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194
УДК 539.42
Fracture loads prediction on notched short glass fibre reinforced polyamide 6 using the strain energy density
F.T. Ib áñez-Gutiérrez1, S. Cicero1, V. Madrazo2, F. Berto3
1 LADICIM (Laboratory of Materials Science and Engineering), University of Cantabria, Santander, 39005, Spain 2 Centro Tecnologico de Componentes, Santander, 39011, Spain 3 Department of Management and Engineering, University of Padova, Vicenza, 36100, Italy
This paper provides an energetic approach useful for the prediction of critical loads on U-notched components without an ideally linear elastic behaviour. The methodology has been applied to 100 fracture specimens of short glass fibre reinforced polyamide 6 (SGFR-PA6), combining four different fibre contents (5, 10, 30 and 50 wt %) and five different notch radii (0.00, 0.25, 0.50, 1.00 and 2.00 mm). The proposal combines the application of the strain energy density criterion with the use of the whole absorbed energy in the tensile test (elastic-plastic area under the stress-strain curve). With all of this, the fracture loads have been well estimated in this type of material.
Keywords: short glass fibre reinforced polyamide 6, strain energy density, notch, fracture assessment
DOI 10.24411/1683-805X-2018-11010
Расчет разрушающих нагрузок для образцов стеклонаполненного полиамида 6 с надрезом на основе критерия плотности энергии деформации
F.T. Ibáñez-Gutiérrez1, S. Cicero1, V. Madrazo2, F. Berto3
1 Университет Кантабрия, Сантандер, 39005, Испания 2 Технологический центр компонентов, Сантандер, 39011, Испания 3 Падуанский университет, Виченца, 36100, Италия
В статье предложен энергетический подход для оценки критических нагрузок на элементы конструкций с U-образным надрезом для неидеального линейно-упругого поведения. Испытания на излом проводили на образцах стеклонаполненного (короткое волокно) полиамида 6 (SGFR-PA6) с разным содержанием волокон (5, 10, 30 и 50 мас. %) и разным радиусом надреза (0.00, 0.25, 0.50, 1.00 и 2.00 мм). Предложенный подход основан на применении критерия плотности энергии деформации с учетом всей поглощенной энергии в ходе испытаний на растяжение (упругопластическая область под кривой растяжения). Проведена оценка разрушающих нарузок для данного типа материала.
Ключевые слова: стеклонаполненный полиамид 6, плотность энергии деформации, надрез, оценка разрушения
1. Introduction
The life of structural components is one of the main concerns that design engineers must address. When a stress riser is present (e.g., cracks or notches), it is very important to evaluate the load that leads to failure: the critical load or fracture load.
For many years, different criteria have been developed to assess both the ductile and the brittle behaviour of notched components. One of the first to be proposed, dating back to 1885, is the strain energy density (SED) criterion [1]. Since then, different SED-based approaches have been formulated (e.g., [2-6]), this being a useful tool in asserting structural
© Ibânez-Gutiérrez F.T., Cicero S., Madrazo V., Berto F., 2018
safety. Sih introduced the concept of the strain energy density factor S as the product of the strain energy density by a critical distance from the point of singularity [7]. In recent years, the local SED criterion [8-10] has been presented as a combination of Sih's point-wise criterion with Neuber's concept of elementary structural volume [11]. As developed in [8], the local SED criterion is based on the strain energy density averaged over a control volume surrounding the notch tip. In plane problems, the control volume becomes a circle or a circular sector with a radius R in the case of cracks or pointed V-notches in mode I or mixed, I + II, mode loading (Fig. 1, a and b) [9].
Fig. 1. Control volume (area) for sharp V-notch (a), crack (b) and blunt V-notch (c) under mode I loading
The SED approach is based on the idea that under tensile stress, failure occurs when the mean value of the elastic strain energy referred to an area W is equal to the critical value Wc, which varies from material to material [8]. According to Lazzarin-Zambardi [8], if the material is ideally brittle, the value of the critical strain energy density follows:
_2
(1)
W = °L c 2 E
where at is the ultimate tensile strength and E is the Young's modulus. When the notch opening angle is zero 2a = 0, the material critical radius Rc can be expressed in terms of the fracture toughness K Ic, the ultimate tensile
strength at, and Poisson's ratio v [12]:
2
for plane strain,
Rc =
(1 + v)(5 -8v)
4n
K
Rc =
5 - 3v
4n
K
for plane stress.
(2)
(3)
In the case of blunt notches, the total strain energy can be determined over the crescent shape volume (Fig. 1, c) and then the mean value of the SED can be expressed in terms of the elastic maximum notch stress amax [13]. By applying this condition, the total strain energy can be obtained over the area ^ (Fig. 1), and the corresponding mean value of the SED follows equation [9]:
W = F (2a) H
2a, Rc
(4)
where F(2a) depends on the notch opening angle, H varies
Table 1
Values of the function H when 2a = 0 for blunted V-shaped notches (coefficients determined numerically with p = 1 mm) [9]
Rc/ P
0.01
0.05
0.10
1.00
v = 0.30
0.5638
0.5086
0.4518
0.1314
v = 0.35
0.5432
0.4884
0.4322
0.1217
v = 0.40
0.5194
0.4652
0.4099
0.1110
with the notch geometry (2a, Rc/p) and amax is the maximum elastic stress at the notch tip. Table 1 gathers different values of the H function for U-shaped notches 2a = 0 [9].
With all of this, the aim of the present research is to estimate the critical load on U-notched short glass fibre reinforced polyamide 6 (SGFR-PA6) by applying the strain energy density criterion. In order to simplify the calculations, the local SED approach explained above is applied. Moreover, as shown below, and due to the non-ideally brittle behaviour of SGFR-PA6, this methodology will be combined with the consideration of the whole (elastic-plastic) absorbed energy of the material (i.e., the total area under the stress-strain curve) so as to provide more accurate predictions.
In recent papers [14, 15], the analysed experimental programme has been used for both the prediction of the notch effect (by the direct application of the theory of the critical distances [16]) and the validation failure assessment validates a structural integrity assessment methodology (based on the use of failure assessment diagrams and the notch effect corrections provided by the line method, respectively.
2. Materials and methods
The experimental programme has been applied to short glass fibre reinforced polyamide 6 (SGFR-PA6) with four
20
H
10
80
170
10
I Short fibre ^orientation \ P
n
2pf
40
II
44
Fig. 2. Tensile (a) and single-edge notch bend test specimens (b). All dimensions are in mm. Thickness is 4 mm, p varying from 0 to 2 mm
Table 2
SGFR-PA6 tensile parameters (average values). E—elastic modulus, a 02—proof stress, a t —ultimate tensile strength, emax —strain under maximum load, v—Poisson's number (x refers to the longitudinal direction)
Fibre content, % Exx, GPa Eyy = Ez, GPa G0.2x, MPa gtx, MPa emaxx, % V = V xy xz V = V yx zx V = V yz yz
5 3.30 3.00 66.90 72.05 2.67 0.39 0.35 0.45
10 3.55 3.15 70.15 78.15 2.84 0.38 0.34 0.46
30 6.45 4.00 105.35 128.00 3.56 0.34 0.21 0.58
50 12.60 5.48 161.15 192.80 2.47 0.30 0.13 0.66
different amounts of fibre content (5, 10, 30 and 50 wt %). Currently, short fibre reinforced thermoplastics are one of the most widely used technical plastics which are standing in for metal parts in engineering components due to their easy fabrication and good mechanical properties [17]. Adding short glass fibres to PAs leads to a higher heat distortion temperature, strength, abrasion resistance and stiffness, despite the fact that properties can be anisotropic (including mould shrinkage, which implies potential distortion) [18]. Regarding PA6—unlike many polymers— this is accomplished without any loss of impact strength, but strain at maximum load is reduced substantially [19]. The extensive use of these materials makes it essential to study their behaviour when containing stress risers which may appear and endanger the structural integrity of the corresponding component. With this purpose, five different notch radii were covered in the present research: 0 (cracklike defects), 0.25, 0.50, 1.00, and 2.00 mm.
The core of the experimental programme is composed of 108 specimens of SGFR-PA6. They were made by injection moulding with the short glass fibres (300 jm length and 10 jm diameter) oriented parallel to the longitudinal axis of the specimens. After injection molding, the mean thickness fibre orientation is 83° [20] and the effective fibre length is 204.2±40.9 |m [20, 21]. The geometry and dimensions are shown in Fig. 2, a. The specimens were dried in an oven at 100°C before testing (tensile or fracture) in order to proceed without moisture. The main mechanical properties were obtained following ASTM D638 [22] as the average values of two tensile tests per fibre content and
orientation. The results are shown in Table 2. Figure 3 gathers one stress-strain curve per fibre content. It can be observed that the higher the fibre content, the lower the linearity of the curve.
Five single-edge notch bend (SENB) specimens were tested per combination of the 5 notch radii and the 4 fibre contents (i.e., 100 total) following ASTM D5045-99 [23]. These fracture specimens were obtained from the central part of the corresponding tensile samples, the geometry being shown in Fig. 2, b. The notches were obtained by machining perpendicularly to the longitudinal direction of the original specimens. Only one test was invalid (30 wt %, notch radius 0.5 mm). Tables 3-6 gather the maximum load (fracture load) reached for each fibre content varying the notch radii. Figure 4 shows some examples of the curves obtained, with certain nonlinearity in those with higher fibre content. On the other hand, on five SENB specimens per fibre content, a crack-like defect (p = 0 mm) was generated by sawing a razor blade. These specimens were used to derive the corresponding fracture toughness (K Ic, see Table 7), which follows equation [23]: \ / a V/2 6 a
K Ic =
BW12
W
1.99 - a/W (1 - a/W )(2.15 - 3.93a/W + 2.7(a/W )2 " (1 + 2 a/W)(1 - a/W)3/2
(5)
3. Estimation of fracture loads
Fig. 3. Strain-stress curves of SGFR-PA6: fibre content 5 (1), 10 (2), 30 (3), 50 wt % (4)
The assessment of the critical load on short glass fibre reinforced polyamide 6 (SGFR-PA6) is here analysed through the application of the strain energy density criterion [9]. By applying this criterion, it is straightforward to obtain the critical strain energy referred to an area, Wc (Eq. (1)), from the mechanical properties gathered in Table 2.
In order to derive the elastic maximum notch stress from Eq. (4), the different SED criterion parameters should be defined. The F factor value depends on the notch opening angle and, in the case of U-blunt notches with 2a = 0, F is equal to 0.785 [9]. In order to obtain the H function, it is necessary to define the material critical radius (Eqs. (2) and (3)). It should be mentioned that when the fracture resistance Kmat, associated to particular material and notch radius, is under the limit established by Eq. (6) [24], it may
Description of SGFR-PA6 notched specimens and maximum load obtained (5 wt %)
Notch radius p, mm Specimen Maximum load Pexp, N Notch radius p, mm Specimen Maximum load Pexp,N
0.25 5-0.25-1 83.80 1.00 5-1-1 120.70
0.25 5-0.25-2 82.20 1.00 5-1-2 99.30
0.25 5-0.25-3 111.00 1.00 5-1-3 119.10
0.25 5-0.25-4 56.40 1.00 5-1-4 122.20
0.25 5-0.25-5 77.10 1.00 5-1-5 106.00
0.50 5-0.5-1 100.10 2.00 5-2-1 151.60
0.50 5-0.5-2 108.40 2.00 5-2-2 135.40
0.50 5-0.5-3 96.00 2.00 5-2-3 126.60
0.50 5-0.5-4 100.60 2.00 5-2-4 149.70
0.50 5-0.5-5 82.70 2.00 5-2-5 125.90
Table 4 Description of SGFR-PA6 notched specimens and maximum load obtained (10 wt %)
Notch radius p, mm Specimen Maximum load Pexp, N Notch radius p, mm Specimen Maximum load Pexp, N
0.25 10-0.25-1 93.10 1.00 10-1-1 124.10
0.25 10-0.25-2 105.20 1.00 10-1-2 116.50
0.25 10-0.25-3 104.50 1.00 10-1-3 141.00
0.25 10-0.25-4 87.80 1.00 10-1-4 125.00
0.25 10-0.25-5 78.60 1.00 10-1-5 119.70
0.50 10-0.5-1 116.20 2.00 10-2-1 173.80
0.50 10-0.5-2 102.10 2.00 10-2-2 166.70
0.50 10-0.5-3 93.40 2.00 10-2-3 167.30
0.50 10-0.5-4 111.10 2.00 10-2-4 146.40
0.50 10-0.5-5 97.70 2.00 10-2-5 153.40
Table 5 Description of SGFR-PA6 notched specimens and maximum load obtained (30 wt %)
Notch radius p, mm Specimen Maximum load Pexp, N Notch radius p, mm Specimen Maximum load Pexp, N
0.25 30-0.25-1 237.80 1.00 30-1-1 231.60
0.25 30-0.25-2 220.20 1.00 30-1-2 251.50
0.25 30-0.25-3 202.50 1.00 30-1-3 287.90
0.25 30-0.25-4 216.40 1.00 30-1-4 302.60
0.25 30-0.25-5 205.40 1.00 30-1-5 246.60
0.50 30-0.5-1 207.10 2.00 30-2-1 305.80
0.50 30-0.5-2 252.40 2.00 30-2-2 284.20
0.50 30-0.5-3 251.80 2.00 30-2-3 269.00
0.50 30-0.5-4 - 2.00 30-2-4 263.70
0.50 30-0.5-5 243.30 2.00 30-2-5 318.30
be considered that plane strain conditions are dominant. In this case, the R, solution for the tested specimens is given by Eq. (2). The fracture resistance is here understood as the material resistance to fracture for a given notch radius,
in stress intensity factor units. On the other hand, when Kmat values are over the limit established by Eq. (7) [24] it may be assumed that plane stress conditions are dominant and the R, solution follows Eq. (3):
Description of SGFR-PA6 notched specimens and maximum load obtained (50 wt %)
Notch radius p, mm Specimen Maximum load p.xp, N Notch radius p, mm Specimen Maximum load p.xp, N
0.25 50-0.25-1 322.00 1.00 50-1-1 389.70
0.25 50-0.25-2 338.00 1.00 50-1-2 394.50
0.25 50-0.25-3 329.40 1.00 50-1-3 402.40
0.25 50-0.25-4 360.60 1.00 50-1-4 395.70
0.25 50-0.25-5 335.00 1.00 50-1-5 389.60
0.50 50-0.5-1 367.40 2.00 50-2-1 432.30
0.50 50-0.5-2 367.90 2.00 50-2-2 414.20
0.50 50-0.5-3 364.60 2.00 50-2-3 426.50
0.50 50-0.5-4 376.90 2.00 50-2-4 431.00
0.50 50-0.5-5 380.10 2.00 50-2-5 427.00
plane strain limit
K
= a
B
V/2
2.5
V
plane stress onset
Kmat =ay( nB >
(6)
12
. (7)
where B is the thickness of the specimen, and ay is the yield strength. In practice, the fracture toughness values can be located between the limits established by Eqs. (6) and (7). For this situation, the Rc solution has been obtained here by interpolating equations (2) and (3).
Fig. 4. Load-displacement curve obtained in some SGFR-PA6 specimens for a fibre content of 10 wt %, p = 0.00 (1), 0.25 (2), 0.50 (3), 1.00 (4), 2.00 (5) (a); for a notch radius of 0.50 mm, fibre content 5 (1), 10 (2), 30 (3), and 50 wt % (4) (b)
Table 1 gathers different H values for some combinations of (Rc/p, v) when 2a = 0. For each combination of (Rc /p, v) the coefficients H are obtained by interpolation. Once all the parameters are known, the proposal here is to combine the SED criterion for brittle materials with the simplified approach proposed in [9, 10]. The complete analytical description can be found in [25]. Then, the maximum notch stress amax is derived from Eqs. (1) and (4) as follows:
a = J-a-—. (8)
max ]j 2x0.785H(2a = 0,Rjp)
Table 8 gathers the values obtained for the different parameters and for the different combination of fibre content and notch radius: the critical radius Rc, the Hfactor values, the critical value of the elastic strain energy Wc and the corresponding maximum notch tip stresses amax. Finite elements simulations have been performed with ANSYS to obtain the critical load PSED, which is that one applied on the specimens generating amax at the corresponding notch tip. The modelling has considered the anisotropy of the material being analysed, with specific properties in width, height and longitudinal directions. Furthermore, the simulation was conducted in purely linear-elastic conditions.
Figure 5, a shows the comparison of the SED prediction PSED with the experimental results Pexp. The experimental data used for each combination of fibre content and notch radius are the average value of those gathered in Tables 36. A strong conservatism of the analysis performed can be observed: the relations between the results predicted PSED and those obtained experimentally Pexp are well below 1. In addition, the higher the fibre content, the lower the accuracy of the model proposed.
The stress-strain analysis assumes fully linear-elastic behaviour of the material in the tensile curve. However, Figure 3 showed evident nonlinearity for the four different materials (four different fibre contents). Hence, the assump-
Description of SGFR-PA6 cracked specimens and corresponding values of KIc (p = 0 mm, crack-like defect)
Specimen K Ic, MPam1/2 Kic,av, MPam1'2 Specimen K Ic, MPam1'2 Kic,av, MPam1'2
5-0-1 2.40 1.84 30-0-1 5.76 4.77
5-0-2 1.64 30-0-2 4.74
5-0-3 1.78 30-0-3 4.90
5-0-4 1.65 30-0-4 4.01
5-0-5 1.73 30-0-5 4.44
10-0-1 2.46 2.13 50-0-1 8.09 8.59
10-0-2 2.28 50-0-2 8.67
10-0-3 1.65 50-0-3 8.56
10-0-4 1.81 50-0-4 8.66
10-0-5 2.47 50-0-5 8.96
Table 8
SED parameters for the different approaches proposed and for each combination of fibre content and notch radius
Fibre content, % Notch radius p, mm Rc, mm Rc/P H Wc, MPa ^max , MPa Wc', MPa ^max , MPa
5 0.25 0.136 0.542 0.190 0.79 132.08 1.09 155.49
0.50 0.136 0.271 0.289 106.89 125.83
1.00 0.164 0.164 0.360 95.84 112.82
2.00 0.271 0.135 0.384 92.79 109.23
10 0.25 0.160 0.640 0.170 0.86 151.37 1.31 186.80
0.50 0.160 0.320 0.269 120.28 148.43
1.00 0.201 0.202 0.336 107.52 132.69
2.00 0.387 0.194 0.342 106.64 131.60
30 0.25 0.400 1.599 0.124 1.27 290.52 3.06 450.95
0.50 0.462 0.925 0.132 281.69 437.24
1.00 0.567 0.567 0.198 229.38 356.04
2.00 0.702 0.351 0.269 196.94 305.69
50 0.25 0.571 2.286 0.131 1.48 424.48 3.13 618.34
0.50 0.630 1.260 0.131 424.48 618.34
1.00 0.685 0.685 0.180 362.42 527.93
2.00 0.785 0.393 0.264 299.30 435.99
tion of the linear approach (SED criterion) is not adequate for this type of material.
Thus, another methodology is here proposed to increase the accuracy of the fracture load prediction. The analysis performed will consider the whole area under the stressstrain (i.e., the total absorbed energy W'). Once the area is
obtained, the corrected maximum notch stress ^m lows:
EWC
0.785H (2a = 0, Rj p)
fol-
(9)
The estimation of the new fracture loads PS'ED
was
finally obtained by repeating the finite element analysis (i.e., obtaining the load that generates the corresponding amax at the notch tip). Table 8 shows the different values of W' and amax obtained, while Fig. 5, b compares the predictions of the fracture load PS'ED with the experimental results Pexp. As shown in Fig. 3, the loss of linearity in the behaviour of tensile specimens is higher when the fibre content increases. Hence, the difference between Wc and W' and therefore,
between amax and a' , increases with the fibre content
max max
^SED^exp
| a
0 § & B
o 0.25 mm o o
a 0.50 mm
□ 2.00 mm
x 2.00 mm
'O 10 20 30 40 50 Fibre content, %
P SED^exp
lb M
ft § o X
ñ o
o 0.25 mm
a 0.50 mm
□ 2.00 mm
x 2.00 mm
0 10 20 30 40 50
Fibre content, %
Fig. 5. Comparison of the critical load values obtained with Wc (a) and W' (b) experimental data
(given that the situation deviates from linear-elastic conditions). By applying this second approach, a clear reduction in the conservatism of the results can be observed. The prediction is accurate not only for high fibre contents but also for low fibre contents. Comparing the fracture loads predicted with the experimental results obtained, the relation is close to 1. Thus, the range of accuracy is much higher here than that observed when notches are treated as cracks and comparable to that obtained through other approaches such as the theory of critical distances [e.g., 14, 26, 27] or the combination of the TCD and failure assessment distances [15, 28, 29]. Summing up, the application of this approach to this type of material is, therefore, both simple and safe.
4. Conclusions
This paper provides and validates a methodology for the assessment of fracture loads in notched specimens of short glass fibre reinforced polyamide 6 (SGFR-PA6) using the strain energy density criterion. The core of the experimental programme is composed of 100 fracture (SENB) specimens combining different notch radii (from 0 up to 2 mm) and different fibre contents (from 5 up to 50 wt %).
The initial assessment, which assumes linear-elastic behaviour, predicts fracture loads that are far from the results obtained experimentally. Given that SGFR-PA6 develops an elastic-plastic behaviour, another methodology has been proposed considering the whole absorbed energy in the
tensile test (elastic-plastic area under the stress-strain curve). With this new approach, the estimation of the critical load is closer to the results obtained within the experimental programme. The prediction of fracture loads on SGFR-PA6 using the methodology proposed gives accurate results, closer to the physics of the problem. Therefore, the local approach is again shown as a powerful tool in fracture assessment.
Acknowledgments
The authors of this work wish to extend their gratitude to the Spanish Ministry of Science and Innovation for the financial support of the Project MAT2014-58443-P, on the results of which this paper is based.
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Francisco Tomas Ibáñez-Gutiérrez, PhD, Researcher, University of Cantabria, Spain, ibanezft@unican.es Sergio Cicero, PhD, Prof., University of Cantabria, Spain, ciceros@unican.es
Virginia Madrazo, PhD, Researcher, Centro Tecnologico de Componentes, Spain, vmadrazo@ctcomponentes.com Filippo Berto, PhD, Prof., University of Padova, Italy, berto@gest.unipd.it, filippo.berto@unipd.it
