The present work investigates the thermodynamic inconsistencies in the definition of the compatibility conditions on stress for constant and nonconstant material functions in one-dimensional modeling of shape memory alloys based on the first principles. In this work, simplifications are provided validating inconsistencies in the earlier proposed non-constant material functions used to satisfy compatibility conditions. It is presented that the inconsistencies originate due to an incorrect definition of the compatibility conditions on stress. In the first step, it is shown that, due to inconsistent definitions of the compatibility conditions, the material functions cannot be derived from the first principles. Consequently, it is presented that the material functions result in an incorrect form of the differential constitutive equation. Furthermore, it is also analyzed that these incorrect definitions on the compatibility conditions result in an inconsistent form of nonconstant material functions as well as the differential equation, which are proposed in earlier models. As a result, in the present work the consistent definition of the compatibility conditions for one-dimensional shape memory alloy models is derived. Next, the new and correct definition for the compatibility conditions is proposed, which is used to derive a new and consistent form of nonconstant material function. Finally, a consistent form of non-constant material function and differential equation are derived from first principles, which satisfy the new definition of compatibility conditions on stress.

中文翻译：

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形状记忆合金一维本构关系的相容性条件和非恒定材料函数的推导

本工作研究基于第一原理的形状记忆合金一维建模中恒定和非恒定材料功能的应力相容性条件定义中的热力学不一致。在这项工作中，提供了简化方法来验证用于满足兼容性条件的较早提出的非恒定材料函数中的不一致。结果表明，不一致是由于对应力的相容性条件定义不正确所致。第一步表明，由于兼容性条件的定义不一致，因此不能从第一原理中导出材料功能。因此，提出了物质功能导致微分本构方程的形式不正确。此外，还分析了这些对相容性条件的不正确定义会导致非恒定材料函数以及微分方程的不一致形式，这是早期模型中提出的。结果，在本工作中，得出了一维形状记忆合金模型的相容条件的一致定义。接下来，提出了对相容性条件的新的正确定义，该定义用于导出新的一致形式的非恒定物质函数。最后，从第一原理推导了非恒定材料函数和微分方程的一致形式，它们满足了应力相容性条件的新定义。