Exact solutions for an elastic damageable hollow sphere subjected to isotropic mechanical loadings

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Highlights

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    Original exact solutions are derived for mechanical fields in a hollow sphere with an elastic damage matrix.

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    These solutions allow us to discuss the relevance of the considered models.

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    Unloading regime is also fully investigated.

Abstract

In this paper, we first recall some available Eshelby-based homogenization schemes applied to microcracked materials. An emphasis is put on models accounting for interacting opened or closed microcracks and their spatial distribution. On the basis of these schemes, we briefly present a class of isotropic damage models. The main part of the study is devoted to the derivation of exact solutions for mechanical fields (damage distribution, displacement, stress fields) in a hollow sphere subjected to a radial loading and made up of an elastic damageable material. The solutions, discussed in link with the different homogenization schemes, may serve as a reference for assessment of numerical predictions of brittle damage models. Interestingly, it is shown that the existence of physically meaningful solutions strongly depends on the model under consideration. Finally, we establish explicit solution to the hollow sphere problem in unloading regime.

Introduction

Nonlinear behavior of quasi-brittle materials such as concrete or rocks is mainly attributed to nucleation and growth of microcracks under mechanical loading (see for example [1]). Modeling of the resulting deterioration phenomena can be suitably performed in the framework of Continuum Damage Mechanics (CDM). This can be done by means of a purely macroscopic approach to which an important literature has been devoted (e.g. [2], [3], [4], [5], [6], [7], [8], [9], [10], etc.).1

A complementary alternative consists in micromechanical studies which aim at deriving the effective behavior and damage of brittle materials based on statistical informations on the microcracks system (e.g. [14], [15], [16], [17], [18], [19]). Most of these models issued from the above studies are based on elementary solutions in Linear Elastic Fracture Mechanics (LEFM) and have significantly contributed to the physical understanding of damage mechanisms: in particular, the damage variable to be used at the macroscopic scale is clearly identified (microcracks density parameter). A series of two review papers by Kachanov [20], [21] can be notably recommended concerning the LEFM-based micromechanics of microcracked media. Mention has to be made of contributions in this field when the solid matrix displays structural anisotropy (see for instance [22], [23], [24], [25], etc.). Other recent developments, in Eshelby-based micromechanics, include studies accounting notably for spatial distribution of microcracks [26], [27], [28], or for poromechanical coupling [29], [30].

From a structural point of view, the difficulty to deal with damage-induced softening regimes in numerical computation is still well recognized as an crucial question which may still deserve significant research effort. This difficulty remains to be a scientific challenge in so far as it appears as a possible limitation for the transfer of damage models in engineering practice. In this respect, it seems highly desirable to rely upon closed form solutions of simple structural problems, to be used as academical benchmark for numerical solutions. This is the main purpose of the present paper which is organized as follows.

First, various homogenization schemes are described in order to derive the effective stiffness of microcracked materials. A standard thermodynamic reasoning allows us to adapt the concept of energy release rate to the damage evolution problem. In practice, implementation of the damage models involves the derivative of the effective stiffness with respect to the damage variable which is provided by the micromechanical analysis. In view of application to the considered structural problem, only isotropic properties are addressed.

The main part of the study deals with the response of a hollow sphere made up of an elastic damage material, and subjected to traction on the external boundary. The model based on the dilute scheme is first implemented and discussed. Thereafter, a more general damage law including the results of both Mori–Tanaka [31] and Ponte-Castaneda and Willis schemes is adopted. The necessary conditions for obtaining a physically meaningful response are identified. When they are satisfied, it is shown that there exists a maximum admissible loading. Beyond this threshold, the softening part of the response is fully described at both the microscopic and macroscopic levels. Unloading phases are also examined.

Section snippets

Brief recall of some available estimates for the effective stiffness of microcraked media

We hereafter propose a short review of the basic ideas of the homogenization theory in view of its application to microcracked materials. The interested reader is referred for instance to [32], [29], [30], [33] or [18].

Basic equations of the considered class of isotropic damage models

In Section 2, were briefly recalled the predictions of various homogenization schemes devoted to the influence of microcrack-induced damage on the overall elasticity Chom(d), with the variable d being a formal descriptor of the current damage state. Indeed, we will only consider the particular microcrack distributions for which the homogenized stiffness depends on a single parameter. For instance, this is the case for randomly oriented microcracks with the identical radius a, the microcracks

General features

We investigate now the response of a hollow sphere having an elastic damageable matrix and subjected to a uniform traction T on its external boundary. The internal (resp. external) radius is denoted by ao (resp. b). The boundary conditions readr=ao:σrr=0r=b:σrr=TThe local constitutive behavior of the solid at the so-called mesoscopic scale6

Conclusion

As a starting point, the paper first provides a brief recall of homogenization schemes applied to isotropic elastic materials weakened by opened or closed microcracks. Implementation of these schemes in the context of evolving damage is then described. The main purpose of the study was to established exact solutions of the mechanical fields in a hollow sphere having an elastic damageable matrix and subjected to traction at external boundary. In the case of the damage model based on the dilute

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