Deformation model for brittle materials and the structure of failure waves

Abstract

Constitutive equations that describe the experimentally observed failure waves are proposed to model inelastic strains of brittle materials. The complete system of equations is hyperbolic, each equation of this system has divergent form. The model is based on the assumption that continual failure is the process of transition from an intact state to a "fully damaged" state described by the kinetics of the order parameter. The structure of stationary traveling compressive waves is analyzed using a simplified model. It is shown that in a certain range of amplitudes, the wave splits into an elastic precursor and a failure wave.Key words: inelastic strain of brittle materials, failure waves, shock-wave structure.In the present paper, constitutive equations for modeling inelastic strains of brittle materials are proposed that can be used to describe the so-called failure waves. The fracture of a brittle material under compressive stresses is characterized by the formation of numerous cracks and has a wave nature [1,2]. Theoretical investigation of the fracture waves is in its infancy [3,4], and no adequate mathematical model has been proposed for a qualitative analysis and numerical study of the processes mentioned above. In the present paper, constitutive differential equations based on the nonlinear theory of inelastic strains [5] are formulated in the form of a hyperbolic system in which each equation has divergent form. Moreover, the model proposed satisfy the laws of nonequilibrium thermodynamics. Models of this type allow the use of well-developed mathematical and effective numerical methods of solving various problems.The model proposed is based on the assumption that an element of a material subjected to continual failure undergoes a transition from an intact state to a "fully damaged" state, which can be characterized by elastic moduli different from those of the intact material. This transition is described by an equation for the order parameter with nonlinear kinetics. Moreover, the model takes into account the inelastic deformation of the material that accompanies continual failure. A similar small-strain model and numerical analyses of some problems in good agreement with experimental data [1] were proposed in [3].In the present paper, the structure of stationary traveling compressive waves is analyzed using a simplified model of continual failure. The investigation technique is similar to the analysis of the shock-wave structure in a medium with relaxation given in [6] and to the method of studying elastoplastic waves in a Maxwell nonlinear medium [5]. In a certain range of amplitudes, the wave splits into an elastic precursor and a failure wave itself, which agrees with the experimentally observed wave structure.1. Complete System of Constitutive Equations. Following [5], we consider the velocity vector u i (i = 1, 2, 3), the elastic deformation gradient c ij (the elastic distortion tensor [5]), the reference density ρ 0 (the density corresponding to an element of the medium reduced to the state of zero ...

A unified first-order hyperbolic model for nonlinear dynamic rupture processes in diffuse fracture zones

“…a Newtonian fluid (see §3c) or viscoplastic fluid, which can be used for modelling of in-fault friction, for example. Yet, in this paper, we do not use an individual distortion evolution equation for each phase, but employ the mixture approach [36,37], and use a single distortion field representing the local deformation of the mixture element, while the individual rheological properties of the phases are taken into account via the dependence of the relaxation time τ 1 on the damage variable ξ as follows: τ1=(false(1−ξfalse)τI+ξτD)−1, where τ I and τ D are shear stress relaxation times for the intact and fully damaged materials, respectively, which are usually highly nonlinear functions of the parameters of state. The particular choice of τ I and τ D that is used in this paper reads τI=τI0exp⁡false(αI−βIfalse(1−ξfalse)Yfalse),1emτD=τD0exp⁡false(αD−βDξYfalse), where Y is the equivalent stress, while τ I 0 , α I , β I , τ D 0 , α D , β D are material constants.…”

“…The continuum model for damage of solids employed in this paper consists of two main ingredients. The first ingredient is the damage model proposed by Resnyansky, Romenski and co-authors [ 36 , 37 ] which is a continuous damage model with a chemical kinetics-type mechanism controlling the damage field ξ ∈ [0, 1] ( ξ = 0 corresponds to the intact and ξ = 1 to the fully damaged state), which is interpreted as the concentration of the damaged phase. Being a relaxation-type approach, it provides a rather universal framework for modelling brittle and ductile fracture from a unified non-equilibrium thermodynamics viewpoint, according to which these two types of fractures can be described by the same constitutive relations (relaxation functions), but have different characteristic time scales, e.g.…”

“…In the latter, the phase transformation is entirely controlled by the free energy functional, which usually consists of three terms: normalΨfalse(bold-italicε,ϕ,normal∇ϕfalse)=Ψ1false(bold-italicε,ϕfalse)+Ψ2false(ϕfalse)+Ψ3false(normal∇ϕfalse), where ε is the small elastic strain tensor, Ψ 1 is the elastic energy which comprises a degradation function, Ψ 2 is the damage potential (usually a double-well potential but also single-well potentials are used [51]), and Ψ 3 is the regularization term. In our approach, only an energy equivalent to Ψ 1 ( ε , ϕ ) is used [37,52], while the phase-transition is described in the context of irreversible thermodynamics and is controlled by a dissipation potential which is usually a highly nonlinear function of state variables 3 [44,53]. Yet, it is important to emphasize that the irreversible terms controlling the damage are algebraic source terms (no space derivatives), which do not affect the differential operator of the model.…”

“…see [ 34 , 35 ]. By contrast, the model developed here originating from [ 36 , 37 ] does not require non-local regularization terms 2 and is formulated based on the thermodynamically compatible continuum mixture theory [ 40 , 41 ] which results in a first-order symmetric hyperbolic governing PDE system and thus is intrinsically well-posed, at least locally in time. Mathematical regularity of the model is supported by the stability of the hereafter presented numerical results, including a mesh convergence analysis (see § 3 ).…”

Space-time adaptive ADER discontinuous Galerkin schemes for nonlinear hyperelasticity with material failure

“…The dynamics of the material damage is driven by a reaction-type source term, that depends on the ratio of equivalent stress to yield stress. This idea goes back to [88,92,87], where the dynamics of failure waves was discussed in the context of hyperelasticity. To account for nonlinear elasto-plastic materials, the GPR model includes a strain relaxation mechanism, see [82,39].…”