We present a geometric theory of nonlinear solids with distributed dislocations. In this theory the material manifold-where the body is stress free-is a Weitzenböck manifold, that is, a manifold with a flat affine connection with torsion but vanishing non-metricity. Torsion of the material manifold is identified with the dislocation density tensor of nonlinear dislocation mechanics. Using Cartan's moving frames we construct the material manifold for several examples of bodies with distributed dislocations. We also present non-trivial examples of zero-stress dislocation distributions. More importantly, in this geometric framework we are able to calculate the residual stress fields, assuming that the nonlinear elastic body is incompressible. We derive the governing equations of nonlinear dislocation mechanics covariantly using balance of energy and its covariance.

This paper presents some developments related to the idea of covariance in elasticity. The geometric point of view in continuum mechanics is briefly reviewed. Building on this, regarding the reference configuration and the ambient space as Riemannian manifolds with their own metrics, a Lagrangian field theory of elastic bodies with evolving reference configurations is developed. It is shown that even in this general setting, the Euler-Lagrange equations resulting from horizontal ͑refer-ential͒ variations are equivalent to those resulting from vertical ͑spatial͒ variations. The classical Green-Naghdi-Rivilin theorem is revisited and a material version of it is discussed. It is shown that energy balance, in general, cannot be invariant under isometries of the reference configuration, which in this case is identified with a subset of R 3 . Transformation properties of balance of energy under rigid translations and rotations of the reference configuration is obtained. The spatial covariant theory of elasticity is also revisited. The transformation of balance of energy under an arbitrary diffeomorphism of the reference configuration is obtained and it is shown that some nonstandard terms appear in the transformed balance of energy. Then conditions under which energy balance is materially covariant are obtained. It is seen that material covariance of energy balance is equivalent to conservation of mass, isotropy, material Doyle-Ericksen formula and an extra condition that we call configurational inviscidity. In the last part of the paper, the connection between Noether's theorem and covariance is investigated. It is shown that the DoyleEricksen formula can be obtained as a consequence of spatial covariance of Lagrangian density. Similarly, it is shown that the material Doyle-Ericksen formula can be obtained from material covariance of Lagrangian density.

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show that time dependency of material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange-d'Alembert's principle with Rayleigh's dissipation functions to derive all the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of deformation gradient $\mathbf{F}=\mathbf{F}_e\mathbf{F}_g$ into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory

Abstract. This paper shows that the stress field in the classical theory of continuum mechanics may be taken to be a covector-valued differential two-form. The balance laws and other fundamental laws of continuum mechanics may be neatly rewritten in terms of this geometric stress. A geometrically attractive and covariant derivation of the balance laws from the principle of energy balance in terms of this stress is presented. Mathematics Subject Classification (2000).

We introduce conformal mixed finite element methods for 2D and 3D incompressible nonlinear elasticity in terms of displacement, displacement gradient, the first Piola-Kirchhoff stress tensor, and pressure, where finite elements for the curl and the div operators are used to discretize strain and stress, respectively. These choices of elements follow from the strain compatibility and the momentum balance law. Some inf-sup conditions are derived to study the stability of methods. By considering 96 choices of simplicial finite elements of degree less than or equal to 2 in 2D and 3D, we conclude that 28 choices in 2D and 6 choices in 3D satisfy these inf-sup conditions. The performance of stable finite element choices are numerically studied. Although the proposed methods are computationally more expensive than the standard two-field methods for incompressible elasticity, they are potentially useful for accurate approximations of strain and stress as they are independently computed in the solution process. nonlinear elasticity. This inf-sup condition can be interpreted as the necessary and sufficient condition for the well-posedness of the linear problems associated to Newtons' iterations for solving nonlinear finite element methods. Based on this interpretation, we also write 6 other inf-sup conditions that are necessary for the stability of Newtons' iterations. Using these conditions, we derive some relations between the dimension of finite element spaces for different unknowns, which are necessary for the stability of Newtons' iterations.By considering 96 choices of simplicial finite elements of degree less than or equal to 2 for obtaining mixed finite element methods, we conclude that 68 choices in 2D and 90 choices in 3D do not satisfy all these inf-sup conditions, in general, and lead to unstable methods. The performance of stable choices in 2D and 3D are studied by solving numerical examples.This paper is organized as follows: The four-field mixed formulation and the associated conformal finite element methods are discussed in Sections 2 and 3. Newtons' iterations for solving nonlinear finite element methods are provided in Section 4. In Section 5, different inf-sup conditions are derived for studying the stability of finite element methods and it is shown that some choices of finite elements lead to unstable methods as they violate the inf-sup conditions. In Section 6, the inf-sup conditions are numerically studied. Moreover, by solving two numerical examples, the performance of stable finite element methods are investigated in 2D and 3D. Finally, some concluding remarks are given in Section 7.