Anisotropic damage in quasi-brittle solids: modelling, computational issues and applications

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Abstract

Crucial issues concerning non-linear behaviour of quasi-brittle materials like some rocks, concrete and certain ceramics are addressed in the paper. In fact, inelastic response for this class of solids results from the evolution of a large number of micro- and mesocracks accompanied with frictional effects regarding closed cracks for compressive loading paths. Progressive microcracking produces volumetric dilatancy, induced anisotropy and further pressure sensitivity on the macroscale.

A continuum damage model attempting to capture salient features of progressive anisotropic degradation and accounting for frictional sliding over internal crack surfaces is synthesised. The model is three-dimensional and micro-mechanically motivated in its essential ingredients. At the same time it is built to provide a tool for structural analysis purpose. The settlement between apparently conflicting requirements of physical pertinency on the one hand and of applicability of the model on the other, is endeavoured through relative simplicity of the approach (a small number of material constants to identify) and its modular character allowing three distinct degrees of complexity.

The first `basic' level concerns modelling of the anisotropic degradation by multiple mesocrack growth generating volumetric dilatancy and non-symmetrical strength effects. The second level consists in accounting for the “normal” moduli recovery with respect to mesocrack-sets constrained to close under predominantly compressive loads (unilateral effect). The damage and frictional blocking/sliding coupled effects are dealt with in a non-classical way at the third level allowing thus to treat highly complex loading paths with rotating loading and damage axes (torsional paths for example). Hysteretic effects for loading and unloading due to plasticity-like mesocrack friction are commented.

Some computational aspects relative to the model (in its basic, extended and coupled version) are then outlined. It is stressed that a fully implicit integration scheme appears naturally compatible one for a class of damage models as it is for the model at stake (in spite of its specific features). The frictional sliding evolution is dealt with in a way close to plasticity integration algorithms. Concomitant growth of damage and frictional sliding is successfully managed by coupled algorithm. The corresponding double integration is facilitated by low interaction degree between damage and sliding criteria.

Significant examples are shown illustrating damage and friction induced non-linear behaviour together with complex hysteretic effects for cyclic loading. They illustrate relevancy of the coupled model for quasi-brittle rocks and efficiency of the algorithms employed.

Introduction

Some important issues concerning inelastic behaviour of quasi-brittle solids (e.g. rock-like materials) are explored and modelled in this paper in the framework of rate-type constitutive theory with internal variables. It is well confirmed nowadays that the non-linear response in such solids under loading results from the evolution of multiple internal micro- and mesocrack-like defects generating progressive degradation of moduli associated with secondary anisotropy effects, volumetric dilatancy and other events. Opening and closure of microcrack-sets under varying loads lead to further complex phenomena like recovery of some degraded moduli in the direction perpendicular to closed crack systems and frictional blocking and/or sliding over the internal crack surfaces.

Recently an interesting tentative of a global vision of the non-linear mechanics of materials has been proposed by Lubarda and Krajcinovic [1], including the classical rate theory of elastoplastic deformation of crystalline solids as well as the non-linear framework for progressively mesofracturing solids. Indeed, even for the latter class of quasi-brittle materials, the continuum damage approach attempting to capture progressive degradation attributable to evolution of multiple defects should be coupled with the plasticity-like approach accounting for irreversible frictional sliding over the internal crack surfaces. The extended framework of damage-elastoplastic constitutive models allows for study of complex, cyclic and non-proportional loading paths, where coupled microcrack-growth and friction related dissipative mechanisms produce strong non-linearity, induced anisotropy and intricate hysteresis.

When considering progressively mesofracturing solid under compressive loadings, i.e. with some crack-sets constrained to closure the initial unloading process is frequently friction-locked, exhibiting a high apparent rigidity. Further unloading may be dissipative if reverse multistage frictional sliding (reverse with respect to a loading branch) becomes active. The inelastic unloading is just a particular effect revealing the above-mentioned coupling of damage by microcracking with a form of plasticity generated by frictional sliding on closed cracks. A strong evidence in this sense can be found for example in Walsh [2].

The problem received some attention in the past in regard to modelling, see e.g. [3], [4], [5], [6], [7], [8], [9], and very recently, [10]. For the most part the texts adduced represent pertinent micromechanical studies leading, for some of them, to models capable to cover a limited range of stress–strain paths (two-dimensional, axisymmetric, etc.). The purpose of this paper is to address, in a synthetic manner, basic issues of the 3D-modelling proposed by the present authors, employing an internal variable formalism for the joint process of anisotropic damage by microcracking and frictional sliding at closed microcracks. The aim of this model is to provide an efficient, macroscopic – whereas strongly micromechanically motivated – approach suitable for boundary-value problems involving non-linear behaviour of quasi-brittle solids.

The approach presented is based on an anisotropic damage model, the “basic version”, proposed by Dragon [11] and extended by Halm and Dragon [12] to include the unilateral effect concerning normal stiffness recovery with respect to a mesocrack system constrained to closure. This extended version, summarized in Section 2, is then completed by the damage and frictional blocking/sliding model depicted above. The coupled model allows to treat complex loading paths with eventually rotating loading and damage axes. The corresponding developments are given in 3 Mesocrack friction induced plasticity, 4 Damage and frictional sliding interaction: fully coupled model. Besides, a somewhat elementary presentation of the coupled model is given in Ref. [13] by Halm and Dragon. Here, an overview covering the three modular segments of the model, i.e. the basic version and the extended one, treated together in Section 2, and the coupled damage-and-friction model completed in Section 4, is presented. The aim here is to set forth a more general survey including a methodology relative to the numerical integration of the constitutive equations proposed. The numerical schemes employed, respectively for the damage model, for the mesocrack-friction plasticity and for the coupled model are given in Section 5. Some selected examples illustrating damage and friction induced non-linear stress–strain behaviour, together with hysteretic effects, are furthermore presented and commented in this section.

The crucial issue of the control of microcrack closure and opening phenomena is addressed through 2 Anisotropic damage and normal unilateral effect, 3 Mesocrack friction induced plasticity, 4 Damage and frictional sliding interaction: fully coupled model, where the stiffness recovery and friction enter into consideration. The central simplifying hypothesis, conveyed through the developments proposed, consists in reduction of any real microcrack-set configuration to an equivalent configuration of three mutually orthogonal systems of parallel cracks characterized by three eigenvectors νk (k=1, 2, 3) and three non-negative eigenvalues Dk of the second-order damage tensor D. In such a manner the damage-induced anisotropy is systematically limited to a form of orthotropy, see also [14].

The problem of transition from volume-distributed damage to surface-localized failure incipience has been amply debated at the end of eighties and the beginning of nineties, see for example [15], [16], [17]. The localization bifurcation in the presence of damage by microcracking inducing net anisotropy effects needs clearly a 3D treatment, two-dimensional projections misrepresenting mostly localization mechanisms (orientation and discontinuity mode). The computational procedure relative to 3D localization detection is given in Ref. [11], where some pertinent results obtained with the basic damage model are amply commented.

Section snippets

Anisotropic damage and normal unilateral effect

This section outlines the salient features of the anisotropic damage model by Dragon [11], Halm and Dragon [12], which forms the framework for further developments presented in 3 Mesocrack friction induced plasticity, 4 Damage and frictional sliding interaction: fully coupled model. An objective of the damage model summarized below is to describe – in a realistic manner applicable to structural calculus – the process of mesocrack-induced anisotropic degradation and relative behaviour of an

Mesocrack friction induced plasticity

The unilateral normal effect included in the model summarized in Section 2 allows a moduli recovery in the direction normal to the closed mesocracks. It fails to capture a shear moduli recovery in the direction parallel to the crack plane, resulting from some blocking of mesocrack lips displacement due to roughness and corresponding friction phenomena. Experimental data involving loading–unloading cycles for specimens subjected to torsion and hydrostatic compression for instance show hysteretic

Damage and frictional sliding interaction: fully coupled model

The model completed in Section 3 incorporating friction-induced blocking and sliding on equivalent mesocrack-sets is valid for a given (`freezed') damage state or for conservative damage evolution (Dk-proportional loading paths). It has proved conclusive in representing multistage loading and unloading dissipative cycles due to blocking and sliding sequences, see [13] for illustrations. In particular a dissipative unloading blocking and sliding sequence could be obtained while for the same

Integration of damage and sliding constitutive relations

This subsection provides an outline of several computational aspects involved in the finite-element implementation of the model presented above. The incremental weak form of the equilibrium equation is formulated for a body Ω  R3 with boundary Ω in the time interval tI=[O,T]. Let fd be the given body forces per unit volume, ud the displacements imposed on the part 1Ω of Ω and Fd the traction vectors prescribed on the complementary part 2Ω. With the time partition : I=Ur=1Ntr,tr+1, the weak

Conclusion

Two dissipative mechanisms, namely damage by microcracking and frictional sliding over internal crack surfaces, are considered in the framework of the three-dimensional model presented, applicable for quasi-brittle rock-like solids. The internal variable formalism employed for the joint process under consideration is based on second-order tensorial representation of damage and sliding effects respectively. The critical issue of the control of microcrack closure and opening phenomena is dealt

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