Computer Methods in Applied Mechanics and Engineering
Apparent and effective physical properties of heterogeneous materials: Representativity of samples of two materials from food industry
Introduction
The effective physical properties of heterogeneous materials strongly depend on the morphology of phase distribution in space [20], [40]. Such microstructural effects are more important when the contrast in the properties of the constituent is high. Much can be done analytically, including bounding and estimation of overall properties, when the composite is made up of inclusions having simple geometries that are embedded in a matrix and as long as the contrast in phase properties is not too high [4], [15], [39], [33]. For higher contrasts, for example in the case of porous materials, the actual morphology of the microstructure plays a dominant role in the final effective property. Modern experimental techniques make it possible to obtain realistic 3D representations of materials microstructures. They include 2D images acquired using electron or light microscopy, 3D images obtained using X-ray microtomography [36], serial sectioning, confocal imaging, or magnetic resonance imaging. These techniques provide the opportunity of directly measuring the complex morphology of the composite materials in 3D at resolutions down to a few microns.
There now exist large-scale computational methods for calculating the properties of heterogeneous materials given a digital representation of their morphology [1], [2], [14], [30], [29]. 3D models have also been directly reconstructed from samples by combining digitized serial sections obtained by scanning electron microscopy [23], or using the technique of X-ray microtomography [12], [37] and laser confocal microscopy [13]. The number of contributions providing direct links between 3D images and finite element computations remains small but is increasing [26]. This is part of the current effort to develop microstructural mechanics with a view to optimizing microstructures for wanted properties [41], [7], [42]. The determination of apparent properties goes through the resolution of boundary value problems on samples of microstructures. The resulting apparent properties depend in fact on the choice of boundary conditions used to impose mean strain, stress, thermal gradient or flux. Three types of boundary conditions are classically used in computational homogenization: kinematic uniform (KUBC), stress uniform (SUBC) and periodic boundary conditions. Periodic boundary conditions have been shown in [21] in the case of a two-phase elastic material modeled as a Voronoi mosaics, to provide correct estimations of the effective properties for smaller volumes than KU and SU boundary conditions.
The first objective of the present work is to propose a computational strategy to estimate the RVE size in the case of real two-phase heterogeneous materials. Contrary to the case of random models of microstructures, the number and size of images of the microstructure of an actual engineering material are often limited. This raises additional difficulties to assess the representativity of the samples and to estimate the physical properties, compared to the approach developed in [21]. Two materials made with the same constituents but having significantly different morphologies are studied in this work. Both morphologies exhibit an interconnected character for both phases. The second objective of this work is then to point out a morphological and/or mechanical parameter that makes it possible to distinguish different interconnected microstructures. The proposed parameter is called here the percolation ratio. It will appear that RVE size and percolation ratio are correlated parameters.
In the present work, 3D confocal images of two materials from food industry are used to predict their effective physical properties, namely linear elasticity and thermal conductivity. The two materials, labelled A and B are made of two phases, namely a hard phase, polycrystalline ice, labelled 1, and a soft and less conductive phase, fat product close to cream, labelled 2. Materials from food industry have gained considerable interest from materials science and mechanical engineering community because of the strong links between physical, mechanical and sensorial properties [19]. Although computational homogenization methods were already applied to polycrystalline ice [10], mechanical analyses of ice creams based on actual morphology of microstructures are not reported in literature [6]. The contrasts in the Young’s modulus and thermal conductivity between both phases are respectively 1000 and 100. The volume fraction of phase 1 in materials A and B is similar, and close to 70%. Material processing techniques differ for both materials, resulting in strongly different microstructures, a rather fine microstructure for material A, and a coarser one for material B. Special care was taken during material processing in order to avoid the presence of porosities in the mixtures. Isolated pores may exist in the sample but they are not taken into account in the proposed simulations. Experimental tests carried out on both materials show that the overall properties of A and B differ significantly: Young’s modulus is found to be twice as high for A as for B. In the present work, the effective elastic and thermal properties are estimated from the available 3D confocal images using computational homogenization techniques. In particular, we try to find out the differences in the morphology of the microstructures that can explain the strongly different elastic behaviour. The attention is drawn on the percolation behaviour of the hard phase inside the mixture. To quantify the precision of the found numerical estimates, the question of the representativity of the finite size samples must be raised and investigated in detail.
The discussion on the estimation of RVE sizes relies on a numerical and statistical approach proposed in [21]. The effective physical properties of random heterogeneous materials can be determined not only by numerical simulations on large volume elements of composite, but also as mean values of apparent properties of rather small volumes, provided that a sufficient number of realizations of the microstructure is considered. The size VRVE must be considered as a function of several parameters: the physical property, the contrast of properties c, the volume fraction of the constituents, the wanted relative precision ϵrel for the estimation of the effective property and the number n of realizations of the microstructure associated with computations that one is able to carry out, and of course on the morphology of spatial phase distribution. The size of RVE was related in [21] to the notion of integral range, denoted A3 which depends on the specific morphological or physical property. The integral range is directly related to the scatter in apparent properties found on volumes of fixed size but containing different realizations of the microstructure of a random material. In most cases, it can be determined only numerically for example by finite element simulations, as done in [21] in the case of a random material model, namely Voronoi mosaics. The integral range is estimated numerically in the present work for the real materials A and B. For that purpose, finite element computations on volumes with increasing sizes extracted from the available samples are performed to determine apparent elastic and thermal properties and their dispersion as a function of domain size. For sufficiently large domains, the mean properties converge towards a single value that is regarded as the effective one.
Section 2 is devoted to the description of the images of the microstructure of materials A and B. The experimental physical properties found for the constituents of the materials and for the materials A and B themselves are presented. Volume fraction and covariance ranges are given for all samples. The computational methods for meshing microstructures and the type of boundary conditions are presented in Section 3. Direct simulations of the elastic and thermal apparent properties of all available samples are provided in Section 4 using KU and SU boundary conditions. The question of the proper RVE size for both materials is discussed in Section 5. Section 6 aims at providing morphological and mechanical arguments justifying why microstructure A leads to stiffer elastic properties and to a more conductive material than microstructure B. The key notion explored is that of percolation of the hard phase within the mixture. Indicators of geometrical and mechanical percolation are defined and estimated using 3D image analysis.
Section snippets
Microstructure and properties of the materials
The two investigated materials A and B are made of two phases, labelled 1 (polycrystalline phase, ice) and 2 (fat polymer phase, cream). Experimental batches of the studied composite materials are produced in blocks of 500 g which can be used for four-point bending tests and confocal imaging. Three samples of each material (SA1, SA2, SA3) and (SB1, SB2, SB3) are studied in the present work. Confocal images of samples SA3 and SB1 are shown in Fig. 1. The material A contains elongated crystals of
Field and constitutive equations
The field equations to be solved numerically in the present work concern the linearized theory of elastic solids, on the one hand, and that of heat transfer, on the other hand. The associated governing equations for a body V are the balance of momentum, on the one hand, and the heat equation on the other hand. They must be fulfilled at any regular material point x ∈ Vin a Cartesian orthonormal coordinate system. The comma ,i denotes partial derivation with respect to the coordinate x
Direct estimation of elastic and thermal properties of sample SA1 to SB3
The finite element meshes designed in Section 3.3 are used to compute elastic and thermal apparent properties of the samples SA1, SA2, SA3, SB1, SB2, SB3 introduced in Section 2.
A matrix notation is used to represent the tensor of elastic moduli
The 6 × 6 matrix of elastic moduli is symmetrical. The apparent elastic properties depend
Effective properties and RVE sizes
The overall physical properties are studied in this part for a large range of volume sizes V and a large number n of small volumes taken out of the whole real specimens of microstructures SAi and SBj. Such volumes can be regular subvolumes of the sample as shown in Section 4.2. Smaller volumes were also extracted randomly from the sample and do not represent a partition of the original sample. To some extent, such random volumes represent different realizations of the studied random material.
Strain localization and percolation phenomena
The objective of this part is to understand from the morphology of the microstructures and the local fields in deformed samples, why the samples of material A have been found to be significantly stiffer than the material B. The previous sections have drawn the attention on the fact that the covariance and integral ranges are larger for material B than for material A. This features explain why the size of the RVE for a given precision will be larger for B than for A. But it does not give any
Conclusions
Confocal images of two materials having the same volume fraction of hard phase 1 but different morphologies, have been used to predict their effective elastic and thermal properties. Direct finite element simulations on the six available samples SAi and SBi show that the apparent properties obtained using KUBC and SUBC boundary conditions are significantly different. The results obtained with SUBC are closer to the experiment. This discrepancy is due to the finite size of the samples, and
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