任意周期介质中线性弹性动力学和弹性静力学的适定均匀化应变梯度模型
Well-posed homogenized strain-gradient models for linear elastodynamics and elastostatics in arbitrary periodic media
AI总结：
研究周期介质中线性弹性动力学和静力学的适定均匀化应变梯度模型，用经典双尺度渐近展开法，经互易恒等式简化计算，引入Boussinesq技巧程序重铸方程，通过特殊情况研究和数值示例验证，能捕捉经典弹性外的效应并保持数学适定性。
AI中文摘要：
本文针对周期介质中的线性弹性静力学和弹性动力学，建立了适定的均匀化应变梯度模型，重点研究弹性波传播。利用经典双尺度渐近展开方法，对\(\mathbb{R}^d\)（\(d = 2, 3\)）中的介质进行二阶周期均匀化，对周期胞元几何形状或材料分布无限制。通过对适当选择的胞元解对应用互易恒等式，得到有效刚度和惯性张量在领先、一阶和二阶的替代表达式，减少了实际需求解的胞元问题数量。由于直接的高阶双尺度均匀化通常会产生不适定的有效算子，引入了一种Boussinesq技巧程序，涉及可调标量权重，将所得四阶偏微分方程重铸为具有所需对称性、符号确定性和强制性的有效应变梯度弹性（SGE）模型。然后通过Hille-Yosida定理建立相应瞬态初值和强迫响应问题的适定性。研究了几个实际相关的特殊情况，包括中心对称胞元、均匀质量密度和均匀弹性，每种情况都产生简化的模型结构。在三个二维周期胞元（正方形、六边形和非中心对称手性晶格）上的数值示例，将所得色散关系与参考Floquet-Bloch计算进行比较，并评估瞬态波传播，证明该模型能够捕捉经典弹性之外的各向异性和色散效应，同时保持数学适定性。
英文摘要：
This work develops well-posed homogenized strain-gradient models for linear elastostatics and elastodynamics in periodic media, with a primary focus on elastic wave propagation. Using the classical two-scale asymptotic expansion method, we carry out second-order periodic homogenization for media in $\mathbb{R}^d$ ($d = 2, 3$), with no restriction on the periodicity cell geometry or material distribution. Reciprocity identities applied to suitably chosen pairs of cell solutions provide alternative expressions for the effective stiffness and inertia tensors arising at the leading, first and second orders, substantially reducing the number of cell problems that must actually be solved. Since direct two-scale homogenization beyond leading order generically yields ill-posed effective operators, a Boussinesq-trick procedure is introduced, involving a tunable scalar weight, to recast the resulting fourth-order partial differential equation as a valid strain-gradient elasticity (SGE) model possessing the requisite symmetry, sign-definiteness and coercivity properties. These properties are then used, via the Hille-Yosida theorem, to establish the well-posedness of the corresponding transient initial-value and forced-response problems. Several practically relevant special cases are examined, including centrosymmetric cells, homogeneous mass density and homogeneous elasticity, each yielding simplified model structures. Numerical illustrations on three two-dimensional periodicity cells (square, hexagonal and a non-centrosymmetric chiral lattice) compare the resulting dispersion relations against reference Floquet-Bloch computations and assess transient wave propagation, demonstrating the model's capacity to capture anisotropic and dispersive effects beyond classical elasticity while preserving mathematical well-posedness.
