We examine the influence of the grain shape on the effective elastic moduli of polycrystalline materials. For that purpose the real material is simulated by a cluster of Wigner-Seitz cells. For clarity each aggregate consists of grains with only one type of shape. Therefore we can classify each cluster by the coordination number of its grains. To determine the elastic moduli a homogeneous deformation is subjected to the surface of the cluster. The solution of this boundary value problem yields the average stress and strain governing inside the material whose interconnection by Hooke's law leads to the sought-for effective constants.The most important result is that with increasing coordination number the elastic moduli decrease.
We determine nonlinear effective elastic constants of polycrystalline materials within the framework of a recently developed method by Kiewel and Fritsche. This method is based on simulating the real system by a cluster of 100–500 crystallites. For simplicity we confine ourselves to macroscopically isotropic materials. The grains are assumed to have the shape of Wigner–Seitz cells of a body-centered-cubic lattice. This type of cluster is topologically similar to the microstructure of a real polycrystal. Therefore, it may be expected to reliably describe the nonlinear effective moduli of the respective materials.
The elastic properties of copper metal with different individual grain orientations exhibiting the same texture are determined. We simulate the real material by two different types of clusters. The first one consists of 365 cubic grains, the second cluster is an arrangement of 181 Wigner-Seitz cells of a body centred cubic (bcc)-lattice. For each type of cluster we let the local grain orientations vary. The displacement field inside these aggregates as a result of a homogeneous deformation acting on the surface of the clusters is calculated. Although the resulting local deformation field for different individual grain orientations varies strongly, the macroscopic elastic moduli are in the frame of this simulation identical for any cluster of the same type, as it has to be for statistically equivalent materials.
The discontinuities of the elastic constants across grain boundaries in polycrystalline aggregates can be considered as sources of "compatibility" strains in the adjacent crystals under the action of an external stress. This effect was studied theoretically using the cluster model of Kiewel and Fritsche. In order to distinguish the contributions of individual grains, the orientation of only one grain was changed whereas all others in the cluster were kept constant. The orientation dependent part of the strain field thus allows to estimate the range over which the influence of an individual grain reaches. The results shows that this part decreases faster than 1/r with the distance from the centre of the considered grain. It has virtually vanished at two or three times the grain diameter.The magnitude of the compatibility strains depends on grain shape. It was found higher in a cluster of cubic grains than in clusters consisting of dodecahedral or cubo-octahedral grains.The compatibility strains vary systematically when the orientation of the considered grain is changed continuously. The influence of even small misorientations < 5 is distinctly visible.The results show that grain shape and orientation correlation must be taken into consideration additional to the texture in order to obtain narrower bounds for the effective elastic constants than those by Voigt and Reuss. They show, however, also that orientation correlations of higher than next-nearest neighbours have virtually no influence.
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