Albrecht Bertram

Elasticity and Plasticity of Large Deformations

Interfacial Fluid Mechanics: A Mathematical Modeling Approach provides an introduction to mathematical models of viscous flow used in rapidly developing fields of microfluidics and microscale heat transfer. The basic physical effects are first introduced in the context of simple configurations and their relative importance in typical microscale applications is discussed. Then, several configurations of importance to microfluidics, most notably thin films/droplets on substrates and confined bubbles, are discussed in detail. Topics from current research on electrokinetic phenomena, liquid flow near structured solid surfaces,evaporation/condensation, and surfactant phenomena are discussed in the later chapters.

An alternative approach to finite plasticity based on material isomorphisms

Abstract A framework for material models describing finite plastic deformations is established by the assumption of isomorphic elastic ranges. The concepts of decomposition into elastic and plastic deformations is not needed, neither intermediate configurations. A comparison with other approaches is given and shows their range of validity.

On frame-indifference and form-invariance in constitutive theory

SummaryThe purpose of this paper is to present an alternative formulation of the principles of Euclidean frame-indifference and indifference with respect to superimposed rigid body motions, i.e., rotations and translations. This is accomplished on the basis of the action of the proper Euclidean group on constitutive relationsinduced by the action of this group on the Euclidean tensors appearing in these relations. The resulting formulation of these concepts can then be used to show that the usual concept of material frame-indifference actually consists of twoindependent concepts, i.e., Euclidean frame-indifference andform-invariance, the latter being generally overlooked as an independent concept. On this basis, one can in addition show that any two of the concepts of Euclidean frame-indifference, form-invariance, and indifference with respect to superimposed rigid body motions, automatically implies the third. As an application of this formulation, we discuss the constitutive relations for a simple (elastic) material and a kinetic gas. In this context, it follows straighforwardly that the latter satisfy Euclidean frame-indifference, as shown by Murdoch [1], but not indifference with respect to superimposed rigid body motion, as shown by Müller [2]. As such, the current formation yields immediately that these are not form-invariant, and so not material frame-indifferent.

The evolution of Hooke's law due to texture development in FCC polycrystals

Initially isotropic aggregates of crystalline grains show a texture-induced anisotropy of both their inelastic and elastic behavior when submitted to large inelastic deformations. The latter, however, is normally neglected, although experiments as well as numerical simulations clearly show a strong alteration of the elastic properties for certain materials. The main purpose of the work is to formulate a phenomenological model for the evolution of the elastic properties of cubic crystal aggregates. The effective elastic properties are determined by orientation averages of the local elasticity tensors. Arithmetic, geometric, and harmonic averages are compared. It can be shown that for cubic crystal aggregates all of these averages depend on the same irreducible fourth-order tensor, which represents the purely anisotropic portion of the effective elasticity tensor. Coupled equations for the flow rule and the evolution of the anisotropic part of the elasticity tensor are formulated. The flow rule is based on an anisotropic norm of the stress deviator defined by means of the elastic anisotropy. In the evolution equation for the anisotropic part of the elasticity tensor the direction of the rate of change depends only on the inelastic rate of deformation. The evolution equation is derived according to the theory of isotropic tensor functions. The transition from an elastically isotropic initial state to a (path-dependent) final anisotropic state is discussed for polycrystalline copper. The predictions of the model are compared with micro-macro simulations based on the Taylor Lin model and experimental data.

Anisotropic continuum damage modeling for single crystals at high temperatures

Abstract In single crystals, the process of creep damage is generally anisotropic. Indeed, the damage evolution does not only depend on the loading conditions, but also on the lattice orientation. And the current state of damage has an anisotropic influence on the effective stress state, so that it is represented by a tensorial damage variable. Based on the continuum damage mechanics theory, a creep damage model for F.C.C. single crystals has been developed and implemented in a three-dimensional anisotropic creep model. It is shown that the resulting material model is capable of describing the orientation dependence of the creep and damage evolution of nickel-based superalloys in the high temperature regime.

Finite thermoplasticity based on isomorphisms

In real processes involving large plastic deformations,a major part of the stress power will be dissipated and induces thermodynamical changes that cannot be neglected in many cases. A framework for the description of elasto-plastic materials based on the notion of isomorphic thermoelastic ranges is developed. Within this framework,general forms of the consistent flow and hardening rules as well as necessary and sufficient conditions for the second law to hold are derived for the rate-independent case. Within this frame,the concept of (unloaded) intermediate configurations is critically reviewed. # 2003 Elsevier Ltd. All rights reserved.

Finite element simulation of metal forming operations with texture based material models

In this paper two different texture-dependent material models based on the Taylor assumption are discussed and applied to the simulation of deep drawing operations of aluminium. From the numerical point of view, large-scale FE computations based on the Taylor model are very time-intensive and storage-consuming if the crystallographic texture is approximated by several hundred discrete crystals. Furthermore, the Taylor model in its standard form, which is based on discrete crystal orientations, has the disadvantage that the anisotropy is significantly overestimated if only a small number of crystal orientations are used. We quantitatively analyse this overestimation of anisotropy and suggest two Taylor-type models which allow us to reduce the sharpness of the crystallite orientation distribution function related to single crystals or texture components. One model is an elastic-viscoplastic Taylor model based on discrete orientations. The sharpness is reduced by modelling the isotropic background texture by an isotropic material law. The other model is a rigid-viscoplastic material one, which is based on continuous model functions on the orientation space. This model allows for a direct incorporation of the scattering around an ideal texture component since the model contains the half-width as a microstructural parameter which can be biased. These models are used to compute yield stresses, R values and earing profiles. The predictions are compared with experimental data.

Modeling of deformation induced anisotropy in free-end torsion

Abstract The main purpose of this work is to develop a phenomenological model, which accounts for the evolution of the elastic and plastic properties of fcc polycrystals due to a crystallographic texture development and predicts the axial effects in torsion experiments. The anisotropic portion of the effective elasticity tensor is modeled by a growth law. The flow rule depends on the anisotropic part of the elasticity tensor. The normalized anisotropic part of the effective elasticity tensor is equal to the 4th-order coefficient of a tensorial Fourier expansion of the crystal orientation distribution function. Hence, the evolution of elastic and viscoplastic properties is modeled by an evolution equation for the 4th-order moment tensor of the orientation distribution function of an aggregate of cubic crystals. It is shown that the model is able to predict the plastic anisotropy that leads to the monotonic and cyclic Swift effect. The predictions are compared to those of the Taylor–Lin polycrystal model and to experimental data. In contrast to other phenomenological models proposed in the literature, the present model predicts the axial effects even if the initial state of the material is isotropic.

Endochronic viscoplastic material models for filled PTFE

To describe the nonlinear material behaviour of polytetrafluoroethylene (PTFE), a viscoplastic material model of the overstress type is proposed based on experimental data. The approach starts with a rheological model without an elastic range, using a rate-independent elastoplastic model with an endochronic flow rule and a nonlinear elastic law in parallel connection with a nonlinear Maxwell model. The generalization to three dimensions is possible for both small and finite strains under the assumption that changes of the density are purely elastic. Numerical simulations show that the model can describe the short and long time material behavior of filled PTFE.

Damage modeling of the single crystal superalloy SRR99 under monotonous creep

The evolution of material damage of single crystal superalloys depends not only on the load conditions, but also strongly on the lattice orientation. Using the theory of continuum damage mechanics, a phenomenological creep damage model for cubic single crystal superalloys is derived. In this model, a symmetric second-order damage tensor is used to describe the anisotropic nature of damage. The damage deactivation and reactivation is represented by an active damage tensor. As the effects of the current state of damage on the deformation process and on the damage development are different in nature, separate effective stresses for creep and damage are defined. Furthermore, a mapped damage active stress is introduced to reflect the influence of material symmetry on the damage growth rate by using an orientation function. Based on microscopic observations, it is assumed that only the principle tensile damage active stresses are responsible for the damage development, and that the anisotropy of the damage evolution mainly depends on the principle directions of the damage active stress and the material symmetry. Within the framework of thermodynamics, the damage evolution law is constructed by considering both the initial material anisotropy and the anisotropic influence of the current state of damage. Based on the effective stress concept, this damage model has been implemented into a three-dimensional anisotropic viscoplastic model and applied to the simulation of the creep damage behavior of the single crystal superalloy SRR99 at 760°C.