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Models of dynamic damage and phase-field fracture, and their various time discretisations

Tomáš Roubíček

Abstract

Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.

Models of dynamic damage and phase-field fracture, and their various time discretisations

Abstract

Several variants of models of damage in viscoelastic continua under small strains in the Kelvin-Voigt rheology are presented and analyzed by using the Galerkin method. The particular case, known as a phase-field fracture approximation of cracks, is discussed in detail. All these models are dynamic (i.e. involve inertia to model vibrations or waves possibly emitted during fast damage/fracture or induced by fast varying forcing) and consider viscosity which is also damageable. Then various options for time discretisation are devised. Eventually, extensions to more complex rheologies or a modification for large strains are briefly exposed, too.

Paper Structure

This paper contains 10 sections, 3 theorems, 70 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Models of damage at small strains
  3. Phase-field concept towards fracture
  4. Various time discretisations
  5. Implicit "monolithic" discretisation in time
  6. Fractional-step (staggered) discretisation
  7. Explicit time discretisation outlined
  8. Concluding remarks -- some modifications
  9. Combination with creep or plasticity
  10. Damage models at large strains

Key Result

Proposition 2.2

Let the ansatz damage-gradient be considered, let also $\varrho,\kappa\in L^\infty(\Omega)$ with ${\rm ess\,inf}\,\varrho>0$ and ${\rm ess\,inf}\,\kappa>0$, $f\in L^1(I;L^2(\Omega;{\mathbb R}^d))$, $g\in L^2(\Sigma;{\mathbb R}^d))$, $u_0\in H^1(\Omega;{\mathbb R}^d)$, $v_0\in L^2(\Omega;{\mathbb R}^ Then the Galerkin approximation $(u_k,\alpha_k)$ exists and, for selected subsequences, we have an

Figures (2)

  • Figure 1: Simulations of a rupture in a two-dimensional specimen loaded by tension in a vertical direction, modelled by the phase-field crack approximation, and subsequent emission of an elastic wave. Seven selected snapshots are depicted. The decoupled energy-preserving time discretisation and P1-finite elements have been used. Courtesy of Roman Vodička (Technical University Košice, Slovakia)
  • Figure 2: Schematic diagram for the viscoelastic Jeffreys rheology (if $\sigma_\text{\sc yld}=0$) which is subjected to damage $\alpha$ in the deviatoric part except undamageable creep (the $G_{_{\rm NH}}$-dashpot) while the Kelvin-Voigt rheology in the spherical (volumetric) part is subjected to damage only under tension but not compression. For $\sigma_\text{\sc yld}>0$, it models (visco)plasticity. Evolution of damage is not depicted.

Theorems & Definitions (9)

  • Definition 2.1: Weak formulation
  • Proposition 2.2: Existence in the linear model
  • Proposition 2.3: Unidirectional damage in nonlinear models
  • Proposition 2.4: Damage with healing in nonlinear models
  • Definition 2.5: Energetic formulation
  • Remark 3.1: Finite fracture mechanics (FFM)
  • Remark 3.2: Coupled stress-energy criterion
  • Remark 3.3: Mixity-mode sensitive cracks
  • Remark 3.4: Various other models