A gradient‐enhanced continuum damage model for fibre‐reinforced materials at finite strains

Abstract

The assumption of purely local continuum damage formulations may imply a loss of well-posedness of the underlying boundary value problem. With regard to numerical methods such as the finite element method, this may lead to mesh-dependent solutions, a vanishing localised damage zone upon mesh refinement, and hence physically questionable results. In order to circumvent these deficiencies, i.e. to regularise the problem, we, in this contribution, apply a non-local gradient-based damage formulation within a geometrically non-linear setting allowing for large deformations. 1 Gradient-enhanced format of the free energy function Let x = ϕ(X, t) describe the deformation of the body, which transforms referential placements X ∈ B 0 to their spatial counterparts x ∈ B t . The deformation gradient is introduced as F = ∇ X ϕ with the Jacobian J = det(F ) > 0. Finally, let the co-factor of F defined as cof(F ) = JF −t . Fibre-reinforcement of the material is considered by assuming two families of fibres to be embedded in the continuum. Their orientation is characterised by referential unit vectors a 0 i , i = 1, 2 with ||a 0 i || = 1. Furthermore, we assume the existence of a free energythat accounts for anisotropic non-linear elastic material response under the influence of scalar damage. The effective free energy of the hypothetical undamaged material is composed additively of an isotropic and an anisotropic contribution, i.e.The isotropic part is specified by a compressible neo-Hookean format with the first invariant I 1 = F : F . The anisotropic part is based on an orthotropic exponential model with N = 2 families of fibres including fibre dispersion according to [2] with E i = κ I 1 + [1 − 3κ] I 4 i − 1 and the invariant I 4 i = a 0 i · F t · F · a 0 i . It is assumed that only the anisotropic part is subjected to damage, whereas the isotropic matrix material remains elastic. In equation (1), κ ∈ [0, ∞) is a scalar internal damage variable, characterising a material stiffness loss of the fibres, while f d (κ) = 1 − d represents an appropriate damage function. Conceptually following the approach by [1], Ψ grd contains the gradient of the non-local damage field variable φ, i.e.Here, the gradient parameter c d = 0 results in a local model, while c d > 0 leads to the regularised gradient-enhanced model. On the other hand, Ψ plty incorporates a penalisation term, which links the non-local damage variable φ to the local damage variable κ. The penalty parameter β d approximately enforces the local damage field κ and the non-local field φ to coincide. The thermodynamic force q driving the local damage process and conjugate to the damage variable d is obtained ascf.[3] for details. Furthermore, we adopt the damage condition Φ d = q − κ ≤ 0, where Φ d < 0 refers to the purely elastic case and Φ d = 0 to possible damage evolution. Based on the postulate of maximum dissipation, a constrained optimisation problem involving the Lagrange multiplier λ can be constructed. This results in the following associated evolution equation for th...

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“…e.g. [20,25,26,28,29]. This concerns the existence and regularity of solutions, let alone the behavior of the damage variables and the displacement field, as the penalty vanishes.…”