On boundary potential energies in deformational and configurational mechanics

This contribution deals with the implications of boundary potential energies, i.e. in short surface, curve and point potentials, on deformational and configurational mechanics. Within the realm of deformational mechanics the surface/curve potentials are allowed in the most general case to depend on the deformation, the surface/curve deformation gradient and the spatial surface normal/curve tangent and are parametrised in the material placement and the material surface normal/curve tangent. The point potentials depend on the deformation and are parametrised in the material placement. From the configurational mechanics perspective the roles of fields and parametrisations are reversed. By considering variational arguments based on the kinematics of deforming surfaces/curves, in particular the relevant surface/curve stresses and distributed forces contributing to (localized) deformational and configurational force balances at surfaces/curves/points, which extend the common traction boundary conditions, are derived. Thereby, dissipative distributed configurational forces that are energetically conjugate to configurational changes are introduced as definitions. The (localized) force balances at surfaces/curves/points together with the contributing stresses and distributed forces within deformational and configurational mechanics display an intriguing duality. The resulting dissipative configurational tractions at the boundary are exemplified for some illustrative cases of boundary potentials.     

A deformational and configurational framework for geometrically non-linear continuum thermomechanics coupled to diffusion

A general framework encompassing both the (conventional) deformational and configurational settings of continuum mechanics is presented. A systematic application of balance principles over a migrating control volume in the undeformed configuration of the continuum body yields the system of governing equations in the bulk, on the surface and on a coherent interface within the continuum. The equations governing the response of the bulk agree with those of the conventional deformational approach. The localised balance equations are expressed in the configurational setting using a pull-back operator and reformulated in terms of the Eshelby stress. The configurational expression of the dissipation elucidates the energy loss associated with configurational changes. The general framework is introduced by considering the problem of coupled deformation, heat conduction and species diffusion within a geometrically non-linear continuum body intersected by a coherent interface. The nature of the coupling is emphasised throughout the presentation and via an example.     

Internal Constraints, Reactive Stresses, and the Timoshenko Beam Theory

It is shown that the kinematical assumptions of the Timoshenko theory for shearable beams can be regarded as internal constraints, some involving the first, others the second deformation gradient; such constrains are thought of as maintained by reaction stresses and hyperstresses of the type occurring in non-simple materials of grade 2. It is discussed how these reactions can be used to better the first approximation of the unknown equilibrium stress field in the three-dimensional body modelled by the Timoshenko beam, an appoximation which is based on the solution of the one-dimensional Timoshenko problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

Multiscale modeling of the dynamics of solids at finite temperature

We develop a general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we develop an additional strategy for defect tracking. This method naturally decouples the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented.     

On energetic changes due to configurational motion of standard continua

The appropriate thermodynamic setting for the combined configurational and deformational motion of standard continua is discussed in this paper. A key ingredient is an absolute (fixed) reference configuration, relative to which configurational and deformational changes (rates) with respect to the material (undeformed) and spatial (deformed) configurations, respectively, can be described in a unified fashion. In particular, we are interested in formulating the local as well as global measures of the energy dissipation due to configurational changes of a given physical system. It is believed that the presentation in this paper provides the following advantages: a unified kinematic and thermodynamic setting of the configurational and deformational motions, in particular the generic balance law accounting for configurational flux, increases the general clarity; the separation of the total dissipation in terms of the intrinsic change in elastic energy and in terms of the material dissipation that is induced by configurational changes becomes transparent. All results are obtained without any restrictions on dynamics, thermomechanical couplings, etc.     

Deformation measurements by digital image correlation: Implementation of a second-order displacement gradient

This paper outlines the procedure for refining the digital image correlation (DIC) method by implementing a second-order approximation of the displacement gradients. The second-order approximation allows the DIC method to directly measure both the first- and second-order displacement gradients resulting from nonlinear deformation. Thirteen unknown parameters, consisting of the components of displacement, the first- and second-order displacement gradients and the gray-scale value offset, are determined through optimization of a correlation coefficient. The previous DIC method assumes that the local deformation in a subset of pixels is represented by a first-order Taylor series approximation for the displacement gradient terms, so actual deformations consisting of higher order displacement gradients tend to distort the infinitesimal strain measurements. By refining the method to measure both the first- and second-order displacement gradients, more accurate strain measurements can be achieved in large-deformation situations where second-order deformations are also present. In most cases, the new refinements allow the DIC method to maintain an accuracy of ±0.0002 for the first-order displacement gradients and to reach ±0.0002 per pixel for the second-order displacement gradients.     

Modifications to the Cauchy–Born rule: Applications in the deformation of single-walled carbon nanotubes

This paper presents a study of the Cauchy–Born (CB) rule as applied to the deformation analysis of single-walled carbon nanotubes (SWNTs) that are modeled as 2-dimensional manifolds. The C–C bond vectors in the SWNT are assumed to deform according to the local deformation gradient as per the CB rule or a modified version thereof. Aspects of the CB rule related to spatial inhomogeneity of the deformation gradient at the atomic scale are investigated in the context of a specific class of extension–twist deformation problems. Analytic expressions are derived for the deformed bond lengths using the standard CB rule as well as modified versions of the standard CB rule. Since the deformation map is conveniently prescribed in this work, it is possible to compare the performance of these deformation rules with the exact solution (i.e. the exact analytic expression for the deformed bond vectors) given directly by the deformation map. This approach provides insights into the CB rule and its possible modifications for use in more complicated deformations where an explicit deformation map is not available. Specifically, it is concluded that in the case of inhomogeneous deformations at the atomic scale for which the CB rule is only approximate (as demonstrated in Section 1 of this paper), the mean value theorem in calculus can be used as a guide to modify the CB rule and construct a more rigorous and accurate atomistic–continuum connection. The deformed bond lengths are used to formulate an enriched continuum hyperelastic strain energy density function based on interatomic potentials (the multi-body Tersoff–Brenner [Tersoff, J., 1988. New empirical approach for the structure and energy of covalent systems. Phys. Rev. B 37, 6991–7000; Brenner, D.W., 1990. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Phys. Rev. B 42, 9458–9471] empirical interatomic potential for carbon-carbon bonds is used in this work). The deformation map (and hence the deformation gradient, the bond vectors and the continuum strain energy density) contains certain parameters, some of which are imposed and others determined as a result of energy minimization in the standard variational formulation. Numerical results for kinematic coupling and binding energy per atom are presented in the case of imposed extension and twist deformations on representative chiral, zig-zag and armchair nanotubes using the CB rule and its modifications. These results are compared with the exact solution based on the deformation map which serves as a basis for evaluating the efficacy of these deformation rules. The ideas presented in this paper can also be directly extended to other lattices.     

Geometrically-based Consequences of Internal Constraints

When a body is subject to simple internal constraints, the deformation gradient must belong to a certain manifold. This is in contrast to the situation in the unconstrained case, where the deformation gradient is an element of the open subset of second-order tensors with positive determinant. Commonly, following Truesdell and Noll [1], modern treatments of constrained theories start with an a priori additive decomposition of the stress into reactive and active components with the reactive component assumed to be powerless in all motions that satisfy the constraints and the active component given by a constitutive equation. Here, we obtain this same decomposition automatically by making a purely geometrical and general direct sum decomposition of the space of all second-order tensors in terms of the normal and tangent spaces of the constraint manifold. As an example, our approach is used to recover the familiar theory of constrained hyperelasticity.     

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.     

Deformation gradients for continuum mechanical analysis of atomistic simulations

We present an expression developed for calculating an atomic-scale deformation gradient within atomistic simulations. This expression is used to analyze the deformation fields for a one-dimensional atomic chain, a biaxially stretched thin film containing a surface ledge, and a FCC metal subject to indentation loading from a nanometer-scale indenter. The analyses presented show that the metric established here is consistent with the continuum mechanical concept of deformation gradient (which is known to have a zero curl for compatible deformations) in most instances. However, our metric does yield non-zero values of curl for atoms near loaded geometric inhomogeneities, such as those that form the ledges themselves and those beneath or adjacent to the indentation contact region. Also, we present expressions for higher order gradients of the deformation field and discuss the requirements for their calculation. These expressions are necessary for linking atomistic simulation results with advanced continuum mechanics theories such as strain gradient plasticity, thereby enabling fundamental, atomic-scale information to contribute to the formulation and parameterization of such theories.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE

IntroductionTherelationsbetweenvariousstresstensors,theequationofmomentumandtheboundaryconditionsofincrementalratetypeforclassicalcontinuummechanicshavebeensystematicallyderivedbyKuanginRef.[1 ] .InRef.[2 ]wehavederivedtherelationsbetweenvariouscouplestresstensorsandtheirratesandpresentedtheequationsofmotionandtheboundaryconditionsofincrementalratetypeofCauchyform ,PiolaformandKirchhoffformforpolarcontinua .InRef.[3 ]wehavepresentednewprinciplesofpowerandenergyrateofincrementalratetypeformi…     

On configurational compatibility and multiscale energy momentum tensors

In this work the continuum theory of defects has been revised through the development of kinematic defect potentials. These defect potentials and their corresponding variational principles provide a basis for constructing a new class of conservation laws associated with the compatibility conditions of continua. These conservation laws represent configurational compatibility conditions which are independent of the constitutive behavior of the continuum. They lead to the development of a new concept termed configurational compatibility, dual to the concept of configurational force. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by the integral is defect-free. It is shown that the Peach-Koehler force can be recovered as one of these invariant integrals. Based on the proposed defect potentials and their corresponding defect energies, two-field multiscale mixed variational principles can be employed to construct multiscale energy momentum tensors. An application is outlined in the form of a mode III elasto-plastic crack problem for which the new configurational quantities are calculated.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE

The purpose is to establish the rather complete equations of motion, boundary conditions and equation of energy rate of incremental rate type for micropolar continua. To this end the rather complete definitions for rates of deformation gradient and its inverse are made. The new relations between various stress and couple stress rate tensors are derived. Finally, the coupled equations of motion, boundary conditions and equation of energy rate of incremental rate type for continuum mechanics are obtained as a special case. Contributed by Dai Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: Dai Tian-min (1931≈)     

On the sensitivity of the rate of global energy dissipation due to configurational changes

The thermodynamic framework for combined configurational and deformational changes was recently discussed by [Runesson, K., Larsson, F., Steinmann, P., 2009. On energetic changes due to configurational motion of standard continua. Int. J. Solids Struct, 46, 1464–1475.]. One key ingredient in this setting is the (fixed) absolute configuration, relative to which both physical and virtual (variational) changes of the material and spatial configurations can be described. In the present paper we consider dissipative material response and emphasize the fact that it is possible to identify explicit energetic changes due to configurational changes for “frozen” spatial configuration (a classical view) and the configuration-induced material dissipation. The classical assumption (previously adopted in the literature) is to ignore this dissipation, i.e. the internal variables are considered as fixed fields in the material configuration. In this paper, however, we define configurational forces by considering the total variation of the total dissipation with respect to configurational changes. The key task is then to compute the sensitivity of the internal variable rates to such configurational changes, which results in a global tangent problem based on the balance equations (momentum and energy) for a given body. In this paper we restrict to quasistatic loading under isothermal conditions and for elastic-plastic response, and we apply the modeling to the case of a moving interface of dissimilar materials.     

An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.     

Thermodynamic and rate variational formulation of models for inhomogeneous gradient materials with microstructure and application to phase field modeling

In this work,thermodynamic models for the energetics and kinetics of inhomogeneous gradient materials with microstructure are formulated in the context of continuum thermodynamics and material theory.For simplicity,attention is restricted to isothermal conditions.The materials of interest here are characterized by(1) first- and secondorder gradients of the deformation field and(2) a kinematic microstructure field and its gradient(e.g.,in the sense of director,micromorphic or Cosserat microstructure).Material inhomogeneity takes the form of multiple phases and chemical constituents,modeled here with the help of corresponding phase fields.Invariance requirements together with the dissipation principle result in the reduced model field and constitutive relations.Special cases of these include the wellknown Cahn-Hilliard and Ginzburg-Landau relations.In the last part of the work,initial boundary value problems for this class of materials are formulated with the help of rate variational methods.     

Micromorphic continuum Part I: Strain and stress tensors and their associated rates

Micropolar and micromorphic solids are continuum mechanics models, which take into account, in some sense, the microstructure of the considered real material. The characteristic property of such continua is that the state functions depend, besides the classical deformation of the macroscopic material body, also upon the deformation of the microcontinuum modeling the microstructure, and its gradient with respect to the space occupied by the material body. While micropolar plasticity theories, including non-linear isotropic and non-linear kinematic hardening, have been formulated, even for non-linear geometry, few works are known yet about the formulation of (finite deformation) micromorphic plasticity. It is the aim of the three papers (Parts I, II, and III) to demonstrate how micromorphic plasticity theories may be formulated in a thermodynamically consistent way.In the present article we start by outlining the framework of the theory. Especially, we confine attention to the theory of Mindlin on continua with microstructure, which is formulated for small deformations. After precising some conceptual aspects concerning the notion of microcontinuum, we work out a finite deformation version of theory, suitable for our aims. It is examined that resulting basic field equations are the same as in the non-linear theory of Eringen, which deals with a different definition of the microcontinuum. Furthermore, geometrical interpretations of strain and curvature tensors are elaborated. This allows to find out associated rates in a natural manner. Dual stress and double stress tensors, as well as associated rates, are then defined on the basis of the stress powers. This way, it is possible to relate strain tensors (respectively, micromorphic curvature tensors) and stress tensors (respectively, double stress tensors), as well as associated rates, independently of the particular constitutive properties.     

An atomistically validated continuum model for strain relaxation and misfit dislocation formation

In this paper, molecular dynamics (MD) calculations have been used to examine the physics behind continuum models of misfit dislocation formation and to assess the limitations and consequences of approximations made within these models. Without compromising the physics of misfit dislocations below a surface, our MD calculations consider arrays of dislocation dipoles constituting a mirror imaged “surface”. This allows use of periodic boundary conditions to create a direct correspondence between atomistic and continuum representations of dislocations, which would be difficult to achieve with free surfaces. Additionally, by using long-time averages of system properties, we have essentially reduced the errors of atomistic simulations of large systems to “zero”. This enables us to deterministically compare atomistic and continuum calculations. Our work results in a robust approach that uses atomistic simulation to accurately calculate dislocation core radius and energy without the continuum boundary conditions typically assumed in the past, and the novel insight that continuum misfit dislocation models can be inaccurate when incorrect definitions of dislocation spacing and Burgers vector in lattice-mismatched systems are used. We show that when these insights are properly incorporated into the continuum model, the resulting energy density expression of the lattice-mismatched systems is essentially indistinguishable from the MD results.