Uditnarayan Kouskiya Amit Acharya
We demonstrate the feasibility of a scheme to obtain approximate weak solutions to (inviscid) Burgers equation in conservation and Hamilton-Jacobi form, treated as degenerate elliptic problems. We show different variants recover non-unique weak solutions as appropriate, and also specific constructive approaches to recover the corresponding entropy solutions.
Siddharth Singh Janusz Ginster Amit Acharya
A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler-Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding, primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.
The fully nonlinear (geometric and material) system of Field Dislocation Mechanics is reviewed to establish an exact analogy with the equations of ideal magnetohydrodynamics (ideal MHD) under suitable physically simplifying circumstances. Weak solutions with various conservation properties have been established for ideal MHD recently by Faraco, Lindberg, and Szekelyhidi using the techniques of compensated compactness of Tartar and Murat and convex integration; by the established analogy, these results would seem to be transferable to the idealization of Field Dislocation Mechanics considered. A dual variational principle is designed and discussed for this system of PDE, with the technique transferable to the study of MHD as well.
Amit Acharya
A methodology for defining variational principles for a class of PDE models from continuum mechanics is demonstrated, and some of its features explored. The scheme is applied to quasi-static and dynamic models of rate-independent and rate-dependent, single crystal plasticity at finite deformation.
A methodology for deriving dual variational principles for the classical Newtonian mechanics of mass points in the presence of applied forces, interaction forces, and constraints, all with a general dependence on particle velocities and positions, is presented. Methods for incorporating constraints are critically assessed. General theory, as well as explicitly worked out variational principles for a dissipative system (Lorenz) and a system with anholonomic constraints (Pars) are demonstrated. Conditions under which a (family of) dual Hamiltonian flows, as well as constants of motion, may be associated with a dissipative, and possibly constrained, primal system are specified.
See Abhishek Arora's blog:
1. Mechanics of micropillar confined thin film plasticity
2. Interface-dominated plasticity and kink bands in metallic nanolaminates
See Siddharth Singh's blog:
Modeling of experimentally observed topological defects inside bulk polycrystals
See Uditnarayan Kouskiya's blog:
Amit Acharya
A formal methodology for developing variational principles corresponding to a given nonlinear
pde system is discussed. The scheme is demonstrated in the context of the incompressible
Navier-Stokes equations, systems of first-order conservation laws, and systems of Hamilton-
Jacobi equations.
Amit Acharya
To appear in J. Mech. Phys Solids
Amit Acharya
An action functional is developed for nonlinear dislocation dynamics. This serves as a fi rst step
towards the application of eff ective field theory in physics to evaluate its potential in obtaining
a macroscopic description of dislocation dynamics describing the plasticity of crystalline solids.
Connections arise between the continuum mechanics and material science of defects in solids,
eff ective field theory techniques in physics, and fracton tensor gauge theories.
The scheme that emerges from this work for generating a variational principle for a nonlinear
pde system is general, as is demonstrated by doing so for nonlinear elastostatics involving a stress
response function that is not necessarily hyperelastic.
