# Modelling of High-Velocity Impact on Woven Carbon Fibre-Reinforced Plastic Laminate
## 1 Introduction
For instance, progressive damage in [15,16,19,20] was incorporated in terms of damage variables, which affected diagonal members of the compliance tensor, with a separate failure criterion controlling different composite failure modes (Hashin failure criterion for fibres and separate criteria for matrix were used in [20]). There are two Hashin models based on LaRC03 failure criteria [21], with progressive damage models only applicable to the diagonal members of the stiffness tensor. A more general approach was published in [18,22], where separate Hashin-based failure criteria were used for fibre tension and compression, matrix tension and compression as well as fibre-matrix interface failure and interlaminar tension and compression failure. In this model, damage variables are applied to all members of the stiffness tensor. It is also worth mentioning a physically based phenomenological model developed by [PERSON] [23]. This model is developed for the unidirectional composites and very complicated to modify and apply to woven composites. [PERSON] and co-workers [24,25] developed strain invariant failure theory (SIFT) for modelling damage. The theory was applied in the static analysis of composite materials, where the damage initiation criteria were defined using the critical values of isotropic strain invariants. The use of the strain invariants contributed to the physical significance of the theory, but the criteria had to be calculated/checked in the number of points in the material, which made this approach very tedious and not effective for application in the dynamic problems. Only a selection of these models is currently available in commercial FEM solvers. One example of such a model is the Chang-Chang Composite Failure Model implementedin LS Dyna and other FEM code. Nevertheless, a significant constraint of the existing damage models for composites is the lack of capabilities for modelling high-velocity impact, especially for the rates of loading where shock waves are formed and propagate within the composite material. Shock wave propagation through solids became attractive for a wide community of physicists only in the closing decades of the last century, starting from detailed investigations of the shock response of aluminium alloy 2024 [26]. Work by [PERSON] [27] and [PERSON] [28] on 6061-T6 predicted that the spall strength should vary in accordance with the one-dimensional stress yield strength according to orientation, but they concluded that directionality has no significant effect on crack formation. Consequently, the numerical models for modelling shock wave propagation were usually developed for the isotropic material formulation; see, for instance, [28]. A constitutive model, capable of modelling shock wave propagation through a material, comprises two parts: an equation of state (EOS), which determines the material response under compression, and a shear part, responsible for shear deformation. This decomposition is relatively simple for isotropic material, since the spherical part of the stress tensor and strain tensor are co-linear and orthogonal to the deviatoric plane. However, in the case of anisotropic material, orthogonality does not hold, because the spherical component of stress can produce a change in shape--deviatoric strain. The first attempt to solve the problem was by [PERSON] [29], but the proposal failed to predict material response accurately [30]. Consequently, this paper describes a continuum-based progressive damage model for composite materials where the strength part of the model is coupled with vector equation of state. This makes the model applicable to high-velocity impact problems, where a shock wave is formed and propagates through the material. Orthotropic damage is included in the constitutive model through the damage effect tensor applied on the principal material stiffness and used in the definition of the effective stress, combined with the hypothesis of strain energy equivalence.