Advances in Extended and Multifield Theories for Continua

The purpose of this work is the formulation of models for selected extended or generalized continua with the help of continuum thermodynamic and rate variational methods. The current approach is based on energy balance, the dissipation principle, as well as frame-indifference (i. e., Euclidean and material). Energetics and kinetics are based on the free energy density and a dissipation potential, respectively. More specifically, attention is focused here on the class of generalized continua whose energetic behavior depends on (i) the first- and second-order gradients of the standard deformation field, (ii) a microstructure field and its gradient, (iii) local inelastic internal variables. This is sufficiently general to include well-known cases such as second-order Mindlin, director, micropolar (Cosserat), microstretch, or micromorphic, continua, as well as gradient inelasticity. Two types of models are identified depending on whether or not the microstructure field involved is modeled as spatial or non-spatial (e. g., intermediate or material) in nature. In particular, this constitutive assumption influences the form of the evolution-field relation for the microstructure field as well as its coupling to standard momentum balance. Given the resulting continuum thermodynamic model relations, the corresponding initial-boundary-value problem is then formulated in rate-variational form. This is based on bulk and surface rate potentials determining a rate functional whose stationarity conditions yield the corresponding evolution-field relations and flux boundary conditions of the model.

The mechanical behavior of cellular structures is dominated by the underlying micro-topology and therefore has to be denoted very complex. For that reason the question of the computational treatment of such structures is not completely cleared yet. One way to do computations for materials with underlying microstructure is to treat them as homogeneous bodies in the framework of extended continuum theories. These theories consist of an extended set of balance equations, extended kinematic strain measures and extended constitutive equations with additional material parameters. Mostly the physical interpretation of those additional material parameters is not demonstrated vividly. So it is very difficult to quantify them by parameter identification based on experimental reference data.

It will be shown how virtual reference data from microscopic computations can be used to determine the extended set of macroscopic material parameters for the linear Cosserat theory.

Engineering materials are non-homogeneous and even discrete, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. We review models dealing with the discrete nature of granular materials and with layered materials with sliding layers. Both cases require the introduction of rotational degrees of freedom; for layered materials it is the layer bending that represents a rotational degree of freedom. We consider the effects of the rotational degrees of freedom from apparent strain localization in simple shearing to new fracture modes.

Strain gradient models and generalized continua are increasingly used to introduce characteristic lengths in the mechanical behavior of materials with microstructure. On the other hand, phase-field models have proved to be efficient tools to simulate microstructure evolution due to thermodynamical processes in the presence of mechanical deformation. It is shown that both methods have strong connections from the point of view of thermomechanical field theory. A general formulation of thermomechanics with additional degrees of freedom is presented that encompasses both applications as special cases. It is based on the introduction of additional power of internal forces introducing generalized stresses. The current knowledge in the formulation of physically non-linear constitutive equations is used to develop strongly coupled elastoviscoplastic material models involving phase transformation and moving boundaries.

This chapter introduces a geometrically nonlinear, continuum thermomechanical framework for case II diffusion: a type of non-Fickian diffusion characterized by the wave-like propagation of a low-molecular weight solvent in a polymeric solid. The key objective of this contribution is to derive the coupled system of governing equations describing case II diffusion from fundamental balance principles. A general form for the Helmholtz energy is proposed and the resulting constitutive laws are derived from logical, thermodynamically consistent argumentation. The chapter concludes by comparing the model developed here to various others in the literature. The approach adopted to derive the governing equations is not specific to case II diffusion, rather it encompasses a wide range of applications wherein heat conduction, species diffusion and finite inelastic effects are coupled. The presentation is thus applicable to the generality of models for non-Fickian diffusion: an area of increasing research interest.

The present contribution discusses a two-scale homogenization procedure for the continuum mechanical modeling of heterogeneous electro-mechanically coupled materials. The direct meso-macro formulation is implemented into an FE²-homogenization environment, which allows for the computation of a macroscopic boundary value problem in consideration of attached heterogeneous representative volume elements at each macroscopic point. The resulting homogenization approach is capable of computing the effective elastic, piezoelectric, and dielectric properties of electro-mechanically coupled materials in consideration of arbitrary mesostructures.

This chapter gives a compact overview of multi-phase continuum mechanics by recourse to the general concepts of mixture and porous media theories. Attention is focused on volumetrically coupled, multi-field formulations arising from the continuum mechanical treatment of multi-phase materials accounting for different physical phenomena. For this purpose, starting with the basics of the macroscopic mixture approach, mixture kinematics, some aspects of electromagnetism and the stress concept are reviewed. Finally, the axiomatic conservation laws of thermodynamics and electrodynamics are fused and represented within the holistic framework of a general multiphasic theory.

Ice formation in porous media results from coupled heat and mass transport and is accompanied by ice expansion. The volume increase in space and time corresponds to the moving freezing front inside the porous solid. In this contribution, a macroscopic model based on the Theory of Porous Media (TPM) is presented toward the description of freezing and thawing processes in saturated porous media. Therefore, a quadruple model consisting of the constituents solid, ice, liquid and gas is used. Attention is paid to the description of capillary suction, liquid- and gas pressure on the surrounding surfaces, volume deformations due to ice formation, temperature distribution as well as influence of heat of fusion under thermal loading. For detection of energetic effects regarding the control of phase transition of water and ice, a physically motivated evolution equation for the mass exchange based on the local divergence of the heat flux is used. Numerical examples are presented to the applications of the model.

This chapter is organized into two parts: The first part is concerned with the experimental determination of thermo-mechanical properties for a cold-box sand, which have a strong influence during the solidification in a sand casting process. To this end the uniaxial behavior for different temperatures up to the casting temperature of an aluminium alloy is investigated. For the tests at room temperature an optical measurement equipment is used, which is ideally suited to measure shear bands. Furthermore, different strain rates are applied to analyze rate dependent behavior. The second part of the chapter is concerned with elasto-plastic modeling for granular materials. Here particular attention is directed to the SD-effect, by a generalized yield function for the non-polar part, and shear band development by a micropolar part for the yield function. In the outlook perspectives for error controlled adaptive strategies for solution of the direct equilibrium problem and the inverse parameter identification problem for micropolar elasto-plastic models are discussed.

Numerical simulation plays an important role in research fields with increasing complexity. Among these are e. g. extended continua, biomechanics, production technology, medical technology and many more. Modern simulation tools often provide both, realistic results and a close link to the physics of the problem. However, increased accuracy is paid by very high computational effort which makes realtime simulation impossible. Nevertheless the latter is urgently needed in important application fields. A good example are surgery training and on-line support during minimally invasive real surgeries. Here, a numerical model can be well used to give otherwise unaccessible information about the stress and strain state in the biological material. The development of such a real-time computation method requires to extend existing model reduction concepts to non-linear solid mechanics based on complex continua. This paper discusses the effectiveness of two singular value decomposition based model reduction methods in this context. Currently, the modal basis reduction method as well as the proper orthogonal decomposition method are widely used for solving linear problems. We will extend these approaches to nonlinear elasticity including large deformations. The performance of the extended concepts is first investigated for the simple geometry of a cantilever beam and then for the very complex system of a human nose. Comparing the results puts us into the position to discuss the suitability of the two model reduction methods for non-linear solid mechanics, especially from the point of view of biomechanics.