Rainer Glüge

Finite element simulation of texture evolution and Swift effect in NiAl under torsion

The texture evolution and the Swift effect in NiAl under torsion at 727 ◦ C are studied by finite element simulations for two different initial textures. The material behaviour is modelled by an elastic-viscoplastic Taylor model. In order to overcome the well-known shortcomings of Taylor’s approach, the texture evolution is also investigated by a representative volume element (RVE) with periodic boundary conditions and a compatible microstructure at the opposite faces of the RVE. Such a representative volume element takes into account the grain morphology and the grain interaction. The numerical results are compared with experimental data. It is shown that the modelling of a finite element based RVE leads to a better prediction of the final textures. However, the texture evolution path is not accounted for correctly. The simulated Swift effect depends much more on the initial orientation distribution than observed in experiment. Deviations between simulation and experiment may be due to continuous dynamic recrystallization.

The Eigenmodes in Isotropic Strain Gradient Elasticity

We present the spectral decomposition of the isotropic stiffness hexadic that appears in Mindlin’s strain gradient elasticity, where the kinematic variable is the second gradient of the displacement field. It turns out that four distinct eigenmodes appear, two of which are universal for all isotropic strain gradient materials, and two depend on an additional material parameter. With the aid of the harmonic decomposition, general interpretations of the eigenmodes can be given. Further, the material parameters are related to commonly employed special cases, namely the cases tabulated in Neff et al. (Int J Solids Struct 46(25–26):4261–4276, 2009) and isotropic gradient elasticity of Helmholtz type.

BANDWIDTH REDUCTION ON SPARSE MATRICES BY INTRODUCING NEW VARIABLES

A sparse matrix bandwidth reduction method is analyzed. It consists of equation splitting, substitution and introducing new variables, similar to the substructure decomposition in the finite element method (FEM). It is especially useful when the bandwidth cannot be reduced by strategically interchanging columns and rows. In such cases, equation splitting and successive reordering can further reduce the bandwidth, at cost of introducing new variables. While the substructure decomposition is carried out before the system matrix is built, the given approach is applied afterwards, independently on the origin of the linear system. It is successfully applied to a sparse matrix, the bandwidth of which cannot be reduced by reordering. For the exemplary FEM simulation, an increase of performance of the direct solver is obtaine.

Does convexity of yield surfaces in plasticity have a physical significance

Convexity of a function or set is an often needed and important mathematical property. In the case of yield functions ϕ ( T ) (or elastic ranges) in terms of stresses, almost all empirical and mechanism-based yield functions have this property. However, requiring positive plastic dissipation does not necessarily exclude non-convex yield functions, which is confirmed by the fact that non-convex yield functions are observed experimentally, although this rarely happens. We therefore ask whether this nice mathematical property reflects a physical material property. This is investigated in an elastic–plastic, small strain, 2D setting. It appears that, at least in this setting, no specific material property can be attributed to the convexity of the yield function.

ONE-DIMENSIONAL MATERIAL THEORY

We consider at first a uniaxial tensile or compression test. For such a test, one often uses samples with rotational symmetry, which are clamped at the (thicker) ends, and have a middle region that is long enough to apply the measuring device so that we can assume a uniaxial state of stress. A cylindrical form shall keep the inhomogeneities as low as possible so that we can assume a homogeneous state of stresses and strains.

VECTOR AND TENSOR ANALYSIS

Let f be a real-valued differentiable function of a real variable. Its linear approximation at x is the differential.

INTRODUCTION TO TENSOR CALCULUS

In the first part we have considered one-dimensional models to describe material behavior. The application of such models, however, is rather limited, since in the general case, both the stress and the strain state are three-dimensional. For the generalization of such concepts to higher dimensions than one, an appropriate mathematical instrumentarium is needed. In mechanics, the appropriate objects for the mathematical description are vectors and tensors. For readers not familiar with such objects, we will next give a brief introduction into tensor algebra and analysis.

FOUNDATIONS OF CONTINUUM MECHANICS

In Continuum Mechanics we consider material bodies which continuously occupy a region in the three-dimensional EUKLIDean space. On its boundary and in its interior, one defines physical quantities as fields. Its points are called material points.