig. 3, b to obtain the expected position d^ for P = 0. The maximum actuator displacement in a fatigue cycle relative to its position at d¡m is defined as dmax and given as

dmax = dmax - d0 . (4)

For a given value of Rd, it is possible to determine the minimum displacement dmin of the actuator in a fatigue cycle relative to its position at d m as

d min = d max Rd. (5)

With the values of dmax and dmin, the cyclic amplitude A is found as

A = (dmax - dmin)/2• (6)

An algorithm was written using the WaveMatrix software to control the test machine. The data entered into the

*

algorithm includes dmax, A, the test frequency f, and the maximum number of cycles N. The values of Rd, N and f for each specimen are presented in Table 1. The data which is inserted into the algorithm is sufficient to define a fatigue test. Also, the measurement frequency At of temperature and RH, during each test, is also given in Table 1. For each cycle, the values of Pmax and Pmin, as well as dmax

Table 1

Displacement ratio, number of cycles, frequency and measurement frequency of temperature and RH in fatigue tests

Specimen Rd N f Hz At, min

FTG-wet-1-09 0.10 1.35 x 106 5 10

FTG-wet-1-10 0.10 2.70 x106 4 10

FTG-wet-1-11 0.10 3.00x106 4 10

FTG-wet-1-12 0.10 3.00x106 4 15

FTG-wet-1-13 0.10 3.00x106 4 15

FTG-wet-1-14 0.48 0.47 x106 5 15

FTG-wet-1-15 0.48 3.00x 106 5 15

FTG-wet-1-16 0.48 3.00x 106 5 15

FTG-wet-1-17 0.48 3.00x 106 5 15

Homogenized material +457-45° 0790°

0° UD

Homogenized material

Fig. 4. Distribution of plies in the DCB finite element model (color online)

and dmin were recorded. The number of cycles desired for each specimen was 3 x 106. In Table 1, it is possible to observe that not all specimens experienced the full number of cycles. An explanation is detailed in Sect. 4.

3. Theory and calculations

In this section, the process for obtaining various required quantities will be described. In Sect. 3.1, details about the finite element analyses (FEAs) used in this study are discussed. In addition, the integrated average of the mode I energy release rate along the delamination front is presented so that GIR, GImax and GImin may be determined. In Sects. 3.2 and 3.3, respectively, the methods for obtaining the delamination length a in each fatigue cycle and the de-lamination length per cycle da/dN are described. Several choices for f (GI) in Eq. (1) are presented in Sect. 3.4.

3.1. Finite element analyses to obtain GI

Finite element analyses of the DCB specimens were carried out using the program ABAQUS [17]. Four plies directly above the interface and three plies directly below it were modeled using the effective material properties found for each ply. In order to reduce computation time and computer memory required, the six outer upper and outer lower plies were modeled as one effective homogenous, anisotro-pic material (see Fig. 2). These plies are relatively far from the delamination/interface. The material properties of all plies are presented in Sect. S2. There were four types of materials used in the FEAs including the UD-fabric, the +45°/-45° weave, the 0°/90° weave and the outer homogenized, anisotropic material. The effective mechanical properties of the latter are presented in Table S10; whereas, the thermal properties are the same as those for the woven plies given in Table S9. In Fig. 4, the distribution of the plies in the finite element model is illustrated. Since the homogenized outer material is truncated in Fig. 4, its thickness is not to scale relative to the other plies. In Sect. S4, a convergence study to determine an optimal mesh is described.

Next, the stress intensity factors resulting from the mechanical loading KP (m = 1, 2, III), as well as the thermal residual curing stresses K^ (m = 1, 2, III), arising from the autoclave process, are considered. For each specimen, seven FEAs were performed to determine the displacement

field in the specimen including six mechanical analyses and one thermal analysis. All seven analyses were carried out in the same manner as described in Sect. S4 with a mesh similar to that shown in Fig. S8. The refined region of the mesh remains the same as that shown in Fig. S8. Since it is placed surrounding the delamination front, a change in the delamination length translates the location of the refined region.

For mechanical analyses, the applied load was taken to be 1 N. Six delamination lengths were used in the fracture toughness and fatigue models. For fracture toughness: flj = 50 mm, a2 = 65 mm, a3 = 80 mm, a4 = 95 mm, a5 = = 110 mm, and a6 = 120 mm; for fatigue: ax = 50 mm, a2 =55 mm, a3 = 60 mm, a4 = 65 mm, a5 = 70 mm, and a6 = 80 mm. These delamination length ranges span the lengths achieved during the tests. Using the conservative mechanical M-integral, which was extended in [18] for the interface shown in Fig. 2, six sets of mechanical stress intensity factors, KP (m = 1, 2, III) were obtained for each specimen, along the delamination front, for each delami-nation length an (n = 1, 2, ..., 6). Of the six sets obtained from the mechanical analyses, it is possible to assemble the data in a three-dimensional space. The data included points [KP, a, x3] for each of the three stress intensity factors (m = 1, 2, III). A three-dimensional surface was fit through the points for each mechanical stress intensity factor as shown in Fig. 5. In addition, a functional relation was obtained relating K^P to a and x3 which is given by

Kf =EAc, ,

¿=o , b

+ E Pj\ b

¿=0 I b

(7)

In Eq. (7), pi0 and p^ are surface fitting parameters, b and a0 are, respectively, the width and initial delamination length of the specimen as shown in Fig. 1. Using any value of a, K^) may be expressed as a function of X3, namely KP(x3). The fitting parameters for the fracture toughness specimens may be found in Tables S13 to S17 and for the fatigue specimens in Tables S23 to S31.

For the thermal analysis, a constant temperature change was applied given by

Aft = ft - 90°.

(8)

In Eq. (8), fti was selected as the value of the temperature at the beginning of the test for each specimen. These values are presented in Tables S41 and S45, for fracture toughness and fatigue delamination propagation tests, respectively. In Eq. (8), the value of 90°C is the maximum temperature measured during the curing process at five thermocouples placed within the composite plate. There was an exothermic reaction. For the thermal analyses, a delamina-tion length of a = 85 mm for the fracture toughness tests and a = 65 mm for fatigue tests were selected. One analysis, instead of six, was sufficient for each specimen; it was found that the delamination length has a negligible effect on the residual curing stress intensity factors K^-1. Using

Fig. 5. Surface fitting examples of Kf as a function of a and x3; two views of Kf (a), Kf Kf

the thermal M-integral [18], a single set of K^ was obtained for each specimen, along the delamination front. From the set obtained from the thermal analyses, a mathematical relation between K^ and x3 was found using a 14th-order polynomial defined as

Km=| dt [j j, (9)

where di are fitting constants. For the fracture toughness specimens, may be found in Tables S18 to S22; for the fatigue specimens, see Tables S32 to S40.

The total stress intensity km may be calcu-

lated as a function of x3 for each cycle in the test as

K^fe) = PKif)( X3) + K«( x3). (10)

The parameter P in Eq. (10) is the load leading to delami-nation propagation in a fracture toughness test or Pnax or Pmin from the fatigue tests. Through the images, the given load P is associated with the delamination length a and used in Eq. (7). With K^ a function of x3, for a given value of a and P, it is possible to obtain the interface energy release rate as a function of x3 as

Gi =-^[Kit)2 + K 2t)2] + -L K (I)2. (11)

H1 H 2 By calculating the phase angles in Eqs. (S7) and (S8) [19], it is shown in Sect. S4 that they are less than 0.2 rad along the delamination front for a wide range of examples. Thus, Gi in Eq. (11) is replaced by GI in this study. In Eq. (11), H1 and H2 which are related to the mechanical properties of the plies on either side of the interface were found to be 7.87 and 10.49 GPa, respectively.

An integrated average for GI (x3), along the delamina-tion front, leads to

Gi =1J Gi(x3)dx3, (12)

b 0

where b is the specimen width. If P in Eq. (10) is the load for delamination propagation in a fracture toughness test, then GI in Eq. (12) is identified as GIR for the corresponding delamination length a. On the other hand, if P in Eq. (10)

is chosen as Pmax or Pmin> GI in Eq. (12) would be GImax

and GImin, respectively, for the appropriate delamination length a. In the sequel, the overbar is omitted and the energy release rates are understood to be averaged through the specimen width.

3.2. Delamination length in each fatigue cycle

In this section, the method for determining the delami-nation length during a particular fatigue cycle is presented. For each cycle in a fatigue test, the values of the maximum and minimum loads and displacements, Pmax, Pmin, dmax and dmin, respectively, were recorded. From these values, the compliance C was calculated in each cycle as

C = dmax ~ dmm. (13)

P - P

max min

To proceed, the delamination lengths from the specimen images, taken during the test, were measured using ImageJ software [20]. The corresponding compliance values from Eq. (13) were known as a result of the synchronization between load, displacement and image number. Points of compliance versus delamination length were plotted as shown as red x in Fig. 6 for specimen FTG-wet-1-09. For these points, an expression was found as

a = gC", (14)

where the parameters g and n are power law fitting parameters. For specimen FTG-wet-1-09, the values of g, n and

a, mm-

70-

60-

50-

Calculated delamination length Visually measured delamination length

0.2

0.3

0.4

0.5 C, mm/N

Fig. 6. Delamination length versus compliance in a fatigue test for specimen FTG-wet-1-09 (color online)

the coefficient of determination R were found as 98.97, 0.3386, and 0.9949, respectively. Values for all other specimens may be found in Table S43. Using the relationship from Eq. (14) and knowledge of the compliance values in each cycle, it was possible to calculate the approximate delamination length in each cycle which is shown as blue dots in Fig. 6. It may be observed that there is good agreement between the measured and calculated values of a.

3.3. Delamination propagation rate da/dN in each cycle

A method is presented for determining delamination length a as a function of the cycle number N, in each cycle of the test. Differentiating this function with respect to N leads to the relation between da/dN and N, in each cycle. In many studies, a seven (or another number) point incremental fit was used to obtain a relation between a and N, in each cycle of the test. This method is described in [21]. In [10], it was shown that the seven point incremental fit leads to much scatter in the relationship between a and N. Instead, the relation [10]

a - Aj( N + Bj)B + A2

(15)

was proposed, where Bj and B2 are arbitrary nondimen-sional parameters. Using these parameters, together with N, a value of (N + Bj)B was defined for each cycle so that points [a, (N + B)B ] were plotted, where a was obtained from Eq. (14). The parameters Aj and A2 with units of mm were determined from a linear fit of these points, together with the coefficient of determination R2.

Using the "solver" function of [22], the value of R for Eq. (15) was calculated with Bj and B2 varying between -10 000 and 10000, until the best value of R2 was obtained. The solver function uses a nonlinear generalized reduced gradient (GRG). The solver increments the values of Bj and B2 so that the value of R2 increases. When the value of R2 ceases to increase, the solver stops and produces the relevant values of Aj, A2, Bj, B2 and R2. This method produces a local maximum for R2 in a range of Bj and B2 and not necessarily the global maximum. Therefore, it is necessary to run the solver multiple times with different initial values for Bj and B2. Moreover, existence of a high value for R2 was not the final step taken to verify

Eq. (15). Equation (15) was compared to the visually obtained measurements of a from the images taken during the test. For specimen FTG-wet-1-09, the values of Aj, A2, Bj, B2, and R2 of Eq. (15) were found to be 5.74 mm, 50.42 mm, -62.36, 0.114, and 0.9975, respectively. For all specimens, see Table S44. In Fig. 7, a plot of delamination length as a function of the cycle number for specimen FTG-wet-1-09 is presented. Points marked with a red x are visually obtained measurements of delamination lengths using the images taken during the test. In addition, the green curve describes Eq. (15). Note that the visual measurements from the test and Eq. (15) are well correlated. Differentiating Eq. (15) by the cycle number N leads to

da=AjBj( n+Bj) m

dN

(16)

3.4. Relation between da/dN and GI

In [10], normalized functions for f (GI) in Eq. (1) were

considered such as

f (Gl) = Gum« --

IR

and

f (GI) -AG\eff Gimax — Girnn ) ,

(17)

(18)

where GImin is defined similarly to GImax in Eq. (17). The parameter GIR in Eq. (17) is taken from the GIR- curve for a specific delamination length; that is, during a fatigue test as a increases, GIR increases until it reaches its steady state value. By substituting Eq. (17) into Eq. (1), it is possible to plot da/dN versus GImax on a log-log scale for various values of the cycle ratio R as shown schematically in Fig. 8a. For each R-ratio, the same asymptotic value (at high values of da/dN) giving the cycle fracture toughness are found for all curves. For low values of da/dN, each curve approaches a different threshold value which increases with R. Next consider substituting Eq. (18) into Eq. (1); one may obtain the plot shown in Fig. 8, b. It may be observed that the same threshold value is obtained for each curve, independent of R. Moreover, each curve approaches a different value (at high values of da/dN), increasing as R decreases. It may be observed that the slopes of the linear part of the curves are high. In [10], the values of m in Eq. (1)

a, mm-

75

Visually measured delamination length Calculated delamination length

55

0.00

0.45

0.90

N, x106 cycles

Fig. 7. Delamination propagation in fatigue test for specimen FTG-wet-1-09 (color online)

da/dN

^Ithri

Jlthr2

^Ithr3 Qmax

Fig. 8. Schematic description of fatigue test data for different cycle ratios on a log-log scale: da/dN versus GImax (a) and AGIeff (b) (courtesy of [10])

were between 7 and 27 for 0.10 < Rd < 0.75. When comparing slopes obtained for metals using stress intensity factors in a Paris relation, the powers for f (GI) must be multiplied by a factor of 2. Thus, for a small change or error in the applied load, the delamination growth rate is very high making this approach difficult for use in a damage tolerance paradigm.

Hence, a function used originally for metals [23] was adapted in [4] by defining the function

f (Gi) = AKi

^Imax

(19)

and used in [8, 9]. In Eq. (19), GIthr is the threshold value of GImax from which there will be no delamination propagation and A is a constant which may be thought of as the cyclic fracture toughness. In [4], the value of m in Eq. (1) was fixed at 2 and various material sets for modes I and II, and mixed mode I/II were examined for UD material. For each material, a value of D in Eq. (1) was fixed and GIthr and A were varied. In [8], values of m varied between 2.4 and 7.5. In [9], a master curve was achieved using Eq. (19) for a large data set, by taking A to be the value of GIc, varying GIthr and determining m = 2.65. With a master curve, all the data collapses about one straight line.

Equation (19) was normalized in [10] resulting in

= V Glmax

f (Gi) = AK 2 =

Ithr

(20)

Imax

where the hat quantities are as in Eq. (17). In [10], Eq. (20) was substituted into Eq. (1) yielding a master curve which is independent of the cyclic load or displacement ratio for the material system described there. A value for GIthr was obtained by examining the test data. By fitting Eq. (1) to the test data, a slope of m = 5.2 was found. It would appear that with these lower slopes for both Eqs. (19) and (20) as compared to the functions in Eqs. (17) and (18), the delami-

nation would propagate more slowly. It was shown in [10], that for a small change in GImax, there is a large change in da/dN, as well as AK2. Hence, a small slope for the latter function does not ameliorate the large change in d a I dN. Moreover, using Eq. (20), the full d a I dN curves were recalculated as functions of GImax and AGIeff as shown schematically in Fig. 8.

4. Results

In Sects. 4.1 and 4.2, respectively, the fracture toughness and fatigue delamination propagation test results are presented. The fatigue thresholds for the two models considered here are determined in Sect. 4.3. In Sect. 4.4, two master curves making use of Eqs. (19) and (20) are obtained. Using the master curve found with Eq. (20), a full set of data is back-calculated in Sect. 4.5.

4.1. Fracture toughness test results

Fracture toughness tests were carried out on a set of 5 DCB specimens. Additional information about the tests is presented in Sect. S5 in the Supplementary material.

The specimens were analyzed by means of the finite element method to determine the stress intensity factors as described in Sect. 3.1. The values of pt0 and pj1 (i = 0, 1, ..., 5 andj = 0, 1, ..., 4) from Eq. (7), and dt (i = 0, 1, ..., 14) from Eq. (9), are presented for each specimen in Tables S13 through S22. Values of load and displacement were recorded during the test producing load-displacement (P-d) curves which are presented in Fig. S9. Since the system compliance was 0.5% less than the compliance of the specimen, the actuator displacement d was used in place of the load-line displacement. Using the appropriate loads and delamination lengths, the total stress intensity factors in Eq. (10) were obtained. It was found that the value of K1(t) was much greater than the other two stress intensity factors. Further details are given in Sect. S4. Thus, it may be concluded that the tests produce nearly mode I deforma-

Fig. 9. Resistance curve GTR versus Aa ■

: FT-wet-1-04 (1),

FT-wet-1-05 (2), FT-wet-1-06 (3), FT-wet-1-07 (4 ), FT-wet-1-08 (5) (color online)

tion. Furthermore, from the FEAs, numerically obtained P-d curves for each specimen were determined. Comparison between the tests and the numerical analyses in the linear region of the curves showed good agreement. By this comparison, the effective mechanical and thermal properties found for the plies are validated.

As mentioned previously, the specimens were measured before the tests (see Table S1). After the tests were carried out, the delamination lengths were measured on both sides of each specimen with the optical mode of an Olympus Confocal Microscope (model no. OLS4100, Tokyo, Japan) and given in Table S42. The difference between these measurements was less than 2 mm conforming to the standards [11, 12] for UD material. In addition, during the tests, values of temperature and RH were recorded and may be found in Table S41. The tests conform to the standards.

Using Eqs. (11) and (12), values of Gl were calculated for each load and delamination length. Note that Gt in Eq. (11) is identified as Gl and the overbar is omitted. These values are averaged over the thickness. This data is plotted as points in Fig. 9. An initiation fracture toughness is shown for Aa = 0 as GIc = 357.9 N/m. As Aa increases, the values of Gir increase, as well, reaching a steady state value of Gss = 727.7 N/m for Aa = 30 mm. Fitting the points between 0 < Aa < 30 mm leads to the power law given by

Gir = 100.2( Aa )0384 + 357.9. (21)

This curve and the extreme points are plotted in Fig. 9. A coefficient of determination R2 of the power law in Eq. (21) was found as 0.82.

4.2. Fatigue tests results

In this section, the fatigue tests carried out as described in Sect. 2.3 are presented. Results and additional explanations are given in Sect. S.6. Recall that two cyclic displace-

ment ratios were used: Rd = 0.10 and 0.48. The aim was for each specimen to undergo 3 x 106 cycles. In Table 1, the number of cycles experienced by each specimen is presented. During the tests on specimens FTG-wet-1-09 and FTG-wet-1-10, there was a power failure. If a test is interrupted, it is not possible to remount the specimen in the

same position that it had been before the interruption. Hence, the number of cycles experienced by each of these specimens is less than 3 x 106. Although specimen FTG-wet-1-14 was subjected to all planned cycles, it appears that there was a temperature change which affected the measured compliance values as shown in Fig. 10. It is possible to note that the room temperature for N ~ 475 000 cycles is about 23.5°C; after that, it decreased suddenly to less than 21.0°C. In addition, the compliance value increased significantly for N = 475 000 cycles. As a result, data from this cycle forwards was not taken into account in calculating the delamination propagation rate. It was found that the specimens tested in this investigation were very sensitive to temperature changes. Another illustration is described in the sequel for specimen FTG-wet-1-15. As a result, data from the first 30 000 cycles of this test was omitted. For all specimens, the compliance values calculated as a function of N are illustrated in Fig. S10.

As described in Sect. 3.2, the delamination length may be obtained using Eq. (14) with the value of the compliance in each cycle. In Table S43, the fitting constants g and n, together with the coefficient of determination R2 of Eq. (14) are presented for each specimen with R2 > 0.97. Moreover, in Sect. 3.3, a method for determining a relation between the delamination length a and the cycle number N in Eq. (15) was described. In Table S44, the fitting constants, A1, A2, B1 and B2, and the coefficient of determination R2 are presented for all specimens with R2 > 0.97.

Next, in a manner similar to that described in Sect. 4.1 for the fracture toughness tests, the stress intensity factors and the maximum and minimum values of the energy release rates GImax and GImin were determined. The coefficients pt 0 and pj1 (i = 0, 1, ..., 5 and j = 0, 1, ..., 4) from Eq. (7), and dt (i = 0, 1, ..., 14) from Eq. (9), are presented for each specimen in Tables S23 through S40. Values of GImax and AGIeff, from Eqs. (17) and (18), respectively, were also determined in each cycle.

As mentioned earlier, a change in the room temperature affected the behavior of specimen FTG-wet-1-15 during the test. All obtained data for this specimen was used to plot a curve for d a I dN as a function of GImax on a log-log scale. This curve, together with that for specimen FTG-

0C

C, mm/N

C

Jump in compliance N =475 000

-0.4

20-

'0.0 0.5 1.0 1.5 2.0 N, x106 cycles

Fig. 10. Compliance C values calculated by Eq. (13) and temperature û values measured during fatigue test of specimen FTG-wet-1-14 (color online)

da /dN, mm/cycle

10-1-103-| io-5H

ft, °C

10

i-7

0.1

0.2

0.3 0.4 0.5 0.7

| a 25 b.

• FTG-wet-1-14 -

• FTG-wet-1-15 24 ■ \ J* 1 I I 1 } \ ft, ,f|___ C

23 ■ \ 1 \ 1 ^ ! W \ /

N. 30 000— V

22 i i -

1 1 i i i i i i _ 21 i i i i ' i i 1 1 1 1 r

C, mm/N -0.5

-0.4 -0.3 -0.2 -0.1 0.0

20

40

60

80 N, x103 cycles

Fig. 11. Temperature effect on results. Plot of da/dN versus GImax for specimens FTG-wet-1-14 and FTG-wet-1-15 (a). Compliance C values calculated using Eq. (13) and measured temperature values during the fatigue test of specimen FTG-wet-1-15 during the first 100 000 cycles (b) (color online)

wet-1-14, are shown in Fig. 11, a. It may be observed that for low values of da/dN the curves approach one another; whereas, for high values of da/dN, they diverge. It should be noted that for specimen FTG-wet-1-14, data is shown only for N < 475 000. Since the tests of these two specimens were performed with Rd = 0.48, it was expected that they would show similar behavior. Next, for specimen FTG-wet-1-15, the temperature is examined in Fig. 11, b during the first 100 000 cycles. For N < 30 000 cycles, the temperature increased from 23.75°C to above 24.5°C. For N > >30 000 cycles, the temperature decreased to about an average value of 23.3°C until the end of the test. This small change in temperature apparently created a reduction in the delamination propagation rate. There is no discontinuity in the compliance values as in specimen FTG-wet-1-14. But the compliance values differ from those of specimen FTG-wet-1-14 during the first 30 000 cycles of the test. As a result of Fig. 11, going forward, the analysis of specimen FTG-wet-1-15 was performed without the data from the first 30 000 cycles. It may be noted that the ASTM standards recommend temperatures of 23 ±3°C. Clearly, the temperatures for this specimen conform to the standards.

Possible explanations for this behavior are presented next. The plate from which the specimens were taken was manufactured by means of a wet-layup. During the curing stage, the maximum imposed temperature was about 85 °C with an exothermic reaction bringing the temperature to 90°C within the composite plate. The data sheet for the epoxy used in these specimens specifies that the maximum curing temperature should be 130°C in order to obtain a glass transition temperature T = 130°C [24]. Dynamic mechanical analysis (DMA) tests were performed where it was found that Tg = 104°C. This lower value of Tg may explain the sensitivity of these specimens to small temperature changes.

Recall that for specimen FTG-wet-1-14 in Fig. 10, a temperature decrease of about 3°C significantly affected the compliance value. In practice, this change affected the minimum load Pmin measured in the corresponding cycle

which decreased by about 0.4 N. As a result the compliance value increased, as may be seen in Eq. (13).

Moreover, the temperature sensitivity may be related to the load cell sensitivity of the Instron testing machine. To this end, a test was carried out by hanging a mass of 1 kg on the load cell and heating and cooling the room between 25 and 21.5°C. It was seen that the load varied between 9.81 N, at the higher temperature, and 9.56 N, at the lower one. These small changes in the load affect, in particular, Pmin and hence, the compliance. It is speculated that both the low glass transition temperature, as well as the load cell sensitivity affected the compliance and, hence, the final results.

Some additional data is given in the Supplementary material. In Table S45, the average cyclic load ratio Rp, for each specimen in the first 10 000 cycles, is shown. It is somewhat lower than Rd. For specimen FTG-wet-1-15, Rp is an average for the first 30 000 to 40 000 cycles. In addition, the delamination lengths at the end of the tests for each specimen, on both sides, and their differences A f, are given, as well. It was seen that Af < 2 mm, in keeping with the fracture toughness test standards. Initial test temperatures are also shown.

Next, values of da/dN versus GImax and AGIeff, from Eqs. (17) and (18), were plotted on a log-log scale as shown in Figs. 12, a and 12, b, respectively. Generally, linear behavior is observed. Such curves may be used to predict delamination propagation when GImax or AGIeff is known for a structure containing the interface studied here. In Fig. 12, the plots include data for all fatigue specimens. It may be observed that the data for high values of da/dN appear to approach an asymptote as also shown in Fig. 8. Similar to Fig. 8, a, in Fig. 12, a, GImax ^ 1; in Figs. 8, b and 12, b for AGIeff, different asymptotic values for large da/dN are approached for each value of R.

Ignoring the data for large values of da/dN (up to 500 cycles), a mathematical expression in the linear range of the curves on a log-log scale, in Figs. 12, a and 12, b, are given by

da /dN, mm/cycle da /dN, mm/cycle

Fig. 12. Plots of da/dN versus GImax (a) and AGIeff (b): FTG-wet-1-09 (1), FTG-wet-1-10 (2), FTG-wet-1-11 (3), FTG-wet-1-12 (4), FTG-wet-1-13 (J), FTG-wet-1-14 (6), FTG-wet-1-15 (7), FTG-wet-1-16 (8), FTG-wet-1-17 (9) (color online)

Ta = ^1GGIrna

dN

and

da dN

= D2 AG

Ieff

(22)

(23)

where the parameters Dt and (i = 1, 2) are power law fitting constants. These parameters for each specimen are presented in Table 2, together with R2 and the number of sampling points. It may be seen that R2 > 0.9877 for all specimen, which is excellent.

In Figs. 12, a and 12, b, it may be observed that the specimens separate into two groups according to the cyclic displacement ratio Rd. Ideally, the curves for each Rd should coincide. In practice, this is not the case, and there is variability between the curves, as reflected by that of the slopes, m1 and m2. In Fig. 12, it may be observed that the data from specimen FTG-wet-1-16 deviates from the rest of the specimens in its group. In Table 2, a slope significantly higher than that of the other specimens with the same value of Rd is noted. We have no explanation for this behavior except that it is normal scatter in fatigue data. Fur-

thermore, it may be seen in Table 2 that the slopes of the curves for Rd = 0.10 are smaller than those for Rd = 0.48, as illustrated in Fig. 8. Averages of the slopes, for each set of specimens, are m(0'1) = 6.10, m2,01) = 5.86 for Rd = 0.10; mi048) = 8.77, m2°'48) = 8.92 for Rd = 0.48. The value of the coefficient of variation (CV) of the slopes may be calculated as

CV = a* (24)

X

where SDV is the standard deviation and X is the average value. The values of CV for Rd = 0.10 are 0.06 for m1 and 0.03 for m2; for Rd = 0.48, they are 0.13 for m1 and 0.16 for m2 An increase in the slopes, as Rd increases was expected and also seen in [10] and Fig. 8. Furthermore, in Fig. 12, a, it is possible to observe that for a given value of GImax, a higher value of da/dN is obtained for specimens with Rd = 0.10 as compared to those with Rd = 0.48. On the other hand, it may be seen in Fig. 12, b that for a given value of AGIeff, a higher value of da/dN is obtained for specimens with Rd = 0.48 as compared to those with Rd = = 0.10.

Table 2

Displacement ratio, fitting constants, R2 and number of sample points for Eqs. (22) and (23) for each specimen

Specimen Rd Eq. (22) Eq. (23) Number of

A m1 R2 D2 m2 R2 sample points

FTG-wet-1-09 0.10 0.0121 5.90 0.9937 0.0414 5.75 0.9929 734

FTG-wet-1-10 0.10 0.0181 5.68 0.9950 0.0602 5.62 0.9958 794

FTG-wet-1-11 0.10 0.0144 6.65 0.9972 0.0431 5.93 0.9967 806

FTG-wet-1-12 0.10 0.0109 5.97 0.9957 0.0376 5.91 0.9938 806

FTG-wet-1-13 0.10 0.0427 6.29 0.9971 0.1347 6.09 0.9973 806

FTG-wet-1-14 0.48 0.0202 8.68 0.9969 1165.6 9.08 0.9896 601

FTG-wet-1-15 0.48 0.0071 7.52 0.9902 22.048 7.14 0.9915 541

FTG-wet-1-16 0.48 0.0070 10.31 0.9953 3487.9 10.67 0.9878 806

FTG-wet-1-17 0.48 0.0123 8.56 0.9975 629.80 8.79 0.9950 806

0.01

N, cycles

Fig. 13. Plot of AGIeff from Eq. (18) versus the cycle number N, for all fatigue specimens: FTG-wet-1-09 (1); FTG-wet-1-10 (2); FTG-wet-1-11 (3); FTG-wet-1-12 (4); FTG-wet-1-13 (J); FTG-wet-1-14 (6); FTG-wet-1-15 (7); FTG-wet-1-16 (8); FTG-wet-1-17 (9); fit, Rd = 0.10 (10); fit, Rd = 0.48 (11) (color online)

4.3. Energy release rate threshold values

In this section, values of GIthr and GIthr from Eqs.(19) and (20), respectively, are determined. It is difficult to experimentally obtain threshold values. However, it is possible to determine approximate values as outlined here and presented in [10].

It may be shown that [10]

GIthr =

AG,

Ieffthr

(1 - Rp)2

Thus, knowledge of AGIeffthr and Rp is sufficient for obtaining GIthr. Moreover, normalized values of GIthr and AGIeffthr, respectively, may be used in Eq. (25), so that

GIthr =

AC?,

Ieff thr

(1 - Rp)2

Thus, the threshold values GIthr and GIthr may be found once AGIeffthr and AGIeffthr are known.

As presented in Fig. 8, b, the values of AGIeffthr or AGIeffthr are each the same for any value of Rd for a par-

test results are plotted versus the cycle number N on a loglog scale as illustrated in Fig. 13. For each value of Rd, a straight line was fit through each group of results. The intersection of the lines yields an initial value A(jI(e)ffthr. This value of AGIeff is the initial value used to determine a final value A(jfffthr when determining a master curve. For each displacement ratio, the expressions for the straight lines are given in Eqs. (S9) and (S10). The parameter R2 for each equation is presented in Table S46. In Fig. 13, the intersection point of the two lines was obtained for values of N = 4.7 x 109 and AGf^ = 0.0492. In the same manner, with a plot of AGIeff versus N as shown in Fig. S11, a value of AG^jjjj. = 33.98 N/m was found. This value of AGIeff was used as an initial value for determining a final

value AG{efffthr.

4.4. Master curves

One major objective of this study is to obtain a master curve to describe the fatigue data which is independent of Rd. This curve may be found by using the fatigue test results presented in Sect. 4.2, together with the initial threshold values given in Sect. 4.3. The method used here follows that described in [10] for AK2 in Eq. (20).

In Fig. 14, da/dN is plotted versus AKj and AK2, from Eqs. (19) and (20), respectively. Initially, the graphs (25) were obtained using the values AGI(e)ffthr, AG^f^ and the initial value of A in Eq. (19). Recall that the value of A is a constant representing the critical cyclic energy release rate. For an initial value of A, GIR in Eq. (21) was chosen for 3mm<Aa<5mm as 500 N/m. The value of GIR to be used in Eqs. (17) and (20) is GIR for the given value of Aa. Note that A is a free parameter and used in the fitting; whereas, GIR changes as the delamination length increases.

Equation (1) is used with f (GI) = AKj or AK2. The parameters D and m are found by fitting the fatigue data to a straight line on a log-log scale with A and AGIeffthr as free parameters for AKj, and AGIeffthr as a free parameter

(26)

ticular material system. To determine AG?]

Ieffthr'

all of the

for AK2. The values of AG,

Ieffthr'

AC?,

Ieff thr

and A were

da /dN, mm/cycle

da /dN, mm/cycle m-1

Fig. 14. Master curves for AK1 in Eq. (19) (a) and AK2 in Eq. (20) (b): FTG-wet-1-09 (1), FTG-wet-1-10 (2), FTG-wet-1-11 (3), FTG-wet-1-12 (4), FTG-wet-1-13 (J), FTG-wet-1-14 (6), FTG-wet-1-15 (7), FTG-wet-1-16 (8), FTG-wet-1-17 (9), master curve (10) (color online)

Table 3

Fitting constants, coefficient of determination, initial and final threshold values AGIeffthr and AGIeffthr for Eq. (1) using AK1 and AK2 as f (GI), respectively

Eq. (1) with AK1 as f (GI)

D m R2 AG(i) AGIeffthr' N/m AG(f) Ieffthr' N/m AGm Ieffthr

3.14 x 10-" 5.1 0.8796 33.98 32.70 0.0449

Eq. (1) with AK2 as f (GI)

D m R2 AG(i) Ieffthr AG(f) Ieffthr, N/m AGm Ieffthr

4.09 x 10-4 4.0 0.8765 0.0492 42.64 0.0586

incremented from their initial values, until the highest value of R was achieved. The values ofD, m using AK1 or AK2 are presented in Table 3. The initial and final values of AGIeffthr and AGIeffthr, respectively, are given in Table 3, as well. The final value of A was 619 N/m.

In Figs. 14, a and 14, b, using AK1 or AK2 as f (GI), respectively, the data was plotted and the best fit for the master curve is shown as a continuous black line. The lines are independent of Rd. However, it may be seen that there is some scatter in the test results, as reflected in the coefficient of determination. Recall that the results obtained from each test do not include the first 500 cycles. The value of the slope m for Eq. (1) using AK2 is lower than the slopes presented in Table 2 for the curves presented in Fig. 12, a for GImax. Recall that for purposes of comparison, the slopes in Eq. (22) require multiplication by a factor of 2. Nevertheless, in [10] for another material system, it was shown that use of AK2 does not result in a smaller increase in da/dN as the delamination propagates. The value of the slope in Eq. (1) using AK1, is higher than the slope for Eq. (1) using AK2 with values of R2 almost identical.

In Table 3, it may be observed that the value of AGfff thr is rather close to that of AGI(e)ffthr. There is a greater percent difference between AG^f^ and A<j(^ffthr. It may be noted that AGfffthl. when using AK1 is not identical to AGfffthl. for AK2. This difference results from the independent fitting processes. In Table S47, the final values of Githr and Githr, for Eq. (1) using AK or AK as f (Gi), respectively, are presented for all specimens. When using AK2 and for Rd = 0.10, the average value of Gfffthr = = 51.8 N/m; for Rd = 0.48, the average value of GI(efffthr = = 139.5 N/m. These values appear reasonable.

4.J. Back-calculation of full da/dN curves

In this section, the values of da/dN are calculated using AK2 of Eq. (20) in Eq. (1) for each of the two displacement ratios. They are then plotted versus GImax and AGIeff for an extended range of da/dN. For the calculation, the average values of the cyclic load ratios Rp presented in Table S45 were used for each set of specimens with the same Rd ratio. The average values are Rp = 0.093 and Rp = 0.447, for Rd = 0.10 and Rd = 0.48, respectively. The former values were used to determine the threshold values from Eqs. (25) and (26). In addition, the value of AGf thr for AK2 from Table 3 was used. In Fig. 15, two plots are presented: a plot in Fig. 15, a of da/dN versus GImax and a plot in Fig. 15, b of da/dN versus AGIeff. The data obtained from the tests shown in Figs. 12, a and 12, b, are also presented in Figs. 15, a and 15, b, respectively. In addition, two continuous curves (black and green) are exhibited which were obtained using Eq. (20) in Eq. (1) for displacement ratios of Rd = 0.10 and Rd = 0.48, respectively. It may be observed that the theory presented in [10] which also appears in Fig. 8 is validated for this material system. In the same manner, using Eq. (19) in Eq. (1), values of da/dN may be recalculated versus GImax and

da /dN, mm/cycle da/dN, mm/cycle

Fig. 15. Recalculation of da/dN using Eq. (20) in Eq. (1), with values ofD and m presented in Table 3 for AK2. Black curves for Rp = = 0.093 and green curves for Rp = 0.447. Plot of da/dN versus GImax (a) and AGIeff (b): FTG-wet-1-09 (1); FTG-wet-1-10 (2); FTG-wet-1-11 (3); FTG-wet-1-12 (4); FTG-wet-1-13 (J); FTG-wet-1-14 (6); FTG-wet-1-15 (7); FTG-wet-1-16 (8); FTG-wet-1-17 (9); fit, Rp = 0.09 (10); fit, Rp = 0.45 (11) (color online)

AGIeff, for each of the two displacement ratios. These results suggest that use of Eqs. (19) and (20) in Eq. (1) allows a full approximate prediction of the results without conducting tests of extended length.

5. Discussion and conclusions

In this section, a discussion and conclusions are presented. This study focused on nearly mode I fracture toughness and fatigue delamination propagation tests carried out with DCB specimens illustrated in Fig. 1. The specimen plies were composed of carbon fibers in an epoxy matrix. The laminate consisted of 18 multidirectional, plain, woven plies and one UD-fabric ply as shown in Fig. 2 which was fabricated by means of a wet-layup process. The woven plies alternated between those with tows in the 0°/90°-directions and +45°/-45°-directions. The delamination was between a UD-fabric (upper) ply and a woven (lower) ply with tows in the +45°/-45°-directions.

Fracture toughness tests on five DCB specimens led to a GIR-curve as shown in Fig. 9. This resistance curve or R-curve is characterized by two values of G. The curve initiates from its fracture toughness value of GIc = 357.9 N/m at Aa = 0. From this value, an increase may be observed until a steady state value Gss = 727.7 N/m is reached at Aa = 30 mm. For the material system discussed in [10], which included only MD woven plies, the values were GIc = = 508.0 N/m and Gss = 711.0 N/m. Moreover, a steady state was reached at Aa = 10 mm.

Fatigue delamination propagation tests on nine DCB specimens for two cyclic displacement ratios Rd = 0.10 and 0.48 provided da/dN data. The compliance C in each cycle was obtained using Eq. (13). Measured values of delami-nation length a were plotted versus matching compliance values as presented as red x in Fig. 6. A relation between a and C was obtained by fitting Eq. (14) to the measured delamination lengths. In this way, values of a in each cycle could be found without additional measurements. These values of a, calculated with Eq. (14), were plotted versus the corresponding values of (N + B1)B where B1 and B2 are non-dimensional parameters to be determined in the fitting of Eq. (15). The methods for determining a in each cycle of a test and obtaining a relation between a and N were found to be effective. In Fig. 6, it may be observed that there is good agreement between the measured and calculated values of a. Moreover, in Fig. 7 it is possible to observe that the visual measurements of a from a test and Eq. (15) are well correlated. A relation between da/dN and N in each cycle was determined by differentiating Eq. (15) to obtain Eq. (16).

The curing process may have a significant effect on the results. In this study, the glass transition temperature Tg was found to be 104°C which is lower than that specified in the data sheet (130°C) for this epoxy. This lower value of Tg may partially explain the sensitivity of the speci-

mens to small temperature changes. Recall that for specimen FTG-wet-1-14 in Fig. 10, a temperature decrease of about 3°C significantly affected the compliance value. Moreover, the temperature sensitivity may be also related to the load cell sensitivity of the Instron testing machine, as described in Sect. 4.2. Hence, it appears that the specimens tested in this study are sensitive to small temperature changes.

Values of da/dN versus GImax and AGIeff, from Eqs. (17) and (18), were plotted on a log-log scale as shown in Figs. 12, a and 12, b, respectively. The data for high values of da/dN appear to approach an asymptote as also shown schematically in Fig. 8. Similar to Fig. 8, a, for large values of da/dN, GImax ^ 1; as for AGIeff, different asymptotic values are approached for each value of the cycle ratio R, as described in Fig. 8, b. In Figs. 12, a and 12, b, ideally the curves for each Rd should coincide. In practice, this is not the case, and there is variability between the curves. The data from specimen FTG-wet-1-16 for Rd = = 0.48 deviates from the rest of the specimens in its group. No explanation was found for this behavior except that it is normal scatter in fatigue data. Furthermore, the slopes of the curves for Rd = 0.10 are smaller than those for Rd = = 0.48, as illustrated in Fig. 8. An increase in the slopes, as Rd increases was expected and also seen in [10] for a different material system.

Master curves were obtained using AK1 and AK2 in Eqs. (19) and (20), respectively, as f (GI) in Eq. (1). These curves, presented in Figs. 14, a and 14, b, respectively, eliminate the dependence of da/dN on the cyclic ratio Rd. To compare the slopes of the curves, for example in Fig. 12, a and Table 2 for GImax to that for AK2 in Fig. 14, b, the former is multiplied by two making the difference even greater. Nevertheless, in [10] for another material system, it was shown that use of AK2 does not result in a smaller increase in da/ dN for a given change in GImax. Thus, the master-curves do not overcome the problem of accelerated delamination propagation in laminates. In addition, the value of the slope in Eq. (1) using AK1, is higher than the slope for Eq. (1) using AK2 with values of the coefficient of determination R2 almost identical. For AK2 and for Rd = = 0.10, the average value of Gff^ = 51.8 N/m; for Rd = = 0.48, the average value of Gfff^ = 139.5 N/m. These values appear reasonable. It may be pointed out that using these results in a damage tolerance paradigm may not be particularly useful. Perhaps the threshold values found here for this material combination may be used in predictions of safety factors.

Finally, using the master curve of, for example, AK2 of Eq. (20) in Eq. (1), back calculation of da/dN as a function of GImax or AGIeff for two cyclic displacement ratios was carried out leading to a range of values beyond that of the tests. The behavior shown in Fig. 8 was validated for this material system in Fig. 15. The back calculation pro-

vides a full range of da/dN values (from high asymptotic values approaching the fracture toughness, to low asymptotic values approaching the thresholds) without the need to conduct extended fatigue tests. Future studies should include tests on the same material system for nearly mode II deformation or mixed modes 1 and 2.

Acknowledgments

We would like to express our appreciation to Mor Mega, Ido Simon, Victor Fourman and Rami Eliasy for their invaluable help during this investigation.

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SUPPLEMENTARY MATERIAL

Section, figure, table and equation numbers are preceded with an S; those without an S are in the body of the paper. In Sect. S1, measurements of the fracture toughness and fatigue delamination propagation specimens are given. The mechanical and thermal properties of the plies are described in Sect. S2. In Sect. S3, the test setup is discussed. Details about the finite element analysis and a convergence study are presented in Sect. S4. In Sects. S5 and S6, elaboration of the fracture toughness and fatigue delamination propagation tests, respectively, is discussed.

S1. Measurements of DCB specimens

Fracture toughness tests were performed on five DCB specimens denoted as FT-wet-1-0j (j = 4, 5, 6, 7, 8); FT represents fracture toughness; wet indicates that the specimens were fabricated by means of a wet-layup; the numeral 1 denotes batch 1; the next number represents the location in the row of the plate from which it was fabricated. For each specimen, the length, width and thickness were measured; these measurements are presented in Table S1. The width and thickness were measured at five locations along the specimen length as shown in Fig. S1. Average values and standard deviations (SDVs) of these five measurements were calculated and are also presented. Note that the width of the specimen was measured by means of a caliper with a resolution of 0.01 mm and the thickness of the specimen was measured by means of a micrometer with a resolution of 0.001 mm. For each specimen, the delamination length a0 was measured from the center of the loading holes to the end of the delamination front on both sides of the specimen: front and back, and marked with a superscript (f) and (b), respectively. These measurements were made with the optical mode of an Olympus confocal microscope (model number OLS4100; Tokyo, Japan) and given in Table S1; its resolution is 0.001 mm. The difference between them A0 is also presented. It may be seen that the differences are less than 2 mm which complies with the fracture toughness standards [1, 2]. The parameter a in

Table S1

Fracture toughness specimen measurements

Specimen hi, mm h2, mm h3, mm h4, mm h5, mm h, mm SDV, mm

FT-wet-1-04 5.01 5.07 5.00 5.01 4.96 5.01 0.04

FT-wet-1-05 4.98 5.02 5.08 5.01 4.90 5.00 0.06

FT-wet-1-06 4.94 5.03 4.97 4.88 4.97 4.96 0.05

FT-wet-1-07 4.94 4.88 4.94 4.86 4.85 4.89 0.04

FT-wet-1-08 4.94 4.87 4.94 4.84 4.88 4.89 0.04

by, mm b2 , mm b3, mm b4, mm b5, mm b, mm SDV, mm

FT-wet-1-04 19.74 19.96 19.64 19.71 19.72 19.75 0.12

FT-wet-1-05 20.01 19.94 20.13 20.31 20.00 20.08 0.13

FT-wet-1-06 19.89 19.96 19.64 19.65 19.97 19.82 0.15

FT-wet-1-07 19.90 20.00 20.10 19.95 20.31 20.05 0.16

FT-wet-1-08 19.90 19.95 19.99 19.72 19.62 19.80 0.16

a0f), mm a0b), mm A0, mm a(f), mm a(b), mm A, mm l, mm

FT-wet-1-04 53.92 54.57 0.65 63.92 64.26 0.34 202

FT-wet-1-05 53.72 54.01 0.29 63.60 64.12 0.52 201

FT-wet-1-06 54.22 53.94 0.28 64.10 63.56 0.54 200

FT-wet-1-07 53.92 53.64 0.28 63.87 63.59 0.28 200

FT-wet-1-08 53.81 53.34 0.47 63.79 63.49 0.30 200

sented in Table S2 and thickness measurements in Table S3. Each specimen is denoted with FTG indicating that it is a specimen tested in fatigue; wet denotes that it is fabricated by means of a wet-layup; the number 1 indicates that this is batch 1; the next number represents the location in the row of the plate from which the specimen was fabricated. It may be noted that all of the specimens tested here, both fracture and fatigue, were fabricated from the same plate. Specimen FT-wet-1-08, from the fracture toughness set, was next to specimen FTG-wet-1-09, from the fatigue delami-nation propagation set, in the plate.

The locations at which measurements were carried out of the width b and thickness h are presented in Fig. S2. The coordinate system in Fig. S2 is positioned at the location where b2 is indicated with the origin on the edge of the specimen next to h23. It is not located there in the figure so that it will be clearly visible. Two width measurements shown as b1 and b4 were made 25 mm from each edge of the specimen (along the x1 - axis). These two measurements were made as indicated in the ASTM fracture toughness standard [1], while the fatigue standard [3] suggests measurement of b2 along the delamination front. The last width measurement b3 was made nearly half-way between b2 and b4. The ASTM standard [ 1] recommends making only one measurement between b1 and b4. In addition, this standard recommends making only one thickness measurement, along the specimen width, in each location at which a width measurement is made and along the centerline of the speci-

Table S1 is the delamination length from the edge of the specimen to the delamination front. This parameter was also measured on both sides (front and back) of the specimen. The difference between these two measurements is denoted as A. In addition, the length of the specimens l was measured with a ruler and complies with the requirement that l be at least 125 mm.

Fatigue delamination propagation tests were carried out on nine DCB specimens: five with the cyclic displacement ratio Rd = 0.1 and four with Rd = 0.48 (see Eq. (2) for a definition of Rd). Specimen width measurements are pre-

Fig. S1. Measurement locations of DCB specimens for fracture toughness tests (color online)

Table S2

Width measurements and averages as presented in Fig. S2 (mm)

Specimen bi b2 b b4 b SDV

FTG-wet-1-09 19.86 19.84 19.84 20.25 19.95 0.20

FTG-wet-1-10 19.93 19.92 19.68 19.66 19.80 0.15

FTG-wet-1-11 19.85 19.38 19.61 20.13 19.74 0.32

FTG-wet-1-12 19.87 19.89 19.94 19.70 19.85 0.10

FTG-wet-1-13 19.86 19.91 19.85 20.09 19.93 0.11

FTG-wet-1-14 19.56 19.77 19.84 19.78 19.74 0.12

FTG-wet-1-15 19.98 19.91 20.01 20.12 20.01 0.09

FTG-wet-1-16 19.98 19.98 19.94 19.83 19.93 0.07

FTG-wet-1-17 20.03 20.05 19.99 20.03 20.03 0.03

men. Since the polytetrafluoroethylene (PTFE) film (13^m thick) forming the delamination may have an effect on the thickness measurement, at the location where b was measured, a thickness measurement was not made there. At each of the locations at which b2, b3 and b4 were measured, three thickness measurements were made along the specimen width and not only at the one prescribed in [1]. The purpose of making three measurements was to account for the change of thickness along the x3-axis, which may vary as a result of the wet-layup manufacturing process. In the ISO fracture toughness standard [2], three width measurements at equal distances along the specimen length are recommended. At the location of the middle one, of these three, performance of three thickness measurements is recommended, similar to those shown in Fig. S2. At the locations of the other width measurements, only one thickness measurement is prescribed. In this investigation, a combination of that recommended in both fracture toughness standards was carried out on the fatigue delamination propagation specimen set so that the measurements are ideal for obtaining the width and thickness of the specimen. These measurements were performed using the same instruments,

Fig. S2. Measurement locations of DCB specimens for fatigue delamination propagation tests (color online)

Thickness measurements and averages as presented in Fig. S2 (mm)

Specimen h21 h22 h23 h2 SDV

FTG-wet-1-09 4.89 4.89 4.95 4.91 0.03

FTG-wet-1-10 4.88 4.86 4.86 4.87 0.01

FTG-wet-1-11 4.81 4.79 4.89 4.83 0.05

FTG-wet-1-12 4.83 4.83 4.85 4.84 0.01

FTG-wet-1-13 4.88 4.85 4.85 4.86 0.02

FTG-wet-1-14 4.74 4.76 4.70 4.73 0.03

FTG-wet-1-15 4.83 4.79 4.88 4.83 0.05

FTG-wet-1-16 4.91 4.93 4.81 4.88 0.06

FTG-wet-1-17 4.86 4.85 4.84 4.85 0.01

h31 h32 h33 h3 SDV

FTG-wet-1-09 4.91 4.94 4.95 4.93 0.02

FTG-wet-1-10 4.79 4.83 4.89 4.84 0.05

FTG-wet-1-11 4.78 4.77 4.75 4.77 0.02

FTG-wet-1-12 4.58 4.68 4.74 4.67 0.08

FTG-wet-1-13 4.76 4.72 4.72 4.73 0.02

FTG-wet-1-14 4.84 4.77 4.75 4.79 0.05

FTG-wet-1-15 4.91 4.89 4.94 4.91 0.03

FTG-wet-1-16 4.85 4.79 4.91 4.85 0.06

FTG-wet-1-17 4.72 4.87 4.82 4.80 0.08

h41 h42 h43 h4 SDV

FTG-wet-1-09 4.85 4.89 4.82 4.85 0.03

FTG-wet-1-10 4.88 4.97 4.90 4.92 0.05

FTG-wet-1-11 4.78 4.81 4.77 4.79 0.02

FTG-wet-1-12 4.77 4.72 4.77 4.75 0.03

FTG-wet-1-13 4.87 4.88 4.86 4.87 0.01

FTG-wet-1-14 4.94 4.80 4.83 4.86 0.07

FTG-wet-1-15 4.92 4.89 4.87 4.89 0.03

FTG-wet-1-16 4.90 4.89 4.98 4.92 0.05

FTG-wet-1-17 4.83 4.90 4.91 4.88 0.04

caliper and micrometer, respectively, for the width and thickness, which were used for measurements of fracture toughness specimens. In addition, for the measurements presented in Tables S2 and S3, the SDV is also given.

For all width and thickness measurements presented in Fig. S2, average values were calculated and are presented in Table S4. The value of the coefficient of variation (CV) given as

where X is the average value of a group of measurements was calculated. The largest value of CV for the width measu-

Table S4

Fatigue specimen measurements as presented in Fig. S2 (mm)

Specimen b h a (f) a0 a (b) a0 l

FTG-wet-1-09 19.95 4.82 53.53 53.92 0.39 200

FTG-wet-1-10 19.80 4.87 53.58 54.01 0.43 200

FTG-wet-1-11 19.74 4.80 54.14 54.00 0.14 200

FTG-wet-1-12 19.85 4.75 53.97 53.87 0.10 200

FTG-wet-1-13 19.93 4.82 53.53 53.92 0.39 200

FTG-wet-1-14 19.74 4.79 53.46 53.82 0.36 200

FTG-wet-1-15 20.01 4.88 53.95 54.07 0.12 200

FTG-wet-1-16 19.83 4.88 53.70 53.79 0.09 200

FTG-wet-1-17 20.03 4.84 53.86 54.28 0.42 199

rements was 0.02. In addition, for each specimen, the SDV value for each group of thickness measurements hfl, hi2 and hi3 (i = 2, 3, 4) are presented, as well, in Table S3. The largest value of CV for the thickness measurements is 0.02, as well.

In addition, the delamination length measurements shown in Fig. S2, at the front and back sides of the specimen, a0f) and a0b), respectively, and the difference between them A0 are also presented in Table S4. The measurements were made from the center of the loading block hole to the delamination front using the optical mode of the confocal microscope. The values of A0 in Table S4 are less than 2 mm as recommended in both fracture toughness standards. Similarly, after the tests were performed, the delamination length was measured on both sides of the specimen; these measurements will be presented in Sect. S6. The length of each specimen was measured with a ruler and is also presented in Table S4.

S2. Mechanical and thermal properties of plies in the laminate

This investigation deals with carbon fiber reinforced polymer (CFRP) composite laminates. Figure 2 in the body of the paper may be consulted for the laminate layup. The constituents include carbon fibers in an epoxy matrix with a small amount of glass fibers in the UD-fabric ply. The mechanical properties of the carbon fibers taken from [4]. The coefficients of thermal expansion (CTEs) were taken

from [5, 6]. Typical properties were taken for the epoxy. From [7], the properties of the glass fibers were obtained. The mechanical and thermal properties are presented in Table S5, where EA and ET are the Young's moduli, GA and GT are the shear moduli, vA and vT are the Poisson's ratios, aA and aT are the CTEs in the axial and transverse directions, respectively. The carbon fibers are transversely isotropic, described by five independent mechanical properties and two independent CTEs. The shear modulus GT is related to ET and vT. The epoxy and glass are both isotropic materials, described by two independent mechanical properties and one independent CTE.

The volume fraction of each constituent in the UD-fabric and woven plies are presented in Table S6. Two samples were provided for each ply and an average value was calculated which was used to obtain the effective material properties by means of the micro-mechanical model, the high-fidelity generalized method of cells (HFGMC) [8]. For the UD-fabric, these properties are presented in Table S7. It may be noted that the glass fibers which are transverse to the carbon fibers were used to increase the Young's modulus of the epoxy. Thus, the ply is effectively transversely isotropic.

In order to calculate material properties of the woven ply by means of the HFGMC micromechanical model, a repeating unit cell (RUC) was chosen as shown schematically in two dimensions as the red box in Fig. S3, a and in three dimensions in Fig. S3, b. The geometric parameters of the RUC are the thickness of the RUC h, the length of the tows a in the xj - and x3 - directions and the length and width of the epoxy pocket g between them. The parameters a and g were measured at 110 random locations of the ply using a Dino Lite digital microscope (model no. AM311ST, AnMo Electronics Corp., Taiwan). The thickness of an entire specimen was measured with a micrometer at 7 locations. The thickness of the UD fabric ply was measured to be 0.145 mm. By simple arithmetic, the ply thickness h was obtained and is presented in Table S8 together with the SDV. An additional geometric parameter, which is dependent on two of the previous ones, is the angle of the yarn P with respect to the xj - direction as shown schematically in Fig. S4. This angle is given by ■( h ^

ß = tan

-1

4 g

(S2)

and is also presented in Table S8.

Table S5

Mechanical properties [4] and CTEs [5, 6] of carbon fibers, epoxy and glass fibers [7]

Material Ea, GPa ET, GPa Ga, GPa GT, GPa VA vT a A, 10-6/°C aT, 10-6/°C

Carbon 230.0 8.0 27.3 3.08 0.26 0.30 -0.41 10.08

Epoxy 2.8 - - - 0.36 - 70.0 -

Glass 69.0 - - - 0.22 - 7.2 -

Table S6

Volume fractions of constituents in the UD-fabric and woven plies

Component UD-fabric Weave

Sample 1 Sample 2 Average Sample 1 Sample 2 Average

Carbon, % 51.90 51.60 51.8 42.67 42.76 42.7

Epoxy, % 39.44 40.54 40.0 53.81 53.85 53.8

Glass, % 4.56 4.10 4.3 - - -

Voids, % 4.10 3.76 3.9 3.52 3.39 3.5

Effective mechanical properties and CTEs of UD-fabric ply

Table S7

Ea, GPa ET, GPa Ga, GPa GT, GPa vA vT aA, 10-6/°C aT, 10-6/°C

104.4 8.74 6.60 3.15 0.30 0.39 2.32 49.70

Fig. S3. Two-dimensional schematic view of a plain weave with an RUC shown in red (a); three-dimensional schematic view of the RUC (b); image of +45°/-45° woven ply (c) (color online)

Using HFGMC with these measurements, together with the constituent properties and volume fractions, the effective mechanical properties and CTEs of the woven ply with tows in the 0°/90°-directions were found; they are presented in Table S9. Further details my be found in [9]. These properties were rotated [10] by 45° about the x2- axis to obtain the effective mechanical properties and CTEs of the ply with tows in the +45°/-45°-direction, which are also presented in Table S9 (see Fig. S3, c).

Rather than modeling each individual ply, six outer, upper and lower plies of the laminate, as presented in Fig. 2, were modeled in the finite element analyses as an effective material with anisotropic, homogenized properties. In this

Table S8

Geometrical measurements of the RUC in the woven ply

Parameter Value SDV

a, mm 1.75 0.05

g, mm 0.25 0.05

h, mm 0.27 0.03

ß 15.1° -

way, computer time and memory were reduced. The mechanical properties of the effective material were calculated by means of HFGMC and are presented in Table S10. The effective CTEs are the same as for each woven ply and presented in Table S9. All other plies were modeled individually in the analyses.

S3. Test setup

The tests (fracture toughness and fatigue) were performed with an Instron hydraulic tensile testing machine (model number 8872; High Wycombe, UK). It includes a fast track 8800 controller as shown in Fig. S5. The load cell has a capacity of250 N with an accuracy of ±0.25% of the load measured for a load greater than 2.5 N. To track the delamination front, a digital camera from LaVision (model Imager pro SX, LaVision GmbH, Gottingen, Ger-

Fig. S4. Schematic view of ß in the RUC

Table S9

Weave with tows in the 0°/90°- and +45°/-45°-directions: effective mechanical properties and CTEs

Effective mechanical properties

Orientation E11 = E33, GPa e22, GPa G13, GPa G21 = G23' GPa V13 =

0°/90° 43.9 4.9 2.3 1.8 0.034 0.046

+45°/-45° 8.4 4.9 21.2 1.8 0.816 0.046

Effective CTEs

Orientation an, 10-6/°C a22,10-6/°C a33, 10-6/°C a23, 10-6/°C a13, 10-6/°C a12,10-6/°C

Both ply types 4.3 72.1 4.3 0 0 0

many), was used, with the accompanying computer and software. Using an external PTU (programmable timing unit) controlled by DaVis computer software [11], the images taken with the camera were adjusted to the load and actuator displacement at the moment of capture. An image of the DCB specimen, taken with the camera of the LaVision system, is shown in Fig. S6. All of the LaVision components were synchronized with a LaVision light source shown in Fig. S5. Each time an image was taken, the light source was turned on. In addition, the LaVision system is synchronized with the Instron; the load value appears on the image of the specimen. Attached to the specimen was millimeter paper used for calibration when the delamination length was determined visually. In Fig. S6, it is possible to see tracking marks for the post-test delamination propagation measurements. On both sides of the specimens, white acrylic paint was applied for easier measurement of the delamina-tion length after the test.

S4. Finite element analyses and convergence study

In this study, DCB specimens were modeled for finite element analyses (FEAs) [12]. Achieving convergence of the results, requires sufficiently fine meshes. For the convergence study, a DCB specimen was modeled with dimensions shown in Fig. 1 of l = 200 mm, b = 20 mm, h = 5 mm and a0 = 50 mm. The loading blocks were not modeled. It was found that the difference in the results with and without loading blocks is negligible. Four different meshes were used in the convergence study. Very coarse, coarse, fine and finest meshes with a description of each presented in Table S11. Twenty noded, isoparametric brick elements were used in the analyses. Along the delamination front, quarter-point elements were used to model the dominant

Table S10

Effective mechanical properties of alternating +45°/-45° and 0°/90° plies

En = E33, GPa E22, GPa G13, GPa G21 = G23, GPa V13 =

30.9 4.9 11.7 1.8 0.32 0.046

square-root singularity. The oscillatory part of the singularity was not modeled.

The laminate layup is presented in Fig. 2. Each of the seven plies, four above and three below the interface, was

Digital camera

Fig. S5. Test system; zoom out view (a) and close-up (b) (color online)

Fig. S6. Image of DCB specimen taken with LaVision digital camera (color online)

Table S11

Four meshes which were used in a convergence study of the DCB specimen

Mesh No. of elements No. of nodes Element size near delamination front, mm3

Very coarse 18 720 85 007 0.145x0.145x0.5

Coarse 82 360 352 290 0.048x0.048x0.5

Fine 201 600 846 733 0.024x0.024x0.5

Finest 310000 1293 831 0.018x0.018x0.5

modeled individually using their effective homogenous, anisotropic material properties given in Tables S7 and S9. The outer six upper and lower plies were modeled as one effective homogenous, anisotropic material with mechanical properties given in Table S10 and the CTEs in Table S9. These plies are relatively far from the delamination/inter-face. Therefore, there is no need to model each ply individually. In total, there were four material types used in the finite element analyses: the UD-fabric ply, the +45°/ -45° woven ply, the 0°/90° woven ply and the group of outer plies. In Fig. 4, the distribution of plies in the finite element model, according to the materials which were used, is presented, with the effective outer material truncated. Thus, the thickness of the outer ply groups are not to scale relative to the other plies.

Two different problems were solved in the convergence study. The first problem examined the DCB specimen with an applied load; in the second, a change in temperature was considered. For the first problem, a line-force perpendicular to the delamination in the x2 - direction (see Fig. S1) along the specimen width was applied to nodes along the upper surface of the upper arm of the DCB specimen. The force was applied at a distance a0 from the delamination front. In addition, the opposite nodes on the lower specimen arm were constrained to be stationary in the xj - and x2 - directions. The applied load and displacement boundary conditions cause the specimen arms to separate in the x2 - direction. For all four meshes, there were 40 elements along the width of the specimen which is in the x3- direction. As a result of this number and the type of element

123 2Fei/3

Fig. S7. Example of nodal point forces for three elements along the load line for a DCB specimen (color online)

used, the displacement boundary conditions were applied to 81 nodes on the lower specimen arm. The total force applied in the model was

FT = 20 N, (S3)

which occurs at the end of the linear region of the load-displacement curves in the tests. This force was divided equally between the 40 elements along the specimen width. The force acting on each element is given by

Fel = ft = 0.5 N, (S4)

nel

where nel is the number of elements across the width of a specimen. The force Fel from Eq. (S4) is divided between the three nodes of an element according to finite element theory. In Fig. S7, the load acting on the center node is 2 Fel/3, while the force acting on the outer two nodes is Fel/6. As a result, the total force on a shared node from two elements is Fel/3.

In the second problem, a temperature difference was imposed with

Aft = ft,. - 90°. (S5)

In Eq. (S5), was taken to be a typical room temperature value of ft,- = 25°C. Thus, Aft = -65°C. In Eq. (S5), the value of 90°C is the maximum temperature which was measured in the laminate during the curing process.

By means of mechanical and thermal M-integrals [9], mechanical and thermal residual stress intensity factors were obtained along the delamination front, for the four meshes. For 0.0375 < x3/ b < 0.9625, the relative difference between the stress intensity factors, for each pair of meshes and for each problem type, was calculated. In Table S12, the largest relative differences (LRD) calculated between the stress intensity factors are presented. One may observe the con-

Meshes Very coarse and coarse Coarse and fine Fine and finest

K2 KIII K2 KIII Ki K2 KIII

Mechanical model

LRD, % 2.0 148.9 9.1 0.09 1.5 1.0 0.07 0.3 0.2

Thermal model

LRD, % 42.7 37.4 8.0 3.7 5.2 0.8 1.5 1.0 0.2

Table S12

Largest relative difference (LRD) between stress intensity factors along the delamination front with 0.0375 < x3/b < 0.9625, for pairs of meshes and for both problems. The data of the meshes is presented in Table S11

Fig. S8. Finite element mesh for DCB specimen used in this investigation with mesh data presented in Table S11 (color online)

vergence of the stress intensity factors as the meshes become finer. The finest mesh was used in the remainder of this investigation. The data of the mesh is presented in Table S11 and the mesh itself is illustrated in Fig. S8.

Using the finest mesh determined in the convergence study, mechanical and thermal finite element analyses were performed using specimen FT-wet-1 -04 with initial and final delamination lengths from the fracture toughness tests

Table S13

Table S14

Constants pi0 (i = 0,1,5) and Pj1 (i = of Eq. (7) for specimens FT-wet-1

0,1,4) -04

Constants pi0 (i = 0,1,5) and pj1 (i = of Eq. (7) for specimen FT-wet-1

0,1,4) 05

Constant K(f), I 4f), MP^mm(mm)-IE K(f), MPWmm Constant K(f), | K2f), MPWmm(mm)-iE K(f), KIII , MPWmm

Poo 0.1676 0.0233 -0.0467 P00 0.1674 0.0232 -0.0446

Pio -0.5801 -0.1628 1.6958 P10 -0.5812 -0.1640 1.6378

P20 0.8781 0.5247 -11.6128 P20 0.8549 0.5318 -11.2065

P30 -0.5945 -0.7231 30.4266 P30 -0.5527 -0.7366 29.3392

P40 0.2955 0.3603 -34.0272 P40 0.2861 0.3704 -32.8023

P50 0.0009 0.0009 13.6109 P50 -0.0072 -0.0016 13.1209

P01 0.1654 -0.2230 -0.2514 P01 0.1562 -0.2179 -0.2465

P11 9.7156 2.2110 0.9411 P11 9.6137 2.1701 0.9165

P21 -29.1336 -8.2727 -1.3149 P21 -28.8535 -8.1355 -1.2706

P31 38.8350 12.1235 0.8767 P31 38.4794 11.9304 0.8471

P41 -19.4170 -6.0617 -0.0001 P41 -19.2395 -5.9650 0.0000

Table S15 Table S16

Constants pi0 (i = 0,1,5) and pj1 (i = 0,1,4) of Eq. (7) for specimen FT-wet-1-06 Constants pi0 (i = 0,1,5) and pj1 (i = 0,1,4) of Eq. (7) for specimen FT-wet-1-07

Constant K(f), | K2f), MPWmm(mm)-,E K(f), KIII , MPWmm Constant K(f), | K2f), MPWmm(mm)-,E K(f), KIII , MPWmm

P00 0.1695 0.0235 -0.0463 P00 0.1734 0.0242 -0.0448

P10 -0.5892 -0.1651 1.6929 P10 -0.6233 -0.1746 1.6651

P20 0.8762 0.5333 -11.5884 P20 0.9529 0.5696 -11.3866

P30 -0.5784 -0.7386 30.3502 P30 -0.6525 -0.7894 29.7923

P40 0.2969 0.3724 -33.9370 P40 0.3150 0.3936 -33.3019

P50 -0.0056 -0.0020 13.5748 P50 0.0077 0.0006 13.3208

P01 0.1639 -0.2264 -0.2560 P01 0.1552 -0.2281 -0.2583

P11 9.9238 2.2504 0.9531 P11 10.0654 2.2643 0.9530

P21 -29.7794 -8.4313 -1.3231 P21 -30.2670 -8.5157 -1.3092

P31 39.7110 12.3623 0.8821 P31 40.4029 12.5028 0.8729

P41 -19.8555 -6.1814 0.0000 P41 -20.2011 -6.2513 -0.0001

Table S17

Constants pi0 (i = 0,1,..., 5) and pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FT-wet-1-08

Constant K(f), | K2f), MPaVmm(mm)-'E K (f) kiii , MPaVmm

p00 0.1733 0.0241 -0.0465

P10 -0.6086 -0.1705 1.7119

P20 0.9002 0.5526 -11.7140

P30 -0.5781 -0.7633 30.6671

P40 0.2790 0.3793 -34.2868

P50 0.0076 0.0021 13.7148

p01 0.1618 -0.2312 -0.2615

P11 10.1134 2.2889 0.9687

P21 -30.3733 -8.5954 -1.3375

P31 40.5206 12.6136 0.8919

P41 -20.2608 -6.3071 -0.0002

of a0 = 53.9 mm and af = 114.8 mm. The final delamina-tion length was measured from the last image taken during the test. For each of the two delamination lengths, FT was taken as the value recorded in the test. The thermal analyses were performed as described here, but with the temperature change in Eq. (S5) appropriate for this specimen; namely, ft = 21.6°C, the initial temperature in the test. All stress intensity factors were calculated along the delamina-

Table S19

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FT-wet-1-05

Constant K(r), K2r), MPWmm(mm)-li K (r) KIII , MP^vmm

d0 -5.3845 0.4263 9.0172

d1 252.25 -48.738 -340.26

d2 -6.0796 x103 1.3662x 103 7.4853 x103

d3 8.7606 x104 -2.0252 x104 -9.5652 x104

d4 -8.0286 x105 1.8841 x105 7.6952 x105

d5 4.9483 x106 -1.1748x 106 -4.1234x 106

d6 -2.1297 x107 5.1031 x106 1.5242 x107

d7 6.5506x 107 -1.5809 x107 -3.9649 x107

d8 -1.4570 x108 3.5357 x 107 7.3082 x107

d9 2.3464 x108 -5.7169x 107 -9.4860 x107

d10 -2.7072 x108 6.6151 x107 8.4732 x107

d11 2.1799x 108 -5.3372x 107 -4.9551 x107

d12 -1.1627 x 108 2.8501 x107 1.7072 x107

d13 3.6894x 107 -9.0486 x106 -2.6261 x106

d14 -5.2706 x106 1.2927 x106 -102.88

Table S18

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FT-wet-1-04

Constant K1(r), K2(r), MPWmm(mm)-IS K(r), KIII , MPWmm

d0 -5.4592 0.4383 9.1482

d1 257.50 -49.530 -345.03

d2 -6.2176 x103 1.3898x 103 7.5773 x103

d3 8.9617x 104 -2.0632 x104 -9.6728 x104

d4 -8.2128 x105 1.9215 x 105 7.7771 x105

d5 5.0617x 106 -1.1990x 106 -4.1656 x106

d6 -2.1784x 107 5.2105 x106 1.5394x 107

d7 6.7004 x107 -1.6147x 107 -4.0037 x107

d8 -1.4904 x108 3.6121 x107 7.3789 x107

d9 2.4001 x108 -5.8415 x107 -9.5771 x107

d10 -2.7691 x108 6.7600 x107 8.5542 x107

d11 2.2298 x108 -5.4545 x107 -5.0024 x107

d12 -1.1893 x 108 2.9128 x 107 1.7235 x107

d13 3.7739 x107 -9.2479 x106 -2.6516 x106

d14 -5.5913 x106 1.3211 x106 -7.5699

tion front by means of the mechanical and thermal M-inte-grals, resulting in Kf (m = 1, 2, III) for the former and Km for the latter. For each of the two delamination lengths,

Table S20

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FT-wet-1-06

Constant K1(r), K2(r), MPaVmm(mmT'e K(r), KIII , MPWmm

d0 -5.3928 0.4279 9.0077

d1 253.25 -48.851 -339.42

d2 -6.1091 x103 1.3703 x103 7.4583 x103

d3 8.8054 x104 -2.0329 x104 -9.5244 x104

d4 -8.0706 x105 1.8924 x105 7.6593 x105

d5 4.9745 x106 -1.1805 x106 -4.1031 x106

d6 -2.1410x 107 5.1292 x 106 1.5164 x 107

d7 6.5858 x 107 -1.5894 x107 -3.9441 x107

d8 -1.4649 x108 3.5551 x107 7.2693 x107

d9 2.3591 x108 -5.7489 x107 -9.4350 x107

d10 -2.7219x 108 6.6527 x107 8.4274 x107

d11 2.1918x 108 -5.3678 x107 -4.9283 x107

d12 -1.1690 x 108 2.8665 x107 1.6980 x107

d13 3.7095 x107 -9.1010x 106 -2.6121 x106

d14 -5.2993 x106 1.3002 x106 -41.821

Table S21

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FT-wet-1-07

Constant K(r), 1 K2r), MPWmm(mm)-iE K (r) kiii , MPaVmm

d0 -5.2189 0.4049 8.6832

¿1 243.09 -47.104 -326.86

d2 -5.8527 x103 1.3210x 103 7.1927 x103

¿3 8.4350 x104 -1.9573 x104 -9.1935 x104

d4 -7.7322 x105 1.8205 x105 7.3972 x105

d5 4.7666 x106 -1.1349 x106 -3.9640 x106

d6 -2.0517 x107 4.9287 x106 1.4654 x107

d7 6.3113 x 107 -1.5268 x107 -3.8118 x 107

d8 -1.4039 x108 3.4143 x107 7.0263 x107

d9 2.2609 x108 -5.5203 x107 -9.1203 x107

d10 -2.6086 x108 6.3874x 107 8.1467 x107

d11 2.1006 x108 -5.1533 x107 -4.7642 x107

d12 -1.1204x 108 2.7518 x 107 1.6415 x107

d13 3.5552 x107 -8.7366 x106 -2.5252 x106

d14 -5.0789 x106 1.2481 x106 -34.833

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FT-wet-1-08

Constant K(r), 1 K2r), MPaVmm(mm)-,e K(r), kiii , MPaVmm

d0 -5.2583 0.4119 8.7479

d1 246.07 -47.536 -329.12

d2 -5.9324 x103 1.3339x 103 7.2335 x103

d3 8.5516 x 104 -1.9783 x104 -9.2387 x104

d4 -7.8392 x105 1.8413 x 105 7.4301 x 105

d5 4.8325 x106 -1.1484x 106 -3.9804 x106

d6 -2.0801 x107 4.9893 x106 1.4712x 107

d7 6.3986 x107 -1.5459 x107 -3.8264 x107

d8 -1.4233 x108 3.4576 x107 7.0525 x107

d9 2.2922 x108 -5.5910 x107 -9.1537 x107

d10 -2.6447 x108 6.4697 x107 8.1762x 107

d11 2.2197x 108 -5.2200 x107 -4.7814 x107

d12 -1.1359 x 108 2.7875 x107 1.6474 x107

d13 3.6045 x107 -8.8501 x106 -2.5344 x106

d14 -5.1492 x106 1.2643 x106 -14.148

the mechanical and thermal stress intensity factors were summed as

Km=Kf+k« (S6)

where Km is the total stress intensity factor. Two phase angles, y and ^ were calculated along the delamination front as [13]

y = tan

I (K (t)Zi'E) R( K (t)Zi'e)

(S7)

y = tan

K (t) K iii

Vk«2+k 2t)2

(S8)

In Eq. (S7), R and I are the real and imaginary parts ofthe quantities in parentheses, i = V-I, K(t) = K1(t) + iK2(t), L is an arbitrary length parameter which was set to 0.2 mm. The oscillatory parameter e = 0.0452, H1 = 7.87 GPa, H2 = = 10.49 GPa, all depending on the mechanical properties

Table S23

Constants pi0 (i = 0,1,..., 5) and pJ1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-09

Constant K(f), 1 K2f), MPaVmm(mm)-,e K(f), kiii , MPaVmm

p00 0.1728 0.0241 -0.0317

p10 -0.6163 -0.1723 1.2753

P20 0.9381 0.5609 -8.6704

p30 -0.6404 -0.7772 22.5618

p40 0.3142 0.3885 -25.1721

P50 0.0046 0.0002 10.0687

p01 0.1581 -0.2295 -0.2594

p11 10.0528 2.2669 0.9600

P21 -30.2104 -8.5236 -1.3234

p31 40.3157 12.5136 0.8819

P41 -20.1582 -6.2569 0.0003

Table S24

Constants pi0 (i = 0,1,..., 5) and pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-10

Constant K(f), 1 K2f), MPaVmm (mm)-'e K(f), kiii , MPaVmm

p00 0.1737 0.0240 -0.0324

p10 -0.6092 -0.1702 1.2996

P20 0.8971 0.5533 -8.8391

P30 -0.5791 -0.7674 23.0079

p40 0.2905 0.3850 -25.6727

P50 0.0011 -0.0006 10.2691

p01 0.1612 -0.2318 -0.2625

p11 10.1401 2.2926 0.9712

P21 -30.4597 -8.6162 -1.3383

p31 40.6430 12.6480 0.8921

p41 -20.3238 -6.3244 0.0001

Table S25

Constants Pi0 (i = 0,1,..., 5) and Pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-11

Constant Kf 1 K2f), MPaVmm(mm)-,E K(f), Km , MPaVmm

P00 0.1780 0.0248 -0.0327

P10 -0.6439 -0.1798 1.3228

P20 0.9959 0.5905 -8.9937

P30 -0.7048 -0.8218 23.4006

P40 0.3538 0.4118 -26.1072

P50 -0.0007 -0.0006 10.4428

P01 0.1625 -0.2423 -0.2742

P11 10.5546 2.3841 1.0110

P21 -31.7464 -8.9841 -1.3876

P31 42.3839 13.2002 0.9250

P41 -21.1925 -6.6003 0.0001

of the plies on either side of the interface. The phase angle ^ represents the in-plane mode mixity; whereas, the phase angle ( represents the out-of-plane to in-plane mode mixity. It was found that both ^ and ( were less than 0.2 rad along the delamination front, except near the specimen edges. Thus, the shear modes are negligible. As a result, the total energy release rate GT is considered to be the mode I energy release rate G:. Although K(t) was dominant, calculations of Gj included all stress intensity factors as given in Eq. (11).

As described in Sect. 3.1, surfaces were fit through the finite element results for each of the five fracture tough-

Table S27

Constants Pi0 (i = 0,1,..., 5) and Pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-13

Constant Kf 1 K2f), MPaVmm(mm)-,E K(f), Km , MPaVmm

P00 0.1768 0.0247 -0.0317

P10 -0.6400 -0.1799 1.2919

P20 0.9795 0.5903 -8.7797

P30 -0.6772 -0.8194 22.8341

P40 0.3368 0.4076 -25.4718

P50 0.0009 0.0013 10.1889

P01 0.1561 -0.2349 -0.2660

P11 10.2950 2.3173 0.9791

P21 -30.9708 -8.7347 -1.3414

P31 41.3507 12.8346 0.8948

P41 -20.6749 -6.4171 -0.0005

Table S26

Constants Pi0 (i = 0,1,..., 5) and Pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-12

Constant Kf 1 K2f), MPaVmm(mm)-,E K(f), Km , MPWmm

P00 0.1805 0.0252 -0.0320

P10 -0.6589 -0.1827 1.3153

P20 1.0117 0.5957 -8.9373

P30 -0.7039 -0.8251 23.2372

P40 0.3493 0.4119 -25.9190

P50 0.0015 0.0001 10.3679

P01 0.1582 -0.2446 -0.2772

P11 10.6825 2.4003 1.0166

P21 -32.1603 -9.0616 -1.3872

P31 42.9550 13.3217 0.9256

P41 -21.4770 -6.6603 -0.0007

ness specimens to obtain a relation between Kf (m = 1, 2, III), the normalized delamination length a/a0 and normalized position along the delamination front x3/b for the mechanical problem (f). The equation for the surfaces is given in Eq. (7). The values of Pi0 and pj1 (i = 0, 1, ..., 5 and j = 0, 1, ..., 4) in this equation are given in Tables S13 through S17.

For the residual stress intensity factors arising from the curing process, lines were fit through the finite element results to obtain a relation between Km (m = 1, 2, III) and normalized position along the delamination front x3 /b. The equation for the lines is given in Eq. (9). The values of di

Table S28

Constants Pi0 (i = 0,1,..., 5) and Pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-14

Constant K(f), K2f), MPaVmm(mm)-,E K(f), Km , MPWmm

P00 0.1785 0.0249 -0.0327

P10 -0.6428 -0.1795 1.3253

P20 0.9820 0.5868 -9.0088

P30 -0.6786 -0.8154 23.4365

P40 0.3384 0.4090 -26.1460

P50 0.0011 -0.0008 10.4584

P01 0.1604 -0.2406 -0.2722

P11 10.4586 2.3607 1.0032

P21 -31.4559 -8.8995 -1.3765

P31 41.9955 13.0778 0.9178

P41 -20.9983 -6.5390 -0.0002

Table S29

Constants pi0 (i = 0,1,..., 5) and pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-15

Constant K(f), 1 K2f), MPWmm(mm)-,E K(f), kiii , MPaVmm

p00 0.1738 0.0244 -0.0314

p10 -0.6268 -0.1779 1.2710

p20 0.9701 0.5868 -8.6391

p30 -0.6907 -0.8195 22.4738

p40 0.3512 0.4124 -25.0718

P50 -0.0039 -0.0019 10.0287

p01 0.1563 -0.2307 -0.2609

p11 10.1236 2.2805 0.9636

p21 -30.4432 -8.5871 -1.3252

p31 40.6403 12.6135 0.8837

p41 -20.3207 -6.3068 -0.0001

(i = 0, 1, ..., 14) in this equation are given in Tables S18 through S22.

For the nine fatigue delamination propagation specimens, the values of pi0 and pi1 (i = 0, 1, ..., 5 and j = 0, 1, ..., 4) from Eq. (7), are given for each specimen in Tables S23 through S31. The value of d (i = 0, 1, ..., 14) from Eq. (9) are given for each specimen in Tables S32 through S40.

S5. Fracture toughness tests

Fracture toughness tests were carried out on five DCB specimens guided by the ASTM [ 1] and ISO [2] standards. These standards were developed for UD laminates. Values

Table S31

Constants pi0 (i = 0,1,..., 5) and pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-17

Constant K(f), 1 K2f), MPaVmm(mm)-,e K(f), kiii , MPaVmm

p00 0.1754 0.0244 -0.0312

p10 -0.6344 -0.1755 1.2743

p20 0.9716 0.5734 -8.6587

p30 -0.6720 -0.7962 22.5160

p40 0.3321 0.3980 -25.1153

P50 0.0030 0.0003 10.0462

p01 0.1551 -0.2329 -0.2640

p11 10.2475 2.2998 0.9713

p21 -30.8342 -8.6669 -1.3299

p31 41.1734 12.7345 0.8867

p41 -20.5868 -6.3676 -0.0001

Table S30

Constants pi0 (i = 0,1,..., 5) and pj1 (i = 0,1,..., 4) of Eq. (7) for specimen FTG-wet-1-16

Constant K(f), 1 K2f), MPWmm(nm)-IS K(f), kiii , MPaVmm

p00 0.1739 0.0243 -0.0318

p10 -0.6231 -0.1742 1.2820

P20 0.9545 0.5682 -8.7153

p30 -0.6622 -0.7887 22.6765

p40 0.3315 0.3959 -25.2991

P50 -0.0008 -0.0012 10.1195

p01 0.1576 -0.2308 -0.2609

p11 10.1001 2.2774 0.9645

p21 -30.3596 -8.5678 -1.3279

p31 40.5179 12.5806 0.8848

p41 -20.2583 -6.2901 0.0004

of load and displacement were recorded during the tests to produce load-displacement curves, as presented in Fig. S9 for each specimen. Each precipitous load drop in the curve indicates unstable delamination propagation. When the load decreased in a continuous fashion, the delamination propagated stably. The first delamination propagation (first load drop) occurred for 42 N < P < 45 N, except for specimen FT-wet-1-04. In that case, the load reached 50 N before abruptly decreasing. In addition, the initial decrease in the

Table S32

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-09

Constant K(r), 1 K2r), MPWmm(mm)-,E K(r), kiii , MPaVmm

d0 -5.3020 0.4139 8.8304

d1 247.6614 -47.8829 -332.4229

d2 -5.9671 x103 1.3434 x103 7.3111

d3 8.6005 x104 -1.9917x 104 -9.3416 x103

d4 -7.8837 x105 1.8532 x105 7.5148 x104

d5 4.8598 x106 -1.1556x 106 -4.0265 x105

d6 -2.0918 x107 5.0196 x106 1.4883 x106

d7 6.4346 x107 -1.5551 x107 -3.8714 x107

d8 -1.4313 x108 3.4779 x107 7.1358 x 107

d9 2.3051 x108 -5.6235 x107 -9.2623 x107

d10 -2.6596 x108 6.5070 x107 8.2734 x107

d11 2.1416 x 108 -5.2500 x107 -4.8383 x107

d12 -1.1423 x108 2.8034 x107 1.6670 x107

d13 3.6246 x107 -8.9005 x106 -2.5646 x106

d14 -5.1781 x106 1.2715 x106 -6.1223

Table S33

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-10

Constant K(r), 1 Kr), MPaVmm(mm)-,E K(r), Km, MPaVmm

d0 -5.4504 0.4254 9.0570

d1 254.7753 -49.2489 -340.5976

d2 -6.1407 x103 1.3819x 103 7.4863 x103

d3 8.8522 x104 -2.0493 x104 -9.5620 x104

d4 -8.1152 x 105 1.9071 x105 7.6903 x105

d5 5.0028 x106 -1.1894 x 106 -4.1199 x 106

d6 -2.1535 x107 5.1668 x 106 1.5227 x107

d7 6.6245 x107 -1.6008 x107 -3.9605 x107

d8 -1.4736 x108 3.5802 x107 7.2996 x107

d9 2.3732 x108 -5.7891 x107 -9.4745 x107

d10 -2.7382 x108 6.6988 x107 8.4627 x107

d11 2.2049 x108 -5.4047 x107 -4.9489 x107

d12 -1.1761 x108 2.8861 x107 1.7051 x107

d13 3.7318x 108 -9.1628 x106 -2.6230 x106

d14 -5.3312 x 108 1.3090 x106 -60.8341

load occurred at a displacement value of about 9 mm, except for specimen FT-wet-1-04. In that case, it occurred at more than 11 mm. For a given value of displacement, dur-

Table S35

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-12

Constant K1(r), K2(r), MPaVmm(mm)-,E K(r), Km, MPWmm

d0 -5.3256 0.4049 8.7896

d1 247.0653 -47.9206 -329.6842

d2 -5.9457 x103 1.3452 x103 7.2518 x 103

d3 8.5720 x104 -1.9934 x104 -9.2673 x104

d4 -7.8606 x105 1.8540 x105 7.4554x 105

d5 4.8470 x106 -1.1558x 106 -3.9948 x106

d6 -2.0867 x107 5.0195 x106 1.4766 x107

d7 6.4199 x107 -1.5549x 107 -3.8410x 107

d8 -1.4282 x108 3.4771 x107 7.0800 x107

d9 2.3001 x108 -5.6218x 107 -9.1899 x107

d10 -2.6540 x108 6.5049 x107 8.2090 x107

d11 2.1372 x 108 -5.2481 x107 -4.8009 x107

d12 -1.1399 x 108 2.8024 x107 1.6543 x107

d13 3.6172x 107 -8.8973 x106 -2.5457 x106

d14 -5.1675 x106 1.2710x 106 183.7385

Table S34

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-11

Constant K1(r), K2(r), MPWmm(mm)-IE K(r), Km, MPaVmm

d0 -5.3021 0.4094 8.7764

d1 247.1880 -47.8327 -329.4974

d2 -5.9554x 103 1.3428 x103 7.2421 x103

d3 8.5861 x104 -1.9913 x104 -9.2503 x104

d4 -7.8725 x105 1.8531 x105 7.4396 x105

d5 4.8538 x106 -1.1557 x 106 -3.9856 x106

d6 -2.0895 x107 5.0204 x106 1.4731 x107

d7 6.4280 x107 -1.5554x 107 -3.8314x 107

d8 -1.4299 x108 3.4788 x107 7.0617x 107

d9 2.3029 x108 -5.6250x 107 -9.1657 x107

d10 -2.6572 x108 6.5089 x107 8.1870 x 107

d11 2.1397 x 108 -5.2515 x107 -4.7878 x107

d12 -1.1413x 108 2.8042 x107 1.6497 x107

d13 3.6215 x107 -8.9030 x106 -2.5382 x106

d14 -5.1735 x 106 1.2718x 106 61.8230

ing the stable delamination propagation phase, it appears that the delamination of specimen FT-wet-1-07 propagated at lower loads than the remainder of the specimens; whereas,

Table S36

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-13

Constant K1(r), K2(r), MPaVmm(mm)-,E K(r), Km, MPaVmm

d0 -5.4302 0.4176 8.9996

d1 252.5470 -48.9442 -338.1650

d2 -6.0797 x103 1.3734x 103 7.4392 x103

d3 8.7638 x104 -2.0353 x104 -9.5070 x104

d4 -8.0349 x105 1.8932 x105 7.6485 x105

d5 4.9538 x106 -1.1803 x106 -4.0983 x106

d6 -2.1325 x107 5.1262 x 106 1.5149x 107

d7 6.5603 x107 -1.5880x 107 -3.9406 x107

d8 -1.4593 x108 3.5513 x 107 7.2636 x107

d9 2.3502 x108 -5.7419 x107 -9.4281 x107

d10 -2.7117x 108 6.6439 x107 8.4216 x 107

d11 2.1837 x 108 -5.3603 x107 -4.9250 x107

d12 -1.1647 x 108 2.8624 x107 1.6969 x107

d13 3.6958 x107 -9.0876 x106 -2.6106 x106

d14 -5.2798 x106 1.2982 x106 0.0312

Table S37

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-14

Constant K(r), 1 K2r), MPaVmm(mm)-,E K(r), KIII , MPWmm

d0 -5.5793 0.4299 9.2321

¿1 260.0183 -50.3048 -346.5374

d2 -6.2642 x103 1.4125 x103 7.6168 x103

¿3 9.0316 x104 -2.0947 x104 -9.7291 x104

d4 -8.2812 x105 1.9493 x105 7.8248 x105

d5 5.1059 x106 -1.2156 x 106 -4.1919 x106

d6 -2.1980 x107 5.2808 x106 1.5493 x107

d7 6.7621 x107 -1.6361 x107 -4.0297 x107

d8 -1.5042 x108 3.6592 x107 7.4273 x107

d9 2.4226 x108 -5.9167 x107 -9.6401 x107

d10 -2.7953 x108 6.8464 x107 8.6106 x107

d11 2.2509 x108 -5.5238 x107 -5.0354 x107

d12 -1.2006 x108 2.9497 x107 1.7349 x107

d13 3.8097 x107 -9.3648 x106 -2.6688 x106

d14 -5.4424 x106 1.3378x 106 -58.4536

for specimen FT-wet-1-05, propagation occurred for higher loads than those of the other specimens.

During the tests, values of temperature and relative humidity (RH) were recorded. In Table S41, the temperature

Table S39

Constants di (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-16

Constant K1(r), K2(r), MP^mm(mm)-iE K(r), KIII , MPWmm

d0 -5.4714 0.4259 9.1018

d1 255.3895 -49.3939 -342.4768

d2 -6.1528 x 103 1.3859 x 103 7.5320 x 103

d3 8.8687 x 104 -2.0546 x 104 -9.6239 x 104

d4 -8.1301 x 105 1.9116 x 105 7.7418 x 105

d5 5.0120 x 106 -1.1920 x 106 -4.1481 x 106

d6 -2.1574 x 107 5.1776 x 106 1.5333 x 107

d7 6.6365 x 107 -1.6040 x 107 -3.9883 x 107

d8 -1.4763 x 108 3.5874 x 107 7.3513 x 107

d9 2.3774 x 108 -5.8005 x 107 -9.5419 x 107

d10 -2.7431 x 108 6.7118 x 107 8.5231 x 107

d11 2.2089 x 108 -5.4152 x 107 -4.9844 x 107

d12 -1.1782 x 108 2.8917 x 107 1.7174 x 107

d13 3.7385 x 107 -9.1807 x 106 -2.6421 x 106

d14 -5.3408 x 106 1.3115 x 106 -0.0156

Constants dt (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-15

Constant K1(r), K2(r), MPaVmm(mm)-,E K(r), KIII , MPWmm

d0 -5.3034 0.4115 8.8226

d1 247.1732 -47.8483 -332.0231

d2 -5.9522 x103 1.3422 x103 7.3051 x103

d3 8.5789 x104 -1.9893 x104 -9.3361 x104

d4 -7.8643 x105 1.8504 x105 7.5114x 105

d5 4.8481 x106 -1.1537 x 106 -4.0250 x106

d6 -2.0868 x107 5.0106 x106 1.4879 x107

d7 6.4194 x107 -1.5522 x107 -3.8704 x107

d8 -1.4279 x108 3.4712 x107 7.1342 x107

d9 2.2996 x108 -5.6125 x107 -9.2603 x107

d10 -2.6533 x108 6.4941 x107 8.2717 x107

d11 2.1366 x108 -5.2395 x107 -4.8374 x107

d12 -1.1396 x 108 2.7978 x107 1.6667 x107

d13 3.6161 x107 -8.8827 x106 -2.5642 x106

d14 -5.1659 x106 1.2690 x106 0.1980

and RH values at the beginning, middle and end of the tests are presented. The ASTM standard prescribes $ = 23 ± 3°C and RH = 50 ± 10%. The temperature values in the middle of the test were not recorded for specimens FT-wet-1-07

Table S40

Constants dt (i = 0,1, 2,..., 14) of Eq. (9) for specimen FTG-wet-1-17

Constant K1(r), K2(r), MPWmm(mm)-,E K(r), KIII , MPWmm

d0 -5.4937 0.4225 9.1157

d1 255.3185 -49.5157 -342.7493

d2 -6.1447 x 103 1.3891 x 103 7.5431 x 103

d3 8.8565 x 104 -2.0582 x 104 -9.6422 x 104

d4 -8.1196 x 105 1.9141 x 105 7.7584 x 105

d5 5.0059 x 106 -1.1932 x 106 -4.1576 x 106

d6 -2.1548 x 107 5.1818 x 106 1.5370 x 107

d7 6.6289 x 107 -1.6051 x 107 -3.9981 x 107

d8 -1.4746 x 108 3.5894 x 107 7.3698 x 107

d9 2.3748 x 108 -5.8034 x 107 -9.5662 x 107

d10 -2.7400 x 108 6.7149 x 107 8.5451 x 107

d11 2.2064 x 108 -5.4175 x 107 -4.9973 x 107

d12 -1.1768 x 108 2.8928 x 107 1.7218 x 107

d13 3.7344 x 107 -9.1843 x 106 -2.6489 x 106

d14 -5.3348 x 106 1.3120 x 106 0.2844

Temperature and relative humidity during fracture toughness tests

Specimen umit> C 1 °C u mid-test' C f °C RHinlt' % RHmid-test ' % RHfin' %

FT-wet-1-04 21.6 22.2 22.4 60.4 59.8 60.1

FT-wet-1-05 22.2 22.7 22.4 57.8 56.8 57.6

FT-wet-1-06 22.6 22.7 22.6 56.9 57.0 57.4

FT-wet-1-07 24.8 - 24.6 - - -

FT-wet-1-08 24.6 - 24.2 - - -

P, N

C, mm/N

50 d, mm

Fig. S9. Fracture toughness tests; load-displacement curves: FT-wet-1-04 (1), FT-wet-1-05 (2), FT-wet-1-06 (3), FT-wet-1-07 (4), FT-wet-1-08 (J) (color online)

and FT-wet-1-08; there were also no values for RH recorded for these two specimens. All the temperature values are in the range specified in the ASTM standard. The RH values are also in the required range except for RHinit and RHfin for specimen FT-wet-1-04. After the tests were carried out, using the optical mode of the Olympus confocal microscope, the delamination length was measured on both sides of each specimen. Those values are presented in Table S42, as and respectively, as well as their difference Af. It may be observed that Af < 2 mm for all specimens as required in the standards. For specimen FT-wet-1-04, the delamination length was not measured after the test using the confocal microscope. Based on these tests, a GIR-curve is presented in Fig. 9.

S6. Fatigue delamination propagation tests

Fatigue delamination propagation tests were carried out on five specimens at a cyclic displacement ratio R = 0.1

Table S42

Fracture toughness tests; final delamination length

Specimen a(f)' mm a(b) ' mm Af, mm

FT-wet-1-04 - - -

FT-wet-1-05 114.60 113.35 1.25

FT-wet-1-06 121.79 121.91 0.12

FT-wet-1-07 134.23 135.06 0.83

FT-wet-1-08 135.26 134.48 0.78

0.0 0.5 1.0 1.5 2.0 N, x106 cycles Fig. S10. Compliance values calculated by Eq. (13) for fatigue delamination tests: FTG-wet-1-09 (1), FTG-wet-1-10 (2), FTG-wet-1-11 (3), FTG-wet-1-12 (4), FTG-wet-1-13 (J), FTG-wet-1-14 (6), FTG-wet-1-15 (7), FTG-wet-1-16 (8), FTG-wet-1-17 (9). Rd = 0.10 (1-J), 0.48 (6-9) (color online)

and four specimens at Rd = 0.48. For all specimens, the compliance values calculated during the test are illustrated in Fig. S10 as a function of the cycle number N. It may be observed that there is scatter in the compliance values between specimens with the same displacement ratio Rd. Note that for specimen FTG-wet-1-14, all compliance values are shown in the graph. As discussed in the paper, only those obtained for N < 475 000 were taken into account in the calculations (see Table 1).

In Table S43, the constants g and n and the coefficient of determination R for Eq. (14), found for each speci-

Table S43

For each specimen, fitting constants and coefficient of determination for Eq. (14)

Specimen g n R2

FTG-wet-1-09 98.97 0.3386 0.9949

FTG-wet-1-10 98.82 0.3368 0.9955

FTG-wet-1-11 100.24 0.3575 0.9953

FTG-wet-1-12 100.84 0.3585 0.9955

FTG-wet-1-13 101.01 0.3730 0.9971

FTG-wet-1-14 106.70 0.4125 0.9926

FTG-wet-1-15 100.71 0.3403 0.9779

FTG-wet-1-16 109.08 0.4134 0.9946

FTG-wet-1-17 103.01 0.3745 0.9942

Table S44

For each specimen, fitting constants and coefficient of determination for Eq. (15) and number of sampling points

Specimen A1, mm A2, mm B B2 R2 Sample points

FTG-wet-1-09 5.74 50.42 -62.36 0.114 0.9975 734

FTG-wet-1-10 14.84 38.99 1.02 0.074 0.9965 794

FTG-wet-1-11 26.09 23.75 361.63 0.053 0.9963 806

FTG-wet-1-12 245.42 -201.65 933.13 0.010 0.9946 806

FTG-wet-1-13 163.55 -119.12 392.39 0.015 0.9985 806

FTG-wet-1-14 5.37 -50.92 0.01 0.103 0.9964 601

FTG-wet-1-15 687.71 -642.30 0.01 0.003 0.9700 541

FTG-wet-1-16 4.59 51.05 0.01 0.010 0.9918 806

FTG-wet-1-17 32.88 20.30 0.01 0.036 0.9944 806

men, are presented. Using the compliance values obtained during each cycle, the delamination length was calculated by means of Eq. (14). In Table S43 for all specimens, it may be observed that R2 is very high. Note that for specimen FTG-wet-1-15, there were 15 delamination length values used to determine g and n; while for the other specimens, there were between 19 to 28.

In Sect. 3.3, a method for obtaining a relation between the delamination length a and the cycle number N was described with a relation given in Eq. (15). For each specimen, in Table S44, the fitting constants A1, A2, B1 and B2 and the coefficient of determination R2 of Eq. (15) are presented. The number of sampling points is also given in Table S44. It may be observed that R2 is very high for all specimens; for specimen FTG-wet-1-15, it is slightly lower than the others. Here there is a lower number of sampling points. When data from all cycles for this specimen were used, a higher value of R , like those for other specimens was achieved. Nevertheless, R2 for this specimen is still excellent. It may be concluded from the high R2 values

that the method for obtaining Eq. (15) is an effective tool for determining a relation between a and N. In addition, from Fig. 7, for specimen FTG-wet-1-09, it may be seen that the visually measured delamination lengths, from the images taken during the test, fit well to the curve described by Eq. (15). Moreover, it may be observed in Table S44 that the values of the fitting constants of Eq. (15) are quite variable. For each specimen, different fitting constants were obtained which led to the best value of R2.

In Table S45, the average cyclic loading ratios Rp for each specimen during the first 10 000 cycles, are shown. For specimen FTG-wet-1-15, Rp is an average for the first 30 000 to 40 000 cycles. It may be observed that the cyclic load ratios differ somewhat from the cyclic displacement ratios. For values of N > 10 000, Rp does not change significantly. In addition, in Table S45, the delamination length at the end of the tests for all specimens, ofP and the front and back sides of the specimen, respectively, are shown. The difference between these lengths is denoted as Af and is also presented in Table S45. The values of Af,

Table S45

Cyclic load ratio for the first 10 000 cycles in fatigue delamination propagation tests, final delamination lengths, their difference and initial temperature in the tests

Specimen Rp, first 10k cycles a(f), mm a(b), mm A f, mm °C

FTG-wet-1-09 0.095 81.35 82.11 0.76 23.8

FTG-wet-1-10 0.092 82.81 83.79 0.98 22.3

FTG-wet-1-11 0.103 81.54 80.42 0.88 22.5

FTG-wet-1-12 0.087 83.31 83.20 0.11 24.3

FTG-wet-1-13 0.086 83.93 84.84 0.91 22.6

FTG-wet-1-14 0.444 74.08 75.23 1.15 22.3

FTG-wet-1-15 0.437 (30k to 40k) 77.14 77.46 0.32 23.8

FTG-wet-1-16 0.455 71.65 71.83 0.18 21.8

FTG-wet-1-17 0.452 76.67 77.47 0.80 22.6

Table S46

Coefficient of determination for Eqs. (S9) and (S10)

Equation No. R2 value

(S9) 0.9529

(S10) 0.8478

are less than 2 mm as required by the two fracture toughness test standards [1, 2]. In addition, the initial temperature in the tests is also shown in Table S45 and is denoted as . This temperature value was used in the thermal analyses described in Sect. 3.1 in the body of the paper. It should be noted that during the fatigue tests, the temperature may change by as much as 1.5°C from its initial value. It does not appear that this difference affects the calculated results of GImax or GImin in each cycle. To examine this point, a thermal analysis was carried out on specimen FTG-wet-1-17 with = 25 °C. The greatest difference in GImin in each cycle between this analysis and that when = 22.6°C was 0.3%. This calculation was carried out for GImin rather than GImax, since the former is more sensitive to temperature changes.

Values of GIthr and GIthr from Eqs. (19) and (20), respectively, were determined. As described in Sect. 4.3, these values may be found from Eqs. (25) and (26), respectively, once AGIeffthr and AGIeffthr are known. To obtain AGIeffthr, all of the test results are plotted versus the cycle number N on a log-log scale as illustrated in Fig. 13 for AGIeff. For each value of Rd, a straight line was fit through each group of results. The intersection of the lines yields an initial value AcG-^ef thr. This value of AGIeff is the initial value used to obtain a final value A(jI(ffthr when determining the master-curve. The expressions for the fitting lines are given as

..........(S9)

AGIeff = 1.6671 N

-0.1581

, N/m

102 104 106 10

10 10'

N, cycles

Fig. S11. Plot of AGIeff versus the cycle number N, for all fatigue specimens: FTG-wet-1-09 (1); FTG-wet-1-10 (2); FTG-wet-1-11 (3); FTG-wet-1-12 (4); FTG-wet-1-13 (J); FTG-wet-1-14 (6); FTG-wet-1-15 (7); FTG-wet-1-16 (8); FTG-wet-1-17 (9); fit, Rd = 0.10 (10); fit, Rd = 0.48 (11) (color online)

Final threshold values Gf for AK1 or AK2 as f (GI)

and Gf

in Eq. (1)

Table S47

Specimen AK1 AK2

GIthr Githr, N/m G(f) GIthr f N/m

FTG-wet-1-09 0.0548 39.9 0.0715 52.0

FTG-wet-1-10 0.0546 39.7 0.0711 51.7

FTG-wet-1-11 0.0559 40.7 0.0729 53.0

FTG-wet-1-12 0.0540 39.3 0.0704 51.2

FTG-wet-1-13 0.0537 39.1 0.0701 51.0

FTG-wet-1-14 0.1455 105.9 0.1898 138.1

FTG-wet-1-15 0.1417 103.1 0.1847 134.4

FTG-wet-1-16 0.1513 110.1 0.1973 143.6

FTG-wet-1-17 0.1496 108.9 0.1952 142.0

AGIeff = 0.4616N

-0.1004

(S10)

for Rd = 0.1 and Rd = 0.48, respectively. The coefficient of determination for Eqs. (S9) and (S10) is given in Table S46. The difference in R2 for the two equations results from the difference between the behavior of specimen FTG-wet-1-16 for Rd = 0.48 and that of the other specimens for the same displacement ratio. Note that the initial threshold values are used to obtain the master-curves. This value is changed until a best fit is obtained between the fatigue data and the master-curve.

The same procedure for obtaining AG^f^, as explained in Sect. 4.3, may be carried out to determine an initial value AG^ef^. In may be noted that

AGIeff = ((GImax ~4GImin )• (S11)

With a plot of AGIeff versus N as shown in Fig. S11, a value of AGf = 33.98 N/m was found. This value of AGIeff was used as an initial value in the master curve

calculation for determining a final value AGI(ffthr. In Table S47, the final values of GItl.r and GItllr as calcu-

lated from Eqs. (25) and (26), respectively, are presented for all specimens. The dimensional results in Table S47 are reasonable. Perhaps they may be used in an industrial setting to prevent failure. Finally, it may be observed that there is a difference between the values of G^thT when calculated by means of the two master curves for AK1 and AK2. This is to be expected since the curves are different as seen in Table 3.

References

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Received November 02, 2018, revised November 02, 2018, accepted November 09, 2018

Ceedeuua 06 aemopax

Tomer Chocron, M.Sc. Student, Tel Aviv University, Israel, tomerchocron@mail.tau.ac.il Leslie Banks-Sills, Professor Emerita, Tel Aviv University, Israel, banks@eng.tau.ac.il