Modelling of Damage in Orthotropic Materials: Including Strain-Softening Effects in Dynamic Problems

Abstract

Damage models are developed within the continuum damage mechanics framework which allows the description of material degeneration with general constitutive equations. The difficulty in the description of damage behaviour increases with increasing complexity of the material behaviour. This is especially true when it comes to composite materials which have an orthotropic material behaviour. The conventional description of damage, i.e. the local continuum damage mechanics description, leads to strain-softening behaviour which is characterised by a decline in stress with simultaneously increasing strain. Due to strain-softening the tangent stiffness becomes negative which forces the wave speed to become imaginary in dynamic problems. Consequently the partial differential equations governing the dynamic problem change from hyperbolic to elliptic and, therefore, the initial boundary value problem no longer has a unique solution. Due to this the physical meaning becomes unrealistic. Strain-softening is limited to an infinitely small area in which waves are not able to propagate in a process called wave trapping. A displacement discontinuity in an area of width zero (localisation zone) develops. The strain becomes infinite in this zone and is accompanied with a zero stress. Areas outside the softening zone are not able to interact with the strain-softening domain. As a consequence the strain-softening domain acts similar to a free boundary at which waves reflect. The implementation of local continua with strain-softening behaviour in finite element codes leads to additional numerical problems. Strain-softening behaviour manifests itself in the smallest area possible which is a single point in analytical considerations. This area is defined by the element discretisation in finite element codes. Therefore, strain-softening leads to a pronounced mesh sensitivity of results in addition to mathematical and physical issues. This work aims to find a solution which removes problems associated to strain- softening. Its aim is to represent material behaviour due to damage realistically and enable numerical results to convergence to a unique solution. The strain-softening problem is the focus of this work. It was investigated using a 1D wave propagation problem described by Bažant and Belytschko [1]. This simple experiment allows for an easy comparison of analytical and numerical results and therefore gives an insight into the problems connected to strain-softening. Furthermore, regularisation methods, specifically nonlocal and viscous methods, were investigated. Regularisation methods add additional terms to constitutive equations which keep the initial boundary value problem well-posed and enable a unique solution independent of the element discretisation. It was found that these methods are indeed capable of regulating the softening problem; however, they add additional difficulties in the description of material behaviour. A new approach to the strain-softening issues, unique at this point of time, was developed in this work which implements damage as an equivalent damage force. This approach is able to keep the initial boundary value problem stable and converge to a unique solution without adding additional terms in the constitutive equations, such as regularisation methods. This new approach to strain-softening was implemented for an isotropic material with scalar damage variable in DYNA3D successfully. Numerical results converged to a unique solution and were physically reasonable. The concept of an equivalent damage force was further developed to orthotropic material behaviour. This made an advanced representation, using an 8th rank damage tensor, necessary. The 8th rank damage tensor is able to represent anisotropic damage and it is also the most general damage representation possible.