Numerical investigation of matrix cracking propagation in cross‐ply laminated composites subjected to three‐point bending load using concurrent multiscale model

The purpose of the present study is to analyze fiber‐matrix debonding and induced matrix cracking formation as two major micromechanical damage modes in cross‐ply composite laminates using a two‐dimensional numerical approach. To this aim, the cross‐ply laminates containing 90‐degree layers are modeled, where the fibers are arranged randomly in transverse plies. Damage modes in this numerical model are simulated by the cohesive surface method. The performed analyses reveal that in the laminates with 90‐degree layers located in the outer positions, the primary micro damage mode is micro matrix cracking which is initiated from the fiber‐matrix debonding damage mode and will be followed by matrix cracking. The main benefit of the present study in comparison to other numerical methods is proposing a virtual test method for damage analysis of different cross‐ply laminates in which, the matrix cracking formation will emerge physically in a random and antisymmetric pattern similar to the experimental observations.     

Cohesive zone length in numerical simulations of composite delamination

Accurate analysis of composite delamination using interface elements relies on having sufficient elements within a softening region known as the cohesive zone ahead of a crack tip. The present study highlights the limitations of existing formulae used to predict numerical cohesive zone length and demonstrates modifications necessary for improved accuracy. Clarification is also provided regarding the minimum number of interface elements within the cohesive zone. Finally, appropriate values of numerical interfacial strength are examined. The results presented will aid the application of mesh design techniques that both preserve numerical accuracy, whilst minimising computational expense.     

Mesh-independent discrete numerical representations of cohesive-zone models

The importance of the cohesive-zone approach to analyse localisation and fracture in engineering materials is emphasised and ways to incorporate the cohesive-zone methodology in computational methods are discussed. Until recently, numerical implementations of cohesive-zone models have suffered from a certain mesh bias. For discrete representations this is caused by the initial mesh design, while for smeared representations it is rooted in the ill-posedness of the rate boundary value problem that arises upon the introduction of decohesion. A proper representation of the discrete character of cohesive-zone formulations which avoids any mesh bias is obtained by exploiting the partition-of-unity property of finite element shape functions. The recently developed cohesive segments method, which is well-suited for simulating the entire process of crack nucleation, growth and coalescence is reviewed. The effectiveness of this approach is illustrated by some examples for crack nucleation and growth in heterogeneous materials and for fast crack growth.     

Mesh‐independent matrix cracking and delamination modeling in laminated composites

The initiation and evolution of transverse matrix cracks and delaminations are predicted within a mesh‐independent cracking (MIC) framework. MIC is a regularized extended finite element method (x‐FEM) that allows the insertion of cracks in directions that are independent of the mesh orientation. The Heaviside step function that is typically used to introduce a displacement discontinuity across a crack surface is replaced by a continuous function approximated by using the original displacement shape functions. Such regularization allows the preservation of the Gaussian integration schema regardless of the enrichment required to model cracking in an arbitrary direction. The interaction between plies is anchored on the integration point distribution, which remains constant through the entire simulation. Initiation and propagation of delaminations between plies as well as intra‐ply MIC opening is implemented by using a mixed‐mode cohesive formulation in a fully three‐dimensional model that includes residual thermal stresses. The validity of the proposed methodology was tested against a variety of problems ranging from simple evolution of delamination from existing transverse cracks to strength predictions of complex laminates withouttextita priori knowledge of damage location or initiation. Good agreement with conventional numerical solutions and/or experimental data was observed in all the problems considered. Published 2011. This article is a US Government work and is in the public domain in the USA.     

Cohesive zone with continuum damage properties for simulation of delamination development in fibre composites and failure of adhesive joints

A new approach is developed to implement the cohesive zone concept for the simulation of delamination in fibre composites or crack growth in adhesive joints in tension or shear mode of fracture. The model adopts a bilinear damage evolution law, and uses critical energy release rate as the energy required for generating fully damaged unit area. Multi-axial-stress criterion is used to govern the damage initiation so that the model is able to show the hydrostatic stress effect on the damage development. The damage material model is implemented in a finite element model consisting of continuum solid elements to mimic the damage development. The validity of the model was firstly examined by simulating delamination growth in pre-cracked coupon specimens of fibre composites: the double-cantilever beam test, the end-notched flexure test and the end-loaded split test, with either stable or unstable crack growth. The model was then used to simulate damage initiation in a composite specimen for delamination without a starting defect (or a pre-crack). The results were compared with those from the same finite element model (FEM) but based on a traditional damage initiation criterion and those from the experimental studies, for the physical locations of the delamination initiation and the final crack size developed. The paper also presents a parametric study that investigates the influence of material strength on the damage initiation for delamination.     

Modeling of the dynamic delamination of L-shaped unidirectional laminated composites

One of the widely used geometrically complex parts in advanced commercial aircraft is the L-shaped composite. Due to the sharp curved geometry, interlaminar opening stresses are induced and delamination occurs under considerable mode-mixities in L-shaped beams. Dynamic phenomena during delamination initiation and propagation of L-shaped beams are investigated using dynamic (explicit) finite element analysis in conjunction with cohesive zone methods. The 2-D model consists of 24 plies of unidirectional CFRP laminate with an initial 1 mm crack at the center of the laminate at the bend. Loading is applied parallel to one of the arms quasi-statically. The loading type yields different traction fields and mode-mixities in the two sides of the crack in which delamination occurs under shear stress dominated loading on one crack tip and opening stress dominated loading on the other. The speed of the delamination under shear dominated loading at one side is 800 m/s and under normal stress dominated loading is 50 m/s. In addition radial compressive waves at the interface are observed. Finally, as the thickness is changed, a different failure mode is observed in which a secondary crack nucleates at the arm and propagates towards the center crack.     

Determination of fracture energy and tensile cohesive strength in Mode I delamination of angle-ply laminated composites

The energy release rate in delamination of angle-ply laminated double cantilever composite beam specimens was calculated using the compliance equation, and interlaminar cohesive strengths were obtained. Instead of the traditional approach of a beam on an elastic foundation, a second-order shear-thickness deformation beam theory (SSTDBT) was considered. The equilibrium equations were obtained using the principle of minimum total potential energy and the system of ordinary differential equations were solved analytically. The problem was solved for [0°]₆ , [±30°]₅, and [±45°]₅ laminates with mid-plane delaminations and the results were verified using experimental evidence available in the literature.     

Influence of the matrix constituent on mode I and mode II delamination toughness in fiber-reinforced polymer composites under cyclic fatigue

Mode I and mode II fracture behaviour under static and dynamic loading was analyzed in two composites made up of the same reinforcement though embedded in two different matrices. Specifically, the delamination energy under static and dynamic loading was obtained for both materials and both fracture modes, i.e. the number of cycles necessary for the onset of fatigue delamination. Subsequently, the crack growth rate (delamination rate) was obtained for different percentages of the critical energy rate. The main goal of the study was to ascertain the influence of the matrix on the behaviour of the laminate under fatigue loading.From the experimental results for the onset of delamination, similar fatigue behaviour was observed at a low number of cycles for both matrices and both fracture modes, while in fatigue at a high number of cycles, a higher fatigue limit was obtained in the composite with the modified resin (higher toughness) for both fracture modes. From the point of view of crack growth rate, both materials behaved similarly for different levels of stress under fatigue and the two fracture modes for small crack lengths (initial growth zone < 5 mm), although the growth rate increased for large crack lengths. This behaviour was the same in both loading modes.     

Accurate simulation of delamination growth under mixed-mode loading using cohesive elements: Definition of interlaminar strengths and elastic stiffness

A methodology for predicting accurately the propagation of delamination under mixed-mode fracture with cohesive elements is proposed. It is shown that changes in the local mode ratio during the evolution of damage under mixed-mode loading can cause errors in the determination of the energy dissipation and result in inaccurate predictions of the global load–displacement response – even under conditions where, according to Linear Elastic Fracture Mechanics, the global mode ratio is constant. To address this difficulty, relations between the interlaminar strengths and the penalty stiffness are proposed which ensure a correct energy dissipation when delamination propagates. The validity of the proposed methodology is demonstrated for different mode ratios by comparison with the corresponding analytical solutions.     

Simulations of delamination in CFRP laminates: Effect of microstructural randomness

Due to their high specific strength and stiffness, fibre-reinforced composite materials are being increasingly used in structural applications where a high level of performance is important (e.g. aerospace, automotive, offshore structures, etc.). Performance in service of these composites is affected by multi-mechanism damage evolution under loading and environmental conditions. For instance, carbon fibre-reinforced laminates demonstrate a wide spectrum of failure mechanisms such as matrix cracking and delamination. These damage mechanisms can result in significant deterioration of the residual stiffness and load-bearing capacity of composite components and should be thoroughly investigated. The delamination failure mechanism is studied in this paper for a double cantilever beam (DCB) loaded in mode I. Several sensitivity studies are performed to analyse the effects of mesh density and of parameters of the cohesive law on the character of damage propagation in laminates. The microstructural randomness of laminates that is responsible for non-uniform distributions of stresses in them even under uniform loading conditions is accounted for in the model. The random properties are introduced with the use of Weibull’s two-parameter probability density function. Several statistical realisations are carried out which show that the effect of microstructure could significantly affect the macroscopic response emphasizing the need to account for microstructural randomness for accurate predictions of load-carrying capacity of laminate composite structures.     

Crack patterns in thin layers under temperature loading Part II: Cyclic loading

The paper presents an analysis of cracking and delamination of an elastic plate bonded to a rigid substrate. The plate is subjected to a cyclic temperature loading. The bond between the plate and the substrate is simulated using rigid-frictional and rigid-cohesive-frictional interface models. The states of adaptation or incremental failure are distinguished in the case of the softening interfacial law. The proposed modelling is validated to predict fragmentation of a concrete pavement subjected to temperature loading. The obtained results can serve as reference solutions for numerical calculations of more complex cases.     

Crack – fiber sensor interaction and characterization of the bridging tractions in mode I delamination

Fiber bridging is regularly encountered in mode I delamination tests of unidirectional fiber reinforced composites. However, characterization of the bridging tractions is rather difficult. One way to indirectly evaluate the bridging traction distribution is to embed a fiber Bragg grating (FBG) sensor close to the crack tip and to measure the distributed strain along this FBG. The strain measurements from the FBG sensor are used to characterize the fiber bridging tractions by an identification method. In this work, the sensor is embedded in a unidirectional carbon/epoxy composite. Firstly, it is treated as an inclusion near the crack plane and a numerical analysis is performed to study its effect on the measured strain field and energy release rate. The results demonstrate that the sensor, located at about two fiber diameters from the crack plane, has a negligible effect on the fracture process. Secondly, among the identified linear, bilinear, and exponential bridging traction distributions, the exponential one is found to be a suitable model. Characterization of the bridging tractions allows to calculate the energy release due to the bridging fibers which is similar to the difference between the initiation energy release and the propagation value . The results also agree with the bridging tractions evaluated from the conventional energy release rate – crack opening displacement method.     

Numerical analysis of the Edge Crack Torsion test for mode III interlaminar fracture of composite laminates

A detailed numerical analysis of the Edge Crack Torsion test was performed to verify its adequacy for the mode III interlaminar fracture characterization of composite laminates. A new data reduction scheme based on specimen compliance was proposed. Three-dimensional finite element analyses including a cohesive damage model were performed in order to evaluate the accuracy of perceived G_IIIc values obtained with the proposed method. Despite some degree of crack length dependency of perceived G_IIIc, acceptable errors could be achieved within a certain crack length range, which is thus recommended for experimental tests.     

An interface element for the simulation of delamination in unidirectional fiber-reinforced composite laminates

Unidirectional fiber-reinforced composite laminates are widely used in aerospace industry for a great variety of structural parts. In order to enhance the exploitation of material reserves, there is a need for the integration of progressive damage scenarios in the design phase. Due to their hazardous effects on the load-carrying capacity of composite structures, this work focusses on the simulation of delaminations. A finite element based on a cohesive zone approach is developed. Two constitutive laws are proposed. One is characterized by linear degradation after delamination onset, the other is governed by exponential softening response. The damage process is history-dependent leading to an irreversible stiffness degradation in damaged zones. The practicability of the proposed model and the assets and drawbacks of the two material laws are shown by some numerical examples.     

Modelling of mode-I stable crack growth under hydrogen assisted stress corrosion cracking

The paper presents a new strategy based on combined analytical and finite element (FE) solution to hydrogen assisted stress corrosion crack growth. The diffusion process is solved analytically through both one-and two-dimensional modelling. These solutions are adopted with two-dimensional FE based cohesive zone model of crack extension study. The results fit well with published experimental data and show improvement over the predictions by full FE approach. The new solution approach helps to reduce time required for simulation/computation. The study has produced a relationship between concentration dependent reduction in cohesive strength and plastic strain rate.     

Analysis of impact-induced damage and delamination in the composite shell of a helmet

The impact energy absorption by helmets is of vital importance to the safety of motorcyclists during accidents. The paper is concerned with the study of damage and delamination, which are the principal modes of failure and energy absorption, in a composite shell and their influence on the impact performance of a helmet. Numerical simulations were conducted with different composite shells made of cross-ply laminate, woven fabric, and glass mat. The effectiveness of the numerical model is established using available experimental results from the literature. Hashin failure criteria and cohesive zone model (CZM) were used for predicting the in-plane damage and delamination in composite plies, respectively. An interface layer having a bilinear relationship between traction and relative displacement was placed between the plies of the composite shell to predict the delamination. The influence of damage and delamination in shells made of composite materials on impact-induced forces is evaluated and their performance is compared with helmet shells made of Acrylonitrile Butadiene Styrene (ABS).     

Interaction between matrix cracking and edge delamination in composite laminates

Matrix cracking and edge delamination are two main damage modes in continuous-fibre composite laminates. They are often investigated separately, and so the interaction between two damage modes has not yet been revealed. In this paper, a simple parallel-spring model is introduced to model the longitudinal stiffness reduction due to matrix cracking and edge delamination together. The energy release rate of edge delamination eliminating the matrix crack effect and the energy release rate of matrix cracking in the presence of edge delamination are then obtained. Experimental materials include carbon- and glass-fibre-reinforced bismaleimide composite laminates under static tension. The growth of matrix cracks and edge delaminations was recorded by means of NDT techniques. Results show that matrix cracks may initiate before or after edge lamination. This depends on the laminate layup, and especially on the thickness of the 90° plies. Edge delamination may also induce matrix cracking. Matrix cracking has a significant effect on the stiffness reduction in GRP laminates. The present model can predict the stiffness reduction in a laminate containing both matrix cracks and edge delaminations. The mixed-mode delamination fracture toughness obtained from the present model shows up to 50% differences compared with O'Brien's model for GRP laminates. However, matrix cracking has a small effect on the mixed-mode interlaminar fracture toughness of the CFRP laminates.     

Prediction of first matrix cracking in micro/nanohybrid brittle matrix composites

The classical Aveston–Cooper–Kelly shear-lag model for predicting the first matrix cracking strength in a brittle matrix composite is extended to the case of a hybrid brittle matrix composite containing both micro-scale and nano-scale fibers. First, closed-form solutions for the stresses in the two types of fibers and the matrix are derived. These are then used along with an energy analysis to predict the matrix cracking stress as a function of relevant material parameters. The analysis is applied to a typical Nicalon-SiC/CVI-SiC ceramic matrix composite containing additional nanofibers, for a wide range of nanofiber properties. A few volume percent of small diameter, moderate-stiffness nanofibers is predicted to provide significant strengthening and reduced crack opening while maintaining acceptable post-cracking fiber stresses. Various issues in the design of such micro/nanohybrid composites are then discussed.