A methodology based on Lie analysis is proposed to investigate the mechanical behavior of materials exhibiting experimental master curves. It is based on the idea that the mechanical response of materials is associated with hidden symmetries reflected in the form of the energy functional and the dissipation potential leading to constitutive laws written in the framework of the thermodynamics of irreversible processes. In constitutive modeling, symmetry analysis lets one formulate the response of a material in terms of so-called master curves, and construct rheological models based on a limited number of measurements. The application of symmetry methods leads to model reduction in a double sense: in treating large amounts number of measurements data to reduce them in a form exploitable for the construction of constitutive models, and by exploiting equivalence transformations extending point symmetries to efficiently reduce the number of significant parameters, and thus the computational cost of solving boundary value problems (BVPs). The symmetry framework and related conservation law analysis provide invariance properties of the constitutive models, allowing to predict the influence of a variation of the model parameters on the material response or on the solution of BVPs posed over spatial domains. The first part of the paper is devoted to the presentation of the general methodology proposed in this contribution. Examples of construction of rheological models based on experimental data are given for setting up a reduced model of the uniaxial creep and rupture behaviour of a Chrome-Molybdenum alloy (9Cr1Mo) at different temperatures and stress levels. Constitutive equations for creep and rupture master responses are identified for this alloy, and validated based on experimental data. Equivalence transformations are exemplified in the context of parameter reduction in fully nonlinear anisotropic fiber-reinforced elastic solids.

中文翻译：

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本构模型的构造和约简的对称分析和等价变换

提出了一种基于李分析的方法，以研究具有实验主曲线的材料的力学行为。它基于这样的思想，即材料的机械响应与隐式对称性相关联，隐式对称性以能量功能和耗散势的形式反映出来，从而导致在不可逆过程的热力学框架中写出本构定律。在本构模型中，对称性分析使人们可以根据所谓的主曲线来公式化材料的响应，并基于有限数量的测量来构建流变模型。对称方法的应用在双重意义上导致了模型的简化：在处理大量测量数据以将其简化为可用于构造本构模型的形式时，通过利用等价变换扩展点的对称性来有效减少有效参数的数量，从而减少了解决边值问题（BVP）的计算成本。对称框架和相关的守恒律分析提供了本构模型的不变性，从而可以预测模型参数变化对材料响应或空间域上BVP求解的影响。本文的第一部分专门介绍此贡献中提出的一般方法。给出了基于实验数据构建流变模型的示例，以建立在不同温度和应力水平下铬钼合金（9Cr1Mo）的单轴蠕变和断裂行为的简化模型。确定了该合金的蠕变和断裂主响应本构方程，并根据实验数据对其进行了验证。在完全非线性各向异性纤维增强的弹性固体中，在参数减少的情况下例举了等效变换。