标题： 刚性和$\mathrm{BD}_{dev}(Ω)$的功能属性 显示英文标题

标题： Rigidity and functional properties of $\mathrm{BD}_{dev}(Ω)$

摘要： 我们对有界偏量变形函数空间$\mathrm{BD}_{dev}$进行了结构分析，该空间出现在塑性和流体力学模型中。 主要结果是确定了波锥中具有常数极向量的$\mathrm{BD}_{dev}$-映射的零化子，并证明了一个刚性定理。 这一结果，结合一个显式的核投影算子，使得可以进行松弛和均匀化问题的迭代膨胀过程，允许积分函数显式依赖于$u$和$\mathcal{E}_d u$。 与$\mathrm{BD}$情况相比，我们的方法克服了许多困难，特别是由于$\mathcal{E}_d$在极向量正交化下缺乏不变性。 讨论了在积分表示和材料科学中的应用。 显示英文摘要

摘要： We provide a structural analysis of the space of functions of bounded deviatoric deformation, $\mathrm{BD}_{dev}$, which arises in models of plasticity and fluid mechanics. The main result is the identification of the annihilator and a rigidity theorem for $\mathrm{BD}_{dev}$-maps with constant polar vector in the wave cone characterizing the structure of singularities for such maps. This result, together with an explicit kernel projection operator, enables an iterative blow-up procedure for relaxation and homogenization problems, allowing for integrands with explicit dependence on $u$ as well as $\mathcal{E}_d u$. Our approach overcomes several difficulties as compared to the $\mathrm{BD}$ case, in particular due to the lack of invariance of $\mathcal{E}_d$ under orthogonalization of the polar directions. Applications to integral representation and Material science are discussed.