用于纤维增强水凝胶的双组分粘弹性孔隙系统：分析与均匀化

中文摘要 AI 辅助

我们研究组织工程支架所用纤维增强水凝胶（FIHs）建模中产生的耦合粘弹性孔隙系统的多尺度行为。该复合材料由周期性纤维支架和填充间隙空间的水凝胶相组成，纤维支架服从准静态线性弹性，水凝胶相被建模为增强了Kelvin-Voigt结构阻尼的Biot线性孔隙弹性介质。两相通过共享界面处的位移和牵引力连续性耦合。纤维支架与水凝胶相均连通，使机械力可在复合材料中传递，间隙流体可直接流经水凝胶网络。从微观（ε尺度）模型出发，我们推导了一致先验估计，并通过Rothe时间离散化论证，分别针对标准Biot流体含量η=0和粘性流体含量η=αδ>0的情况建立了适定性。随后，我们在ε→0的极限下利用周期展开法进行严格的双尺度均匀化。除常规的有效弹性、储能、耦合和渗透系数外，均匀化本构定律还包含由微观粘弹性松弛产生的非局部时记忆项。所有有效系数和记忆核均根据微观几何与材料参数明确表征。

英文摘要

We study the multiscale behavior of a coupled visco-poroelastic system arising in the modelling of fibre-reinforced hydrogels (FIHs) used in tissue engineering scaffolds. The composite material consists of a periodic fibre scaffold, which is governed by quasi-static linear elasticity, and a hydrogel phase saturating the interstitial space, which is modelled as a Biot linear poroelastic medium enhanced with Kelvin--Voigt structural damping. The two phases are coupled through continuity of displacement and traction across their shared interface. Both the fibre scaffold and the hydrogel phase are connected, so that mechanical forces can be transmitted through the composite and interstitial fluid can flow directly through the hydrogel network. Starting from a microscopic ($\varepsilon$-scale) model, we derive uniform a-priori estimates and establish well-posedness via a Rothe time-discretisation argument for both the case of standard Biot fluid content $η=0$ and the case of viscous fluid content $η=αδ>0$. We then perform a rigorous two-scale homogenization in the limit $\varepsilon \to 0$ using periodic unfolding. In addition to the usual effective elasticity, storage, coupling, and permeability coefficients, the homogenized constitutive laws contain nonlocal-in-time memory terms generated by the microscopic viscoelastic relaxation. All effective coefficients and memory kernels are explicitly characterized in terms of the microscale geometry and material parameters.

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preprint, 51 pages, 3 figures