Nonlinear elasticity

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.

In this paper we formulate and solve the initial-boundary value problem of accreting circular cylindrical bars under finite extension. We assume that the bar grows by printing stress-free cylindrical layers on its boundary cylinder while it is undergoing a time-dependent finite extension. Accretion induces eigenstrains, and consequently residual stresses. We formulate the anelasticity problem by first constructing the natural Riemannian metric of the growing bar. This metric explicitly depends on the history of deformation during the accretion process.

In this paper we formulate the initial-boundary value problem of accreting circular cylindrical bars under finite torsion. It is assumed that the bar grows as a result of printing stress-free cylindrical layers on its boundary while it is under a time-dependent torque (or a time-dependent twist) and is free to deform axially. In a deforming body, accretion induces eigenetrains, and consequently residual stresses. We formulate the anelasticity problem by first constructing the natural Riemannian metric of the growing bar.

For a given class of materials, universal deformations are those that can be maintained in the absence of body forces by applying only boundary tractions.  Universal deformations play a crucial role in nonlinear elasticity.

Dear colleagues,

We invite you to see the preprint of our new paper "Accretion and Ablation in Deformable Solids with an Eulerian Description: Examples using the Method of Characteristics" which will appear in Mathematics and Mechanics of Solids. Recent work has proposed an Eulerian approach to the surface growth problem, enabling the side-stepping of the issue of constructing the reference configuration. However, this raises the complementary challenge of determining the stress response of the solid. To resolve this, the approach introduced the elastic deformation as an additional kinematic descriptor of the added material, and its evolution has been shown to be governed by a transport equation. Here, we applied the method of characteristics to solve concrete simplified problems motivated by surface growth in biomechanics and manufacturing (https://journals.sagepub.com/doi/10.1177/10812865211054573)

Universal (controllable) deformations of an elastic solid are those deformations that can be maintained for all possible strain-energy density functions and suitable boundary tractions. Universal deformations have played a central role in nonlinear elasticity and anelasticity. However, their classification has been mostly established for homogeneous isotropic solids following the seminal works of Ericksen. In this paper, we extend Ericksen's analysis of universal deformations to inhomogeneous compressible and incompressible isotropic solids.

Universal deformations of an elastic solid are deformations that can be achieved for all possible strain-energy density functions and suitable boundary conditions. They play a central role in nonlinear elasticity and their classification has been mostly accomplished for isotropic solids following Ericksen's seminal work. Here, we address the same problem for transversely isotropic, orthotropic, and monoclinic solids.