Computer Methods in Applied Mechanics and Engineering
Homogenization-based multiscale crack modelling: From micro-diffusive damage to macro-cracks
Introduction
The majority of natural and engineering materials are materials in which deformation and failure processes take place at multiple scales and are therefore called multiscale materials (by multiscale we mean multiple length scales). At the macroscopic level of observation, it is reasonable to consider materials as homogeneous, traditionally modelled by phenomenological constitutive laws. At lower observation levels (meso and/or micro) heterogeneities appear which are very troublesome to be taken into account in phenomenological constitutive models. There are three approaches by which heterogeneous materials can be modelled. The first approach is known as either direct numerical simulation (DNS) or brute-force fullscale simulation, in which the heterogeneities from the fine scale are explicitly modelled in the coarse scale model. Although this guarantees accuracy, the computational effort is impractical (at least for current computer technology) which limits the applicability of the DNS approach. The second approach is based on the homogenization concept and has emerged as a valuable tool to model heterogeneous materials in an efficient way. The third approach, known as the concurrent multiscale method, somehow resemble domain decomposition methods. For a detailed taxonomy of multiscale methods, refer to [1].
Homogenization-based multiscale modelling techniques can either be numerical homogenization [2] or computational homogenization [3], see [4] for a detailed discussion. In numerical homogenization schemes, a macroscopic canonical constitutive model e.g., a visco-plasticity model, is assumed with parameters determined by fitting the data produced by finite element (or any other numerical method) computations of a microscopic sample where all the microstructure is explicitly modelled. In the literature, those numerical homogenization techniques are known as unit cell methods. Due to the assumption of the form of the macroscopic constitutive law, the methods become less appropriate for highly nonlinear problems.
According to computational homogenization techniques, to every macroscopic material point there is an associated microscopic sample (with all relevant heterogeneities) which provides the macroscopic constitutive behavior. When implemented in a finite element (FE) framework, the method is known as an FE2 [5] scheme. Although the method is computationally expensive, it has been proved to be a valuable and flexible (due to the lack of an assumption on the macroscopic constitutive model) tool for analyzing a wide range of heterogeneous materials with complex microstructures with highly nonlinear behavior see, among others, [6], [7], [8], [9] for a recent review. An open source program for homogenization problems has recently been made available [10].
A concept of crucial importance in homogenization methods is the representative volume element (RVE). There is not a single and exact definition of the RVE for an arbitrary heterogeneous material. That might explain the existence of various definitions of the RVE, see [11] for a recent review. In this contribution, we consider a microscopic sample to be an RVE when (i) an increase in its size does not lead to considerable differences in the homogenized properties, (ii) the micro-sample size is large enough so that the homogenized properties are independent of the microstructural randomness and (iii) the RVE size should be small enough so that the separation-of-scales principle holds. An implicit assumption usually made in FE2 modelling is the existence of an RVE. This is a correct assumption in linear and hardening regimes [11]. However, this is no longer the case in softening regimes when a standard homogenization method is used [11].
In the past two years homogenization schemes for adhesive cracks (or material layers) [12], [13], [14], [15] and cohesive cracks [16], [17] have been developed in which a traction–separation law is obtained based upon finite element computations at the microscale wherein the complex microstructure of the material have been explicitly modelled. By doing homogenization for adhesive cracks, in [14] the existence of an RVE for softening materials (under tensile and mixed-mode loading) which exhibit diffusive damage has been reported. The same result has been recently presented in [15] for fibre-epoxy material that shows discrete cracking. Very recently, in [17] the authors have proved the existence of an RVE for softening materials, for both adhesive and cohesive cracks, by deriving a traction–separation law from the microscopic inelastic stresses and strains. However the method has been applied only to materials with a simple microstructure undergoing discrete cracking.
In [18], [1], the authors have developed the MAD (Multiscale Aggregating Discontinuities) method in which a macroscopic crack is determined as equivalent to a bunch of microscale cracks. The method is however restricted to cases where the separation-of-scales principle is violated for the micro-sample had to match the macro-element to which it is linked. Computational multiscale methods to model cracks wherein the microstructural length scale is of the same magnitude as the macroscopic length scale are also given in [19], [20].
In [21], the existence of an RVE for quasi-brittle materials (under tensile loading) with random complex heterogeneous microstructure exhibiting diffusive damage has been confirmed based upon a special averaging scheme, the failure zone averaging scheme, which filters out the linear contribution of the micro-sample.
In this manuscript departing from the result reported in [21], we are going to derive cohesive laws, which are objective with respect to micro-sample size, for softening materials with a random heterogeneous microstructure subjected to tensile, shear and mixed-mode loading. It is confirmed that an RVE does exist for softening materials with microstructures undergoing diffusive damage. We also present a computational framework to incorporate those objective cohesive laws in an iterative FE2 setting which is an extension of the approach given in [17] for micro-discrete cracking failure to micro-diffusive damage fracture. Although similar in some aspects, the major difference between our work and [17] is twofold. Firstly, the microstructure in this paper exhibits diffusive damage i.e., we homogenize a macroscopic cohesive law from a microscale localization band. Secondly, the representativeness of the macroscale cohesive law is obtained for materials with a random microstructure and under various loading conditions.
The structure of the paper is as follows. In Section 2, the investigated microstructures and the utilized continuum damage model are given. Section 3 presents the standard averaging scheme, the proposed failure zone averaging technique followed by a series of numerical simulations that confirm the existence of an RVE for softening materials. The next section, Section 4, describes the energetic equivalence theorems to link macroscopic and microscopic models. The multiscale algorithm for adhesive and cohesive cracks is given in Section 5 followed by three numerical examples given in Section 6.
Section snippets
Microstructures
In this paper, two microstructures are considered. The first one is a simple voided microstructure (radius of the void equals 5 mm) of which three samples with dimensions 20 × 20 mm2, 40 × 20 mm2 and 40 × 40 mm2 as shown in Fig. 1 will be studied. The second type of microstructure that is analyzed in this contribution is a random heterogeneous material which is a three-phase material with matrix, aggregates and an interfacial transion zone (ITZ) surrounding each aggregate, see Fig. 2. Two samples of
Objective macroscopic cohesive laws
Considering a macro-crack with the outward unit normal denoted by n and the unit tangent vector represented by s. A micro-sample Ωm, which is a rectangle of dimension w × h, wherein the underlying microstructure is explicitly modelled is associated to every integration point on the macro-crack. Let us consider the case in which n coincides with see Fig. 4 for the microscale orthogonal coordinate system. Since the deformation modes of the macro-crack include mode I opening and mode II
Energetic equivalence theorem
Fracture can occur inside the bulk of the material or along the interface between two different materials. For the former, the fracture surface is termed a cohesive crack whereas for the latter, the term adhesive (or interfacial) crack is used. For the case of cohesive cracks, homogenization is applied to a finite element model representing the bulk material around the crack. In the case of adhesive cracks, the adopted micro-model represents the material in the adhesive layer. This section
Computational multiscale crack modelling
In the previous sections it has been shown that traction–separation laws which are objective to the micro-model size are obtained for materials with simple underlying microstructure and complex random microstructure under various loading conditions including tensile, shear and mixed-mode loading. These cohesive laws can be utilized in two ways, either in sequential homogenization schemes or in semi-concurrent homogenization methods e.g., in a FE2 setting. According to the former, a micro-model
Adhesive crack
Fig. 22 gives a benchmark problem for verifying the proposed multiscale scheme. The aim is to check whether the macroscopic response is objective to the micro-sample size. The material parameters of the micro-model are given in Section 2.2. The macro-bulk material is an elastic material with Young’s modulus of 25,000 N/mm2 and Poisson’s ratio of zero. Note that in contrary to the cohesive crack, for an adhesive crack, the macroscopic bulk is not related to the microstructure of the adhesive
Conclusions
In this contribution macroscopic cohesive laws (both mode I and mode II), which are independent of the micro-sample size, were obtained for quasi-brittle materials with a random heterogeneous microstructure under various loading conditions for both cohesive and adhesive cracks. This was achieved by extracting only the active inelastic responses occuring in the micro-sample (rather than the whole responses in the micro-sample as in standard homogenization schemes) to determine the equivalent
Acknowledgements
The financial support from the Delft Center for Computational Science and Engineering (DCSE) is gratefully acknowledged. The first author gratefully appreciate the discussion with Dr. Clemens Verhoosel.
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