Shuguang Li

General unit cells for micromechanical analyses of unidirectional composites

Two typical idealised packing systems have been employed for unidirectionally fibre reinforced composites, viz. square and hexagonal ones. A systematic approach has been adopted and it involves the use of only the translational symmetry transformations. There are a number of important advantages resulting from this. The unit cells so derived are capable of accommodating fibres of irregular cross-sections and imperfections asymmetrically distributed around fibres such as microcracks and local debonding in the system, provided the regularity of the packing and imperfections is present. Furthermore, all the unit cells established can be subjected to arbitrary combinations of macroscopic stresses or strains unlike most available unit cells in the literature which can only deal with individual macroscopic stress or strain components. Boundary conditions for these unit cells have been derived from appropriate considerations of the conditions of symmetry transformations. Applications of macroscopic stresses or strains as the loads to the unit cells have been described in such a way that they can be implemented in a straightforward manner and the effective properties of the composite can be evaluated following a standard procedure.

Unit cells for micromechanical analyses of particle-reinforced composites

Unit cells are established in this paper for micromechanical analyses of particle-reinforced composites. A range of typical packing systems are examined in a systematic manner for each of them. Only the translational symmetry transformations are employed in establishing these unit cells. There are a number of important advantages resulting from this. The unit cells so derived are capable of dealing with problems involving reinforcing particles of irregular geometries and local imperfections such as debonding between the particles and the matrix, and microcracks in the matrix, provided the regularity of the packing and the orientation of the particles and the imperfections is maintained within the material. Furthermore, all the unit cells established can be subjected to arbitrary combinations of macroscopic stresses or strains using a single set of boundary conditions unlike most available unit cells in the literature, with which individual macroscopic stress or strain components may have to be analysed using different boundary conditions because of the use of reflectional symmetries. Boundary conditions for the unit cells proposed in this paper are derived from appropriate considerations of the conditions resulting from translational symmetry transformations. Applications of loads in terms of macroscopic stresses or strains and thermal loading to the unit cells are described in such a way that they can be implemented in a straightforward manner and the effective properties of composites can be evaluated following a standard and simple procedure without a numerical averaging process. The implementation of the unit cells in the micromechanical finite element analysis of particle-reinforced composites has been demonstrated fully. Spherical particles are assumed and both the particle and the matrix are assumed to be linear elastic materials and the bonding between the particle and the matrix to be perfect. 3D brick elements have been employed to generate the meshes for analysing the unit cells corresponding to various packing systems. The effective properties of the composite represented by the unit cells have been obtained through the analyses and they have been discussed and compared with results in the literature. Stress distributions in the particle and surrounding matrix have been examined. Some interesting characteristics of the different packing systems have been elaborated.

A continuum damage model for delaminations in laminated composites

Delamination, a typical mode of interfacial damage in laminated composites, has been considered in the context of continuum damage mechanics in this paper. Interfaces where delaminations could occur are introduced between the constituent layers. A simple but appropriate continuum damage representation is proposed. A single scalar damage parameter is employed and the degradation of the interface stiffness is established. Use has been made of the concept of a damage surface to derive the damage evolution law. The damage surface is constructed so that it combines the conventional stress-based and fracture-mechanics-based failure criteria which take account of mode interaction in mixed-mode delamination problems. The damage surface shrinks as damage develops and leads to a softening interfacial constitutive law. By adjusting the shrinkage rate of the damage surface, various interfacial constitutive laws found in the literature can be reproduced. An incremental interfacial constitutive law is also derived for use in damage analysis of laminated composites, which is a non-linear problem in nature. Numerical predictions for problems involving a DCB specimen under pure mode I delamination and mixed-mode delamination in a split beam are in good agreement with available experimental data or analytical solutions. The model has also been applied to the prediction of the failure strength of overlap ply-blocking specimens. The results have been compared with available experimental and alternative theoretical ones and discussed fully.

Mode separation of energy release rate for delamination in composite laminates using sublaminates

Abstract Individual energy release rates for delamination in composite laminates do not exist according to two- or three-dimensional elastic theory due to the oscillatory characteristics of the stress and displacement fields near the delamination tip (Sun, C.T., Jih, C.J., 1987. Engng. Fracture Mech. 28, 13–20; Raju, I.S., Creus Jr., J.H., Aminpour, M.A., 1988. Engng. Fracture Mech. 30, 383–396.) In this paper, sublaminates governed by transverse shear deformable laminate theory are adopted to model such delamination. Oscillatory singular stresses around the delamination tip are avoided as a result. Instead, stress resultant jumps are found in the sublaminates across the delamination tip. It transpires that mode I, II and III energy release rates can then be obtained using the virtual crack closure technique. The results produced by this approach for a homogeneous double cantilever beam and an edge-delamination in a non-homogeneous laminate show good agreement with those available in the literature. The approach produces both total and individual components of energy release rate for delamination, which converge as the sublaminate division is refined and the sizes of the delamination tip elements decrease.

Modelling Interlaminar and Intralaminar Damage in Filament-Wound Pipes under Quasi-Static Indentation

A pragmatic modelis proposed to predict interlaminar and intralaminar damage, i.e., delamination and matrix cracking, in filament wound pipes. The pipe is modelled as an assembly of sublaminates connected along their interfaces. The initiation of delamination and transverse matrix cracking is predicted based on stress-based failure criterion. Delamination propagation is governed by the energy release rates. After the occurrence of damage, constraint between sublaminates is removed to model the delamination and a ply discount model is used to account for the materialdegradation effects of matrix cracking. The establishment of a leakage path is comprised by jointed matrix cracking in each layer and delamination at each interface. Finite element analysis has been carried out to simulate the quasi-static indentation of filament-wound composite pipes. Despite the simplification of treating each damage mechanism independently (i.e., no direct interactions), good agreement has been achieved between experimentalresul ts and the predictions of the model for the load–indentation relationship, the evolution of multiple delaminations and the formation of a leakage path.

On the unit cell for micromechanical analysis of fibre-reinforced composites

This paper is concerned with the unit cell for micromechanical analyses of unidirectionally fibre-reinforced composites. A systematic consideration has been made to the symmetries present in idealised fibre-matrix systems. The symmetries enable the stresses, strains and displacements to be mapped from a unit cell to the whole field of interest. Appropriate boundary conditions of the unit cell have been derived from these symmetry considerations for micromechanical analyses. The loads to the unit cell and the responses of it in terms of macroscopic stresses or strains have been addressed in such a way that the effective properties of the material can be obtained from micromechanical analyses of the unit cell in a standard manner. No systematic account of this kind is available in the literature while the use of the unit cell for analyses of the microscopic behaviour of composite materials is becoming popular, in particular using finite elements, in order to obtain in–depth understanding of the performance of these materials. Ineffective introductions of the unit cell and incorrect applications of boundary conditions are often encountered. These have been clarified in this paper so that correct and effective use of the unit cell for micromechanical analyses can be achieved.

The background to the third world-wide failure exercise

In 2004, the authors completed the First World-Wide Failure Exercise dealing with benchmarking recognised failure criteria under two-dimensional, in-plane loadings. Based on the success and the lessons learnt, and using the same strategy, the authors have organised the ‘Second World-Wide Failure Exercise’. It aims at filling key gaps identified and establishing the status of 12 theoretical methods for predicting failure in fibre-reinforced composite materials subjected to three-dimensional or (triaxial) states of stress. This paper gives an account of the background to the Second World-Wide Failure Exercise, the process of completing the first part (Part A) and a summary of key conclusions.

A comparison between the predictive capability of matrix cracking, damage and failure criteria for fibre reinforced composite laminates: Part A of the third world-wide failure exercise:

This paper provides a set of concluding remarks on Part A of the third world-wide failure exercise where a comparison has been made between the capabilities of 12 different mathematical models for predicting the evolution of matrix cracking, damage and failure in continuous fibre-reinforced polymer composites when subjected to multi-axial loading. The originators (or their collaborators) of those theories have employed their methods to 13 carefully selected challenging problems (test cases) addressing the cracking and damage evolution arising from ply thickness, lay-up sequence, size effects and a variety of loading conditions (biaxial, bending, thermal loading and loading-unloading) of a number of unidirectional and multi-directional glass and carbon epoxy laminates. These covered eight different lay-ups consisting of 0°, [0°/90°/0°], [0°/90°8/0°], [0°/90°]s, [±45°]s, [±50°]s, [30°/90°/−30°/90°]s and a family of [0°m/45°m/90°m/−45°m]s, [45°/0°/90°/−45°]s and [0°/45°/−45°/90°]s quasi-isotropic laminates. Key features in each theory are identified including: types of damage models employed, whether linear or nonlinear analysis was carried out, reliance on software and numerical methods and identification of modes of damage. The results of stress–strain curves, crack density and damage curves have been superimposed and bar charts were constructed to show similarities and differences between the predictions of the various theories.

A CONTINUUM DAMAGE MODEL FOR TRANSVERSE MATRIX CRACKING IN LAMINATED FIBRE-REINFORCED COMPOSITES

In this paper, the effects of damage in the form of transverse matrix cracking in fibre–reinforced laminates of arbitrary layup are considered in the context of continuum damage mechanics. A complete model for the damage process is accomplished by establishing an appropriate damage representation and a damage growth law. Talrejas damage representation has been modified and significant simplifications have been achieved in defining the damage–related material constants for this particular form of damage in a convenient way. The modified damage representation is lamina–based while Talrejas damage representation is, in the context of this paper, laminate–based. The assumptions introduced to simplify the damage representation are examined and justified. Employing the concept of a damage surface, an incremental damage growth law is formulated. A complete damage model is achieved by combining the damage representation and the damage growth law. The model results in a new laminate theory which describes the deformation of laminates as well as the development of the damage process in the form of crack multiplication. This enables practical predictions to be made of the behaviour of laminated structures made of fibre–reinforced composites experiencing transverse matrix cracking.

Application of a delamination model to laminated composite structures

Abstract A model for progressive interlaminar delamination is presented for laminated composite structures. Instead of a cumbersome 3D description, a computationally efficient 2D technique is adopted which models the laminated structure as an assembly of sublaminates connected through their interfaces. Constraints between sublaminates are removed to represent the presence of delaminations. The use of laminate theory results in jumps in stress resultants across the delamination tip and this helps to avoid dealing with the singular stress field at the delamination front. A stress-based failure criterion is used to predict delamination initiation. Delamination propagation is analysed by adopting a fracture mechanics approach. The major intralaminar damage mode, matrix cracking, is also included in the present analysis. This is detected by a stress-based failure criterion and a ply discount model is used to account for the effects of material degradation. Finite element analysis has been carried out to assess the deformation and the delamination development in a range of typical structures: a double cantilever beam, a cross-ply laminate and some filament-wound composite pipes. Good agreement has been achieved between the predictions and available experimental data. A study of the effect of mesh size shows that a relatively coarse mesh gives sufficiently accurate results. These examples give a useful indication of the versatility and feasibility of the present approach for real structural applications.