III loading

Journal Pre-proofs Time-dependent fracture of epoxy resin under mixed-mode I/III loading Attasit Wiangkham, Piyamon Poapongsakorn, Prasert Aengchuan, Nitikorn Noraphaiphipaksa, Chaosuan Kanchanomai PII: DOI: Reference:

S0167-8442(19)30331-3 https://doi.org/10.1016/j.tafmec.2019.102445 TAFMEC 102445

To appear in:

Theoretical and Applied Fracture Mechanics

Received Date: Revised Date: Accepted Date:

20 June 2019 25 October 2019 17 December 2019

Please cite this article as: A. Wiangkham, P. Poapongsakorn, P. Aengchuan, N. Noraphaiphipaksa, C. Kanchanomai, Time-dependent fracture of epoxy resin under mixed-mode I/III loading, Theoretical and Applied Fracture Mechanics (2019), doi: https://doi.org/10.1016/j.tafmec.2019.102445

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Time-dependent fracture of epoxy resin under mixed-mode I/III loading Attasit Wiangkham1, Piyamon Poapongsakorn2, Prasert Aengchuan1, Nitikorn Noraphaiphipaksa3, and Chaosuan Kanchanomai4 1 – School of Manufacturing Engineering, Institute of Engineering, Suranaree University of Technology, Muang, Nakhon Ratchasima 30000, Thailand 2 – Department of Mechanical Engineering, School of Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK 3 – Nitikorn Research Partner Co., Ltd., Ladsawai, Lumlukka, Pathumthani 12150, Thailand 4 – Center of Materials Engineering and Performance, Department of Mechanical Engineering, Faculty of Engineering, Thammasat University, Klong Luang, Pathumthani 12120, Thailand

Abstract In this study, fracture behavior of epoxy resin under mixed-mode loading between opening mode (mode I) and out-of-plane shearing mode (mode III) was investigated using an adjustable loading fixture set with a single-edge notched tension (SENT) specimen. Effects of specimen thickness and loading rate were investigated. Two different thicknesses of specimen were employed (4 and 8 mm) And, the fracture toughness tests were performed at two different loading rates of 0.1 and 1000 mm/min. Fracture toughnesses of epoxy resin were calculated using finite element (FE) analysis based on both linear-elastic fracture mechanics (LEFM) and elastic-plastic fracture mechanics (EPFM). It was found that under mixed-mode I/III and pure mode III loadings, fracture occurred in a ductile manner with severe plastic deformation. Therefore, LEFM is no longer valid, and EPFM approach was applied in this study to characterised fracture behavior of epoxy resin under mixed-mode I/III loading. Two different fracture mechanics parameters based on EPFM, e.g. critical Jintegral (Jc) and critical crack-tip displacement, were calculated. Jc results revealed that the effect of specimen thickness was significant at low loading rate. However, at high loading rate the effect of specimen thickness was marginal. While, critical crack-tip displacement results indicated that fracture occurred at the same level of crack-tip displacements regardless of specimen thickness. An observation of fracture surfaces agreed with the crack-tip displacement results, i.e., a similar fracture surface topology for both thick and thin specimens tested at the same loading rate. Keywords: Epoxy resin; mixed-mode fracture; J-integral, crack-tip displacement; FE analysis.

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1. Introduction Epoxy resin is one of the most widely used thermosetting polymers for engineering purposes. Its applications are such as high-strength adhesive joint in various engineering structures and matrix materials in composites. In these structures, present of small defects or discontinuities caused by manufacturing process are highly possible. During loading, such discontinuities may grow and coalescence as a main crack and lead to failure from unstable crack propagation, i.e., fracture. In order to design epoxy-based joint and composite structures, it is important to understand fracture behavior of neat epoxy resin. Fracture of a crack can occur under three different loading modes, i.e., opening mode (mode I), sliding or in-plane shearing mode (mode II), and tearing or out-of-plane shearing mode (mode III). Many researchers have been investigated fracture under pure mode I loading of epoxy resin [1-4] as well as its composites [5-6]. It is found that fracture behavior under pure mode I loading was significantly influenced by stress state at the crack tip as well as localised plastic deformation, such as crack blunting, crazing, stretched zone, and shear lip [3, 5]. In addition, crack tip stress state and deformation can be affected by strain rate or loading rate due to viscoelastic behavior of epoxy resin. Due to a wide range of applications of epoxy-based adhesive joint and composite, epoxy resin may undergo complex loading during services. Thus, failure from fracture under mixed-mode loading is likely. Although a number of researches on mixed-mode fracture of epoxy-based composites have been done, only limited number of works are dedicated to studying fracture behavior of neat epoxy resin under mixed-mode loading. Kanchanomai and Rattananon [7] used various types of single edge-notched bending (SENB) specimen to study fracture behavior of neat epoxy resin under mixed-mode I/II loading. In their work, critical stress intensity factor (K) which is a linear-elastic fracture mechanics (LEFM) parameter was used to describe fracture behavior under mixed-mode loading. Due to the viscoelastic behavior of the epoxy resin, effects of loading rate and specimen thickness on the mixed-mode I/II fracture were observed. Jamali et al. [8] investigated fracture of epoxy under mixed-mode I/II loading using compact tension shear (CTS) specimen. Two LEFM parameters, e.g. stress intensity factor (K) and strain-energy released rate (G), were employed to characterise fracture of epoxy resin under mixed-mode loading. They found that fracture behavior and surface morphology of epoxy under mixed-mode I/II loading was significantly affected by the thickness of specimen.

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A few researches on mixed-mode I/III fracture of epoxy resin have been recently done. Fracture mechanism of epoxy under mixed-mode I/III loading was investigated in a study of Ren et al. [9]. They used K to describe mixed-mode I/III fracture behavior and found that mixedmode crack propagation was dominated by mode I component. However, in their work fracture toughness for each of loading mode was not determined. Ayatollahi and Saboori [10] designed a new fixture set for fracture test of the single edge-notched tension (SENT) specimen under mixed-mode I/III loading. They have used their new fixture set for studying fracture of epoxy and its composites under mixed-mode I/III loading [11-12]. However, in their works, mixed-mode I/II fracture was characterised based only on LEFM. Since there is no standard practice for determination of mixed mode fracture toughness, various types of loading configuration have been developed to investigate the mixed-mode fracture. For mixed-mode I/II fracture, the techniques of CTS as well as asymmetric notch and/or loading SENB tests were widely used. While for mixed-mode I/III fracture, various types of specimen and test method were employed, e.g., circumferentially notched cylindrical bars [9], single edge notched tension (SENT) with an adjustable loading angle fixture [10], semi-circular bend (SCB) with inclined crack [13], edge notched disc bend (ENDB) with an inclined crack in three-point bending loading [14], asymmetric disc bend (ADB) specimen containing an inclined crack [15]. It can be seen that the understanding of fracture behaviors and mechanisms of epoxy resin under mixed-mode I/III loading is very limited and only LEFM parameters have been used to describe fracture of epoxy resin under mixed-mode loading. Therefore, in this study mixed mode I/III fracture of epoxy resin was investigated. Fracture toughness specimen and test configuration used in this study were in accordance with the procedure proposed by Ayatollahi and Saboori [10]. The effect of specimen thickness was evaluated using two different thicknesses of specimen (4 and 8 mm). Moreover, to investigate the time-dependent effect of epoxy resin on the fracture behavior, fracture toughness tests were carried out at two different loading rates of 0.1 and 1000 mm/min. To characterise fracture behavior under mixed-mode I/III loading, both linear-elastic fracture mechanics (LEFM) parameter, i.e., critical stress intensity factor (K) and elastic-plastic fracture mechanics (EPFM) parameters, i.e., critical J-integral (Jc) and crack tip displacement upon fracture were calculated using finite element (FE) analysis. Fracture surfaces of epoxy resin after mixed-mode I/III

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fracture toughness tests were observed using a scanning electron microscope (SEM) and mixedmode I/III fracture mechanisms were discussed.

2. Material and experimental procedure The thermoset epoxy resin system used in this study was supplied by Aditya Birla Chemicals (Thailand) Ltd. The system consists of a DGEBA (diglycidylethers bisphenol-A) resin and an aliphatic amine hardener. The resin and the hardener were initially mixed at a weight ratio of 100:35 according to manufacturer recommendation. Then the mixture was poured into a square plate mold (W x L = 100 mm x 100 mm) and cured at ambient temperature for 24 h prior to a postcuring process in an air circulating oven at 80°C for 4 h. After postcuring, epoxy resin was cooled down at ambient temperature. In the final step, the epoxy plate was taken out from the mold and machined into specimen shape. Fracture toughness test under mixed-mode I/III loading was carried out according to Ayatollahi and Saboori [10]. A single edge-notched tension (SENT) specimen with an adjustable loading angle fixture was employed. SENT specimen was machined to achieve the final dimensions as illustrated in Fig. 1 (a). The method for preparing a precrack on the fracture toughness specimen was according to the ASTM D5045-14 [16]. Firstly, a notch was introduced in the specimen by sawing. Then, a sharp crack tip was generated at the notch root by tapping on a fresh razor blade. The total depth of the notch and the sharp crack is the initial crack length, ao. Initial crack length and crack tip radius were observed and measured using an optical microscope. The average crack tip radius produced by tapping a razor blade was 0.05 mm. The fixture set for mixed-mode I/III fracture toughness test used in this study are illustrated in Fig. 1 (b). To generate various modes of loading, a loading angle () was adjusted by changing the location of the loading pin. According to the present design, pure mode I loading (  0°), pure mode III loading (  90°), and two mixed-mode I/III loadings (  65° and 78°) can be achieved. The fracture toughness tests were performed on a universal testing machine, Instron 5565, with 5-kN load cell. The specimen and fixture setup are shown in Fig. 1 (c). To investigate the effect of stress state on fracture behavior of epoxy resin, two different specimen thicknesses of 4 and 8 mm were tested at two different loading rates of 0.1 and 1000 mm/min. Four different testing conditions are abbreviated as LL (low thickness and low loading rate), LH (low thickness and high loading rate), HL (high

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thickness and low loading rate), and HH (high thickness and high loading rate), as defined in the nomenclature list. After the test, fracture surfaces were then observed using a scanning electron microscope (SEM).

3. Finite element analysis Since there is no empirical calculation method for determination of fracture toughness under mixed-mode loading, in this study, FE analysis was used in order to calculate fracture toughness under mixed-mode I/III loading conditions. 3D finite element analysis was performed using a commercial FE software, Abaqus. The SENT specimen together with the loading fixture, the connecting bolts were included in the FE model (Fig. 2). Mechanical properties of epoxy resin used in FE analysis were determined from the tensile tests performed according to ASTM D638 [17]. The tensile true stress-strain curves for loading rates of 0.1 and 1000 mm/min were illustrated in Fig. 3. Table 1 shows mechanical properties at two different loading rates used in FE analysis. A crack tip was modelled as a U notch with a radius of 0.05 mm corresponding to the actual crack tip radius generated by a razor blade cut. The loading fixture was modeled as a rigid body, while, the connecting pin made of a high strength steel with an elastic modulus of 210 GPa and a Poisson’s ratio of 0.33. Contact pairs considered in the FE model are (i) the contact between the fixture and the specimen, and (ii) the contact between the specimen and the connecting bolts. The small-sliding contact was assumed for the contact between the fixture and the specimen. While, the tied contact formulation was applied between the specimen and the connecting bolts due to the fact that the sliding of this contact was trivial. In this study, as FE results from the assumptions of friction contact and frictionless contact were marginal due to a relatively more severe condition at the crack tip, frictionless contact was assumed. The fixture which is a rigid body was tied to a reference point located at the center of each loading hole thus the translation and the rotation of the fixture was defined by the displacement constraints of the reference point. For the reference point of the lower fixture, no translation and rotation were allowed, while, for the reference point of the upper fixture, only the translation in the loading direction (y-direction) was allowed as schematically shown in Fig. 2 (a).

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Fig. 2 (b) illustrated the meshed FE model. The fixture model was meshed as 3D rigid element, while the connecting bolt model was meshed as 20-node hexahedral element. For the specimen, two types of mesh were introduced, i.e., 15-node wedge element around the crack front, and 20-node hexahedral element in other areas (Fig. 2 (c). To achieve the square root singularity of the stress/strain field at the vicinity of the crack tip, mesh refinement near the crack front was done, and the singular element with the midside node at the quarter point was used at the crack tip vicinity. A mesh convergence test was performed on both the fixture and the specimen in order to determine a proper number of elements for the model. Total numbers of elements on each of the fixture and the connecting bolts were 3431 and 1696, respectively. While, for the specimen, a total number of 19968 elements were employed.

4. Results and discussion 4.1 Load-displacement relationship Load-displacement relationships obtained from fracture toughness tests at various loading conditions are shown in Fig. 4. Critical fracture loads for the tested specimens were summarised in Table 2. The load-displacement plots could be divided into three regions, i.e., (i) linear relationship region in an early stage, (i) nonlinear relationship region, and (iii) a drop of load upon propagation of the crack. For all specimens tested under pure mode I loading, only linear relationship region was observed before a load drop, and the displacement at fracture was small comparing to those tested under mixed-mode loadings. Moreover, fracture load under pure mode I loading increased with increasing specimen thickness and decreasing loading rate. Under mixed-mode and pure mode III loadings, specimens tested under low loading rate (LL and HL) underwent severe nonlinear deformation indicating ductile fracture (Fig. 4 (a) and (b)), while nonlinear deformation was trivial for LH and HH specimens tested under high loading rate indicating brittle fracture (Fig. 4 (c) and (d)). The observations corresponded to the fact that the deformation of polymer is a time-dependent behavior. As the loading rate increased, time for the deformation process decreased. Therefore, specimens tested at higher loading rate (LH and HH) failed under a relatively brittle manner, while, large scale deformation was observed in those tested at lower loading rate (LL and HL).

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In addition, thin (LL and LH) specimens are more compliant than thick (HL and HH) specimens, in particular, under mixed-mode and pure mode III loadings. It was observed during the test under mixed-mode I/III and pure mode III loadings that thin specimens slightly bended. Bending of a specimen is possible when a specimen is subject to out-of-plane shear loading ( > 0°). The effect of specimen thickness on the specimen compliance was clear under mixed-mode and pure mode III loadings but not likely under pure mode I loadings.

4.2 Critical stress intensity factors (KIQ and KIIIQ) Critical stress intensity factors (KIQ and KIIIQ) under mixed-mode loading were calculated using linear-elastic FE analysis. Noted that KIQ and KIIIQ, defined in the ASTM D5045-14, are conditional fracture toughnesses which may not equal to plain-strain fracture toughnesses (KIc and KIIIc). A critical load (PQ) used for calculation of K was determined according to ASTM D5045-14 [15] as summarised in Table 2. To validate FE model, a strain gage was installed on the back surface of a specimen tested under pure mode I loading. Strain gage installation site is illustrated in Fig. 5 (a). Noted that the direction of strain gage was parallel to loading direction. Strain collected from the experiment was plotted against applied load and compared to the result from the FE analysis (Fig. 5 (b)). The results from the experiment and FE analysis show a good agreement. In the study of fracture toughness under mixed-mode I/III loading, the loading angle was adjusted to achieve different modes of loading. At the loading angle of 0, the crack is perpendicular to load line, thus it is under opening mode, i.e. pure mode I loading. The higher the loading angle, the more the combination of out-of-plane shear (mode III loading). To study the 3D effect on variations of fracture parameters, an FE analysis for the 4-mm specimen with a reference load of P = 500 N, a crack ratio of a/W = 0.5 was performed. In order to simplify the fracture parameters in this study, the geometry factor, Yi which is the dimensionless form of stress intensity factor is employed as given below.

𝐾𝑖𝑊𝑡

𝑌𝑖 = 𝑃

, 𝑖 = 𝐼, 𝐼𝐼,𝐼𝐼𝐼

𝜋𝑎

(5)

where Ki is the stress intensity factor, W is the specimen width, t is the specimen thickness, P is the applied load, and a is the crack length. Through-thickness variations of fracture parameters were presented in Fig. 6 as plots between the geometry factors, Yi versus the normalised crack front

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location, z/t. Noted that t represents the specimen thickness and z represents the position of each point along the crack front. The results in Fig. 6 (a) indicates that the distribution of mode I geometry factor, YI, along the crack front for the specimen under mode I loading ( = 0°) are relatively uniform. However, near the free surfaces, a slight drop of YI was observed as a result of a transition of the stress state from plane strain at the mid-section to plane stress at the free surface. Under mode I loading ( = 0°), YII and YIII remain zero throughout the thickness. Under mixed-mode loadings ( = 60, 78, and 90°), YII and YIII vary along the through-thickness direction as seen in Fig. 6 (b) – (d). In-plane and out-of-plane shearing modes, i.e., mode II and mode III, are known to be coupled and involves in a generation of 3D singular stress state at the crack tip. Since stronger coupled fracture modes exists in the area close to the free surface of the specimens, higher effective mode III and mode II stress intensity factors occurs at both sides of the crack front [18]. Although YII values are relatively large at the free surface, they become zero at the mid-section of specimen (z/t = 0) all mode mixities as theoretically predicted elsewhere [19]. While, YI and YIII attain the maximum values at the mid-section of the specimen. It was also suggested in the literature [19] that the mid-section is the critical location at the onset of mixed-mode fracture. Therefore, in order to investigate the critical fracture behaviour under an effect of the combined loading between opening mode (mode I) and out-of-plane shearing mode (mode III), numerical results at the mid-section of the specimen (z = 0) were mainly considered throughout this study. As the mode mixity was varied in this study, the combination between mode I and mode III loading is described by mode mixity angle () which is defined as 𝛽 = tan ―1 (𝐾𝐼𝐼𝐼𝑄 𝐾𝐼𝑄)

(6)

where KIQ and KIIIQ are the critical stress intensity factors obtained from FE analysis at a corresponding critical load, PQ, under mode I and III loadings, respectively. The mode mixity angles achieved in this study are summarised in Table 3. It is noted that at pure mode I loading β is 0. β increases with an increase of out-of-plane shear combination and becomes 90 at pure mode III. Critical stress intensity factors (KIQ and KIIIQ) at various mode mixity angles are shown in Fig. 7 (a) and (b). Under pure mode I loading, an effect of loading rate on the fracture toughness was observed, i.e., fracture toughness under low loading rate (LL and HL) were higher than those under high loading rate (HL and HH) (Fig. 7 (a)). However, an effect of loading rate was not found under

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pure mode III loading. Under pure mode I loading, crack is subjected to normal stress, so the deformation in the through-thickness direction, i.e., Poisson contraction, is possible. Crack tip deformation is known to be an important factor reducing the severity of a crack tip and enhancing fracture resistance of materials [3]. With increasing loading rate, less crack tip deformation can be expected due to a decrease in time for the deformation process and thus results in a low fracture resistance. On the other hand, under pure mode III loading the Poisson contraction effect is not likely because crack is subjected to pure shear stress. Therefore, the fracture toughness was not affected by a change in loading rate (Fig. 7 (b)). As the loading mode changed from pure mode I loading to mixed-mode I/III loading, the critical stress intensity factors (KIQ and KIIIQ) increased. Then, the loading mode changed from mixedmode I/III loading to pure mode III loading, KIQ decreased, while, KIIIQ continuously increased. Under mixed-mode loading, the combination of mode III loading tends to obstruct an opening of the crack due to distortion of the crack front. Thus, under mixed-mode loading, high KIQ is required to drive the crack. Fig. 8 illustrates the critical stress intensity factors obtained from mixed-mode I/III fracture tests compared with some 3D prediction criteria, e.g., maximum tangential stress (MTS) [20], Maximum tangential strain energy density (MTSED) [21], and Richard [22] criteria. Significant differences between the experimental and the predicted results for mixed-mode loading were observed. All criteria are likely to underestimate the critical stress intensity factors of the epoxy resin. A similar discrepancy was reported by many researchers [10, 22, 23]. It is believed that this discrepancy in involved by the nonlinear behaviour occurring under out-of-plane shear loading (Fig. 4) which is not taken into account by the mentioned fracture criteria.

4.3 Crack tip plastic zone As crack is a discontinuity in materials, stress concentration at the crack tip is much higher than in other areas. Once the stress at the vicinity of the crack tip is higher than the yield strength of the material, plastic deformation occurs. The fracture process of a cracked body is significantly affected by the shape and size of the plastically deformed zone. Several simple analytical models have been proposed to estimate the role of the plastic zone. Irwin [24] proposed a model for determination of a plastic zone size ahead the crack tip assuming elastic-perfectly plastic behaviour.

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Dugdale [25] and Barenblatt [26] proposed the plastic strip yield model used to estimate the effect of plastic strip on the stress distribution around crack tip under mode I loading. The estimation of the plastic zone size by Irwin and Dugdale models were based on the width of the plastic zone along the crack line. However, it is more often than not found that the largest dimension of the plastic zone is not along the crack line but along a line shifting from the crack line at some angle. Moreover, the shape and size of a plastic zone at the vicinity of the crack tip strongly depends on material properties, state of stress, and specimen geometry [27]. Therefore, in order to estimate the size and shape of the plastic zone ahead the tip of a crack, FE analysis was applied in the present work. Based on elastic-plastic behavior, equivalent stress distributions upon corresponding critical loads for various modes of loading are shown in Fig. 9. Note that the stress distribution considered is on the midplane of the thickness and grey color represents the area where equivalent stress value is higher than the yield stress of the epoxy resin at a corresponding loading rate, i.e., crack tip plastic zone. While, the blue color indicated the area with an elastic deformation. As mentioned previously that the size of a plastic zone is influenced by material properties, the loading rate effect was considered in the FE analysis since material properties of epoxy resin are dependent of the loading rate. In this study, the yield stresses of the epoxy resin are 41.0 and 71.5 MPa for the loading rates of 0.1 and 1000 mm/min, respectively. As the mode mixity angle increased, the shape of the plastic zone changed from a butterfly shape to a circular shape. The size of plastic zone was also varied with changes in mode mixity angle and specimen thickness. Although a number of works have been reported the analytically-obtained plastic zone size of epoxy resin under pure mode I, there is a lack of the determination of plastic zone size of epoxy resin under mixed-mode I/III loading. The size of a plastic zone (ri) at various angles in front of the crack tip was graphically obtained from the FE results in Fig. 9 by measuring the distance between the crack tip and the edge of the plastic zone, as schematically shown in Fig. 10. The critical plastic zone size (rc) was then determined as the maximum ri value. The rc results are summarised in Table 3. The rc under pure mode I in this study is comparable to those analytically obtained from Irwin’s model in the literatures [28, 29]. The numbers shown in the parentheses are the plastic zone size ratio (R) which is calculated from a ratio between critical plastic zone size (rc) and specimen thickness (t). Noted that R < 1 when a critical plastic zone size is smaller than the specimen thickness, R = 1 when a critical plastic zone size is equal to the specimen thickness, and R > 1 when a critical plastic zone size is larger than the specimen thickness.

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The effect of specimen on the plastic zone size of epoxy resin can be observed in this study. At the loading rate of 0.1 mm/min, the plastic zone ratios (R) for thin specimen (LL) are significantly higher than those for thick specimen (HL) indicating that more severe plastic deformation occurred in the thin specimen. As the loading rate increased, the effect of specimen thickness, however, became insignificant as the plastic zone ratios (R) for thin and thick specimens (LH and HH) were similar. Localised plastic deformations at the vicinity of the crack tip such as crack tip blunting and Poisson contraction can be influenced by geometries of the cracked body. Due to less material constraint, a thinner specimen may experience more severe localised plastic deformation. Moreover, for a viscoelastic material, such localised plastic deformations are dependent of time. Under a lower loading rate where the time for deformation process was longer, the effect of time-dependent localised plastic deformation became significant for a thinner specimen. As the loading rate increased the time for deformation process became shorter; time-dependent localised plastic deformation was marginal. Therefore, the effect of specimen thickness was minor at high loading rate. Under pure mode I loading, with a lower fracture toughness, the critical plastic zone sizes (rc) are smaller than those under mixed-mode I/III and pure mode III loading. The effect of mixed-mode loading on the critical plastic zone size is however still controversial. Although it was suggested in some literatures that rc is independent of the mode mixity [30, 31], some have found that the size of the plastic zone is dependent of the mode mixity [32, 33]. It can be seen that the critical plastic zone sizes for low loading rate (LL and HL) are substantially larger than those for high loading rate as the epoxy resin became more compliant at low loading rate. Moreover, under mixed-mode and pure mode III loading the critical plastic zone sizes of LL specimens are larger than the geometry limitation, which is the thickness of specimen in this case. For other cases, although rc are smaller than the geometry limitation (R <1), substantial deformation was observed under mixed-mode I/III and pure mode III loadings. Therefore, due to larger deformation at the crack tip vicinity, linear elastic fracture mechanics is no longer valid, and critical stress intensity factor is not suitable to describe fracture behavior of epoxy resin in this study.

4.4 Elastic plastic fracture mechanics (EPFM) Elastic plastic fracture mechanics (EPFM) is a fracture mechanics approach to analyse the crack problem of elastic-plastic materials with a relatively large plastic zone. In EPFM approach, the

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strain energy fields or opening displacement near the crack tips are used to describe severity of a crack. J-integral is a fracture mechanics parameter based on strain energy fields at the vicinity of a crack tip, while, crack-tip opening displacement (CTOD) is based on opening displacement of a crack tip. When the energy or opening exceeds a critical value, i.e., critical J-integral or critical CTOD, the crack will grow. In this study critical J-integral (Jc) under various modes of loading were calculated using elastic-plastic FE analysis. Jc results determined from FE analysis are shown in Fig. 11. As the loading rate decreased, Jc increased due to the effect of time-dependent deformation. Moreover, the effect of specimen thickness was clear at low loading rate (LL and HL), while, at high loading rate (LH and HH) the effect of specimen was marginal. Due to a longer time for the deformation process at low loading rate, the thinner specimen tends to experience more severe deformation than the thicker specimen. Since deformation at the crack tip vicinity has a significant effect on fracture behavior of elastic-plastic materials, fracture toughness at low loading rate is likely to be dependent of the thickness of specimen. However, under high loading rate deformation is trivial for both thin and thick specimens. Therefore, fracture behavior at high loading rate is not significantly influenced by the specimen thickness. In this study, as mixed-mode loading was applied the crack tip was deformed in both opening and out-of-plane shear directions. Therefore, crack-tip displacements cannot be measured experimentally. In order to investigate the effect of loading mode on the fracture behavior of epoxy resin, FE analysis based on elastic-plastic fracture mechanics was applied. Crack-tip displacements were evaluated based on the definition of 90 line intersection as illustrated in Fig. 12 [34]. Displacements of the two nodes (node A and B) at the intersection of the 90 line were calculated. Noted that the nodes considered are on the midplane of the thickness. Crack-tip displacement was determined in two directions, i.e., x-direction representing crack-tip displacement under opening mode (mode I), and z-direction representing crack-tip displacement under out-of-plane shearing mode (mode III). Relationships between crack-tip displacement and mixed-mode angle at the loading rate of 0.1 and 1000 mm/min were shown in Fig. 13 (a) and (b), respectively. The crack-tip displacements at low loading rate were larger than those at high loading rate since epoxy was more compliant at low loading rate. Crack-tip displacement under mode I loading increased as the mode mixity increased from pure mode I loading to mixed-mode I/III loading and then decreased as the mode mixity

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increased toward pure mode III loading. While, crack-tip displacement under mode III loading increased continuously with an increase in the combination of mode III loading. For specimens with different thicknesses crack-tip displacements were, however, nearly identical, i.e., the effect of specimen thickness was marginal for crack-tip displacement results. Therefore, under the same loading conditions, crack-tip displacement of 4-mm specimen may be suitable for describing fracture behavior of the present epoxy resin.

4.5 Fracture surface observation Some samples of 4-mm specimen tested under different mode mixities at 1000 mm/min are presented in Fig. 14 (a). The fracture patterns under different mixed mode I/III loading conditions, i.e., tilting and twisting of the crack plane, can be observed and are in good agreements with the results reported in the literatures [23, 35, 36]. In-plane shearing mode (mode II) involves tilting of a crack plane, while out-of-plane shearing mode (mode III) results in twisting of a crack plane. In order to investigate the mode III crack propagation behaviour, twisting of the crack plane of each specimen was observed by measuring a twisting angle of the crack plane. It is seen in Fig. 6 that the combination of in-plane shear (mode II) loading exists throughout the thickness direction, except at the mid-section of specimen. Therefore, the measurement of crack twisting angle,  under mixedmode I/III loading has to be done at the mid-section of specimen as illustrated in Fig. 14 (b). The measurement was performed by capturing a photo of the fracture specimen from its front view with sufficient magnification and then using the image processing software package to determine the crack twisting angle, . The  results are summarised in Table 3 as well as plotted against the mode mixity angle,  in Fig. 14 (c). As the combination of mode III loading increases, the twisting angle,  increases. The results in this work correspond to those reported by Aliha et. Al [37]. Moreover, it can be observed that the crack twisting angle,  is independent of the specimen thickness, but depends on mode mixity and loading rate. After the fracture toughness test, the fracture surfaces were observed using a scanning electron microscope. SEM micrographs of the fracture surfaces were shown in Fig. 15 - 17 for specimens tested under pure mode I loading (Fig. 15), mixed-mode loading (Fig. 16), and pure mode III loading (Fig. 17). Solid and dashed lines represent crack front location and mid-section line, respectively. Under pure mode I loading (Fig.15), fracture of LL and HL specimens was in a ductile

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manner with a rough fracture surface indicating plastic deformation. An evidence of Poisson’s contraction (shear lip) was also found in the thin specimen (LL). Moreover, during the test at low loading rate, crack tip blunting was also observed. Since localised plastic deformation in the vicinity of the crack tip, such as shear lip and crack tip blunting, can reduce the severity of the crack tip [3], fracture toughness of specimens with severe plastic deformation tends to increase (Fig. 11). However, as the loading rate increased (LH and HH specimens), a smooth fracture surface with no plastic deformation was observed. Brittle fracture is likely at high loading rate and led to a low fracture toughness (Fig. 11). Under mixed-mode I/III and pure mode III loadings, fracture surfaces (Fig. 16 and 17) show an irregular pattern. Fig. 18 illustrates a schematic drawing of a fracture surface under mixed-mode I/III loading. The observation area indicated in Fig. 18 is the area where SEM micrographs in Fig. 16 and 17 were taken. It can be observed from Fig. 16 and 17 that under mixed-mode I/III and pure mode III loadings crack propagation started from one point near a corner and spread out until fracture. The propagation path of the crack is indicated by radiating lines or “River lines”, as schematically shown in Fig. 18. Due to torsion from mode III loading, small cracks formed at a slight tiling angle from the original crack plane. Then the small cracks coalesced together and so formed steps on the fracture surface between each crack [38, 39]. Moreover, an evidence of substantial plastic deformation under mixed-mode I/III and pure mode III loadings (Fig. 16 and 17) was found comparing to the fracture surface under pure mode I loading (Fig. 15). In addition, fracture surface topology of specimens tested under the same loading rate (LL and HL for low loading rate, or LH and HH for high loading rate) was quite similar. This agrees with the crack-tip displacement results (Fig. 13) showing that under the same loading rate, fracture specimens failed under the similar crack-tip displacements regardless of specimen thickness.

5. Conclusions In this study, fracture behavior of epoxy resin under mixed-mode I/III loading was characterised with the effects of specimen thickness and loading rate. It was found that plastic zone size at the crack tip under mixed-mode I/III and pure mode III loading was large comparing to specimen geometries and Linear Elastic Fracture Mechanics (LEFM) was not valid. Elastic Plastic Fracture Mechanics (EPFM) was then applied. Critical J-integral and crack-tip displacements were

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calculated using FE analysis. From the critical J-integral results (Jc) the effect of specimen thickness was substantial for the tests at low loading rate where time-dependent deformation was possible. It was observed during the fracture toughness test that bending of thin specimens involved the fracture process while bending was not likely for thick specimens. Moreover, deformation under bending was also time-dependent and thus enhanced by reducing loading rate. However, at high loading rate the effect of specimen thickness was insignificant as less plastic deformation was possible. Crack-tip displacements in opening and out-of-plane shear directions were determined using 90° intersection line method. It was found that at the same loading conditions, i.e. same mode mixity and loading rate, specimens failed at a similar level of crack displacements regardless of specimen thickness. From the observation of fracture surface, a smooth crack nucleation region, which was an evidence of crack formation under mode I loading was found and followed by “river lines” pattern which was an evidence of coalescence of small tilted cracks produced by mode III loading. Moreover, the observation of fracture surface agreed with the crack-tip displacement results as no obvious difference was observed between the fracture surfaces of thick and thin specimens tested at the same loading condition.

Acknowledgement The authors would like to thank Aditya Birla Chemicals (Thailand) for supplying materials used in this research, free of charge. The first author would like to thank Suranaree University of Technology for the Prestige Postgraduate grant.

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Nomenclature Abbreviations EPFM

Elastic-plastic fracture mechanic

LEFM

Linear-elastic fracture mechanic

SENT

Single-edge notched tension specimen

LL

Specimen with thickness of 4 mm tested at 0.1 mm/min

LH

Specimen with thickness of 4 mm tested at 1000 mm/min

HL

Specimen with thickness of 8 mm tested at 0.1 mm/min

HH

Specimen with thickness of 8 mm tested at 1000 mm/min

List of symbols a

Crack length

JC

Critical J-integral

Ki

Stress intensity factor for mode i (i = 1, 2, 3)

KIQ

Mode I critical stress intensity factor

KIIIQ

Mode III critical stress intensity factor

PQ

Critical load

rc

Plastic zone size at a critical load

R

Plastic zone size ratio (rc /t)

t

Specimen thickness

W

Specimen width

Yi

Dimensionless geometry function for mode i (i = 1, 2, 3)

z

Distance in z-direction (through-thickness direction)

Greek letters



Mode mixity angle



Fracture initiation angle



Loading angle

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Table 1 Mechanical properties of epoxy resin at different loading rates. Loading rate (mm/min)

Properties

0.1

1000

Modulus (GPa)

3.16±0.08

4.15±0.18

Yield strength (MPa)

41.07±0.98

71.48±3.96

Tensile strength (MPa)

55.15±2.55

71.48±3.96

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Table. 2 Critical fracture load for tested specimens. Loading rate

Loading angle

(mm/min)

(°)

No.

Critical load, PQ (N) 4 mm

8 mm

1 2 3

935.451 1150.368 1066.863

578.301 563.158 555.445

1 2

1581.996 1537.120

656.236 783.820

3

1515.332

547.852

78

1 2 3

1661.547 1711.784 1651.550

655.873 695.980 651.550

90

1 2 3

1582.885 1489.500 1602.350

621.744 639.253 602.350

0

1 2 3

105.680 124.188 129.484

99.497 78.744 71.878

65

1 2 3

1381.610 1384.663 1498.173

422.522 521.250 450.680

78

1 2 3

1453.639 1347.385 1502.150

449.074 552.352 503.522

90

1 2 3

1379.088 1774.255 1661.651

631.264 604.780 612.326

0

65 0.1

1000

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Table 3 Summary of mixed-mode I/III loading conditions and plastic zone sizes for various loading conditions.

()

Mode mixity angle for 4-mm specimen (β)

0

0

65

20

78

35

90

90

Loading angle

Plastic zone size, rc (mm)

Fracture initiation angle, γ (°)

LL

LH

LL

LH

2.49 (0.62) 7.38 (1.85) 6.26 (1.57) 6.40 (1.60)

0.07 (0.018) 0.53 (0.13) 0.71 (0.18) 0.81 (0.20)

0 47.5 52.0 56.6

0 57.8 64.0 66.7

Mode mixity angle for 8-mm specimen (β) 0 15 30 90

Plastic zone size, rc (mm)

Fracture initiation angle, γ (°)

HL

HH

HL

HH

2.36 (0.29) 7.88 (0.99) 8.89 (1.11) 6.82 (0.85)

0.10 (0.013) 1.23 (0.15) 1.45 (0.18) 1.58 (0.20)

0 48.0 54.6 56.0

0 57.2 61.1 62.5

*The number in the parentheses is the plastic zone size ratio (R), which is the ratio between plastic zone size and specimen thickness (R = rc/t).

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Specimen

ao

(a)

(b)

Fig. 1 Drawing of (a) specimen and (b) fixture for mixed-mode fracture toughness test, and (c) specimen and fixture setup.

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Precrack

X

X X

Y Z

(a)

Y

Z

(b)

Z

(c)

Fig. 2 Details for finite element model for mixed-mode I/III fracture toughness test: (a) load and boundary conditions, (b) mesh on fixture components and specimen, and (c) mesh refinement in the crack tip vicinity.

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Fig. 3 True stress-strain curves of epoxy resin tested at two different loading rates.

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(a)

(b)

(c)

(d)

Fig. 4 Load-displacement relationship obtained from fracture toughness tests at various loading angles () for (a) LL, (b) HL, (c) LH, and (d) HH specimens.

25

Load

Fig. 5 FE analysis validation; (a) installation of strain gage, and (b) relationship between strain on the back surface and applied load.

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Fig. 6 The distribution of geometry factors (YI, YII, and YIII) along the crack front for different modemixities.

27

(a)

(b)

Fig. 7 Critical stress intensity factors under (a) mode I loading, and (b) mode III loading for various mode-mixity angles.

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Fig. 8 Comparison between experimental data and predictions of different fracture criteria for mixedmode I/III loading.

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LL ( = 0°)

LL ( = 20°)

LL ( = 35°)

LL ( = 90°)

HL ( = 0°)

HL ( = 15°)

HL ( = 30°)

HL ( = 90°)

LH ( = 0°)

LH ( = 20°)

LH ( = 35°)

LH ( = 90°)

HH ( = 0°)

HH ( = 15°)

HH ( = 30°)

HH ( = 90°)

Y

4 mm Z

Fig. 9 Distribution of von Mises stress on the mid-thickness plane of the SENT specimens with different thicknesses tested at various loading conditions.

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Fig. 10 Determination of critical plastic zone size at fracture load by FE analysis.

31

Fig. 11 Critical J-integral for various mode-mixity angles.

32

Y Z

Z

X

Y

Fig. 12 Determination of crack-tip displacements using 45° method.

33

(a)

(b)

Fig. 13 Crack tip displacement in x- and z-directions.

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Fig. 14 Fracture initiation angles; (a) fractured samples tested at loading angles, (b) definition of fracture initiation angle, and (c) crack initiation angle at different mode mixities.

35

LL 0o

LH 0o

HL 0o

HH 0o

1 mm Fig. 15 Fracture surfaces of specimens tested under pure mode I loading ( = 0°)

36

LL 35o

LH 35o

HL 30o

HH 30o

1 mm Fig. 16 Fracture surfaces of specimens tested under mixed-mode loading.

37

LL 90o

LH 90o

HL 90o

HH 90o

1 mm Fig. 17 Fracture surfaces of specimens tested under pure mode III loading ( = 90°).

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Fig. 18 Schematic drawing of fracture surface under mixed-mode I/III loading.

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HIGHLIGHTS 

EPFM is applied to characterise fracture of epoxy under mixed-mode I/III loading.



Critical J-integral and crack tip displacements are dependent of loading rate.



Specimen thickness effect can be neglected when critical crack tip displacements is considered.



Mode III loading induces severe plastic deformation and leads to high fracture resistance.



Evidences of mode III fracture, i.e., a twisted crack surface and river lines, are observed.

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Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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