Abstract It is known from experimental findings that three-dimensional effects can have a strong influence on magnetic shape memory behavior. Such phenomena are, however, often neglected in MSMA constitutive models, as they only become meaningful under complex loading conditions. The extensions of our original modeling framework, cf. Bartel et al. (2020) , to include 3D-effects is threefold: (i) vector-valued microstructural variables are now elements in R 3 , i.e. no longer parameterizable in polar coordinates, (ii) a third tetragonal martensite variant may form/vanish by switching from/back into both other variants, and (iii) a more general and robust algorithmic treatment is necessary. The latter includes the implementation of a staggered Augmented Lagrangian scheme to handle the now much larger and numerically more advanced sets of equality and inequality constraints. In this context, two extended model formulations are presented. The first considers a first-order, two-variant laminate approach (rank-one convexification), in which domain magnetizations, interface orientations etc. are now three-dimensional vectors. The second model is based on a convexification approach, for which the incorporation of the third martensitic variant is quite natural. Numerical examples are investigated to test the generalized modeling framework. Firstly, it is confirmed that both extended models recover the solution of the previously established two-dimensional model for a simple loading case. Secondly, response predictions for more complex loading scenarios (non-proportional bi-axial stresses, orthogonal magnetic field), motivated by experiments, are investigated. It is found that capturing the formation, elimination and mutual interaction of all martensitic variants as well as general three-dimensional magnetization vector orientations is of key importance under these conditions. The extended convexification model and modified algorithmic formulation are shown to reliably handle even such general cases.

中文翻译：

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基于能量松弛的磁性形状记忆合金建模框架——均质加载条件下的三维效应模拟

摘要 从实验结果可知，三维效应对磁性形状记忆行为有很强的影响。然而，这些现象在 MSMA 本构模型中经常被忽略，因为它们只有在复杂的负载条件下才有意义。我们原始建模框架的扩展，参见。巴特尔等人。(2020) ，包括 3D 效果有三方面：(i) 向量值微观结构变量现在是 R 3 中的元素，即不再可在极坐标中参数化，(ii) 第三个四方马氏体变体可能通过从/back 到其他两个变体中，并且 (iii) 需要更通用和更强大的算法处理。后者包括实施交错的增广拉格朗日方案，以处理现在更大、数值更先进的等式和不等式约束集。在这种情况下，提出了两个扩展模型公式。第一个考虑一阶、二变体层压方法（秩一凸化），其中畴磁化、界面取向等现在是三维向量。第二个模型基于凸化方法，其中第三种马氏体变体的结合是非常自然的。数值例子被研究以测试广义建模框架。首先，确认两个扩展模型都恢复了先前建立的二维模型对于简单加载情况的解决方案。第二，研究了由实验驱动的更复杂负载情况（非比例双轴应力、正交磁场）的响应预测。发现在这些条件下捕获所有马氏体变体的形成、消除和相互作用以及一般的三维磁化矢量方向至关重要。扩展的凸化模型和修改后的算法公式被证明可以可靠地处理这种一般情况。