标题： 连续单轴多晶材料的有效输运谱理论 显示英文标题

标题： Spectral theory of effective transport for continuous uniaxial polycrystalline materials

摘要： 在20世纪80年代早期的开创性工作确立了两相随机介质的均质化输运系数的存在性和表示方法之后，我们建立了一个数学框架，该框架为单轴多晶材料的体输运系数提供了Stieltjes积分表示，涉及自伴随机算子的谱测度，这些算子是确定性算子和随机投影算子的组合。 我们证明同样的数学框架也能够描述双组分复合材料，只需简单地替换随机投影算子，从而使这两个不同物理系统的数学描述直接相互对应。 详细的分析表明，在每种情况下出现的算子确实是在$L^2$类型的希尔伯特空间上自伴的，为Golden和Papanicolaou于1983年建立的形式谱理论框架提供了严格的理论基础。 亥姆霍兹定理的一个抽象扩展也导致了有效参数的逆的积分表示，例如有效电导率和电阻率。 以Sobolev型希尔伯特空间形式表述的有效参数问题为Bergman和Milton首先建立的方法提供了严格的理论基础。 我们证明这两种表述之间是一一对应的等距映射。 从这种Stieltjes积分得出的严格界限以及对材料几何的部分了解，通过多晶材料有效参数的数值计算得到了回顾和验证。 显示英文摘要

摘要： Following seminal work in the early 1980s that established the existence and representations of the homogenized transport coefficients for two phase random media, we develop a mathematical framework that provides Stieltjes integral representations for the bulk transport coefficients for uniaxial polycrystalline materials, involving spectral measures of self-adjoint random operators, which are compositions of non-random and random projection operators. We demonstrate the same mathematical framework also describes two-component composites, with a simple substitution of the random projection operator, making the mathematical descriptions of these two distinct physical systems directly analogous to one another. A detailed analysis establishes the operators arising in each setting are indeed self-adjoint on an $L^2$-type Hilbert space, providing a rigorous foundation to the formal spectral theoretic framework established by Golden and Papanicolaou in 1983. An abstract extension of the Helmholtz theorem also leads to integral representations for the inverses of effective parameters, e.g., effective conductivity and resistivity. An alternate formulation of the effective parameter problem in terms of a Sobolev-type Hilbert space provides a rigorous foundation for an approach first established by Bergman and Milton. We show that the correspondence between the two formulations is a one-to-one isometry. Rigorous bounds that follow from such Stieltjes integrals and partial knowledge about the material geometry are reviewed and validated by numerical calculations of the effective parameters for polycrystalline media.