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УДК 539.422.5
Анализ разрушения иолиметилметакрилата при нагружении типа I с использованием модифицированных энергетических критериев
M.R. Ayatollahi1, S.M.J. Razavi1, M. Rashidi Moghaddam1, F. Berto2
1 Иранский университет науки и технологии, Тегеран, 16846, Иран 2 Падуанский университет, Виченца, 36100, Италия
В статье обсуждаются два энергетических подхода к предсказанию траектории трещины и разрушающей нагрузки в деталях с трещиной нормального отрыва. Проведено экспериментальное и теоретическое исследование трещиностойкости компактных образцов на растяжение и двухконсольных образцов из полиметилметакрилата. В условиях нагружения типа I траектории роста трещины, а также значения истинного сопротивления разрушению для данных образцов сильно различаются. Два модифицированных критерия — плотность энергии деформации и усредненная плотность энергии деформации — используются для оценки траектории трещины и разрушающей нагрузки для хрупких материалов с учетом влияния Т-напряжения. Показано, что различия в траекториях трещины и в сопротивлении разрушению разных образцов из полиметилметакрилата с трещиной связаны с величиной и знаком Т-напряжения.
Ключевые слова: модифицированные энергетические критерии, Т-напряжение, хрупкое разрушение, зона процесса, траектория трещины
Mode I fracture analysis of polymethylmetacrylate using modified energy-based models
M.R. Ayatollahi1, S.M.J. Razavi1, M. Rashidi Moghaddam1, and F. Berto2
1 Fatigue and Fracture Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16846, Iran 2 University of Padova, Vicenza, 36100, Italy
The paper presents two energy-based approaches to predict the fracture trajectory and the fracture load in components containing a mode I crack. The fracture behavior of polymethylmetacrylate (PMMA) samples was investigated experimentally and theoretically for compact tension and double cantilever beam test specimens. The crack growth trajectories and the values of apparent fracture resistance in these two specimens were considerably different although both were under pure mode I loading. Two energy-based methods, i.e., the strain energy density and the averaged strain energy density criteria were modified to estimate the fracture trajectory and the fracture load in brittle materials respectively by considering the T-stress effects. The difference between the crack trajectories and the fracture resistances of different cracked specimens of the same material (PMMA) was found to be related to the magnitude and the sign of T-stress.
Keywords: modified energy-based criteria, T-stress, brittle fracture, process zone, crack trajectory
1. Introduction
Different models have been proposed in the past for describing the physical procedure associated with the onset of crack growth in brittle and quasi-brittle materials under static loading. When a cracked body is subjected to far-field loads, a volume of material at the vicinity of the crack tip is subjected to very large strains. Mesomechanical models for brittle fracture suggest that large strains near the crack tip generate a small zone within which the material is severely damaged. For brittle and quasi-brittle materials, the localized damage can be observed as microcracking,
crazing, etc. At the onset of fracture, this damage zone, often called the fracture process zone or the control volume, is not capable of sustaining additional stresses or energies, and hence the crack propagation is expected to initiate from the boundary of the damage zone and not from the original crack tip. In the case of sharp cracks, the damage zone is approximated by an area of circular shape which its radius is equal to a critical distance rc from the crack tip. Mesome-chanical fracture models which consider the presence of crack tip damage zone are often based on a critical stress or a critical energy in the neighborhood of crack tip. The sin-
© Ayatollahi M.R., Razavi S.M.J., Rashidi Moghaddam M., Berto F., 2015
gular terms of elastic stresses at the critical distance rc from the crack tip are sometimes not sufficiently accurate particularly for materials having relatively large fracture process zone sizes. In such cases, the addition of the second term of Williams series expansion, called the T-stress, to the singular terms can provide more accurate approximation of the crack tip stresses.
Under pure mode I loading, cracked components and structures may fracture along curvilinear paths and not necessarily along the line of initial crack [1-6]. Crack curving is more likely to occur when the value of T-stress is considerably high. As a constant stress parallel to the crack, the T-stress is independent of the distance from the crack tip. Several researchers have reported that the application of T-stress in stress-based failure models provides more reliable predictions for the onset of brittle fracture [7-10].
A generalized maximum tangential stress criterion was proposed by Smith et al. to study the effect of T-stress in mixed mode brittle fracture [7]. According to the generalized maximum tangential stress criterion, mixed mode brittle fracture is significantly influenced by the T-stress, while mode I fracture is insensitive to the value of T-stress. However, some researchers have reported that there is a considerable difference between the experimental results of mode I fracture resistance when different test specimens are used for a given brittle material [11-16]. Ayatollahi and Sedi-ghiani proposed an energy-based criterion which could consider the influence of specimen geometry on the critical mode I stress intensity factor in brittle and quasi-brittle materials [6].
Lazzarin and his co-researchers suggested the averaged strain energy density criterion which uses the energy averaged in a volume of radius rc (which depends on the material properties) to estimate the fracture load for different types of notched specimens [17-23]. According to the averaged strain energy density criterion, material failure occurs when the mean value of the deformation energy within the control volume reaches a critical value. The critical energy and the radius of the control volume are both material properties. Lazzarin applied his criterion to a variety of materials and different notch geometries (U- and V-shape notched specimens) to investigate their failure behavior. According to experimental findings, Lazzarin et al. proposed that rc depends on fracture toughness and the ultimate tensile strength of material [18]. When the notch is blunt, the control volume is assumed to be of a crescent shape and rc is its width as measured along the notch bisector line. Berto et al. extended the use of averaged strain energy density criterion to the specimens under mixed mode loading conditions [19]. For instance, they have reported that for iso-static graphite, the radius of the control volume depends on the mesomechanical parameters like grain size. These parameters do not depend on the in-plane mode mixity and the notch acuity, and can be simply evaluated based on the results of pure mode I fracture [19].
In this paper, generalized forms of minimum strain energy density criterion and averaged strain energy density criterion are developed to consider the effect of T-stress on fracture behavior of PMMA. For this purpose, two commonly used test samples are selected for conducting the fracture tests, i.e. the compact tension and the double cantilever beam specimens. These specimens have a simple geometry and loading configuration. PMMA is often considered as a brittle material that its optical transparency allows the direct observation of crack trajectory in fracture experiments. Brittle fracture is a major mode of failure in brittle materials like PMMA. Therefore, the fracture resistance is an important parameter required for failure analysis in engineering components made of brittle materials. In addition, for applying the crack growth retardation methods an approximation of the crack growth path from the existing cracks is required. In this paper, the fracture trajectory and the fracture loads of these two PMMA test specimens are investigated experimentally and theoretically. It is shown that the specimen geometry can strongly influence the fracture resistance and the crack growth path under pure mode I loading.
2. Experiments
PMMA (polymethylmethacrylate or Perspex) has been recognized by numerous researchers as a favorite model material for conducting fracture experiments. The brittle fracture behavior at room temperature, the convenience of machining and creating a sharp crack and the optical transparency are among the advantages of PMMA in brittle fracture experiments. As shown in Fig. 1, the test specimens were prepared from PMMA in two different geometry shapes, namely, the compact tension specimen (width of W= 30 mm, height of h = 30 mm) and the double cantilever beam specimen (width of W = 150 mm, height of h = = 30 mm), both with a thickness t = 10 mm. Considering the relatively large value of thickness with respect to the other geometrical dimensions, the plane strain condition is assumed in the computational investigations. For creating the cracks, first a very thin strip saw blade of thickness 0.2 mm was used to create a notch with the initial depth of slightly less than 15 and 75 mm respectively for compact tension and double cantilever beam specimens. Then, the notch tip was sharpened by pressing a razor blade carefully to make the final length of each crack equal to 15 and 75 mm
Fig. 1. A scheme of specimen geometries: compact tension (a) and double cantilever beam specimens (b)
for compact tension and double cantilever beam specimens. The crack length ratio a/W was therefore equal to 0.5 for all of the specimens. Moreover, using some standard tensile test specimens (ASTM D638), the elastic modulus E and the ultimate tensile strength at of PMMA were determined as E = 2.9 GPa and at = 55 MPa. According to reference [24], the value of Poisson's ratio for PMMA was considered to be equal v = 0.35.
Three samples for each specimen geometry were tested under quasi-static loading with a constant displacement rate of 0.1 mm/min at room temperature and the load-displacement curves of the specimens were obtained. The load-displacement curves were all linear up to the fracture load and the specimens were failed by sudden fracture. The critical load corresponding to fracture initiation was recorded for each experiment. The fracture load was used to calculate the critical mode I stress intensity factor, KIf (material resistance against brittle fracture) and the corresponding T-stress from finite element analysis.
3. Fracture model
The elastic stress field around the crack tip can be expressed as a set of infinite series expansions:
1
y/Íñr'
0 1 . „ . 30 cos---sin 0 sin-
2 2 2
' 0 1 . „ . 30
cos— + — sin 0sin-
2 2 2
1 . 0 30 —sin 0 cos—
2 2
„ . 0 1 . „ 30 -2 sin — + — sin 0 cos — 2 2 2
1 . 0 30 —sin 0 cos-
2 2
0 1 . „ . 30
cos---sin 0 sin-
2 2 2
J *
Un
~T T O(r12)$
0 + O(r12)
0 O(r12)
(1)
azz = v(a*x + ayy) for plane strain, a zz = 0 for plane stress, where (r, 0) are the polar coordinates with the origin located at the crack tip, K and Kn are the mode I and mode II stress intensity factors, respectively and axx, a^, azz and a are the stresses in the Cartesian coordinate system. The first term is singular and dependent of K and Kn. The first non-singular term is called the T-stress. The higher order non-singular terms O(r^2) are often negligible very close to the crack tip, v in Eq. (1) is the Poisson's ratio.
3.1. Path of crack extension
The minimum strain energy density concept is used in this study for predicting the path of crack growth. The strain
energy density factor S for in-plane loading can be expressed in terms of the stress components as
S = r
dW dA
r
IG
K+1
+ CTyy ) -CT„CT+ CT
yy'
xx yy
xy
(2)
where dW/dA is the strain energy density function and G is the modulus of rigidity which is equal to E/ (2(1+ v)), k is defined as 3 - 4v for the plane strain problems and (3 - v)/(1 + v) for the plane stress ones.
According to the minimum strain energy density criterion [25, 26], the crack growth initiates in the direction 60 along which the strain energy density factor S at a critical distance rc from the crack tip is a minimum. This can be written mathematically as
dS
30
= 0,
d 2 S
0=0o
302
> 0.
(3)
0=00
Sih [25, 26] considered only the singular term of stress in his criterion. In this paper, the effect of T-stress is also taken into account in crack tip stresses. By substituting the stress components for plane strain conditions Eq. (1) into Eq. (2) and manipulating the resulting expressions, one may write
(16nG ) S = axK\ + a2 K^ + a3 KIKII +
+ a4 (>/2rcr )KIT + a5 (V2rcr )KIIT + a6 (2nr )T2 (4) in which
aj = (k- cos 0)(1 + cos 0),
a2 = [k(1 - cos0) + cos0(1 + 3 cos0)],
a3 = 2 sin 0(2 cos 0 - (k -1)),
a4 = 2cos-j(cos(20) - cos0+ (k- 1)),
(5)
K + 1
a5 =-2sin^(cos(20)+cos0+(K+1)), a6 = ^
To normalize the effect of T-stress, a dimensionless parameter called the biaxiality ratio B is used and the critical radius from the crack tip rc is presented in the dimension-less form of a, as
Tyfna Tyfna
B = -
K
eff
Vk2+K2
and a =
(6)
where a is the crack length. By replacing Eq. (4) into Eq. (3), the direction of fracture initiation 0O is found from
bjKl + b2 K2I + ¿3KiKn + b4 (BaKff )Kl + + b5( BaKf Kn = 0, (7)
where
bj = sin 0O (2 cos 0O - k +1), b2 = -sin0O(6cos0O -k + 1),
b3 = 2(2cos(20O) - (k- 1) cos 0O), (8)
0
b4 = - sin —2^ (5 (cos(20O ) + cos 0O ) + (k +1)), 0
b5 = - cos (5 (cos(20O ) - cos 0O ) + (k + 3)).
-180° -90° 0° Angle I
180°
(N
, 3
co ^ ?
VO ^
b l
v/\ f\s
-180° -90° 0° Angle (
90°
180°
Fig. 2. Variations of the normalized strain energy density factor versus 6 under pure mode I loading for v = 0.35: positive (a) and negative (b) values of Ba = 3 (1), 2 (2), 1 (3), 0 (4)
The terms involving Ba in Eq. (7) represent the contribution of T-stress in the near crack-tip strain energy density. Equations (7) and (8) suggest that the crack trajectory under mixed mode loading depends not only on KI and Kjj but also on the sign and magnitude of T-stress and on the material properties (v and rc) as well. For pure mode I loading when Kn = 0, Eq. (4) and Eq. (7) are simplified to
(16 nG ) 5 = a K2 + a4 (BaKx ) K1 + a6 ( BaKI )2
(16nG)5 _ , ,2-,
v 7 -[a1 + a4( Ba) + a6( Ba)2],
(9)
Kt
bxK\ + b4 (BaK1) K1 = 0 ^ b1 + b4 (Ba) = 0. (10) The angular variations of the normalized strain energy density factor (16nG)sjKl (Eq. (9)) versus the angle 6 are shown in Fig. 2 for different values of Ba when v = 0.35 under pure mode I loading. It is seen that Ba has a significant influence on the strain energy density factor. In general, increasing the absolute value of Ba raises the normalized strain energy density factor.
For Ba < 0 the strain energy density factor is always minimum at 6 = 0 (Fig. 2, b). But, this is not the case for Ba > 0 where the strain energy density factor sometimes becomes minimum at angles other than 6 = 0 (Fig. 2, a).
Figure 3 illustrates the variations of the angle of minimum strain energy density or the fracture initiation angle 60 in terms of Ba for v = 0.35. The diagram shown in Fig. 3 is obtained by solving Eq. (10). It can be seen from this figure that for Ba < 0.22, the minimum strain energy density factor occurs in the initial plane of crack (6 = 0), however, when Ba is too large (i.e. greater than 0.22), the minimum strain energy density factor is no longer along 6 = 0. Subsequently, it can be deduced that for higher values of T-stress or for large process zone sizes rc, it is probable that the crack kinks out of its initial direction. Similar investigations have been performed by some other researchers theoretically and experimentally to explore parameters that affect the crack trajectory [7, 13, 14]. It should be noted that the critical value of Ba depends on the material Pois-son's ratio. For v = 0.35, the critical Ba is equal to 0.22, as described earlier.
3.2. Fracture resistance (generalized averaged strain energy density criterion)
As described earlier, the averaged strain energy density criterion is an energy-based approach for failure analysis of structures and components containing crack or notch. Theoretically, the stresses and strain energy density tend toward infinity in the vicinity of the crack tip. However, in practice the energy in a small volume surrounding the crack tip has a finite value, and failure occurs when the energy within a control volume of radius rc reaches a critical value [27].
The relation between the strain energy density function
and the strain energy density factor is as follows:
dW S 1 _ . 2
-[ a1 K2 + a2 K2 + a3 K Kn +
dA r 16nrG
+ a4V2nrKir + a542nrKllT + a6 (2nr )T2 ]. For pure mode I ( Kn = 0), Eq. (11) simplifies to: dW _ 5 _ 1 dA " " X
(11)
16nrG
x [a1 K2 + a^yl2nrK{T + a6(2nr )T2]. (12)
The elastic deformation energy within a region of radius rc (Fig. 4) around the crack tip is: rc dW
E(rc) :
f dW A dA :
A dA
n
0 -n
dr d0:
_ N1KI + N4 KT + N6T
4 KiT
(13)
Fig. 3. Angle of minimum strain energy density versus Ba under mode I loading for v = 0.35
Fig. 4. The critical volume around the crack tip
in which
N =
(2K-l)rc V2rc3/2 (5k - 7)
16G
,N4 =
15GVrc
N =
= nrc2( K +1)
(14)
16G
For a sharp crack, the area on which the integration is carried out is a circle of radius rc
rc n
A(rc) = JJ rdrd0 = nrc2. (15)
0 -n
The elastic deformation energy, averaged on the area A( rc), turns out to be:
N K2 + N4 K{T + N6T2]. (16)
E=EM =.
A(rc) nrc
According to the generalized averaged strain energy density criterion, material failure occurs when the mean value of the deformation energy reaches a limit value which is a characteristic of the material:
E = Ecr. (17)
Let's consider a mode I specimen with negligible T-stress from which the fracture toughness KIc is obtained. At the onset of fracture, one may write KI = KIc and T = 0. Therefore, by replacing these values in Eq. (16), the elastic deformation energy averaged on the area A(rc), is determined as
E =
1
nrc
NiK2,
(18)
where KIc is called mode I fracture toughness, which should be obtained experimentally using a cracked specimen in which the T-stress (or B) is zero or negligible.
By introducing Eq. (18) into Eq. (16), the onset of crack extension can be found from
N K2 = N1 Kf + N4 K if T + N6T2 (19)
or in terms of Ba from
KTc = KTf +
16(5k- 7) 15n(2K-1) K+1
(BaKf) Kf +
2(2k-1)
(BaKf )2.
(20)
If KIc is considered to be a constant material property, Eq. (20) suggests that KIf is not constant but depends on Ba (T-stress and rc). Using Eq. (20), the mode I fracture resistance of material for different geometries and loading
conditions is written in terms of KT„ as
Kb
1
, 16(5 k-7) K + 1
1 + —--- Ba +-
15n(2K-1) 2(2k- 1)
(21)
(Ba)2
3.3. Numerical procedure for crack path prediction
In this study, a numerical code was developed to predict the path of crack growth by an iterative procedure based on the crack tracking model. The direction of crack growth in each iteration is determined using the equations described in Sect. 3.1. The developed numerical code is linked to finite element (software ABAQUS to provide a two-dimensional simulation of the crack growth procedure. The values of stress intensity factors and T-stress required for the crack tracking model are calculated automatically by ABAQUS and are used as input data for the crack growth code. An interaction integral method built in ABAQUS software was used for obtaining stress intensity factors and the T-stress directly from software [28]. The crack extension is simulated by using a constant prespecified incremental length of crack growth Aa (Fig. 5).
By substituting the numerical values of stress intensity factors and T-stress obtained in each step into Eqs. (6) and (7), the crack growth angle 0O can be determined. The crack length for the next iteration is considered to be
af = a i-1 + A a. (22)
The geometry of finite element model is redefined in every step of iterative crack growth computation. The new geometry is remeshed and the previous computational steps are repeated until the crack tip reaches the specimen boundaries.
A typical mesh pattern used for simulating the crack growth is shown in Fig. 6. The elastic material properties of PMMA which was mentioned before were also consider-
Fig. 5. Incremental crack growth process
ed in the finite element modeling. Considering the large value of thickness versus other dimensions, the entire specimens were assumed to be under plane strain conditions. The 8-node biquadratic plane strain quadrilateral elements were used to mesh the finite element model. The singular elements were used in the first ring of elements surrounding the crack tip for producing the square root singularity of stress/strain field. A mesh convergence study was also conducted to assure the sufficiency of the number of elements in finite element modelling. All the specimens were loaded under pure mode I loading conditions so the crack growth initiation was modeled using the mode I relations. For the double cantilever beam specimen which had a large value of T-stress, the crack growth trajectory was deviated as described in Sect. 3.1. In this case, the incremental crack propagation was simulated using the mixed mode relations.
4. Results and discussion
The crack growth code described in Sect. 3.3 is now extended to study the fracture trajectories for the compact tension and double cantilever beam specimens using the
generalized strain energy density criterion. Also, the generalized averaged strain energy density criterion described in Sect. 3.2 is employed to predict the value of mode I fracture resistance of PMMA for the two specimens used in experiments. The theoretical results are then compared with the experimental data. The details of experimental results including the specimen dimensions, the biaxiality ratio B, the fracture loads, the critical mode I stress intensity factor (or fracture resistance KIf) and the measured fracture initiation angles are listed in Table 1 for both compact tension and double cantilever beam specimens. The fracture loads obtained from the experiments were applied to the finite element model and the critical values of crack tip parameters (KIf and T-stress) were determined. It is seen from Table 1 that B (or the T-stress) has a negligible value in the compact tension specimen. Therefore, the fracture resistance KIf = 1.4 MPa • m05 obtained from this specimen can be considered as material fracture toughness KIc of the tested PMMA sample. It is noteworthy that the experiments performed on the compact tension specimen comply with the ASTM standard E1820, too. Schmidt [29] has suggested
Table 1
The values of fracture resistance, fracture initiation angle, biaxiality ratio B and Ba for the tested PMMA specimens (a/W = 0.5), all dimensions in (mm)
Specimen type W h a B Ba Fracture load Fc, N Kif, MPa • m05 Fracture initiation angle 60 (theoretical) Fracture initiation angle 60 (experimental)
322 1.41 0 0
Compact 30 30 15 0.51 0.06 320 1.40 0 0
tension 315 1.38 0 0
Avg. = 319 Avg. = 1.40 Avg. = 0 Avg. = 0
83 1.24 -55 -53
Double cantilever 150 30 75 4.80 0.25 88 1.31 -55 -55
beam 90 1.34 -55 - 60
Avg. = 87 Avg. = 1.30 Avg. = -55 Avg. = -56
Fig. 7. Sample PMMA specimens broken under pure mode I loading
Eq. (23) to estimate the size of critical distance rc for brittle and quasi-brittle materials:
r =-
2n
KT
\2
(23)
Based on the experimental value of mode I fracture toughness (KIc = 1.4 MPa • m05) and the tensile strength (a t = 55 MPa) of the tested PMMA, the critical distance for this material is obtained from Eq. (23) as rc = 0.1 mm. This value of critical distance was used for calculating Ba from Eq. (6) as given in Table 1.
4.1. Crack trajectory
As described in Sect. 3.1, even in pure mode I loading where stress/strain fields are initially symmetric relative to the crack line, crack growth might take place along curved paths and not necessarily along the direction of the original crack. Based on the generalized strain energy density criterion, crack curving is expected to occur for specimens having large T-stresses. The prediction of the crack growth path
is one of the topics of interest in many practical applications. To predict the path of crack growth in a cracked specimen, first the fracture initiation angle at the crack tip should be determined. Figure 7 shows two sample PMMA specimens failed under pure mode I loading. It is seen from this figure that unlike the compact tension specimen, in which the crack extension takes place along the initial crack, crack growth in the double cantilever beam specimen is in a non-coplanar manner and kinks from the original crack line. The average fracture initiation angles measured from the broken samples are 0° for the compact tension specimen and -56° for the double cantilever beam specimen (see Table 1). These angles were also predicted using the generalized strain energy density criterion by solving Eq. (10) for each of the two specimens. As presented in Table 1, the fracture initiation angle 0O is 0° for the compact tension specimen both from theory and from the experiments. However, for the double cantilever beam specimen the angle 0O is no longer zero and the crack kinks out of its initial line. The crack kinking can be related to the value of Ba = 0.25 which is larger than the critical value of 0.22 as elaborated in Sect. 3.1. Meanwhile, there is very good agreement between the experimentally measured value of 0O = -56° and the theoretically predicted value of 0O = -55° as given in Table 1.
There are some theoretical and computational methods for crack path estimation [30-35]. The incremental crack growth method is one of the favorite methods to investigate the crack trajectory by propagating the crack using a multistep process. In this research, the crack growth direction of each step was determined by using the generalized strain energy density criterion. The crack growth angle 0O was determined by substituting the numerical results obtained for stress intensity factors and T-stress in the previous step
Fig. 8. Some of the steps of crack growth simulation for a double cantilever beam specimen under pure mode I loading
Yk
Crack tip
— N 1
--------O
Fig. 9. Crack tip coordinates in the Cartesian system
into Eqs. (6) and (7). Different stages of crack propagation for the double cantilever beam specimen simulated by the crack growth code are shown in Fig. 8.
The crack tip coordinates in the Cartesian system is shown in Fig. 9. This local coordinate system was introduced to present the results of crack growth path more clearly. Figure 10 shows a comparison between the theoretical curves obtained from the generalized strain energy density criterion for the crack growth path and the experimental results observed in each of the two specimens. It is seen from Fig. 10 that there is good agreement between the experimental results and the theoretical estimates obtained for the fracture trajectories. Moreover, as mentioned earlier and shown in Fig. 10, the crack initiation angles for both specimens are also well predicted by the generalized strain energy density criterion in the very first step of crack extension.
4.2. Critical stress intensity factor
Figure 11 illustrates the variations of normalized critical stress intensity factor (KIf /KIc) versus Ba obtained from Eq. (21) for different values of Poisson's ratio under plane strain conditions. It is seen from this figure that according to the generalized averaged strain energy density criterion, the critical mode I stress intensity factor, KIf is considerably dependent on the T-stress (or Ba).
Two different trends are seen in Fig. 11 for negative and positive values of T-stress. For negative values of T-stress, increasing Ba generally results in rising the critical stress intensity factor KIf, while for positive values of T-stress higher values of Ba decrease the critical stress intensity factor. The peak point in each diagram is dependent on the value of Poisson's ratio.
The generalized averaged strain energy density criterion accounts for the effect of T-stress as a secondary important parameter in calculation of the fracture resistance of materials. Based on classical models of brittle fracture, the higher order terms of stress expression in Eq. (1) can be neglected because the elastic stresses in the vicinity of crack tip are assumed to be ruled by the singular stress term alone. However, the micromechanical models of brittle fracture often consider a fracture process zone around the crack tip within which the material is damaged. The radius of this zone is approximated by the critical distance rc. Along the boundary of this area, the singular terms of stresses dimin-
Y, mm Y, mm
Fig. 10. Crack growth paths for PMMA specimens: compact tension (a), double cantilever beam (b)
ish and the T-stress is no longer negligible. Therefore, the T-stress may play an important role particularly for materials with larger critical distances rc.
Here, the generalized averaged strain energy density criterion is used to estimate the values of fracture resistance obtained from the experiments. Therefore, first the fracture curve (KIf/KIc) related to v = 0.35 (for PMMA) is derived. Then, the results are multiplied by KIc = = 1.4 MPa-m05 obtained from fracture tests on the compact tension specimen. The variations of theoretical estimates for the fracture resistance KIf based on the generalized averaged strain energy density criterion are shown in Fig. 12 in terms of Ba when v = 0.35. The experimental results obtained from fracture tests on compact tension and double cantilever beam specimens are shown in this figure. A comparison between the fracture resistance predicted by the generalized averaged strain energy density criterion (Eq. (21)) and the test data shows that the generalized averaged strain energy density criterion provides very good estimates for the experimental results obtained from the PMMA specimens. It can be seen that the double cantilever beam specimens (with a higher value of T-stress) have lower critical mode I stress intensity factor in comparison with the compact tension specimens (with a lower value of T-stress).
1.25
0.50-1-,-,-,-
-1.0 -0.5 0.0 0.5 1.0
Ba
Fig. 11. Variations of normalized fracture resistance for different values of Ba under plane strain conditions. v = 0.1 (1), 0.2 (2), 0.3 (3), 0.4 (4)
Fig. 12. Variation of fracture resistance versus Ba for PMMA. CT—compact tension specimen, DCB—double cantilever beam specimen, GASED—generalized averaged strain energy density
The fracture resistance of brittle materials are now commonly calculated using some standards like ASTM D5045 [36] and ASTM E399 [37] which are based on the mode I critical stress intensity factor corresponding to the fracture load of materials. The results presented in this paper suggest that using the stress intensity factor alone to describe the fracture resistance of a specified brittle material sometimes can be inaccurate. In other words, when the fracture resistance is calculated using only the singular term of crack tip stresses, it may overestimate or underestimate the real value of material fracture resistance. However, further investigations are still required to assess the same criteria in a wider range of Ba.
4. Conclusions
In this paper, two criteria based on the strain energy density around the crack tip were used to investigate the effect of T-stress on the fracture behavior of PMMA under pure mode I loading. It was shown that the generalized averaged strain energy density and the generalized strain energy density criteria which take into account both Kj and the first non-singular stress term (T-stress) could provide very good estimates for the experimental results obtained from fracture tests on the PMMA specimens. Both the path of crack growth and the fracture resistance were dependent on the T-stress. The values of fracture resistance obtained from the compact tension and double cantilever beam specimens were respectively 1.4 and 1.3 MPa-m05. The generalized strain energy density criterion was successfully used for predicting the path of crack growth in both the compact tension and double cantilever beam specimens. The generalized averaged strain energy density criterion was also able to estimate the fracture resistance of the same specimens well. The effect of T-stress on mode I brittle fracture is more considerable for materials having larger fracture process zones.
References
1. Larsson S. G., Carlsson A.J. Influence of non-singular stress and specimen geometry on small-scale yielding at each tip in elastic-plastic
materials // J. Mech. Phys. Solids. - 1973. - V. 21. - No. 4. - P. 263277.
2. Rice J.R. Limitations to the small scale yielding approximation for crack-tip plasticity // J. Mech. Phys. Solids. - 1974. - V. 22. - No.1.-P. 17-26.
3. Betegon C., Hancock J. W. Two-parameter characterization of elastic-plastic crack-tip fields // J. Appl. Mech. - 1991. - V. 58. - No. 1. -P. 104-110.
4. Du Z.Z., Hancock J. W. The effect of non-singular stresses on crack tip constraint // J. Mech. Phys. Solids. - 1991. - V. 39. - No. 4. - P. 555567.
5. Ayatollahi M.R., Sedighiani K. Crack tip plastic zone under mode I, mode II and mixed mode (I + II) conditions // Struct. Eng. Mech. -2010. - V. 36. - P. 575-598.
6. Ayatollahi M.R., Sedighiani K. Mode I fracture initiation in limestone by strain energy density criterion // Theor. Appl. Fract. Mech. - 2012. -V. 57. - No. 1. - P. 14-18.
7. Smith D.J., Ayatollahi M.R., Pavier M.J. The role of T-stress in brittle fracture for linear elastic materials under mixed-mode loading // Fatigue Fract. Engng. Mater. Struct. - 2001. - V. 24. - No. 2. - P. 137150.
8. Aliha M.R.M., Ayatollahi M.R., Smith D.J., Pavier M.J. Geometry and
size effects on fracture trajectory in a limestone rock under mixed mode loading // Eng. Fract. Mech. - 2010. - V. 77. - No. 11. - P. 22002212.
9. Aliha M.R.M., Ayatollahi M.R. Geometry effects on fracture behavior of polymethylmethacrylate // Mater. Sci. Eng. - 2010. - V. 527. -No. 3. - P. 526-530.
10. Ayatollahi M.R., Aliha M.R.M. Mixed-mode fracture in soda-lime glass analyzed by using the generalized MTS criterion // Int. J. Solids Struct. - 2009. - V. 46. - No. 2. - P. 311-321.
11. Kumar B., Chitsiriphanit S., Sun C.T. Significance of K-dominance zone size and nonsingular stress field in brittle fracture // Eng. Fract. Mech. - 2011. - V. 78. - No. 9. - P. 2042-2051.
12. Sun C.T., Qian H. Brittle fracture beyond the stress intensity factor // J. Mech. Mater. Struct. - 2009. - V. 4. - No. 4. - P. 743-753.
13. Liu S., Chao Y.J. Variation of fracture toughness with constraint // Int. J. Fract. - 2003. - V. 124. - No. 3. - P. 113-117.
14. Chao Y.J., Liu S., Broviak B.J. Brittle fracture: Variation of fracture toughness with constraint and crack curving under mode I conditions // Exp. Mech. - 2001. - V. 41. - No. 3. - P. 232-241.
15. Davenport J.C.W., Smith D.J. A study of superimposed modes I, II and III on PMMA // Fatigue Fract. Engng. Mater. Struct. - 1993. -V. 16. - No. 10. - P. 1125-1133.
16. Khan K., Al-Shayea N.A. Effect of specimen geometry and testing method on mixed mode I-II fracture toughness of a limestone rock from Saudi Arabia // Rock Mech. Rock Eng. - 2000. - V. 33. - No. 3. -P. 179-206.
17. Lazzarin P., Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches // Int. J. Fract. - 2001. - V. 112. - No. 3. - P. 275298.
18. Lazzarin P., Berto F., Elices M., Gomez J. Brittle failure from U- and V-notches in mode I and mixed, I + II, mode: A synthesis based on the strain energy density averaged on finite-size volumes // Fatigue Fract. Engng. Mater. Struct. - 2009. - V. 32. - No. 8. - P. 671-684.
19. Berto F., Lazzarin P., Marangon C. Brittle fracture of U-notched graphite plates under mixed mode loading // Mater. Des. - 2012. -V. 41. - P. 421-432.
20. Lazzarin P., Berto F., Ayatollahi M.R. Brittle failure of inclined keyhole notches in isostatic graphite under in-plane mixed mode loading // Fatigue Fract. Engng. Mater. Struct. - 2013. - V. 36. - No. 9. -P. 942-955.
21. Berto F., Lazzarin P. Recent developments in brittle and quasi-brittle failure assessment of engineering materials by means of local approaches // Mater. Sci. Eng. R. - 2014. - V. 75. - P. 1-48.
22. Lazzarin P., Campagnolo A., Berto F. A comparison among some recent energy- and stress-based criteria for the fracture assessment of
sharp V-notched components under mode I loading // Theor. Appl. Fract. Mech. - 2014. - V. 71. - P. 21-30.
23. Ayatollahi M.R., Berto F., Lazzarin P. Mixed mode brittle fracture of sharp and blunt V-notches in polycrystalline graphite // Carbon. -2011. - V. 49. - P. 2465-2474.
24. Ayatollahi M.R., Aliha M.R.M., Saghafi H. An improved semi-circular bend specimen for investigating mixed mode brittle fracture // Eng. Fract. Mech. - 2011. - V. 78. - No. 1. - P. 110-123.
25. Sih G.C. Strain-energy-density factor applied to mixed mode crack problems // Int. J. Fract. - 1974. - V. 10. - No. 3. - P. 305-321.
26. Sih G.C. Some basic problems in fracture mechanics and new concepts // Eng. Fract. Mech. - 1973. - V. 5. - No. 2. - P. 365-377.
27. Beltrami E. Sulle condizioni di resistenza dei corpi elastici // Rend. R. Ist. Lombardo di Scienze, Lettere e Arti. - 1885. - V. 18. - No. 1. -P. 145-155.
28. Simulia Inc. ABAQUS 6.10 Manual. Providence, RI 02909, 2010.
29. Schmidt R.A. A Microcrack Model and Its Significance to Hydraulic Fracturing and Fracture Toughness Testing // Proc. 21st US Symp. Rock Mechanics, 1980. - P. 581-590.
30. Maiti S.K.M., PrasadK.S.R. A study on the theories of unstable crack extension for the prediction of crack trajectories // Int. J. Solids Struct. -1980. - V. 16. - No. 6. - P. 563-574.
31. Sumi Y. Computational crack path prediction // Theor. Appl. Fract. Mech. - 1985. - V. 4. - No. 2. - P 149-156.
32. Aliha M.R.M., Hosseinpour Gh.R., Ayatollahi M.R. Application of cracked triangular specimen subjected to three-point bending for investigating fracture behavior of rock materials // Rock Mech. Rock Eng. - 2012. - V. 46. - No. 5. - P. 1023-1034.
33. Ayatollahi M.R., Razavi S.M.J., Chamani H.R. A numerical study on the effect of symmetric crack flank holes on fatigue life extension of a SENT specimen // Fatigue Fract. Engng. Mater. Struct. - 2014. -V. 37. - No. 10. - P. 1153-1164.
34. Ayatollahi M.R., Razavi S.M.J., Chamani H.R. Fatigue life extension by crack repair using stop-hole technique under pure mode I and pure mode II loading conditions // Proc. Eng. - 2014. - V. 74. - P. 18-21.
35. AyatollahiM.R., Razavi S.M.J., Yahya M.Y. Mixed mode fatigue crack initiation and growth in a CT specimen repaired by stop hole technique // Eng. Fract. Mech. - 2015 (in press).
36. ASTM D5045, Standard Test Methods for Plane-Strain Fracture Toughness and Strain Energy Release Rate of Plastic Materials. Annual Book of ASTM Standards, Vol. 08.03.
37. ASTME399, Standard Test Methods for Plane-Strain Fracture Toughness of Metallic Materials, Annual Book of ASTM Standards, Vol. 03.01.
Поступила в редакцию 29.09.2015 г.
Ceedeuua 06 aemopax
Majid R. Ayatollahi, PhD, Prof., Director, Iran University of Science and Technology, Iran, m.ayat@iust.ac.ir S.M.J. Razavi, MSc., Iran University of Science and Technology, Iran, mj_razavi@yahoo.com
Morteza Rashidi Moghaddam, PhD Student, Iran University of Science and Technology, Iran, morteza_rashidi@mecheng.iust.ac.ir Filippo Berto, Prof., University of Padova, Italy, berto@gest.unipd.it
