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.

基于能量的原理提供了材料建模的既定策略，从而可以导出常微分方程方面的演化方程。然而，存在多种材料模型也需要考虑非局部效应以捕捉微观结构演化。在这种情况下，微观结构的演变由偏微分方程描述。在这篇文章中，我们展示了汉密尔顿原理如何为推导所有状态变量的瞬态场方程提供一种物理上合理的策略。因此，我们首先演示哈密顿原理如何推广刚体的静止作用原理。此外，我们表明汉密尔顿原理背后的基本思想不限于等温机械过程。相比之下，我们提出了适用于耦合问题和耗散微结构演化的扩展哈密顿原理。例如，我们演示了热机械耦合问题的所有状态变量（即位移、温度和内部变量）的场方程如何由扩展的哈密顿泛函的平稳性产生。给出了与其他原理的关系，如虚功原理和昂萨格原理。最后，示例性材料模型演示了如何将扩展的哈密顿原理用于热机械耦合弹性、梯度增强、速率相关和速率无关材料。我们展示了热机械耦合问题的所有状态变量的场方程，即位移、温度和内部变量，是如何从扩展的哈密顿泛函的平稳性中产生的。给出了与其他原理的关系，如虚功原理和昂萨格原理。最后，示例性材料模型演示了如何将扩展的哈密顿原理用于热机械耦合弹性、梯度增强、速率相关和速率无关材料。我们展示了热机械耦合问题的所有状态变量的场方程，即位移、温度和内部变量，是如何从扩展的哈密顿泛函的平稳性中产生的。给出了与其他原理的关系，如虚功原理和昂萨格原理。最后，示例性材料模型演示了如何将扩展的哈密顿原理用于热机械耦合弹性、梯度增强、速率相关和速率无关材料。