A new variational approach for the thermodynamic topology optimization of hyperelastic structures

We present a novel approach to topology optimization based on thermodynamic extremal principles. This approach comprises three advantages: (1) it is valid for arbitrary hyperelastic material formulations while avoiding artificial procedures that were necessary in our previous approaches for topology optimization based on thermodynamic principles; (2) the important constraints of bounded relative density and total structure volume are fulfilled analytically which simplifies the numerical implementation significantly; (3) it possesses a mathematical structure that allows for a variety of numerical procedures to solve the problem of topology optimization without distinct optimization routines. We present a detailed model derivation including the chosen numerical discretization and show the validity of the approach by simulating two boundary value problems with large deformations.

Rigid-perfectly-plastic materials. Two extremal principles

This chapter introduces the notion of rigid-perfect plasticity, and provides proofs of the extremum principles of applied loads and velocities. The variational principles are applied to generate the upper- and lower-bound theorems for collapse loads in rigid-perfect plasticity models. The method of sliding rigid blocks is presented for the solution of practical problems in forming operations and structural collapse. To facilitate such modelling the hodograph graphical device is also discussed.

Constraints in thermodynamic extremal principles for non-local dissipative processes

Phenomena treated by non-equilibrium thermodynamics can be very effectively described by thermodynamic variational principles. The remarkable advantage of such an approach consists in possibility to account for an arbitrary number of constraints among state or kinetic variables stemming, e.g., from conservation laws or balance equations. As shown in the current paper, the variational principles can provide original evolution equations for the state variables implicitly respecting the constraints. Moreover, the variational approach allows formulating the problem directly in discrete state variables and deriving their evolution equations without the necessity to solve partial differential equations. The variational approach makes it also possible to use different kinetic variables in formulation of dissipation and dissipation function.