Variational Foundations and Generalized Unified Theory of RVE-Based Multiscale Models

A unified variational theory is proposed for a general class of multiscale models based on the concept of Representative Volume Element. The entire theory lies on three fundamental principles: (1) kinematical admissibility, whereby the macro- and micro-scale kinematics are defined and linked in a physically meaningful way; (2) duality, through which the natures of the force- and stress-like quantities are uniquely identified as the duals (power-conjugates) of the adopted kinematical variables; and (3) the Principle of Multiscale Virtual Power, a generalization of the well-known Hill-Mandel Principle of Macrohomogeneity, from which equilibrium equations and homogenization relations for the force- and stress-like quantities are unequivocally obtained by straightforward variational arguments. The proposed theory provides a clear, logically-structured framework within which existing formulations can be rationally justified and new, more general multiscale models can be rigorously derived in well-defined steps. Its generality allows the treatment of problems involving phenomena as diverse as dynamics, higher order strain effects, material failure with kinematical discontinuities, fluid mechanics and coupled multi-physics. This is illustrated in a number of examples where a range of models is systematically derived by following the same steps. Due to the variational basis of the theory, the format in which derived models are presented is naturally well suited for discretization by finite element-based or related methods of numerical approximation. Numerical examples illustrate the use of resulting models, including a non-conventional failure-oriented model with discontinuous kinematics, in practical computations

Transition to plasticity in continuum-atomistic modelling

Continuum‐atomistic modeling denotes a mixed approach combining the usual framework of continuum mechanics with atomistic features like e.g. interaction potentials. Thereby, the kinematics are typically characterized by the so called Cauchy‐Born rule representing atomic distance vectors in the spatial configuration as an affine mapping of the atomic distance vectors in the material configuration in terms of the local deformation gradient. The application of the Cauchy‐Born rule requires sufficiently homogeneous deformations of the underlying crystal. The model is no more valid if the deformation becomes inhomogeneous. By virtue of the Cauchy‐Born hypothesis, a localization criterion has been derived in terms of the loss of infinitesimal rank‐1 convexity of the strain energy density. According to this criterion, a numerical yield condition has been computed for two different interatomic energy functions. Therewith, the range of the Cauchy‐Born rule validity has been defined, since the strain energy density remains quasiconvex only within the computed yield surface. To provide a possibility to continue the simulation of material response after the loss of quasiconvexity, a relaxation procedure proposed by Tadmor et al. [1] leading necessarily to the development of microstructures has been used. Alternatively to the above mentioned criterion, a stability criterion has been applied to detect the critical deformation. For the study in the postcritical region, the path‐change procedure proposed by Wagner and Wriggers [2] has been adapted for the continuum‐atomistics and modified

Localization analysis of mixed continuum-atomistic models

Continuum-atomistic models combine atomistic features like pair potentials and lattice-structures with classical field approaches from continuum mechanics. Thus they may be considered as a mixed scale method, whereby the atomistic scale is typically in the order of nm, whereas the continuum scale is in the order of mm. In this contribution we investigate the influence of the atomistic features on the continuum response, especially with respect to the failure characteristics in terms of the localization tensor

Localization Analysis of Mixed Continuum-atomistic Models

Continuum-atomistic models combine atomistic features like pair potentials and lattice-structures with classical field approaches from continuum mechanics. Thus they may be considered as a mixed scale method, whereby the atomistic scale is typically in the order of nm, whereas the continuum scale is in the order of mm. In this contribution we investigate the influence of the atomistic features on the continuum response, especially with respect to the failure characteristics in terms of the localization tensor

On higher gradients in continuum-atomistic modelling

Continuum-atomistic modelling denotes a mixed approach combining the usual framework of continuum mechanics with atomistic features like e.g. interaction or rather pair potentials. Thereby, the kinematics are typically characterized by the so-called Cauchy-Born rule representing atomic distance vectors in the spatial configuration as an affine mapping of the atomic distance vectors in the material configuration in terms of the local deformation gradient. The application of the Cauchy-Born rule requires sufficiently homogeneous deformations of the underlying crystal. The model is no more valid if the deformation becomes inhomogeneous. Nevertheless the development of microstructures with inhomogeneous deformation is inevitable. In the present work, the Cauchy-Born rule is thus extended to capture inhomogeneous deformations by the incorporation of the second-order deformation gradient. The higher-order equilibrium equation as well as the appropriate boundary conditions are presented for the case of finite deformations. The constitutive law for the Piola-Kirchhoff stress and the additional higher-order stress are represented for the simplified case of pair potential-based energy density functions. Finally, a deformation inhomogeneity measure is introduced and studied for a particular non-homogeneous simple-shear like deformation

Localization Analysis of Mixed Continuum-atomistic Models

Continuum-atomistic models combine atomistic features like pair potentials and lattice-structures with classical field approaches from continuum mechanics. Thus they may be considered as a mixed scale method, whereby the atomistic scale is typically in the order of nm, whereas the continuum scale is in the order of mm. In this contribution we investigate the influence of the atomistic features on the continuum response, especially with respect to the failure characteristics in terms of the localization tensor

Studies of validity of the Cauchy-Born rule by direct comparison of continuum and atomistic modelling

The Cauchy-Born rule is widely used as a standard method in continuum mechanics in order to construct descriptions of material behaviour using atomistic information. The main objective of the present work is to investigate the validity of this kinematic assumption, i.e. to determine the state when a transition to non-affine deformations is possible due to instabilities of the underlying atomic system. To this end, the results of the Cauchy-Born rule are compared with the results of direct molecular dynamics calculations on the one hand and with the results obtained by using the acoustic tensor of continuum mechanics on the other hand