二维耦合本构律的最小协变量基计算
Computation of minimal covariants bases for 2D coupled constitutive laws
AI总结：
该研究针对多种二维耦合本构律，构建了各向同性与半各向同性的最小协变/不变整性基，提出清理算法实现最小性，给出复形式到张量形式的转换公式，方法可推广至任意n个二维本构张量。
摘要：
我们针对最常见的二维本构张量及可能存在耦合的定律，构建了各向同性和半各向同性不变代数（更广义而言是协变代数）的最小整性基，涉及的定律包括压电定律、光弹性定律、Eshelby张量与弹性张量、复黏弹性张量、Hill弹塑性模型，以及最高到12阶的（全对称）组构张量。文中对协变量这一推广了不变量的概念进行了解释并阐明其应用动机，结果表明它在实际应用中价值更高。我们详细说明了得到这些结果所需的全部工具，并提出了一种清理算法以实现各向同性情形下的最小性。借助文中给出的显式转换公式，不变量和协变量先以复数形式表达，再转换为张量形式。所提出的方法还适用于任意n个二维本构张量的组合。
英文摘要：
We produce minimal integrity bases for both isotropic and hemitropic invariant algebras (and more generally covariant algebras) of most common bidimensional constitutive tensors and -- possibly coupled -- laws, including piezoelectricity law, photoelasticity, Eshelby and elasticity tensors, complex viscoelasticity tensor, Hill elasto-plasticity, and (totally symmetric) fabric tensors up to twelfth-order. The concept of covariant, which extends that of invariant is explained and motivated. It appears to be much more useful for applications. All the tools required to obtain these results are explained in detail and a cleaning algorithm is formulated to achieve minimality in the isotropic case. The invariants and covariants are first expressed in complex forms and then in tensorial forms, thanks to explicit translation formulas which are provided. The proposed approach also applies to any $n$-uplet of bidimensional constitutive tensors.
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