非线性磁弹性中作为低能Γ-极限出现的板降维模型
A reduced model for plates arising as low energy $Γ$-limit in nonlinear magnetoelasticity
AI总结：
本文研究非线性磁弹性中板的降维问题，通过计算低能von Kármán体系下磁弹性能的Γ-极限建立降维模型，并证明在外加载荷和耗散作用下近似极小化序列及准静态演化解收敛到该降维模型。
摘要：
我们研究非线性磁弹性中板的降维问题。该模型采用混合欧拉-拉格朗日表述，因为磁化定义在实际空间中变形后的集合上。我们通过按照线性化von Kármán体系重新标度弹性能来考虑低能构型。首先，当板的厚度趋于零时，通过计算磁弹性能的Γ-极限，我们确定了一个降维模型。这将第一作者先前在不可压缩情形下得到的结果推广到了可压缩情形。然后，我们引入由机械力和外部磁场给出的施加载荷，并证明在固支边界条件下，总能量的几乎极小化序列收敛到降维模型中相应能量的极小化元。随后，我们研究由时间相关施加载荷和与速率无关的耗散驱动的准静态演化。我们证明，在体相中近似增量极小化问题的解收敛到降维模型的能量解。这一结果本着演化Γ-收敛的精神为后者提供了进一步的合理性论证。
英文摘要：
We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed set in the actual space. We consider low-energy configurations by rescaling the elastic energy according to the linearized von Kármán regime. First, we identify a reduced model by computing the $Γ$-limit of the magnetoelastic energy, as the thickness of the plate goes to zero. This extends a previous result obtained by the first author in the incompressible case to the compressible one. Then, we introduce applied loads given by mechanical forces and external magnetic fields and we prove that, under clamped boundary conditions, sequences of almost minimizes of the total energy converge to minimizers of the corresponding energy in the reduced model. Subsequently, we study quasistatic evolutions driven by time-dependent applied loads and a rate-independent dissipation. We prove that solutions to the approximate incremental minimization problem at the bulk converge to energetic solutions to the reduced model. This result provides a further justification of the latter in the spirit of evolutionary $Γ$-convergence.
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