# Modelling of High-Velocity Impact on Woven Carbon Fibre-Reinforced Plastic Laminate
## 2 Constitutive Model  
The constitutive model was developed in the framework of irreversible thermodynamics with internal variables. Based on the evident brittle properties of the composites, the elastic strains were assumed to be small, and the plastic deformation was neglected in the current version of the model. Consequently, energy dissipation of a deformation process is solely driven by damage. This was incorporated in the model as a damage effect tensor operating on the material elastic stiffness tensor (see, for instance, [32; 33]). These assumptions significantly simplified the constitutive model formulation and allow for the model to be defined in the current configuration. Due to the pronounced heterogeneity and anisotropy of composite materials, a number of internal damage parameters need to be defined in order to represent distinct damagingand failure modes (see, for instance, [34]), such as the tensile damage parameter, the shear damage parameter, the interface damage parameter, etc. The evolution of these parameters is then defined in the separate constitutive laws. This approach is used in the definition of the constitutive model proposed in this work. Damage in this material model was defined by using the principle of strain energy equivalence, originally derived by [PERSON] and [PERSON] [35; 36] as a generalization of the pioneering work of [PERSON] [37]. A linear relationship between the effective and nominal Cauchy stress was defined by damage effect tensor:  
\[\overline{\sigma}=\mathbb{M}(\mathbf{\omega}):\mathbf{\sigma} \tag{1}\]  
where \(\overline{\sigma}\) denotes effective Cauchy stress tensor, which \"acts\" on the virgin material, and \(\mathbb{M}(\mathbf{\omega})\) is a fourth-order damage effect tensor, which is a function of a second-order damage tensor \(\mathbf{\omega}\). The energy equivalence principle produces a similar relationship between the nominal and effective elastic strains, i.e., \(\overline{\varepsilon}_{e}=\mathbb{M}^{-1}:\varepsilon_{e}\), which, combined with Equation (1), leads to the relationships between the damage stiffness tensor \(\mathbb{C}\) and its virgin counterpart, \(\overline{\mathbb{C}}\), as:  
\[\mathbb{C}(\mathbf{\omega})=\mathbb{M}^{-1}(\mathbf{\omega}):\overline{\mathbb{C}}: \mathbb{M}^{-1}(\mathbf{\omega}) \tag{2}\]  
To maintain symmetry of the effective stress tensor, the damage effect tensor was defined by a product type symmetrisation (see, for instance, [38]), in the following form:  
\[\mathbb{M}(\mathbf{\omega})=(\mathbf{I}-\mathbf{\omega})^{-\frac{1}{2}}(\mathbf{I}-\mathbf{\omega })^{-\frac{1}{2}}\qquad\qquad\qquad\mathbb{M}_{\exists\text{jkl}}=(\delta_{ \text{ik}}-\omega_{\text{ik}})^{-\frac{1}{2}}\Big{(}\delta_{\text{jl}}-\omega _{\text{jl}}\Big{)}^{-\frac{1}{2}} \tag{3}\]  
which has a diagonal form in the principal directions of damage, where \(\omega_{\text{ik}}\), \(i=k=1,2,3\) are principal values of the damage tensor. The damage effect tensor given in Equation (3) can now be substituted in Equation (2), leading to the stiffness tensor of a damaged material as:  
\[\mathbb{C}= \begin{bmatrix}\mathbb{C}_{11}(1-\omega_{1})^{2}&\mathbb{C}_{12}(1 -\omega_{1})(1-\omega_{2})&\mathbb{C}_{31}(1-\omega_{3})(1-\omega_{1})&0&0&0 \\ &\mathbb{C}_{22}(1-\omega_{2})^{2}&\mathbb{C}_{23}(1-\omega_{2})(1-\omega_{3}) &0&0&0\\ &&\mathbb{C}_{33}(1-\omega_{3})^{2}&0&0&0\\ &&&\mathbb{C}_{44}(1-\omega_{2})(1-\omega_{3})&0&0\\ \text{symm}&&&&\mathbb{C}_{55}(1-\omega_{1})(1-\omega_{3})&0\\ &&&&\mathbb{C}_{65}(1-\omega_{1})(1-\omega_{2})\end{bmatrix}  \tag{4}\]  
The stiffness in Equation (4) is used in the definition of Helmholtz free energy and Gibbs energy and further in the definition of the dissipation rate.