Experimental and numerical investigation of fracture of abs polymeric material for different sample's thickness using a new loading device.
The main problem with predicting the failure of materials is the evaluation of fracture toughness properties under mixed mode loading conditions. Various approaches have been taken to develop test specimens for combined loading conditions, but the main research has been conducted on mode-I and mode-II fracture behavior of materials. A number of test methods have been proposed by many researchers to determine fracture toughness for three modes of loading (I, II, and III) and also under mixed-mode conditions. The double cantilever beam (DCB) and the end notched flexure (ENF) specimens have been used for mode-I and mode II tests, respectively [1-5]. A crack rail shear (CRS) specimen has been proposed to determine the mode-III (tearing) critical strain energy release rate . However, the fracture is usually not a result of pure mode-I or pure mode-II loading, and the fracture occurs in the mixed-mode loading conditions. For this reason, the study of mixed-mode fracture toughness is of great significance. Also for the mixed mode tests, beam type specimens were mostly used to obtain mixed mode-I/II critical energy release rates.
Various attempts have been made to characterize fracture toughness under mixed-mode loading conditions, but beam type specimens have been mostly used [7-10], Some of these include: the mixed-mode flexure (MMF) test, the end load split (ELS) specimen, the single leg bending (SLB) specimen, the crack Lap shear (CLS) test, the edge delamination tension (EDT) specimen, and the asymmetric double cantilever beam. However, a common problem with all these test methods which limits their usefulness is that a wide range of mixed-mode ratios cannot be tested. The mixed-mode bending (MMB) test has been proposed by combining the schemes used for DCB and ENF tests, which can produce a wide range of the ratios of mode-I and mode-II components by varying the length of lever arm of the specimen. However, to obtain results for values of fracture toughness for pure mode-I, pure mode-II, and mixed-mode loading conditions, different beam type specimens are required. It is therefore necessary to develop other test methods to evaluate the fracture parameters of materials under all in-plane loading conditions starting from pure mode-I to pure mode-II. In recent years, a modified version of Arcan specimen has been made for the mixed-mode fracture test of adhesively bonded joints, which allows pure mode-I, pure mode-II, and almost any combination of mode-I and mode-II loading to be tested using the same test specimen configuration [11-14], However, the modified Arcan fixture also has some limitations including asymmetry in its structure that leads to an unwanted third mode in the experiments as well as limitations on the loading angle intervals [15, 16]. It should be noted that structural asymmetry increases with increasing the specimen thickness; meanwhile the prevailing theory generally is plain strain (Fig. 1). Recently, several researchers have reported this problem by analyzing the modified Arcan fixture using different methods [17-20]. Therefore, in this study, a new fixture was used that can solve these problems (as shown in Fig. 2). The fracture tests using the new fixture have been carried out on ABS polymer and the stress intensity factors of mode-I, mode-II, and mixed-mode were calculated. Because of the perfect plane conditions within the all thicknesses, experiments were carried out for three different thicknesses to study the effect of thickness on the fracture toughness using the plane strain theory.
The purpose of fracture toughness testing is to determine the value of the critical stress intensity factor, or plane strain fracture toughness [K.sub.C]. This material property is used to characterize the resistance to fracture in the design of structural members. ASTM standards E399 and D5045 give some guidance for plane strain mode-I fracture toughness for metals and plastics, respectively. No standard requirements exist for the validity of linear elastic fracture mechanics and plane-strain conditions for tests with polymeric materials under mode-II and mixed-mode loading conditions. Therefore, it may be necessary to develop tests tailored for use with polymeric materials to investigate the role of mixed-mode loading conditions. This investigation seeks to extend understanding of the polymeric materials fracture behavior under mixed-mode loading conditions through numerical and experimental analysis. Using finite element results, correction factors were applied to the specimen and a polynomial fit was proposed to evaluate the stress intensity factors of the specimen with a crack subjected to mixed-mode loading conditions. The research conducted for this study assumed that the polymeric material under consideration was homogenous and linear elastic. The main objective of this study was to determine the fracture toughness [K.sub.IC] and [K.sub.IIC] for the polymeric materials under consideration for a wide range of mixed-mode loading with assumption of linear elastic conditions according ASTM D5045 standards, in which linear elastic fracture mechanics and plan strain conditions are the primary requirements. During this research several issues remained beyond the scope of this work. However, the effect of direction of the anisotropy in the specimens on the performance of material must also not be ignored. To accurately understand the durability of polymeric materials, it is necessary to have knowledge of the effect of mechanical anisotropy influences on the magnitudes of fracture toughness values. For example, the anisotropy may affect the applicability of linear elastic fracture mechanics criterion and increase the fracture toughness. Also, some degree of future work should focus on investigating alternate approach using elastic-plastic fracture mechanics in the polymeric materials and incorporating this information along with the influence of anisotropy into finite element models.
The value of the critical stress-intensity factor [K.sub.C] for a material can be measured by testing standard cracked specimens, such as the compact tension specimen. The stress intensity factor at the crack tip of a compact tension specimen is given by:
[K.sub.C] = [P.sub.c]/wt [square root of [pi]a] f(a/w), (1)
where [P.sub.c] is the fracture load, a stands for crack length, w represents the specimen width, t is the specimen thickness, and f(a/w) is the nondimensional stress intensity factor. It is important to mention that the nondimensional stress intensity factor, f(a/w), is calculated by numerical analysis and the quantity of critical load is obtained by the experimental results. Linear elastic fracture mechanics and plane strain conditions are the primary requirements. According ASTM standards E399 and D5045, the computed [K.sub.C] value may turn out to be the critical stress intensity factor, or fracture toughness only if all the validity requirements were met. Linear elastic fracture mechanics is the primary requirement, that is, linear elastic material behavior of the cracked specimen on the load-displacement diagram. Second, the conditions for plane strain at the crack-tip must also be checked; if specimen thickness is not sufficient to ensure this stress state, then it is rather more plane-stress than plane-strain, and the fracture toughness will be larger than the actual plane-strain value. The thickness and crack length must be sufficient to ensure the plane-strain conditions; ASTM requirements are: ASTM requirements are; 0.45(a/w < 0.55 and t, a [greater than or equal to] 2.5[([K.sub.C]/[[sigma].sub.y]).sup.2], where [[sigma].sub.y] is the yield strength of the material being tested.
The stress intensity factors on the tip of a crack of loaded specimen for the new fixture were calculated using the following equations [22, 23]:
[K.sub.I] = [P.sub.c]/wt [square root of [pi]a] [f.sub.I](a/w)
[K.sub.II] = [P.sub.c]/wt [square root of [pi]a] [f.sub.II](a/w), (2)
where [K.sub.I] and [K.sub.II] represent mode-I and mode-II stress intensity factors, respectively. Furthermore, stress intensity factors can be related to the strain energy release ratios (J-Integral) for liner elastic materials :
[G.sub.I] = [K.sub.I]/E'
[G.sub.II] = [K.sub.II]/E',
[G.sub.T] = [G.sub.I] + [G.sub.II], (3)
where E' = E for plane stress, E' = E/(1 - [v.sup.2]) for plane strain conditions, E is modulus of elasticity, v is Poisson's ratio. The calculated strain energy release rate values are indications of cracked polymeric material durability, by quantitatively showing how much energy must be put into the specimen to create fracture.
In the experiments, the cracked samples of ABS polymeric material have been tested under tension, ([P.sub.c]) and the critical stress intensity factors can be calculated by Eqs. 2 that the quantity of parameter f(a/w) was obtained using numerical analysis.
FINITE ELEMENT ANALYSES
The stress intensity factors [K.sub.I], [K.sub.II] and [K.sub.III] play an important role in linear elastic fracture mechanics. They characterize the influence of the load or deformation on the magnitude of crack tip stress and strain fields and measure the propensity of the crack propagation or the crack driving forces. Furthermore, the stress intensity factor can be related to the energy release rate (the J-integral) for a linear elastic material:
J = (1/8[pi])[K.sup.T][B.sup.-1]K, (4)
where K = [[[K.sub.I] [K.sub.II] [K.sub.III]].sup.T] and B is called the prelogarithmic energy factor matrix. To calculate the stress intensity factors, interaction integral method is commonly used.
The method is applicable to cracks in isotropic and anisotropic materials. Based on the definition of the J-integral, the interaction integrals can be expressed as :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
where [GAMMA] is an arbitrary contour, q is a unit vector in the virtual crack extension direction, n is the outward normal to [GAMMA], [sigma] is the stress tensor, and u the displacement vector. The subscript aux represents three auxiliary pure mode-I, mode-II, and mode-III crack-tip fields for [alpha] = I, II, and III, respectively. The domain form of the interaction J-integral is :
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where [lambda](s) is the virtual crack advance and dA is the surface element. In the interaction J-integral method [25-27], the two-dimensional auxiliary fields are introduced and superposed on the actual fields. Several contours integral evaluations are possible at each location along a crack. In a finite element model, each evaluation could be thought of as the virtual motion of a block of material surrounding the crack tip (in two dimensions) or surrounding each node along the crack line (in three dimensions). Each contour provides an evaluation of the contour integral. The number of possible evaluations is the number of such rings of elements. Figure 3 shows contours surrounding the crack tip.
The 2D modeling the sample and fixture of the experiment was done through the ABAQUS finite element software under a constant load of 1000 N. The eight nodes collapsed quadrilateral elements in plane strain element model and the mesh was refined around crack tip, so that the smallest element size found in the crack tip elements was approximately 0.25 mm. Linear elastic finite element analysis was performed under a plane strain condition using 1/[square root of r] stress field singularity. To obtain a 1/[square root of r] singularity term of the crack tip stress field, the elements around the crack tip were focused on the crack tip and the mid-side nodes were moved to a quarter point of each element side.
As mesh pattern of the entire apparatus and specimen can be observed in Fig. 4, the total number of elements applied in this model is approximately 6500 elements in both new loading device and specimen. The models are loaded in various angles and the stress intensity factors can be calculated, and then the nondimensional stress intensity factors in the first and second modes are calculated using Eqs. 2, and finally the values of critical stress intensity factors can be calculated using the values of critical loads obtained from experimental results in Eqs. 2.
To study the effect of mixed mode loading conditions, the loading angles were changed by intervals of 6[degrees] and also the fracture parameters at loading angles closed to pure mode-II were studied at intervals 3[degrees].
TEST METHOD AND SET-UP
Since the Arcan specimen was developed by Arcan  has similar dimensions to the specimen used in this investigation. Arcan test specimen was originally designed for use with fiber-composite materials , but has been modified by many researchers for use with adhesives and isotropic materials [28-31]. In modified Arcan device, a number of holes and rod inserts were added to stiffen the specimen-fixture connection (Fig. 1). The pinhole locations on the outer edge of the fixture provide a range of loading angles. The specimen is attached to the fixture by three pins at each end. Flowever, modified Arcan fixture has several problems which limit its usefulness to obtain reliable results [15, 16]. One problem is the specimen-fixture connection that causes bending or unwanted third mode loading conditions [13, 17, 18], Another limitation of modified Arcan is the failure of holes and rod inserts added to stiffen the specimen-fixture connection .
In this study, a new loading device was made for the mixed-mode fracture test of specimens, which allows mode-I, mode-II, and almost any combination of mode-I and mode-II loading to be tested with the same test specimen configuration (Fig. 5). The test specimen was developed to produce a uniform state of plane-stress in solid specimens, where the grips and the butterfly specimen are cut out of a single plate. Thus, no joints were necessary between the butterfly specimen and the grips. The main component of the apparatus is a butterfly specimen, which is inserted on either side to two half-circular grips. The grips are connected to a universal testing machine at the top and bottom, respectively. The grips together with the butterfly specimen form a circular disk with two anti-symmetric cutouts. The scaled locations on the outer edge of the fixture provide a range of loading angles. The specimen is loaded by pulling apart grips of the fixture at a pair of grip on the opposite sides of a scaled radial line. By varying the loading angle [alpha], all mixed-mode conditions starting from pure mode-I to pure mode-II can be created and tested. It is a simple test procedure, clamping/unclamping the specimens is easy to achieve and only one type of specimen is required to generate all loading conditions. Therefore, disadvantages presented in the previous mixed-mode fracture toughness test methods can be avoided. The new fixture has perfect symmetry which provides a uniform stress state, creates the pure plane strain conditions, and eliminates the unwanted mode-III loading conditions.
The fracture test was conducted by controlling the constant displacement rate of 0.5 mm/min, and the fracture loads and displacements were recorded. All tests were carried out using a zwick/z10 testing machine. Tests were repeated at least three times for each loading angle and sample thickness. The load-displacement curves generated by the test machine were used to determine maximum loads and displacement.
RESULTS AND DISCUSSION
According ASTM standard D5045 for plastics, the stress intensity factors ahead of the crack tip were calculated using the Equations [K.sub.I] = [P.sub.c]/wt [square root of [pi]a][f.sub.I](a/w) and [K.sub.I] = [P.sub.c]/wt [square root of [pi]a][f.sub.I](a/w). Where [P.sub.c] is the fracture load, a is crack length, w is the specimen width, t is the specimen thickness, and f(a/w) is a geometrical factor. In turn [K.sub.IC] and [K.sub.IIC] are obtained using geometrical factors or nondimensional stress intensity factors [f.sub.I](a/w) and [f.sub.II](a/w), respectively, which are obtained through finite element analysis of test specimen. Numerical analyses were carried out using the interaction J-integral method to assess geometrical factors or nondimensional stress intensity factors [f.sub.I](a/w) and [f.sub.II](a/w) for polymeric material under consideration.
For the purpose of comparison, in Fig. 6, the finite element results together with those calculated by ASTM D5045 for a homogeneous structure are presented. The authors' finite element results for the nondimensional stress intensity factors of the homogeneous model were very close to ASTM D5045 results.
ASTM standards E399 and D5045 give some guidance for plane strain mode-I fracture toughness for metals and plastics, respectively. Linear elastic fracture mechanics and plan strain conditions are the primary requirements. In this study, for the purpose of comparison, in Fig. 7, the 2D finite element results together with those calculated for 3D models are presented for sample's thickness of 10 mm, in which according to perfect symmetry of new loading device, there is very good agreement.
For appropriate specimen thickness selection, the value of nondimensional stress intensity factors was investigated according to the thickness of the sample. The results of analyses were shown in Figs. 8 and 9. For specimens with the thickness of less than 10 mm and more than 20 mm the plane strain conditions are not predominant due to the nonuniform distribution of stress on the samples. However, the finite element results of dimensionless stress intensity factors with models for sample's thickness of 10, 15, and 20 mm, in which according to perfect symmetry of new loading device (Fig. 5), there is very good agreement and have been chosen for the experiments.
To assess stress intensity factors [K.sub.I] and [K.sub.II] using Eqs. 2, geometrical factors or nondimensional stress intensity factors [f.sub.I](a/w) and [f.sub.II](a/w) for both pure mode-I and pure mode-II loading were determined. In Figs. 10 and 11, the values of [f.sub.I](a/w) and [f.sub.II](a/w) have been presented for a/w = 0.5 and for thicknesses of 10, 15, and 20 mm versus loading angle, respectively. It can be observed that the values of all three graphs are matched. Consequently, the 10-mm thickness of the sample according to the numerical analysis can satisfy the plane strain conditions. These results were predictable as the theory that was used in numerical analysis in each thickness was based on plane strain condition.
Figure 12 shows the quantities of the mode-I and mode-II of nondimensional stress intensity factors for specimen thickness of 10 mm and a/w = 0.5. The result shows that in loading angle of about 78[degrees] the effect of mode-I and mode-II is the same and for loading angles [alpha] [less than or equal to] 78[degrees] the mode-I fracture is dominant and as the mode-II loading contribution increases, the mode-I stress intensity factor decreases and the mode-II stress intensity factor increases. For [alpha] [greater than or equal to] 78[degrees], mode-II fracture becomes dominant.
By drawing upon the results of Fig. 12, the polynomials of third degree were fitted among nondimensional stress intensity factors for various loading conditions, thickness of 10 mm and a/w = 0.5 to prognosticating the quantities of [f.sub.I](a/w) and [f.sub.II](a/w) can be derived as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Figures 13 and 14 show the relationship between the stress intensity factors of mode-I and mode-II which are presented according to the various ratios of a/w for the fixture, respectively. The a/w ratio was varied between 0.3 and 0.8 at 0.1 intervals. Also it can be seen, that as a/w ratio increases, nondimensional stress intensity factor in mode-I increases while mode-II remains almost constant.
The strain energy release rates were calculated using Eqs. 3. The relationship between the mixed-mode ratios of strain energy release rates and the loading angles [alpha] is shown in Fig. 15. It is indicated that very high ratios of mode-I to mode-II are dominant for loading angles close to pure mode-I loading. The ratios of strain energy release rates close to pure mode-II loading exhibit the opposite trend. As expected, it is confirmed that by varying the loading angle of the new loading device, pure mode-I, pure mode-II, and a wide range of mixed-mode loading conditions can be created and tested. The variation of [G.sub.I] and [G.sub.II] versus mixed ratio ([G.sub.II]/[G.sub.T]) is presented in Fig. 16. It can be seen that for ratios less than [G.sub.II]/[G.sub.T] = 51% the strain energy release rate for mode-I is dominant, and after that there is an opposite trend.
In Figs. 17 and 18, strain energy release rates [G.sub.I] and [G.sub.II] obtained by Eqs. 3 and the total strain energy release rate obtained by [G.sub.T] = [G.sub.I] + [G.sub.II] for thickness of 10 mm are compared for a constant value of the load. It is seen that for loading angles [alpha] [less than or equal to] 78[degrees] the mode-I strain energy release rate is dominant and as loading angle increases, [G.sub.I] decreases while [G.sub.II] increases. For [alpha] [greater than or equal to] 78[degrees] mode-II fracture becomes dominant. It is also observed that with the increase in crack length the strain energy release rate by the modes-I and II increases approaching the loading area of mode-I; however, it remains almost constant near the loading area of mode-II. The total strain energy release rate under mixed-mode loading condition decreases with the loading angle. Therefore, the increase in the mode-II loading contribution leads to a reduction in the total strain energy release rate.
Fracture tests were conducted by controlling the constant displacement rate of 0.5 mm/min. Subsequently, the fracture loads and displacements were recorded. Tests were repeated at least three times for each loading angle and sample thickness. Overall, 27 specimens were tested in this survey. The load-displacement curves generated by the test machine were used to determine maximum loads and displacement. Various thicknesses were used for mode-I, mode-II, and mixed-mode loading conditions, as shown in Table 1. The average values of critical fracture loads were used to determine the critical mixed-mode stress intensity factors and strain energy release rates. An example of force-displacement curves for three sample thicknesses according to 45[degrees] mixed-mode loading is presented in Fig. 19.
Critical stress intensity factor values [K.sub.IC] and [K.sub.IIC] were obtained using the Eqs. 2 and critical fracture loads were obtained from load-displacement curve. Table 2 summarizes the average values of experimentally determined critical stress intensity factors mode-I and mode-II under various loading conditions. Using the new designed loading device specimen under pure mode-I loading, the average fracture toughness was measured as [K.sub.IC] = 4.32 [MPa*[m.sup.1/2]] for the sample thickness of 10 mm, [K.sub.IC] = 4.10 [MPa*[m.sup.1/2]] for 15 mm, and [K.sub.IC] = 4.07 [MPa*[m.sup.1/2]] for 20 mm, respectively. For pure mode-II, the average fracture toughness was [K.sub.IIC] = 1.42 [MPa*[m.sup.1/2]] for thickness of 10 mm, [K.sub.IIC] = 1.26 [MPa*[m.sup.1/2]] for 15 mm and [K.sub.IIC] = 1.35 [MPa*[m.sup.1/2]] for 20 mm, respectively.
The calculated critical strain energy release rates [G.sub.C] show quantitatively how much energy must be put into the specimen to create fracture surfaces. Table 3 shows [G.sub.IC], [G.sub.IIC], and [G.sub.TC] = [G.sub.IC] + [G.sub.IIC], obtained by Eqs. 3 using experimental data. [G.sub.IC] decreases while Guc increases with an increase in mode-II loading contribution. The opening-mode and shearing-mode critical strain energy release rates were found to be approximately 8800.3 and 945.1 J/[m.sup.2] for the sample thickness of 10 mm, 7946.5 J/[m.sup.2], and 749.9 J/[m.sup.2] for the sample thickness of 15 mm, 7813.2 J/[m.sup.2], and 966.7 J/[m.sup.2] for the sample thickness of 20 mm, respectively. It can be seen that [G.sub.IC] is larger compared to [G.sub.IIC] indicating that the cracked specimen is tougher in mode-I and weaker in mode-II loading conditions. Table 3 also shows the total strain energy release rate, [G.sub.TC], under various loading conditions, which decreases with the loading angle. Therefore, the results confirmed that the maximum fracture toughness occurs at mode-I loading condition. Also, it can be seen that the quantities of [G.sub.IC], [G.sub.IIC], and [G.sub.T] in different thicknesses are approximately the same. Therefore, it can be assumed that the thickness of the 10 mm specimen can satisfy the plane strain conditions for ABS polymer, without the need to have thick specimens.
Luna et al.  investigated fracture characteristics of ABS thick three-point bend specimens. They found that the value of the essential work at 20[degrees]C is in the range 3.4-4.7 kJ/[m.sup.2]. The values for fracture toughness of PC/ ABS blend for sample thickness of 10 mm obtained from different methods by Lu et al.  reported 4.10 kJ/[m.sup.2] from COD method, 4.17 kJ/[m.sup.2] from ASTM E813-81 method, and 4.48 kJ/[m.sup.2] from new J method. Kwon and Jar  conducted a series of double-edge-notched Iosipescu mode I and mode II fracture tests in products made of ABS. Value of fracture toughness for mode II fracture of ABS was found to be around 32.3 kJ/[m.sup.2] that was about 2.5 times of the value for mode I fracture of 13.1 kJ/[m.sup.2]. In this study, the mixed-mode fracture behavior for ABS polymeric material specimens was investigated based on experimental and numerical analyses using new loading device. The opening-mode and shearing-mode critical strain energy release rates for sample thickness of 10 mm were found to be approximately 8.8 kJ/[m.sup.2] and 9.5 kJ/[m.sup.2], respectively. Both mode-I and mode-II fracture toughness for the new test methodology specimen were found more than those for three-point bend and double-edge-notched Iosipescu specimens. The reason for these differences in fracture toughness may be related to the mode of crack growth and the analysis and test methodology. Although further work is required to obtain accurate information on evaluation of fracture toughness of ABS polymeric material under mixed mode loading conditions, the higher value of mode-I and mode-II fracture toughness can be obtained from the new test methodology specimen rather than three-point bend and double-edge-notched Iosipescu specimens.
CONCLUSION AND SUMMARY
To calculate critical stress intensity factors in this procedure, it is necessary to use numerical methods along with obtained results from experimental tests. A linear elastic finite element analysis was performed under plane strain conditions using the ABAQUS finite element code. The stress intensity calibration was obtained by the finite element analysis using the finite correction factors method. The finite element results indicated that for loading angles close to pure mode-I loading, a high ratio of mode-I to mode-II is dominant and the opposite trend is exhibited for loading angles close to pure mode-II loading. It confirms that by varying the loading angle of a specimen, pure mode-I, pure mode-II, and full range of mixed-mode loading conditions can be created and tested. According to the obtained results, the quantities of the stress factors of mode-I do not undergo any changes from 0[degrees] to 45[degrees] loading angles, but as the loading angle changes from 45[degrees] to 90[degrees] the mode-I stress strain factors decrease and the mode-II stress strain factors increase. In the mixed-mode situation, until loading angle 78[degrees], mode-I is dominant, and from the loading angle 78[degrees] to 90[degrees], mode-II is the effective factor in fracture. It is seen that for the ratio less than [G.sub.II]/ [G.sub.T] = 51% the strain energy release rate for mode-I is dominant, and after that there is an opposite trend. By studying the numerical and experimental fracture results of ABS polymer specimens for three thicknesses, it was found that the thickness of the 10 mm specimen satisfied the plane strain condition, without the need to have thick specimens. Average fracture toughness under pure mode-I loading was found to be [K.sub.IC] = 4.32 [MPa*[m.sup.1/2]], and for pure mode-II, the average fracture toughness was found to be [K.sub.IC] = 1.42 [MPa*[m.sup.1/2]].
NOMENCLATURE a Crack length B Prelogarithmic energy factor matrix E Modulus of elasticity f Geonometrical factor or nondimensional stress intensity factor [f.sub.I] Mode-I geometrical factor or nondimensional stress intensity factor [f.sub.II] Mode-II geometrical factor or nondimensional stress intensity factor G Strain energy release rate [G.sub.I] Mode-I or opening mode strain energy release rate [G.sub.II] Mode-II or shearing-mode strain energy release rate [G.sub.T] Total strain energy release rate [G.sub.C] Critical strain energy release rate [G.sub.IC] Mode-I critical strain energy release rate [G.sub.IIC] Mode-II critical strain energy release rate [G.sub.TC] Total critical strain energy release rate J J-integral K Stress intensity factor [K.sub.I] Mode-I or opening-mode stress intensity factor [K.sub.II] Mode-II or shearing mode stress intensity factor [K.sub.C] Critical stress intensity factor [K.sub.IC] Mode-I critical stress intensity factor [K.sub.IIC] Mode-II critical stress intensity factor n Outwards normal to [GAMMA] P Applied load [P.sub.c] Critical load q Unit vector in the virtual crack extension direction t Specimen thickness u Displacement vector w Specimen width [alpha] Loading angle on new design loading device v Poisson's ratio [GAMMA] Arbitrary contour [sigma] Stress tensor [[sigma].sub.y] Yield strength [lambda] Virtual crack advance dA Surface element CLS Crack lap shear CRS Crack rail shear DCB Double cantilever beam EDT Edge delamination tension ELS End load split ENF End notch flexure MMB Mixed mode bending MMF Mixed-mode flexure SLB Single leg bending
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Abuzar Es'hagi Oskui, (1) Naghdali Choupani, (1) Elyas Haddadi (2)
(1) Department of Mechanical Engineering, Institute of Polymeric Materials/Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran
(2) Technical and Vocational University, P.O. Box: 51745-135, Tabriz, Iran
Correspondence to: Naghdali Choupani; e-mail: firstname.lastname@example.org
Published online in Wiley Online Library (wileyonlinelibrary.com).
TABLE 1. Critical mixed-mode fracture loads [P.sub.c] [N] with crack length 15 mm and for different sample's thickness. Loading angle 0[degrees] 45[degrees] 90[degrees] t = 10 mm 1 2554 2753 3950 2 2573 2945 3899 3 2892 2996 3920 Avg. 2673 2898 3923 Std. 190 128 26 t = 15 mm 1 3479 4530 4999 2 4010 4615 5395 3 3895 4633 5157 Avg. 3795 4593 5183 Std. 279 55 199 t = 20 mm 1 4894 5631 7763 2 5305 5205 7721 3 4798 5357 7812 Avg. 4999 5398 7765 Std. 269 216 46 TABLE 2. Average mixed-mode critical stress intensity factors data [K.sub.C] [Mpa x [m.sup.1/2] with crack length 15 mm and for different sample's thickness. Loading angle 0[degrees] 45[degrees] 90[degrees] t = 10 mm [K.sub.IC] 4.32 3.31 -- [K.sub.IIC] -- 0.74 1.42 t = 15 mm [K.sub.IC] 4.10 3.51 -- [K.sub.IIC] -- 0.79 1.26 t = 20 mm [K.sub.IC] 4.07 3.10 -- [K.sub.IIC] -- 0.70 1.35 TABLE 3. Average mixed-mode critical strain energy release rates Gr [J/[m.sup.2]] with crack length 15 mm and for different sample's thickness. Loading angle 0[degrees] 45[degrees] 90[degrees] t = 10 mm [G.sub.IC] 8800.3 5165.8 -- [G.sub.IIC] -- 257.9 945.1 [G.sub.TC] 8800.3 5423.8 945.1 t = 15 mm [G.sub.IC] 7946.5 5810.7 -- [G.sub.IIC] -- 294.6 749.9 [G.sub.TC] 7946.5 6105.4 749.9 t = 20 mm [G.sub.IC] 7813.2 4545.9 -- [G.sub.IIC] -- 233.9 966.7 [G.sub.TC] 7813.2 4779.8 966.7
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|Author:||Oskui, Abuzar Es'hagi; Choupani, Naghdali; Haddadi, Elyas|
|Publication:||Polymer Engineering and Science|
|Date:||Sep 1, 2014|
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