使用扩展 Noether 形式体系应用于力学中的能量方法
Energy based methods applied in mechanics by using the extended Noether's formalism
AI总结：
本文通过张量代数和连续统力学命名法展示扩展 Noether 形式体系，推导广义连续统损伤力学（特别是应变梯度弹性）中的场方程，克服了传统方法的应用局限。
摘要：
物理系统由场方程建模；这些是空间和时间中的耦合偏微分方程。场方程通常由平衡方程和本构方程给出，其中前者是公理化给出的，而后者是热力学推导的。这种方法在热力学和电磁学中很有用，但一旦我们将其应用于广义连续统的损伤力学中，就会出现挑战。为了推导控制方程，一种替代方法基于称为扩展 Noether 形式体系的变分框架。它的形式化引入依赖于数学概念，限制了其作为场论在应用力学中的使用。在这项工作中，我们通过使用张量代数和通常的连续统力学命名法，展示了扩展 Noether 形式体系的强大功能。我们展示了广义连续统损伤力学中场方程的推导，特别是在应变梯度弹性情况下。
英文摘要：
Physical systems are modeled by field equations; these are coupled, partial differential equations in space and time. Field equations are often given by balance equations and constitutive equations, where the former are axiomatically given and the latter are thermodynamically derived. This approach is useful in thermomechanics and electromagnetism, yet challenges arise once we apply it in damage mechanics for generalized continua. For deriving governing equations, an alternative method is based on a variational framework known as the extended Noether's formalism. Its formal introduction relies on mathematical concepts limiting its use in applied mechanics as a field theory. In this work, we demonstrate the power of extended Noether's formalism by using tensor algebra and usual continuum mechanics nomenclature. We demonstrate derivation of field equations in damage mechanics for generalized continua, specifically in the case of strain gradient elasticity.
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