arash_yavari's blog

It is known that the balance laws of hyperelasticity (Green elasticity), i.e., conservation of mass and balance of linear and angular momenta, can be derived using the first law of thermodynamics and by postulating its invariance under superposed rigid body motions of the Euclidean ambient space---the Green-Naghdi-Rivlin theorem. In the case of a non-Euclidean ambient space, covariance of the energy balance---its invariance under arbitrary time-dependent diffeomorphisms of the ambient space---gives all the balance laws and the Doyle-Ericksen formula---the Marsden-Hughes theorem.

We study particle dynamics under curl forces. These forces are a class of non-conservative, non-dissipative, position-dependent forces that cannot be expressed as the gradient of a potential function. We show that the fundamental quantity of particle dynamics under curl forces is a work 1-form. By using the Darboux classification of differential 1-forms on R² and R³, we establish that any curl force in two dimensions has at most two generalized potentials, while in three dimensions, it has at most three.

Universal displacements are those displacements that can be maintained for any member of a specific class of linear elastic materials in the absence of body forces, solely by applying boundary tractions. For linear hyperelastic (Green elastic) solids, it is known that the space of universal displacements explicitly depends on the symmetry group of the material, and moreover, the larger the symmetry group the larger the set of universal displacements.

In this paper, we formulate a continuum theory of solidification within the context of finite-strain coupled thermoelasticity. We aim to fill a gap in the existing literature, as the existing studies on solidification typically decouple the thermal problem (the classical Stefan's problem) from the elasticity problem, and often limit themselves to linear elasticity with small strains.

For a given material, \emph{controllable deformations} are those deformations that can be maintained in the absence of body forces and by applying only boundary tractions. For a given class of materials, \emph{universal deformations} are those deformations that are controllable for any material within the class.

For a given class of materials, universal deformations are those deformations that can be maintained in the absence of body forces and by applying solely boundary tractions. For inhomogeneous bodies, in addition to the universality constraints that determine the universal deformations, there are extra constraints on the form of the material inhomogeneities—universal inhomogeneity constraints. Those inhomogeneities compatible with the universal inhomogeneity constraints are called universal inhomogeneities.

We study the geometric phases of nonlinear elastic $N$-rotors with continuous rotational symmetry. In the Hamiltonian framework, the geometric structure of the phase space is a principal fiber bundle, i.e., a base, or shape manifold~$\mathcal{B}$, and fibers $\mathcal{F}$ along the symmetry direction attached to it. The symplectic structure of the Hamiltonian dynamics determines the connection and curvature forms of the shape manifold. Using Cartan's structural equations with zero torsion we find an intrinsic (pseudo) Riemannian metric for the shape manifold.

In this paper we revisit the mathematical foundations of nonlinear viscoelasticity.

In this paper, we present a large-deformation formulation of the mechanics of remodeling. Remodeling is anelasticity with an internal constraint---material evolutions that are mass and volume-preserving.  In this special class of material evolutions, the explicit time dependence of the energy function is via one or more remodeling tensors that can be considered as internal variables of the theory. The governing equations of remodeling solids are derived using a two-potential approach and the Lagrange-d'Alembert principle.