Progressive damage analyses of angle‐ply laminates exhibiting free edge effects using continuum damage mechanics with layer‐wise finite element method

Capability of continuum damage mechanics (CDM) to predict the damage mechanism evolution of composite laminates has rarely been carried out, and most of the previous CDM works mainly focused on the overall response of the laminates. In this paper, progressive damage and overall response of the composite laminates under quasi‐static, monotonic increasing loading are investigated using three‐dimensional (3D) CDM implementation in a finite element method that is based on the layer‐wise laminate plate theory. In the damage formulation, each composite ply is treated as a homogeneous orthotropic material exhibiting orthotropic damage in the form of distributed microscopic cracks that are normal to the three principal material directions. The progressive damage of different angle‐ply composite laminates under quasi‐static loading that exhibit free edge effects is investigated. It is shown that using CDM global behaviour and various damage mechanisms affected by the complex nature of free edges can be qualitatively well predicted.     

Effective ply and constituent elastic properties for cracked laminates

Symmetric laminates with cracked plies were studied utilizing finite element unit cell models. The laminate unit cell models were developed based on periodicity which assumes uniformly spaced cracks. A volume-averaging scheme was developed for evaluating the volume-averaged stresses, strains and effective residual properties in the cracked plies. Sensitivities of residual properties to several material parameters were examined. The predictions of the finite element model were found to be in good agreement with published experimental results. Finally, an approach is described for computing the effective fiber and matrix properties, from the effective lamina properties, as required for progressive failure analysis using multicontinuum theory. It was found that, even though the fibers are undamaged, their effective continuum properties in the transverse direction must be degraded to achieve constituent properties that coincide with the damage mode and the residual effective ply properties.     

Modelling Mesh Independent Transverse Cracks in Laminated Composites with a Simplified Cohesive Segment Method

A methodology is proposed for modelling transverse matrix cracks in laminated composites in a three-dimensional explicit finite element analysis framework. The method is based on the introduction of extra degrees of freedom to represent the displacement discontinuity and the use of a cohesive zone model to determine damage evolution and crack propagation. The model is designed for the analysis of matrix cracks in laminates made of uni-directional fibre-reinforced plies, allowing several assumptions to be made which greatly simplify the algorithm. This was implemented in the commercial software Abaqus/Explicit as a user-defined element subroutine (VUEL). The methodology was verified via the analysis of open-hole tension tests considering both ±45˚ and quasi-isotropic layups. The results were found to be in qualitative agreement with experimental observations in terms of the nucleation and propagation of matrix cracks, the progressive delamination behaviour and the evident interactions between these damage mechanisms.     

Damage in composite laminates: Effects of transverse cracks

Two different methods of solution are used to study the effects of transverse cracks in cross-ply composite laminates. The results of an approximate analytical solution are compared with those obtained using a finite element analysis in order to study the effects of transverse cracks on the degradation of elastic and thermal coefficients as well as stress distributions. In particular, it is shown that transverse cracks cause significant degradation of the Poisson's ratio and shear modulus of the laminates, and also affect some stress distributions in a peculiar manner. Theoretical results are compared with existing experimental results where appropriate.     

Mesh-independent modeling of both distributed and discrete matrix cracking in interaction with delamination in composites

A finite element model is presented for failure analysis of composite laminates with the phantom node method for matrix cracking and interface elements for delamination. The phantom node method allows for mesh-independent representation of straight intraply cracks in laminates. In laminates two different phenomena that both involve such cracks are distinguished, namely distributed matrix cracking and discrete splitting, where the transition between the two is related to delamination. It is investigated how both phenomena and their transition can be represented in a single computational framework. Objectivity of the results with respect to element size and the introduced crack spacing parameter is examined.     

Study of layup influences on the nonlinear behavior of composites by evaluation of ply stiffness reduction

The influence of the stacking sequence on the nonlinear response of composite laminates is investigated. It is shown that a layup dependency solely emerges from damage evolution mechanisms, whereas damage initiation and viscoelastic and viscoplastic strain accumulation are not affected by the layup. This is a result of a proposed procedure that enables the evaluation of the stiffness reduction on lamina level. The residual ply stiffness components can be determined at large deformations and for various laminates under in-plane loading conditions. A finite element study is utilized to characterize the properties of a ply containing discrete cracks. The relationship between transverse and shear stiffness reduction is derived from the FE results. This allows the combined determination of the residual lamina moduli from an axial laminate stiffness. The analysis approach is validated by angle-ply specimens with different layups.     

A generalized micromechanical approach for the analysis of transverse crack and induced delamination in composite laminates

The previously developed micromechanical approaches for the analysis of transverse cracking and induced delamination are limited for laminates with specific lay-ups such as cross-ply and specific loading conditions. In this paper a new micromechanical approach is developed to overcome such shortcomings. For this purpose, a unit cell in the ply level of composite laminate including transverse cracking and delamination is considered. Then, the governing equations for the stress and displacement fields of the unit cell are derived. The obtained approximate stress field is used to calculate the energy release rate for the propagation of transverse cracking and induced delamination. To show the capability of the new method, it is employed for the analyses of general laminates with [0/90]_s, [45/−45]_s, [30/−30]_s and [90/45/0/−45]_s lay-ups under combined loadings to calculate the energy release rate due to the transverse cracking and induced delamination. It is shown that the obtained energy release rates for transverse cracking and delamination initiation are in good agreement with the available results in the literature and finite element method. Furthermore, the occurrence priority of further transverse cracks and/or delamination at each damage state of the laminates will be discussed.     

Nonlinear static analysis of composite laminated plates with evolving damage

Considering geometric nonlinearity and damage evolution, the static response characteristics of laminated composite plates subjected to uniformly distributed loading are investigated using finite element approach based on the first-order shear deformation theory. The damage evolution is modeled employing generalized macroscopic continuum theory within the framework of irreversible thermodynamics. The governing nonlinear equations are solved using Newton–Raphson iterative technique. The resulting finite element-based continuum damage model enables to predict the progressive damage and failure load. A detailed parametric study is carried out to investigate the influences of damage evolution, boundary conditions, span-to-thickness ratio, and lamination scheme on the static response of laminated plates undergoing moderately large deformation. It is revealed that the in-plane stretching forces owing to geometric nonlinearity significantly influence the failure load, damage, and stress distribution for immovable thin laminates.     

Transverse matrix cracks in cross-ply laminates: Stress transfer, stiffness reduction and crack opening profiles

Summary Cross-ply laminates with transverse plies containing through-width matrix cracks across the thickness of the transverse plies are studied using an energy-based approach,complementary to that of Hashin. An upper bound on the effective axial stiffness of the cracked laminates with a uniform distribution of transverse cracks is derived. The equations governing the field variables in a typical RVE are derived using thelayerwise laminate theory of Reddy and are solved using the finite element method. The predicted reduction in the effective axial modulus is in good agreement with experimental results, and it approaches a fixed value with increase in crack density for laminates with bothstaggered andnon-staggered cracking. Laminates with staggered cracks showed a greater reduction in effective modulus at lower crack densities. The stress distribution and mechanics of load transfer is examined in detail, at two crack densities including the characteristic damage state. The crack opening profile has been normalized in a special way in terms of the crack density, layup parameters and material properties.     

Residual Stiffness of Cracked Cross-Ply Composite Laminates Under Multi-Axial In-plane Loading

In this paper, the Equivalent Constraint Model (ECM) together with a 2-D shear lag stress analysis approach is applied to predict residual stiffness properties of polymer and ceramic matrix [0/90 _n /0] cross-ply laminates subjected to in-plane biaxial loading and damaged by transverse and longitudinal matrix cracks. It is found that the longitudinal Young’s modulus, shear modulus and major Poisson’s ratio undergo large degradation as the matrix crack density increases, with Poisson’s ratio appearing to be the most affected by transverse cracking. In cross-ply laminates with thick 90° layer strip-shaped delaminations begin to initiate and grow from the tips of matrix cracks at the 0°/90° interface. These delaminations contribute to further stiffness degradation of such laminates, and hence have to be taken into account in failure analysis models. The thickness of the 90° layer plays an important role; the thicker the 90° layer, the bigger stiffness reduction suggesting a size (volume) effect at ply level. In SiC/CAS cross-ply laminates reduction in the longitudinal modulus occurs mainly due to transverse cracks, while the shear modulus appears to be the most affected by the presence of longitudinal cracks. The shear modulus reduction ratio predicted previously by a semi-empirical formula is, in the most of cases, within 10% of the current ECM/2-D shear lag approach value. In some cases, though, the error of the semi-empirical finite element expression can be as big as 20% since it fails to capture damage mode interaction.     

Effects of damage thresholds on stresses in composite laminates under transverse impact

Studies on stresses and damage in fiber reinforced polymeric matrix composite laminates subjected to transverse impact are conducted by a 3D finite element analysis. The stress analysis is carried out by developing a constitutive equation including damage variables, therefore, effects of damage and damage thresholds on the stresses in the laminates can be investigated. Effects of damage threshold of matrix materials on stresses suggest suitable matrix materials for composite laminates, which could improve damage tolerance of the composite laminates, and resistance of the composite laminates to impact could be improved significantly by increasing the damage threshold.     

A cubic spline implementation of non-linear shear behaviour in three-dimensional progressive damage models for composite laminates

An experimental study was carried out to characterise the constitutive response of carbon fibre-reinforced epoxy laminates. While maintaining essentially linear behaviour in the fibre and transverse directions, this material displays significant non-linear shear stress–strain behaviour to rupture. It is shown that the well known Hahn-Tsai non-linear shear model does not provide an acceptable fit for the strain range examined and so a novel approach was derived where a cubic spline interpolation method was used to capture the non-linear shear behaviour. The well known ply discount model, based on Hashin’s failure criteria, was also used to predict fibre and transverse matrix damage in the laminates. The spline approach is coupled with maximum strain failure criteria to predict the response in the in-plane and out-of-plane shear directions. The material Jacobian matrix is fully defined, thus allowing a full implicit material model to be implemented. Hence, the model is suitable for both implicit and explicit finite element codes. It is shown that the model accurately predicts the response of the material for load cases in which shear stresses dominate. The performance of the model is demonstrated by considering a number of laminate configurations and failure of an open-hole tension specimen.     

Analysis of stiffness loss in cross-ply composite laminates

The behaviour of laminated composite plates beyond first-ply failure has been the subject of much research work. It is well known that generally, the load-bearing capability of laminated composite plates can remain significant despite the presence of some damage in the plies. Traditionally, the ply-discount method has been used among analysts and designers, although the approach is generally regarded as too conservative. It is therefore desirable to develop models for the prediction of the mechanical properties of damaged composite laminates at various applied loads, and to be able to correlate the changes in properties with the amount of damage and cracking within each constituent ply. Generally, if the models are to be useful as predictive tools, they must be capable of not only sufficiently describing the damage state but also the nature of the damage evolution with loading. This ‘evolution law’ is often obtained through fracture analysis, although it should be noted that the diffused nature of cracks and the multiplicity of failure modes in composites in general greatly complicates the analysis. The problem of transverse matrix cracking in cross-ply laminates under uniaxial tension is considerably simpler because it is essentially dominated by mode I fracture. Thus it is necessarily the first step for any model aiming to predict stiffness losses in composite laminates. In this paper, a constitutive model of the damage state for composite laminates, first proposed by Allen et al., is used with a damage evolution criterion based on strain energy to predict the stiffness loss due to matrix cracking in cross-ply laminated composite plates. Although the constitutive model does not require the determination of many constants, the state of damage is described by a vector of internal state variables (ISV), which contains information on the crack geometry and fracture modes. A series of parametric finite element analyses was performed to determine the effects of relative ply thicknesses, crack density and crack opening profile on the vector of ISVs. A computer algorithm was written for the analysis of cross-ply laminates based on the damage evolution criterion proposed in this work. The results of the analysis compare favourably with experimental measurements of progressive stiffness loss in damaged cross-ply graphite-epoxy laminates obtained from other researchers.     

Recent developments on damage modeling and finite element analysis for composite laminates: A review

Under complex environments such as continuous or cyclic loads, the stiffness degradation for the laminated composites such as the carbon fiber reinforced polymer matrix composites is an important physical and mechanical response to the damage and failure evolution. It is essential to simulate the initial and subsequent evolution process of this kind of damage phenomenon accurately in order to explore the mechanical properties of composite laminates. This paper gives a comprehensive review on the general methodologies on the damage constitutive modeling by continuum damage mechanics (CDM), the various failure criteria, the damage evolution law simulating the stiffness degradation, and the finite element implementation of progressive failure analysis in terms of the mechanical response for the variable-stiffness composite laminates arising from the continuous failure. The damage constitutive modeling is discussed by describing the evolvement of damage tensors and conjugate forces in the CDM theory. The failure criteria which interpret the failure modes and their interaction are compared and some advanced methods such as the cohesive theory which are used to predict the damage evolution properties of composites are also discussed. In addition, the solution algorithm using finite element analysis which implements progressive failure analysis is summarized and several applicable methods which deal with the numerical convergence problem due to singular finite element stiffness matrices are also compared in order to explore the whole failure process and ultimate load-bearing ability of composite laminates. Finally, the multiscale progressive failure analysis as a popular topic which associates the macroscopic with microscopic damage and failure mechanisms is discussed and the extended finite element method as a new finite element technique is expected to accelerate its practical application to the progressive failure analysis of composite laminates.     

纤维增强复合材料层合板缺口尺寸及形状效应数值模拟

通过考虑基体裂纹、纤维断裂、层内劈裂和层间脱层等破坏形式,建立三维有限元模型研究含中心圆孔和中心裂缝的准各向同性复合材料层合板([45/0/-45/90]_(2S))在拉伸载荷下的缺口尺寸效应及缺口形状效应。模拟结果显示:随着缺口尺寸的增大,层合板的破坏强度逐渐降低,然而,在本文研究范围内含中心裂缝的层合板破坏强度始终高于对应的含中心圆孔的层合板破坏强度。进一步分析有限元模拟结果表明,含中心裂缝的层合板亚临界损伤发生得更早,并且亚临界损伤范围更大,亚临界损伤会大大缓解缺口尖端的应力集中,从而使含中心裂缝层合板表现出更高的破坏强度。     

Intra-laminar cracking in CFRP laminates: observations and modelling

Matrix cracks in composite laminates are almost invariably the first damage mechanisms in composite laminates and are the precursor to more serious forms of damage. The results presented in this paper include experimental investigations into the occurrence of these cracks in a wide range of laminates containing various angle-plies. Ultrasonic detection of these cracks using polar backscattering is found to be successful and shows promise as a method which could be applied outside the laboratory. Further work includes finite element simulations which lead to an understanding of aspects of the accumulation of damage in composite laminates.     

Three-dimensional stress analysis of cracked satin woven carbon fiber reinforced/polymer composites under tension at cryogenic temperatures

We present a thermo-mechanical stress analysis for five harness satin (5HS) woven carbon fiber reinforced polymer (CFRP) composite laminates with cracks under tension at cryogenic temperatures. The three-dimensional finite element model assumes the cracks to be located in the transverse fiber bundles and to span the thickness of the fiber bundles. Numerical calculations are carried out, and the Young’s modulus and stress distributions near the crack fronts for two-layer and infinite-layer woven composite laminates are obtained and shown in graphical form. Results of this analysis demonstrate the effects of the residual thermal stresses and cracks on the mechanical behavior.     

Fracture toughness of transverse cracks in graphite/epoxy laminates at cryogenic conditions

Stress singularity of a transverse crack normal to ply-interface in a composite laminate is investigated using analytical and finite element methods. Four-point bending tests were performed on single-notch bend specimens of graphite/epoxy laminates containing a transverse crack perpendicular to the ply-interface. The experimentally determined fracture loads were applied to the finite element model to estimate the fracture toughness. The procedures were repeated for specimens under cryogenic conditions. Although the fracture loads varied with specimen thickness, the critical stress intensity factor was constant for all the specimens indicating that the measured fracture toughness can be used to predict delamination initiation from transverse cracks. For a given crack length and laminate configuration, the fracture load at cryogenic temperature was significantly lower. The results indicate that fracture toughness does not change significantly at cryogenic temperatures, but the thermal stresses play a major role in fracture and initiation of delaminations from transverse cracks.     

3D constitutive model of anisotropic damage for unidirectional ply based on physical failure mechanisms

A 3D anisotropic continuum damage model is developed for the computational analysis of the elastic–brittle behaviour of fibre-reinforced composite. The damage model is based on a set of phenomenological failure criteria for fibre-reinforced composite, which can distinguish the matrix and fibre failure under tensile and compressive loading. The homogenized continuum theory is adopted for the anisotropic elastic damage constitutive model. The damage modes occurring in the longitudinal and transverse directions of a ply are represented by a damage vector. The elastic damage model is implemented in a computational finite element framework, which is capable of predicting initial failure, subsequent progressive damage up to final collapse. Crack band model and viscous regularization are applied to depress the convergence difficulties associated with strain softening behaviours. To verify the accuracy of the damage model, numerical analyses of open-hole laminates with different lay-up configurations under tension and compression were performed. The numerical predictions were compared with the experimental results, and satisfactory agreement was obtained.     

The interaction between matrix cracks and delaminations during quasi-static impact of composites

This paper discusses the investigation of the impact process for beam-like composite specimens. The research is focused on the initiation of delaminations at matrix cracks and on the interaction of delaminations with matrix cracks. The research is based on experiments and on finite element calculations. This study improves the fundamental understanding of the impact process in composite laminates, and gives clear indications on the relative influence of different material properties (transverse strength, interlaminar fracture toughness) on the damage development during impact.