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An interaction function was constructed based on the axial compression/tension and shear loads of orthotropic plates. The coefficients of the polynomial function were determined by uniaxial test results. Buckling interaction and failure interaction formulae under combined axial tension/compression and shear loads were established. Based on the uniaxial load test results of orthotropic plates, the buckling load and bearing capacity under any proportion of the combined loads could be predicted by using the proposed interaction formulae. The buckling interaction curves and failure envelopes predicted by the proposed interaction formulae were in excellent agreement with the test results.

Fracture plane based failure criteria for fibre-reinforced composite materials under three-dimensional stress state are presented. The failure function is taken as a polynomial expansion in terms of the stress components on fracture plane, which provides a general mathematical technique for constructing Mohr’s fracture hypothesis-based criteria. The polynomial expansion is then truncated at the quadratic terms to approximately describe the failure function. Besides the basic strengths of UD laminates, the fracture plane angles under transverse tension/compression and pure shear are introduced to calibrate the failure criteria, since Mohr’s concept can be described completely and exactly only by both the basic strengths and the fracture plane angles. According to experimental evidences, the interaction between matrix-dominated and fibre-dominated failure modes is also considered in the present study. No empirical or artificially defined parameters are included in the criteria. Experimental verification for different kinds of unidirectional composites under various stress states demonstrates that the proposed failure criteria have a good predictive ability.

Puck's action plane failure criteria have already proven their capability in the first and second world-wide failure exercises (WWFE-I and WWFE-II). However, Puck and his co-workers have only recommended inclination parameters for GFRP/Epoxy and CFRP/Epoxy. These UD composites generally have high transverse compressive-to-tensile strength ratios YC/YT, and are regarded by them as intrinsically brittle materials. Therefore, Puck's inter fibre fracture (IFF) criteria might not be directly applied to other types of UD composites with low YC/YT ratios. In particular, Puck's original IFF criteria will result in unrealistic predictions if YC/YT is extremely low. In the present study, Puck's original IFF criteria are extended to all types of UD composite materials to solve the problem. Composites are divided into three categories, namely semi-brittle materials, brittle materials, and intrinsically brittle materials. Depending on the material category, three different algorithms are proposed for determination of the two parameters in Puck's IFF criteria, namely the resistance of the action plane against transverse tensile stressing, R ⊥ A t , as well as the inclination parameter of contour lines of the fracture body, p ⊥ ⊥ . The present criteria are theoretically evaluated for composites with low YC/YT ratios, while the predictions of the present criteria are compared with the test results of composites with high YC/YT ratios such as thermoset GFRP and CFRP. Theoretical and experimental assessment demonstrates the reasonableness of the extension of Puck's IFF criteria.

A physically based failure criterion distinguishing the different effects of hydrostatic tensile and compressive stress is developed for homogeneous and isotropic materials. It is reasonably assumed that failure is related to the shape and volume change of isotropic materials at the macroscopic scale. Shape distortion and volume dilation (i.e., hydrostatic tension) are capable of producing material failure, while volume contraction (i.e., hydrostatic compression) has an impeding effect upon failure initiation. The different roles of hydrostatic tension and compression may lead to differences of the applied failure functions. In the present study, two failure formulas are proposed, considering the different effects of hydrostatic stress upon yielding/fracture of homogeneous and isotropic materials separately. The good agreement of the present theory with a large number of experimental data is observed, and the present theory shows a wide range of applicability.

A failure criterion for isotropic materials is developed in the present study. The fundamental hypotheses of the Mohr-Coulomb criterion are re-examined and re-evaluated. The failure function of Mohr’s envelope is firstly expanded into a polynomial in terms of the stress components (σn, τn) on the failure plane, then truncated at the second order due to the fine curve-fitting results of quadratic functions with experimental data. The parameters of the nonlinear failure functions are calibrated by the three basic mechanical properties—the uniaxial tensile and compressive strengths, T and C, as well as the pure shear strength, S. Theoretical and experimental evaluation for various isotropic materials demonstrates that the present failure criterion provides no worse results for ductile materials and far better results for brittle materials compared with the linear Mohr-Coulomb criterion. The strength theory shows good agreement with physical reality and has a wide range of applicability.

Failure criteria for unidirectional composite materials based on Hashin’s theory are developed. Although Hashin’s criteria are very popular in composite structural applications due to their simplicity of concept and ease of use, they do not always fit experimental results well. In the present study, the fundamental hypotheses proposed by Hashin are re-examined and re-evaluated. Hashin’s criteria for the tensile fiber mode and tensile matrix mode are modified based on physical considerations. The interaction between the fiber and matrix failure modes is also taken into account. Experimental verification for different kinds of unidirectional composites under various stress states demonstrates that the proposed failure criteria have good improvements over Hashin’s criteria, especially in the tensile fiber and matrix mode region. The present theory is in good agreement with physical reality and has a wide range of applicability.

Low-velocity impact on the free edge of composite stiffener is considered as a critical factor on the loss of compression strength. This paper proposes a phenomenologically-based mechanical finite element model for the Compression-After-Edge-Impact (CAEI) failure prediction of the composite panel stiffened with T-shaped stiffeners, of which the web is subjected to edge impact damage. Based on the specific failure mechanisms during an edge impact, localized crushing failure, which is insignificant during a classic skin impact, however is considered as a key point of the proposed model for the simplification of impact-induced damage. With the help of a composite damage model comprising both continuum damage mechanics model and surface-based cohesive contact model, a good correlation between the experimental and numerical results is obtained, and therefore validates correctness and effectiveness of the proposed mechanical model. Furthermore, the results reveal that the propagation of fiber compressive failure plays a major role in the compressive failure of the impacted web, of which complete fracture determines the ultimate failure load of the T-stiffened composite panel.

A phenomenological-based mechanical finite element model is developed for the prediction of ultimate compression loads and failure modes of wing relevant composite panels stiffened with I-shaped stiffeners, of which the edge is subjected to impact loading. The initiation of intralaminar failure including fiber and inter-fiber fracture is evaluated by Puck criteria, and its evolution is assumed to be controlled by equivalent strains. The interlaminar failure, namely delamination, is simulated by employing a surface-based cohesive contact model. For R-sections of the impacted stiffener, interfacial delamination, which may cause premature failure and to some extent reduce the load carrying capacity, is efficiently considered. A good correlation between the experimental and numerical results shows correctness and effectiveness of the proposed mechanical method for predicting CAEI response. Furthermore, numerical results reveal that not only failure propagation of the impacted site but that of the other undamaged side dominate the compressive mechanism of the edge impacted stiffener, of which complete fracture determines the ultimate failure load of the I-stiffened composite panel.

Experiments along with the finite element model were used to analyze the influences of thickness ratio of flange and skin (hereinafter referred to as thickness ratio) of T-joint on the reinforcing effect of Z-pin. Results showed that the thickness ratio significantly influences the reinforcement effect of Z-pin. Whether or not Z-pin should be applied to a specific T-joint can be decided by the thickness ratio. For T-joints with large thickness ratios (larger than 0.32 for the studied T-joint), the application of Z-pin is an effective way to increase the carrying capacities of the T-joints, while for T-joints with small thickness ratios (smaller than 0.32 for the studied T-joint), the reinforcing effect of Z-pin is not obvious. Besides, the corresponding load value under large displacement in the load-displacement curves of T-joints with different thickness ratios are similar.

A phenomenological-based micro–macro mechanical method, comprising the coupling of a 3D continuum damage mechanics (CDM) model with an original micromechanical FE model, is presented to predict low velocity impact response of composite laminates. The key point of the proposed method is the definition and computation of saturation crack density, which is used to scale impact-induced delamination instead of via expensive cohesive zone model based FEM. In order to gain the specific value of the saturation crack density, a micromechanical FE model is established based on the RVE technique. Additionally, the initiation of intralaminar failure including fiber fracture (FF) and inter-fiber fracture (IFF) is evaluated by Puck criteria, and its evolution, with considering effects of the fracture plane angle, is assumed to be controlled by equivalent strains on the fracture plane rather than the classic material principal axis plane. A good agreement between the experimental and simulated results shows correctness and effectiveness of the developed micro–macro mechanical method for predicting impact response and damage using a relatively coarse mesh. Furthermore, significant reduction of computational cost, compared with existing techniques, can be achieved with the novel mesh-independent method.

The purpose of this paper was to investigate the effect of low velocity edge impact damage on the damage tolerance of wing relevant composite panels stiffened with both T-shaped and I-shaped stiffeners under uniaxial compression load. Six stiffened composite panel configurations, including four specimens for each configuration, were manufactured and tested. Before Compression-After-Impact (CAI) tests, the key dimensions of specimens were measured and a vertical drop-weight testing device was used to impact on critical locations such as the skin and the free edge of a stiffener. Different damage types and shapes were discovered from different locations of impact after careful inspection by visual and ultrasonic C-scan. The experimental results reveal the compression failure mechanism that local buckling, subsequent damage propagation and final fracture of the edge impacted stiffener are triggers of the final failure of a stiffened composite panel, which as well determine the ultimate load carrying capacity. In addition, under identical edge impact levels, the damage tolerance behavior of T-stiffened composite panel is distinctly superior to that of I-stiffened composite panel, which results in more cautious design regarding edge impact damage tolerance of the panel stiffened with I-stiffeners.

The presence of sufficient elements within the fracture process zone (FPZ) ahead of the crack tip is crucial for accurate analysis of composite delamination in using cohesive elements. However, it may lead to extremely high computational costs in the finite element analysis of large-scale structures. Using coarse meshes can reduce such costs, but accurately determining interface strength is important for improving the accuracy of the analysis. In this paper, a novel method is proposed which can determine a more suitable interface strength for the simulation of Mode I delamination than existing methods found in open literature. Meanwhile, the proposed method can predict FPZ length more accurately when coarse meshes are used. Furthermore, this study found that constant cohesive law is more suitable for simulating delamination in the use of coarse meshes than bi-linear cohesive law. Finally, for composite materials, this method can be applied to various structure thicknesses and material properties.

Composite laminates have been extensively used in the aerospace and automotive industries due to their excellent stiffness and strength. Fiber bridging is a unique phenomenon of layered composite materials that increases the fracture toughness of the composites several times. This characteristic enables fiber bridging to be an important crack shielding mechanism during delamination and arouses wide attention from researchers. Most research is focused on studying the mechanism of fiber bridging and establishing the traction separation relation of bridging for delamination simulation, while for a given composite material, a simple and direct approach to identify whether fiber bridging could occur during the delamination is still lacking. This paper provides an efficient method to deal with this problem. It is based on the weak fiber/matrix interface assumption so that fibers can be pulled out easier to bridge the cracked faces. Eleven groups of DCB tests, both with and without fiber bridging, are used to validate the feasibility of this approach. Besides, this approach can also be extended to Mode II or mixed Mode I/II test.

Different failure modes are not explicitly accounted for in the Tsai–Wu failure criterion for composite materials, and the factor F12 in the expression cannot be determined by the basic strength values of the materials. In view of these two shortcomings of the Tsai–Wu criterion, an improved criterion was proposed based on the reasonable assumption that composites exhibit infinite strength under pure hydrostatic pressure. Some coefficients in the quadratic tensor expression of the Tsai–Wu failure criterion are redetermined using the basic strength value of the material, including the coefficient F12, under four different stress states (fiber tension and compression, matrix tension and compression). The reconstructed Tsai–Wu failure criterion can distinguish the failure modes based on different coefficient values under different stress states. In the progressive damage analysis of composite materials, the stiffness can be reduced in different manners based on each failure mode. Experimental verification for different kinds of unidirectional composites under various stress states was conducted, demonstrating that the improved Tsai–Wu failure criterion has a better prediction ability and accuracy than the original criterion.

To accurately predict the delamination behavior, bridging effect should be introduced into the numerical model so that an effective method to determine the bridging law is required. The main effort of the current study is to propose an efficient procedure for determining the mixed-mode I/II bridging law of composite laminates. The crack opening and shear displacements at the pre-crack tip are derived via a numerical method based on equivalent crack length approach. In addition, the formula for energy release rate can also be obtained based on equivalent crack length approach. Then, the bridging law can be derived according to the J-integral approach. The derived bridging laws have been implemented into the cohesive zone model (CZM) for numerical simulation, good agreement with experimental results validate the feasibility of the proposed procedure for determining mixed-mode I/II bridging law. Furthermore, this procedure does not require the measurement of the crack length and relative displacements at the pre-crack tip during the test.

Through-thickness compressive stress of composite laminates can inhibit the interlaminar shear failure which has been proved by experiments. However, common delamination damage initiation criterion cannot well describe this mechanism. For this issue, in this article, based on the insight of the failure mechanism, an interlaminar failure function is constructed at first. Then, a delamination failure criterion considering the effect of through-thickness compression on interlaminar shear failure is established based on logical deductions from a polynomial expansion of the failure function. In the criterion, a constant coefficient with physical meaning is used to describe the effect of through-thickness compression on the interlaminar shear failure, and the estimation method of the coefficient is also given in this study. Here, this failure criterion can be considered as a general form to describe the delamination between laminae, by defining different coefficient, it can be converted to other criteria such as the conventional secondary stress failure criterion or the Christensen failure criterion. Finally, by comparing the numerical calculation and experimental results, the delamination failure criterion proposed in this study exhibited a good prediction ability for the interlaminar damage under different loading conditions.

This article presented a simple procedure to determine the Mode I bridging law for both unidirectional and multidirectional composite laminates. It was based on an ‘Elastic Restrained Beam Model’ which assumed that the boundary conditions at the end of the cracked arms of the double cantilever beam (DCB) are elastically restrained by torsion springs. In this way, the closed-form solution of the DCB has been derived, without the need for a crack-length correction used in the corrected beam theory (CBT). We also derived very accurate data reduction formulas for the energy release rate (GI) and the opening displacement at the pre-crack tip (δ*). Then, the bridging law was obtained by numerical differentiation of GI with respect to δ* using the J-integral approach. Two major advantages of the proposed procedure are that it does not require the measurement of the crack length and separation δ* during the test. The derived bridging laws were implemented into the cohesive zone model (CZM) for numerical simulations and the results were proved to have a great agreement with experimental results, which validated that the proposed procedure is suitable for the determination of Mode I bridging law.

A micro-macro mechanical finite element model is developed for the prediction of the ultimate strength and failure modes of Composite-Metal-Weld (Comeld) joints. A micromechanical finite element model is firstly proposed, based on the periodic RVE technique, to gain the equivalent stiffness and strength of the protrusion layer by FEM. A suitable unit cell model, which can represent the protrusion layer through applying a unified periodical displacement boundary condition, has been established according to the geometric distribution of protrusions on the metal surface. With the predicted equivalent properties of the protrusion layer which can be considered as a new homogenized material layer in the global model, failure modes and ultimate strength of three different Comeld joints subjected to static tensile loads are predicted numerically. A good agreement between the experimental and simulated results shows correctness and effectiveness of the developed micro-macro mechanical method for predicting the load bearing capacity of Comeld joints. Moreover, Comeld parameters like the density and height of protrusions and the lap size of joints need to be adjusted to achieve higher joint strength.

An improved analytical method is presented to predict the damage area of composite laminates induced by low velocity impact, which avoids expensive nonlinear three-dimensional FEM analysis. The key point of the proposed method is the definition and analysis of the initial interlaminar shear strength for the prediction of the interlaminar shear failure due to impact. The initial interlaminar shear strength is predicted by a micromechanical model based on Eshelby’s equivalent inclusion method. The simplified energy-balance model and quasi-static assumption are used to estimate the maximum contact force. In addition, an approximate solution is proposed for the bending of a clamped symmetric angle-ply laminated rectangular plate subjected to a central concentrated force. Finally, the above techniques are combined to formulate a methodology predicting the impact induced damage area. The analytical results agree well with the test data.

This paper presents the development and validation of a mesoscale numerical model for predicting damage and failure of bolted joints in laminated composites with different configurations and geometries. Double-shear and single-shear composite bolted joints with different widths and end distances were analyzed. The composite material model combined smeared crack models for all types of intralaminar failure mechanisms, and an interface discrete cohesive-zone model for interlaminar failure. Three dimensional (3D) phenomenological invariant-based failure criteria using in-situ ply strengths were used for the prediction of intralaminar damage onset, together with mechanism-based continuum damage models (longitudinal bi-linear damage model and transverse 3D smeared crack model) for intralaminar damage propagation. An interface cohesive-zone model with consistent initiation and evolution criteria based on the Benzeggagh-Kenane (B-K) law was used to simulate delamination. Particularly, for the first time, the model considered the inherent cohesive-frictional behavior in the intralaminar transverse failure and delamination, namely the frictional sliding in diffuse micro-cracks during damage propagation and in localized meso-cracks after complete fracture. In the 3D explicit finite element models, a fiber-aligned mesh was used for the discretization of composite plies. A multi-zone modelling strategy was used to make the computational cost acceptable for engineering applications. Detailed comparisons between the numerical results and the experimental data previously obtained were performed, with focus on the macroscopic mechanical behavior and mesoscale failure mechanisms. A good correlation between tests and analyses was found. Additionally, several complex failure mechanisms were revealed by the numerical simulations, which could not be clearly identified in previous experimental studies.

The measurement of initiation and propagation fracture toughness is important for characterizing the delamination behavior of composite laminates, especially in the case with distinct R-curve behavior which can be caused by fiber bridging, shear deformation of matrix, etc. Four-Point End Notched Flexure (4-ENF) is one of the widely adopted tests to measure the mode II toughness values, while the friction between the crack surfaces may have a significant impact on the results. In this paper, a simple procedure to determine the cohesive law in 4-ENF test with consideration of the friction and R-curve effect is proposed. Firstly, determining the equivalent crack advance according to the compliance variation of the specimen is performed, without the need for monitoring the crack length during the test. Next, the Energy Release Rate in mode II ( G II ) and crack sliding displacement ( δ t ) are derived based on the point-friction assumption and beam theory. Finally, the cohesive law is obtained via the J-integral approach. To obtain suitable values of the penalty stiffness and interface strength, the influence of these parameters on the prediction of 4-ENF test is studied. Comparisons of the numerical results and the experimental data show a good correlation. The effect of variation of friction coefficients on derived cohesive stress and initiate energy release rate is discussed as well.

A new cohesive damage model was proposed to analyze the Mode I, Mode II, and mixed-mode bridging micromechanisms of a single Z-pin in the quasi-isotropic laminate. One zero-thickness cohesive element was embedded into each pair of adjacent elements in the damage zone to model the multi-directional Z-pin failure including splitting and rupture, snubbing damage in the resin-rich region, and interfacial debonding during bridging. The FE predicted bridging laws and damage modes of the Z-pin under Mode I, 30° and 60° mixed-mode, Mode II loading all agreed well with the experimental results. Besides, it was found that the splitting damage of the Z-pin and the snubbing damage in the resin-rich region were non-negligible.

This paper is to develop a micromechanics-based model on predicting the effective transverse stiffness of composites reinforced with periodically dispersed cylindrical fibers. Six types of packing patterns are analyzed and their morphology characteristics are accounted for by Green’s function. It is noted that fiber packing configuration plays a great effect on the effective elasticity. The developed analytical model is assessed by the available FEM computations, and illustrates the dependence of the equivalent stiffness on the periodic microstructures. Additionally, it should be noted that the present model can be used to study some important issues, e.g. damage variables, matrix plasticity and so on.

This paper deals with a new micromechanics model of particulate-reinforced composites (PRCs) describing the evolution of debonding damage, matrix plasticity and particle size effect on the deformation. A ductile interphase was considered in the frame of incremental damage theory to analyze the dependence of elastic–plastic–damage behavior on particle size. Progressive debonding damage was controlled by a critical energy criterion for particle–matrix interfacial separation. The equivalent stresses of the matrix and interphase were determined by field fluctuation method. The influences of progressive debonding damage, particle size and interphase properties on the overall stress–strain response of PRC were explained simultaneously. Due to the existence of a ductile interphase, stress transfer and plastic initiation in PRC become very complicated, and thus a unit-cell (UC) based FEM was used to simulate their evolutions and demonstrate the role of the interphase. Finally, particle size effect on the mechanical behaviors of composites was interpreted.

The cohesive strength is an important parameter in numerically modeling composite delamination via CZM (cohesive zone model) based FEM. A micromechanical model is proposed to predict the cohesive strength based on the periodic RVE technique. A periodic displacement boundary condition has been presented on the assumption that the RVE is orthotropic in the sense of overall response. The cohesive strengths of T700/QY8911 and AS4/PEEK laminates at various fibers cross-angles are gained by FEM. With the predicted cohesive strengths the FEM simulations on Mixed-Mode bending (MMB) and seven-point bending test are presented, and the results are in fair agreement with experimental observation.

A modified Hertz contact law is proposed for the contact problem of a laminated plate indented by a rigid sphere, which involves the effects of the thickness, in-plane dimensions and boundary conditions of the plate. The model shows that the deflection difference between the center and the edge of the contact area is a key factor dominating the effects of these variables on the force–indentation response. The modified contact law is assumed to follow the same mathematical formulation as the Hertz law. However, the contact stiffness is no longer constant but varies as a function of the contact force due to the thickness effect. The predictions of the force–indentation response show improved agreement over early work by Yang and Sun. Over the present analysis range, the thickness effect may be negligible if the plate thickness is greater than 2 mm. On the other hand, the thickness has a more significant effect on the force–indentation response if the plate thickness is less than 2 mm. Moreover, the in-plane dimensions and the boundary conditions of the plate show little influence on the force–indentation response within the present scope of analysis.

Quasi-static indentation tests on composite laminates have been performed. Thermal deply and C-scan techniques are used to observe the internal damage characteristics of laminates induced by static indentation. The results indicate that the dent depth increases slowly and almost linearly as the force is lower than its maximum. The observation shows that the damage induced at this loading stage includes matrix cracking and delamination only. Once the force exceeds the maximum, the dent depth increases sharply while the force remains almost unchanged. Deply test results demonstrate that the transition to rapid increase in dent depth is due to the fiber breakage produced at this loading stage. The results also indicate that the damage resistance of a composite laminate can be characterized by the maximum contact force. On the basis of this principle and test results, it is concluded that the damage resistance of toughened composite is much higher than that of brittle one.

The Green’s functions have not been studied in open literatures for the bending problem of an anisotropic plate with an elliptic hole subjected to a normal concentrated force and a concentrated moment. In this paper, the problem is investigated and the Green’s functions are first obtained by using the complex potential approach. The techniques of conformal mapping transformation and analytic continuation are used to derive the closed-form complex stress functions. The Green’s functions obtained have some potential applications in the analysis of composite structures such as the modification of the displacement compatibility model for notched stiffened composite panels and the formulation of a new method for interlaminar stress analysis around holes of laminates.

Stress resultants and moments around the holes of infinite unsymmetrical composite laminates under remote uniform loading are analyzed using the complex potential approach developed by Chen and Shen [Mech. Res. Commun. 28 (4) (2001) 423; 28 (5) (2001) 513]. In the analysis, the authors’ previous work has been modified to determine the complex constants Bk appeared in the complex potential functions of Chen and Shen [Mech. Res. Commun. 28 (5) (2001) 513]. Herein Bk are the unknown quantities related to the remote loading conditions. The effect of bending extension coupling has been discussed for several laminates such as [0/90]T, [45/−45]T and [0/90/45/−45]nT. Results indicates that the coupling between bending and extension significantly affects the stresses around the hole of an unsymmetrical laminate containing one sub-laminate only. Such the effect rapidly decreases as the number of the sub-laminates increases. The results confirm the validity of the developed complex potential approach.

An investigation has been conducted to predict the tensile and compressive strengths of fiber-dominated composite laminates with notches. Research shows that the strength of a notched fiber-dominated laminate can be evaluated from the failure prediction for the 0° plies in the laminate. On the basis of this study and the frequently used ‘characteristic length’ concept, a lay-up-independent failure criterion, the load-bearing ply-failure criterion (LBPF), is proposed, which relates the laminate strength to the failure of the principle load-bearing ply. Two characteristic lengths are obtained for tension and compression, respectively. Results indicate that, in most cases, the two characteristic lengths are properties of a composite material system, regardless of laminate configurations. Once these two parameters are estimated from preliminary tests on notched multidirectional laminates of the material, they can be used to predict the notched strength of fiber-dominated laminates of the same material. Good consistency is observed between test results and predictions.