FE2 computational homogenization for the thermo-mechanical analysis of heterogeneous solids

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Abstract

This paper presents a two-scale thermo-mechanical analysis framework for heterogeneous solids based on a computational homogenization technique. The evolution of the mechanical and thermal fields at the macroscopic level is resolved through the incorporation of the microstructural response. Within the proposed multi-scale approach, the temperature dependent non-linear thermo-mechanical response is accounted by solving a boundary value problem at the micro-scale, the results of which are properly averaged and transferred to the macro level in a consistent way. The framework does not require explicitly determined homogenized material properties (e.g. macroscopic thermal expansion coefficients) since no constitutive equations are required for the macroscopic stresses and heat fluxes at the macro level. A nested finite element solution procedure with an operator-split implementation is outlined and the effectiveness of the approach is demonstrated by illustrative key examples.

Introduction

In a wide spectrum of engineering structures and assemblies, premature failure of components is frequently due to severe thermal loading conditions in the form of rapid temperature changes (thermal shocks) and temperature cycles. Variations of the operating temperature of a material is mostly accompanied by physical and geometrical changes at some scale. In heterogeneous systems, local thermal expansion mismatches and thermal anisotropy of different constituents naturally triggers the appearance of internal stresses. Under highly transient external thermal loading conditions, the resulting heterogeneous temperature distribution may lead to a complicated mechanical response and a non-uniform mechanical and physical property degradation accompanied by irreversible geometrical changes. The altered distribution of mechanical properties dictates the macroscopic response as the external mechanical loading is further varied as well. Therefore, a strong coupling between the evolving microstructure and the macroscopic response arises. Moreover, microstructural configurational changes may trigger a significant interaction between the mechanical and thermal fields, for instance, in the form of a reduced heat transfer across a damaged interface. Due to the aforementioned complications, the thermo-mechanical analysis of heterogeneous material systems constitutes a challenging task.

Numerous homogenization techniques have been developed to predict the effective mechanical and thermophysical properties of materials with complex microstructures. Early research work [1], [2] resulted in bounds for the effective material properties which are particularly suitable for relatively simple geometries and a restricted class of constitutive models for the phases. More general asymptotic homogenization approaches were exploited for the determination of the mechanical and thermal constitutive tensors of composites with a periodic microstructure. Starting with the work of Guedes and Kikuchi [3], the use of computational techniques within the homogenization theory has received considerable attention including applications to other field problems [4], [5], [6], [7]. Recently the focus has shifted to extend the theory and solution algorithms to the non-linear and inelastic range [8], [9], [10]. A sub-class of computational homogenization techniques which addresses both the influence of the microstructure and the coupling with the resulting macroscopic response is presented in [11], [12], [13], [14], [15]. In this multi-scale approach, the macroscopic material response is obtained from the underlying microstructure by solving a boundary value problem defined on a representative volume element (RVE) of the material. The detailed treatment with underlying principles is described in [16]. At present, this method has been applied successfully for (I) stress and structural damage analysis [13], [15], [17], (II) the mechanical analysis of structured thin sheets (shells) [18], (III) failure analysis of cohesive interfaces [19] and (IV) heat conduction analysis in heterogenous solids [20]. In the present paper, the multi-scale framework is extended to the fully coupled thermo-mechanical analysis of heterogeneous material systems including an appropriate solution algorithm.

The interaction of the thermal and mechanical fields within a multi-scale modeling framework requires comprehensive treatment and depends on the characteristic micro-failure mechanisms. For typical high temperature resistant materials (e.g. refractory ceramics), micro-failure mechanisms such as debonding of the grain–grain interfaces, evolve at an apparent microstructural level, which motivates a two-level treatment as adopted in the following sections. Furthermore, damage and failure at the micro level takes place without significant inelasticity (e.g. plasticity), which implies that the effect of mechanical energy dissipation on the thermal field is negligible and therefore not taken into account in the analysis. Similarly, the reduction of the heat flow as a result of mechanical damage is not explicitly considered although the constructed framework can easily accommodate such coupling effects. Before proceeding further, the essential points on which the proposed framework differs from the existing approaches can be summarized as:

  • The proposed approach does not require macroscopic material properties such as the homogenized coefficient of thermal expansion as opposed to other alternative homogenization schemes.

  • The framework has the flexibility to include a non-linear and temperature dependent thermo-mechanical material response at the microstructural level, which is transmitted to the macro level in a consistent way. Furthermore, for different combinations and types of constitutive equations at the micro level, re-derivation of certain effective quantities (expressions) is not necessary.

  • The presented solution algorithm resolves the interaction of the fields in a proper way and can accommodate thermo-mechanically induced coupling mechanisms accompanying microstructural evolution.

The presentation of the paper is organized as follows. After the introduction of the assumptions in the next section, the thermo-mechanical analysis problems at micro and macro levels are formulated in Section 3. Then, the scale transition structure is summarized in Section 4. Thereupon, a two-scale solution strategy is presented which leads to an operator-split nested finite element solution algorithm further detailed in Section 5. Illustrative examples are presented in Section 6, followed by the discussion and conclusion section.

Section snippets

Preliminaries

In this paper, a first order theory is adopted for both the mechanical and thermal homogenization procedure which hinges on the principle of scale separation [13], [15], [20]. A first order theory for mechanical homogenization implies that the macroscopic deformation gradient varies mildly and therefore deformation localization (softening) is excluded from the considerations. However, for typical high temperature resistant structures, the onset of softening is practically the end of service

Micro-scale problem

The evolution of thermal and mechanical fields at the micro level is defined and monitored on a representative volume element (RVE) provided with the essential physical and geometrical information about the microstructural components. Particularly for materials with random microstructures, the choice of the RVE is a delicate task. The difficulty arises due to the fact that RVE should be statistically representative of all microstructural heterogeneities and at the same time remain small enough

The macro–micro scale transition

Within the framework of a first order multi-scale analysis, the actual microscopic displacement and temperature fields at a location x in the current configuration can be decomposed without loss of generality asum(x)=(FM-I)·(X-X1)+uf(x),θm(x)=θmr+MθM·(x-x1)+θf(x)in a macroscopic contribution and a fluctuation field (subscript ‘f’) that represents the fine scale deviations with respect to the average fields as a result of the variations in material properties within the RVE. In (5a)

Two-scale numerical solution framework

Since analytical solutions are limited to relatively simple geometries and constitutive relations, a general approximate solution procedure is pursued on the basis of the finite element method at both scales (FE2). Both mechanical and thermal boundary conditions are parameterized in a (pseudo-)time setting and applied incrementally. Furthermore, in case of transient thermal problems, a proper numerical time integration scheme is introduced to convert the spatially discretized rate equations

Two-scale homogenization examples

The proposed two-scale thermo-mechanical framework has been implemented in a commercial FE environment (MSC MARC), and is next demonstrated by two example problems. The selected problems are 2-D, preferred due to computational cost reasons, although the presented framework is applicable to 3-D cases as well.

Summary and conclusion

Motivated by the results obtained in [13], [15], [20], a two-scale analysis framework for the thermo-mechanical analysis of heterogeneous solids has been presented. Thermo-mechanical approaches at both scales are treated consistently and linked by a rigorous scale bridging procedure. Using an extended computational homogenization framework, macroscopic thermal and mechanical excitations are passed to the micro level through appropriate RVE boundary conditions. The resulting microscopic response

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