FAILURE PREDICTION AND DAMAGE MODELLING OF MATRIX CRACKING IN - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS FAILURE PREDICTION AND DAMAGE MODELLING OF MATRIX CRACKING IN QUASI-ISOTROPIC LAMINATES AT THE PLY LEVEL G.M. Vyas 1 *, S.T. Pinho 1 , P. Robinson 1 1 Department of Aeronautics, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK * Corresponding author (gaurav.vyas04@imperial.ac.uk) Keywords : Matrix cracking, Stiffness reduction, Stress field calculated by comparing the energy release rates for 1 Introduction matrix cracking and delamination. Unlike earlier Matrix cracking under quasi-static loading is an models, to allow for the prediction of crack important failure mode in quasi-isotropic laminates. formation in non-90° plies the effects of shear stress Cracking can result in a significant loss of structural are included and the ply of interest is assumed to be stiffness and lead to other forms of damage, such as supported by two different sublaminates. The model delamination. Early attempts for the modelling of is able to predict the stiffness reduction of the matrix cracking based on shear-lag theories over laminate as matrix cracks accumulate and accurately predicted the saturation crack density. With regards calculate the saturation crack density, after which to crack opening/sliding approaches, although direct delamination is induced. finite element modelling of matrix cracking is The combined failure criteria and damage model are feasible at the micro-scale, it can be a costly coded as a user defined material subroutine in an approach in terms of time and computational effort, explicit commercial finite element package as the exact geometry of the laminate must be (Abaqus\Explicit VUMAT) and are shown to modelled. Furthermore, the measured crack opening compare favourably with the experimental data and/or sliding displacements are then input into a available in the literature. macro-scale model which must be run separately. At the ply level, continuum damage models are 2.1 Failure detection preferred as the fibres and matrix are not The model uses the failure indices previously individually modelled, allowing results to be more outlined by Pinho et al. [5]. The failure index for readily obtained. matrix failure is a modified version of the Mohr- Models using variational mechanics [1-3] are more Coulomb criterion, adapted for UD composites in a conservative, but until recently have been limited to similar manner to Puck et al. [6]: (S°/90°) s layups with either in-plane uniaxial or (1) biaxial loading. Other analytical and numerical models in the literature require the use of iteration to obtain the current crack density and cannot therefore be readily implemented into explicit finite element codes. where and are the transverse and longitudinal is the in-situ shear components of the traction, 2 Proposed Model and in-plane transverse tensile shear strength, A model for matrix cracking based on the concepts are the transverse and longitudinal in-situ transverse previously used in variational and stress transfer shear strengths. is the normal component of the approaches [4] is refined and combined with the traction and and are the slope coefficients for LaRC05 failure criteria [5] to detect the onset of transverse and longitudinal shear strength, matrix cracking and subsequently degrade properties respectively. for generic in-plane loading in S°/90° laminates. Furthermore, the saturation crack density is

2.2 Stress and Displacement Fields The determination of the stress and displacement fields follows the derivation outlined by The derivation of the normal through the thickness Farrokhabadi et al. [4]. stress components assumes that continuity of is The dimensions of the representative volume enforced across the interface between the thick and element of the lamina shown in Fig. 1 are given by thin sub-laminae due to perfect bonding at the , and , where is the width of , as a interface , with the remote stress, the lamina. Generalised plane strain is assumed for boundary condition, as well as satisfying the the displacement field, which implies the equilibrium equations: displacement field is: (7) (2) denotes the average strain along the fibre where direction of the cracked lamina, which is assumed to be equivalent to that of the uncracked lamina. It is assumed that after cracks have formed, two perturbation stresses appear in the -direction at the In the following derivation, and are defined as: interface between the thick and thin sub-laminae of the 90° lamina, and . The through the (8) thickness shear stress components of the cracked lamina are assumed to be piecewise linear for each sub lamina and are independent of the -direction due to the generalised plane strain assumption: (3) and Using the relation the stresses for each sub-lamina, the through the thickness displacements are derived as: (4) (9) where the superscripts (1) and (2) denote the thick and thin sub-lamina respectively and 90 indicates a remote stress applied on the undamaged lamina. From equilibrium with the above equations, the following stresses are derived: (5) The transverse displacements are derived from the using through thickness displacements (6) . Integrating in the cracked and uncracked regions with respect to z and simplifying:

PAPER TITLE Similarly, an expression for can be obtained (10) by using the horizontal displacement at , and rearranging the resultant expression. As before, is found using continuity: (13) The in-plane displacements in the fibre direction are derived using the relation . Integrating in both regions gives: The determination of requires the displacement in the x-direction at , . (11) Using the x-symmetry, rigid body displacements in x and rigid body rotations in z can be prevented imposing . As with the other displacement components, continuity infers and allows for the calculation of : (14) where , and are the displacements at the interface and are determined as follows. By symmetry, the vertical displacement at can be set to zero, which prevents rigid body displacements Using averaged values for all displacements and in z and rigid body rotations in x and y. This allows stresses, the in-plane stress-strain relations are: . At , for the determination of continuity of displacements implies that (15) , allowing for the determination of : (12) and in each sub- The averaged values for lamina in equation are then subtracted from each other, leading to the differential equations: 3

(16) (19) The constants are material dependent and given by: (17) The evolution of the variables in equation 19 is given by the compliance relationships used in the variational models by Nairn & Hu [1] and Hashin [2]. Following the work of McCartney [7], the damaged compliance matrix is then given by: (20) Where k and k’ are given by: (21) The boundary conditions required for the solution of the differential equations are: (18) 2.3 Damaged compliance matrix and saturation crack density The saturation crack density is calculated by comparing the energy release rates for matrix cracking and delamination. The intersection of the two curves is defined as the saturation value, after which delamination will occur instead of additional 3 Results crack accumulation. An example comparing the proposed variational The modelling of the reduction in laminate stiffness approach to a shear lag approach for the calculation after damage requires the definition of two functions of the damage parameter, , given by: of saturation crack density is shown in Fig. 2. Predictions for the change in modulus of cross-ply laminates of glass/epoxy [8] and AS4/3502 [9] are