Effect of delamination face overlapping on strain energy release rate calculations

The strain energy release rate is often used as a fracture mechanics parameter to describe delamination propagation and onset in composites, and is conveniently evaluated using finite element analysis. A common problem encountered in analysis is overlapping (interpenetration) of the delamination faces if these faces are not constrained. This paper examines the effect of overlapping on strain energy release rate calculated using the virtual crack closure method in conjunction with 3-dimensional anisotropic finite element analysis. A bilinear gap element was used between the delamination faces to prevent these faces from overlapping. Three problems were studied: (1) laminates with an embedded crack and embedded delaminations; (2) laminates with free edge delaminations; and (3) end-notched flexure specimens. The results of this investigation indicated that overlapping has a significant effect on the component values of G_I and G_II as well as on the total strain energy release rate. It was also found that end effects can create non-uniform energy release rates along the crack front in edge delamination problems due to twisting of the sublaminate with unsymmetrical lay-up, so that 3-dimensional finite element analysis is required.     

Fracture load predictions for adhesive joints

An engineering approach to fracture load predictions for adhesive joints is presented. The approach is based on the premise that the in-situ strength of the bondline can be characterized by the fracture envelope (critical energy release rate as a function of the mode of loading), for a specific adhesive system. By using the J integral for large deformations together with large-deformation beam theory, a simple closed-form expression is obtained for the energy release rate per unit area extension when a crack propagates in the bondline of a generalized adhesive joint (adhesive sandwich). This technique, together with a published method for mode partitioning, enables fracture load prediction by comparing the calculated fracture parameters with the critical ones from the fracture envelope. The approach is shown to predict fracture loads accurately for a variety of joints including the cracked lap shear (CLS), the single lap shear (SLS) and the double strap (DS) joint.     

Experimental evaluation of stress field around crack tip by caustic method

The caustic method is an optical technique which is useful to determine stress intensity factor values. In this paper, the caustics method was applied to specimens which have an oblique crack, various thicknesses and an open notch to investigate the stress field around the crack tip. The results are summarized as follows:

1. The caustic method is a useful technique to determine the stress intensity factor values of the specimens which have an oblique crack or various thickness and an open notch.

2. The conventional theory of measurement concerning this method is effective when the initial curve r₀ is larger than the minimum initial curve r₀^min which was obtained in this study. It is observed that the values of r₀^min decrease as the ratio of K_II to K_I increases under mixed-mode loading, the one increases with an increase of thickness and notch opening angle.

3. The 3D stress field exists in the vicinity of crack tip; however, the stress state is nearly plane strain deformation in the case of mode I loading. In the case of mixed-mode loading, the stress state approximates to plane stress deformation as the ratio of K_II to K_I increases.

4. A method based on the distribution of the three-dimensional (3D) stress field is proposed to expediently yield the values of K_I using the caustic method in the case of r₀<r₀^min.

Wide range data for crack tip parameters in two disc-type specimens under mixed mode loading

Disc-type specimens are among favorite test samples for determining mode I and mixed mode fracture toughness in brittle materials like rocks, brittle polymers, ceramics, etc. In this research, the finite element method is used to analyze two disc-type specimens: a semi-circular disc specimen containing an edge crack and subjected to three-point-bend loading (SCB specimen), and a centrally cracked circular disc subjected to diametral compressive loading, often called the Brazilian disc specimen. The crack parameters K_I, K_II and T are calculated for different mode mixities from pure mode I to pure mode II. Although the stress intensity factors K_I and K_II are presented mainly for validation of the analyses, they are also used for determining the crack angle corresponding to pure mode II for each specimen. It is shown that in general the T-stress increases for larger crack angles. While the T-stress in the Brazilian disc specimen is always negative for any combinations of mode I and mode II, the sign of T-stress in the SCB specimen depends on the mode mixity. A very good agreement is shown to exist between the calculated results for T and those very limited data presented in other papers.     

Transformation zones, crack shielding, and crack-growth resistance of Ce-TZP/alumina composite in mode II and combined mode II and mode I loading

Crack-tip transformation zones, crack shielding and crack-growth-resistance (R-curve) behaviors of a transformation-toughened ceria-partially stabilized zirconia–alumina (Ce-TZP/alumina) composite were studied in mode II and combined mode I and mode II loading using compact-tension-shear (CTS) specimens. The mode II and mode I stress intensities for both the initial straight cracks and the subsequent kinked cracks were assessed by the method of caustics using geometrically equivalent specimens of polymethyl methacrylate (PMMA). The angle of formation of the transformation zones as well as of extension of the cracks increased systematically with increasing ratio of the mode II and the mode I stress intensities and approached a value of θ^*=−72° in pure mode II loading. This angle was close to the angle for maximum hoop tension in the stress field of a mode II crack (θ^*=−70.5°). A crack-initiation toughness envelope was constructed on a K_I–K_II diagram using the critical loads for incremental crack extension. The crack-initiation toughness in pure mode II loading was less than the corresponding toughness in mode I loading. This result was consistent with calculations that indicated no shielding from the asymmetric and elongated zones developed in mode II loading. The fracture toughness measured for the kinked cracks at long kink lengths approached the maximum fracture toughness measured for a mode I crack.     

Double cantilever beam testing of multidirectional laminates

Several difficulties in the double cantilever beam (DCB) tests of multidirectional laminates often prevent valid measurements of the mode I critical strain energy release rate G_Ic. In this paper, several DCB specimens were analysed with 3D finite element models. The results showed that the undesired effects of residual stresses and of mode-mixity can be minimised. An interlaminar stress based fracture criterion predicts that the G_Ic of multidirectional specimens is typically 10–40% higher than the G_Ic of unidirectional [0°]_n laminates. This agrees with the few valid experimental data available.     

Characteristics of brittle fracture under general combined modes including those under bi-axial tensile loads

In this paper, the brittle fracture initiation characteristics under general combination of the opening mode (Mode I), sliding mode (Mode II) and tearing mode (Mode III) were investigated both theoretically and experimentally.

First, the perfectly brittle fracture tests were conducted on specimens of PMMA (Polymethylmethacrylate) for all possible combinations of the fracture modes including respective pure modes. The experimental fracture strengths were compared with those predicted by the fracture criteria which are represented in terms of: (1) maximum tangential stress, [σ_gq]_max, extended to general combined modes, (2) maximum energy release rate at the propagation of a small kinked crack, [G^k(γ)]_max, and (3) newly derived maximum energy release rate at the initiation of a small kinked crack, [G(γ)]_max. It was found that the [G^k(γ)]_max or [G(γ)]_max criterion was very effective to predict both the direction of initial crack propagation and the fracture strength. These energy release rates are expressed in closed forms, and the interaction curves of the brittle fracture strength under arbitrary combinations of Modes I, II and III were derived.

Next, for fracture accompanied by plastic deformation, tests were carried out on specimens of mild steel (SM 41) imposing bi-axial tensile loads at various low temperatures. Then, brittle fracture with plastic deformation occurs under a combination of Modes I and II. In the case of brittle fracture with small scale yielding, the [G(γ)]_max criterion predicts well the direction of initial crack propagation but estimates only lower fracture strength than the experimental one. In the cases of brittle fracture with large scale yielding and under general yielding, it was found from the fracture tests that the direction of initial crack propagation was nearly normal to the resultant vector of the crack opening displacements in the opening and sliding modes at the notch tip. To this type of fracture, the modified COD criterion predicts well the direction of initial crack propagation, but lower fracture strength.

When brittle fracture occurs under the influence of plastic deformation, in such cases as the last three mentioned above, the actual fracture strength is higher than what the most reliable criterion predicts and it increases as deformation in Mode II becomes larger.     

Critical discussion of the single-fibre pull-out test: does it measure adhesion?

With a comprehensive finite-element model the interface failure process of the single-fibre pull-out test, for the measurement of fibre/matrix adhesion, is investigated on the basis of a fracture-mechanics debonding criterion. Special emphasis is placed on the interface local mixed-mode load, which is shown to have an important influence on the debonding process and is taken into account by a fracture ellipsoid criterion. Additional features investigated are residual thermal stresses, specimen geometrical details (wetting meniscus, drop shape) and a simplistic model of fibre/matrix interfacial friction. For medium debonding lengths the energy release rate runs through a plateau range that can be approximated by a simple analytical approach and can be observed experimentally with a very stiff loading configuration. The mixed-mode state in the plateau range is uniform and dominated by mode 2, but its actual value is quite uncertain. From experimental experience the actual adhesion failure is closely connected with the interface local normal load, while local shear load induces submicroscopic friction and matrix inelasticity which strongly reduce the interface sensitivity, resulting in G_1c<G_2c. G_1c seems to be more significant for adhesion. The interpretation of the plateau range may provide the total critical energy release rate, G_c, for the debonding process, but from a region where mode II prevails. G_c will therefore be far from G_1c, reducing the significance of the tests results for characterization of adhesion.     

Fracture toughness of weft-knitted fabric composites

The mode I inter-laminar fracture toughness of advanced knitted textile composites was investigated. Two complex weft-knitted glass fabrics were selected for the study: a triple rib knit and a Milano knit were impregnated with a tough epoxy resin and tested using a double cantilever beam geometry. For both knitted composites, the influence of the growth direction was studied by investigating crack propagation in both the wale and course directions. The fracture toughness was quantified by determining the critical strain energy release rate (G_IC) using the modified beam theory. The specimens had to be stiffened with layers of glass woven composites added on top and bottom of the beams. This was necessary in order to avoid plastic deformation of the beams and crack deviation out of the inter-laminar plane. The results clearly showed that knitted fabric composites have exceptional inter-laminar fracture toughness properties, namely, more than 7000 J/m². The origin of the high G_IC values, which are superior to woven or UD laminates, lies in the very complex fabric architecture. The three-dimensional loop structure induces various energy consuming mechanisms, which do not occur in other composites. Toughening mechanisms such as crack branching, friction, yarn bridging and breakage were identified using scanning electron microscopy.     

Crack surface relative displacement analysis of mixed-mode fracture mechanics problems

In this paper a unique criteria, crack surface relative displacement, is used to evaluate mixed-mode (mode I and mode II) fracture mechanics problems. Using a conic-section simulation of a crack surface, relationships among the energy release rate G, the stress intensity factors (K₁ and K₂), and crack surface relative displacement are developed. Because the crack surface relative displacement criterion makes direct use of the displacements on the crack surface, instead of the stress field in the region of the crack tip, it simplifies numerical analysis of crack problems. A finite element model of a slant-center-cracked plate is employed to demonstrate the applicability of crack surface relative displacement to mixed-mode problems. The numerical results obtained agree well with analytical solutions. In addition, it is illustrated that similar to K₁, K₂, and G (J in LEFM), crack surface, relative displacement can serve as a fracture criterion for general mixed-mode I and II fracture mechanics problems.     

Relationship between microstructure and fracture behavior of bioabsorbable HA/PLLA composites

Hydroxyapatite particles of four different shapes, that is, micro, nano, spherical and plate, were used to fabricate hydroxyapatite filled poly(l-lactic acid) (HA/PLLA) composites. Effects of HA particle shape on the fracture behavior of HA/PLLA were investigated by mode I fracture testing, fracture surface measurement and scanning electron microscopy. It was found that the micro-HA/PLLA has the highest critical energy release rate, G_IC, with the largest surface roughness, while G_IC of the nano-HA/PLLA was lowest corresponding to the smallest surface roughness. The micro-HA/PLLA composites exhibited interfacial debonding and local ductile deformation of the PLLA matrix, indicating higher fracture energy and therefore, the highest G_IC. On the other hand, the nano-HA/PLLA composites showed brittle fracture surface due to nano-scale interaction between PLLA fibrils and primary HA particles, corresponding to lower fracture energy and hence the lowest G_IC.     

Mixed-mode two-dimensional crack problem by fractal two level finite element method

Fractal two level finite element method (F2LFEM) has been extended to calculate the mixed mode stress intensity factor in a two-displacement cracked body. The complete eigenfunction expansion of displacement by Williams is employed for the global interpolation function, the factors K_I and K_II can be easily computed for any arbitrary loading on any boundary. Results are obtained for some slant crack problems in finite sheets and are compared with known results where available.     

Fracture mechanics‐based progressive damage modelling of adhesively bonded fibre‐reinforced polymer joints

A quasi‐static progressive damage model for prediction of the fracture behaviour and strength of adhesively bonded fibre‐reinforced polymer joints is introduced in this paper. The model is based on the development of a mixed‐mode failure criterion as a function of a master R‐curve derived from the experimental results obtained from standard fracture mechanics joints. Consequently, the developed failure criterion is crack‐length and mode‐mixity dependent, and it takes into account the contribution of the fibre‐bridging effect. Energy release rate values for adhesively bonded double‐lap joints are obtained by using the virtual crack closure technique method in a finite element model, and the numerically obtained strain energy release rate is compared to the critical strain energy release rate given by the mixed‐mode failure criterion. The entire procedure is implemented in a numerical algorithm, which was successfully used for predicting the strength and R‐curve response of adhesively bonded double‐lap structural joints made of pultruded glass fibre‐reinforced polymers and epoxy adhesives.     

Influence of in-plane fibre orientation on mode I interlaminar fracture toughness of stitched glass/polyester composites

The influence of in-plane fibre orientation on the mode I interlaminar fracture toughness, G_Ic of unstitched and stitched glass/polyester composites is investigated in this paper. The G_Ic of planar specimens depends on the fibre orientation, θ in the layers adjacent to the fracture plane, in addition to the property of matrix material. The mode I fracture toughness and fracture behavior of unstitched and stitched 0/0, 30/−30, 45/−45, 60/−60, 90/90 and 0/90 interfaces of unidirectional fibre mats (UD) and 30/−30, 45/−45 and 90/90 interfaces of woven roving mats (WRM) are studied. WRM layer orientation is represented by the direction of warp fibres. Stitching is done by untwisted Kevlar fibre roving of Tex 175 g/km at the stitch densities (number of stitches per unit area) of 10.24 and 20.48 stitches/inch². The specimens having same stitch density, but different stitch distributions are prepared, and the influence of stitch distribution on G_Ic is studied. Double cantilever beam (DCB) tests are carried out and the G_Ic is determined using modified beam theory. The G_Ic of both unstitched and stitched specimens increases with increase in orientation angle, θ upto 45° above which it decreases. The G_Ic values of unstitched 45/−45 delamination interface is around 2.4 times that of the unstitched 0/0 interfaces. The influence of fibre orientation on G_Ic is clearly observed in unstitched specimens, whereas in the stitched specimens, stitching plays an important role in improving the G_Ic and suppresses the influence of fibre orientation; degree of suppression increases with increasing stitch density. When the value of θ is above 45°, transverse cracks are observed in the delamination interface surrounded by UD layers; while in the delamination interface surrounded by WRM layers, transverse cracks are not initiated irrespective of the fibre orientation angle.     

An application of the single specimen technique to a double cantilever beam specimen to determine GIC from a single test record

The single specimen technique is effectively used in classical fracture mechanics for evaluating the J-integral from a single test record. It was previously shown using a cracked lap shear specimen that the technique can be applied to composite structures and the energy release rate of composite laminates can be determined from a single test specimen. The objective of this investigation is to determine the η_el form of a double cantilever beam (DCB) specimen, and to compare it with that of a compact tension (CT) specimen. The result showed that while η_el is linearly related to crack length for the CT specimen, it is proportional to the reciprocal of the crack length for the DCB composite specimen. Also, G_Ic determined from the compliance method was compared with that determined from the η_el approach. The result showed that the η_el form determined from a single beam analysis can be used to determine G_Ic from a single test record.     

The nonlinear energy method for mixed mode fracture

In a linear elastic analysis of mixed mode fracture problem, a general G-K relation has been obtained based upon the maximum energy release rate criterion. The expression of nonlinear energy release rate for specimen with an arbitrarily oriented crack subjected to biaxial loading has been formulated. The numerical results which indicate the biaxial effect are presented.     

Fracture toughness of multilayer silicon nitride with crack deflection

The fracture resistance of a multilayer silicon nitride consisting of alternate dense and porous layers was investigated by a single-edge-V-notched beam (SEVNB) technique. Since silicon nitride whiskers were aligned parallel to the laminar direction in the porous layer, the crack deflected macroscopically along the whiskers, resulting in high apparent K_I values, 15–25 MPa m^1/2. The crack then propagated in mode I, and was arrested when K_I was reduced to the fracture resistance without the crack deflection effects. These fracture resistance behaviors were well-explained in terms of the notch-insensitivity and the shielding effects of pull-out of the aligned whiskers.     

Fracture Analysis of a Special Cracked Lap Shear (CLS) Specimen with Utilization of Virtual Crack Closure Technique (VCCT) by Finite Element Methods

Some of the most important characteristics due to a fracture investigation of a special specimen are taken into account. Debonding considerations for a composite/steel cracked lap shear (CLS) specimen by utilization of finite element methods (FEM) as well as a virtual crack closure technique (VCCT) approach have been investigated. Strain energy release rate, delamination load case and direct cycle fatigue analysis have taken into consideration in this study, and the corresponding simulations have been done by ABAQUS/Standard. Linear elastic fracture criteria are used for validation of numerical results from the simulation. For comparison of three different categories of analysis, some special characteristics such as effective energy release rate ratio, bond state, time at bond failure and opening behind crack tip at bond failure have been illustrated. In this work, a detailed analysis of a special CLS specimen debonding by using VCCT and FEM is presented and varied results for validation of this kind of combination are obtained and have been discussed.     

Mechanics and mechanisms of delamination in a poly(ether sulphone)—Fibre composite

The interlaminar fracture behaviour of AS4/PES (poly(ether sulphone)) composite has been investigated in Mode I, Mode II and for fixed Mode I to Mode II ratios of 0·84, 1·33 and 2·13. The data obtained from these tests have been analysed using several different analytical approaches. The results obtained show that in Mode I the interlaminar crack growth in double cantilever beam (DCB) specimens is accompanied by fibre bridging behind the crack tip and by splitting at the crack tip, and in Mode II by the formation of a damage zone at the crack tip. These failure mechanisms are shown to increase the value of the interlaminar fracture energy considerably as the crack propagates through the composite, i.e. a rising ‘R-curve’ is measured. It is shown also that the value of the interlaminar fracture energy at crack initiation in Mode I, G_CI (init), increases as the length of the initial precrack is increased. The lowest G_IC (init) value obtained for the poly(ether sulphone) (PES) composite in this study is 0·8 kJm⁻², and this value was ascertained from a specimen with the precrack being grown by about 2 mm ahead of the initial crack (a₀ = 23 mm, a_p = 25 mm). The typical Mode II steady-state propagation energy, G_IIC (s/s-prop), value obtained for the specimens was about 2·0 kJm⁻². The length of the initial precrack had no significant effect on the G_IIC (init) and G_I/IIC (init) values. The Mode II tests gave values of G_IIC (init) = 1·25 kJm⁻² and of G_IIC (s/s-prop) = 1·85 kJm⁻². Finally, the failure loci for the PES composite have been constructed and theoretical expressions to describe these data considered.     

基于Puck理论的复合材料层合板横向剪切失效分析

针对复合材料层合板三维失效分析问题,建立了一种基于Puck失效准则的分析模型。针对Puck失效理论中的基体失效,分别采用遍历法和分区黄金分割法（PGSS）的一维搜索算法预测了不同应力状态下基体失效的断裂面角度,并对比分析了两种算法的计算精度和计算效率。研究表明,PGSS具有较高的搜索精度和搜索效率。在ABAQUS有限元分析平台下,编写了基于Puck失效准则的VUMAT显式用户自定义材料子程序,对复合材料层合板横向（G₂₃）剪切性能进行了数值预测和渐进失效分析,并与试验载荷-位移曲线、DIC测得的应变场及破坏模式进行了对比。分析结果表明:当前的分析模型能较好地预测复合材料层合板G₂₃剪切试验的力学响应和破坏模式。