三次非线性磁致伸缩的完整性基
Integrity bases for cubic nonlinear magnetostriction
摘要：
所谓的智能材料是一种材料，是一个或多个多物理耦合的座位。在这些材料的本构律的发展中，无论是在局部还是在全局尺度上，关键点之一是根据构成材料的晶体的对称性类或代表性体积元素的对称性，在给定的阶数和给定的对称性类下，从矢量、张量中制定自由能量密度（或焓）。本文以立方对称介质中磁弹耦合现象所涉及的应力-磁化耦合（$σ$，m）作为研究的支撑。几项研究确实表明，在应力作用下，某些软磁材料的磁化率和磁致伸缩具有非单调敏感性。模拟这种现象需要在吉布斯自由能量密度中引入二阶应力项。在两个变量的应力和磁化的多项式公式是优选的张量公式。对于给定的材料对称类，这允许更容易地表达在$σ$和m中的任何二度处的自由能量密度（即，在所谓的张量公式的任何本构张量阶处）。一个严格的和系统的方法是必不可少的，以获得高程度的磁-机械耦合项，并建立一个自由的能量密度函数在任何顺序是不变的立方（八面体）集团的作用。为了这个目的，在不变量理论的理论和计算机工具，允许三次非线性磁弹性的数学描述，介绍。然后，在适当的（方向保持）和完整的三次群下，提出了对（m，$σ$）不变代数的最小完整性基。适当的三次群的最小完整性基础由60个不变量组成，而完整的三次群的最小完整性基础（磁弹性感兴趣的一个）由30个不变量组成。这些不变量是制定在一个（坐标自由）内在的方式，使用广义叉积写他们中的一些。在（m，$σ$）中对给定的多次独立不变量进行计数。因此，它表明，它是可能的，没有错误的所有材料参数的描述耦合磁弹性行为的完整性的基础上。该技术被施加到导出的一般表达式$Ψ$ $\star$（$σ$，m）的自由能量密度在磁畴规模表现出立方对称性。恢复了各向同性介质的经典结果。
英文摘要：
A so-called smart material is a material that is the seat of one or more multiphysical coupling. One of the key points in the development of the constitutive laws of these materials, either at the local or at the global scale, is to formulate a free energy density (or enthalpy) from vectors, tensors, at a given order and for a class of given symmetry, depending on the symmetry classes of the crystal constituting the material or the symmetry of the representative volume element. This article takes as a support of study the stress and magnetization couple ($σ$, m) involved in the phenomena of magnetoelastic coupling in a cubic symmetry medium. Several studies indeed show a non-monotonic sensitivity of the magnetic susceptibility and magnetostriction of certain soft magnetic materials under stress. Modeling such a phenomenon requires the introduction of a second order stress term in the Gibbs free energy density. A polynomial formulation in the two variables stress and magnetization is preferred over a tensorial formulation. For a given material symmetry class, this allows to express more easily the free energy density at any bi-degree in $σ$ and m (i.e. at any constitutive tensors order for the so-called tensorial formulation). A rigorous and systematic method is essential to obtain the high-degree magneto-mechanical coupling terms and to build a free energy density function at any order which is invariant by the action of the cubic (octahedral) group. For that aim, theoretical and computer tools in Invariant Theory, that allow for the mathematical description of cubic nonlinear magneto-elasticity, are introduced. Minimal integrity bases of the invariant algebra for the pair (m, $σ$), under the proper (orientation-preserving) and the full cubic groups, are then proposed. The minimal integrity basis for the proper cubic group is constituted of 60 invariants, while the minimal integrity basis for the full cubic group (the one of interest for magneto-elasticity) is made up of 30 invariants. These invariants are formulated in a (coordinate free) intrinsic manner, using a generalized cross product to write some of them. The counting of independent invariants of a given multi-degree in (m, $σ$) is performed. It is shown accordingly that it is possible to list without error all the material parameters useful for the description of the coupled magnetoelastic behavior from the integrity basis. The technique is applied to derive general expressions $Ψ$ $\star$ ($σ$, m) of the free energy density at the magnetic domains scale exhibiting cubic symmetry. The classic results for an isotropic medium are recovered.
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