An extended Hamilton principle as unifying theory for coupled problems and dissipative microstructure evolution

Abstract

An established strategy for material modeling is provided by energy-based principles such that evolution equations in terms of ordinary differential equations can be derived. However, there exist a variety of material models that also need to take into account non-local effects to capture microstructure evolution. In this case, the evolution of microstructure is described by a partial differential equation. In this contribution, we present how Hamilton’s principle provides a physically sound strategy for the derivation of transient field equations for all state variables. Therefore, we begin with a demonstration how Hamilton’s principle generalizes the principle of stationary action for rigid bodies. Furthermore, we show that the basic idea behind Hamilton’s principle is not restricted to isothermal mechanical processes. In contrast, we propose an extended Hamilton principle which is applicable to coupled problems and dissipative microstructure evolution. As example, we demonstrate how the field equations for all state variables for thermo-mechanically coupled problems, i.e., displacements, temperature, and internal variables, result from the stationarity of the extended Hamilton functional. The relation to other principles, as the principle of virtual work and Onsager’s principle, is given. Finally, exemplary material models demonstrate how to use the extended Hamilton principle for thermo-mechanically coupled elastic, gradient-enhanced, rate-dependent, and rate-independent materials.

Thermodynamic topology optimization for sequential additive manufacturing including structural self‐weight

“…The thermodynamic topology optimization was established in a series of previous papers. 5,6,12 This approach is based on the Hamilton principle 13 which is also common in material modeling. The extended Hamilton functional for the topology optimization of continua with dissipative microstructure…”

“…The densities are computed element-wise by the differential equation in (13). In this equation, the driving force d determines how the density of an element changes.…”

Efficient and robust numerical treatment of a gradient-enhanced damage model at large deformations

“…Its stationarity conditions agree with the 2 nd Law of Thermodynamics and Onsager's principle by construction if physically reasonable ansatzes for the strain energy density and the dissipation function are chosen. For details on the extended Hamilton principle and its relation to thermodynamics and other modeling strategies, such as the principle of virtual work or the principle of the minimum of the dissipation potential, we refer to [36].…”

“…It can be shown, cf. [36], that it constitutes as the following condition: for the quasi-static case, the sum of the Gibbs energy G, which is also referred to as total potential, and the work due to dissipative processes D tends to be stationary:…”

“…A detailed investigation of the extended Hamilton principle as unifying theory for coupled problems, i.e. thermo-mechanical processes, and dissipative microstructure evolution has been presented in [36].…”