论文标题
在$γ-$融合锂离子电池的变分模型
On $Γ-$Convergence of a Variational Model for Lithium-Ion Batteries
论文作者
论文摘要
考虑了用于模拟锂离子电池（包括化学和弹性效应）的单一扰动相位场模型。基础能量由$$I_ε[u，c]：= \int_Ω\ left（\ frac {1}εf（c） +ε\ | \ | \ nabla c \ |^2 + \ |^2 + \ frac {1} + $ \ mathbb {c} $具有双重井潜力，是一种对称的正定第四阶张量，$ c $是归一化的锂离子密度，$ u $是材料位移。集成剂包含在流体流体和固体相变理论中产生的能量功能中的元素。对于严格的星形，Lipschitz域$ω\ subset \ mathbb {r}^2，$证明，$γ-\ lim_ {ε\ to 0} i_is = i_0，$ y $ i_0 $在pairs $（u，c）$ f（u，c）$ f（c）$ f（c）= 0 $ e仅$ ef（c）= 0 $ e（u）= 0 $ e（u）= 0 $ e（u）= u）= u u），并且此外，$ i_0 $的特征是与$ c的尖锐界面相比，$ c的尖锐界面上的各向异性界面能量密度不可或缺。
A singularly perturbed phase field model used to model lithium-ion batteries including chemical and elastic effects is considered. The underlying energy is given by $$I_ε[u,c ] := \int_Ω\left( \frac{1}ε f(c) + ε\|\nabla c\|^2 + \frac{1}ε\mathbb{C} (e(u)-ce_0) : (e(u)-ce_0)\right) dx, $$ where $f$ is a double well potential, $\mathbb{C}$ is a symmetric positive definite fourth order tensor, $c$ is the normalized lithium-ion density, and $u$ is the material displacement. The integrand contains elements close to those in energy functionals arising in both the theory of fluid-fluid and solid-solid phase transitions. For a strictly star-shaped, Lipschitz domain $Ω\subset \mathbb{R}^2,$ it is proven that $Γ- \lim_{ε\to 0} I_ε= I_0,$ where $I_0$ is finite only for pairs $(u,c)$ such that $f(c) = 0$ and the symmetrized gradient $e(u) = ce_0$ almost everywhere. Furthermore, $I_0$ is characterized as the integral of an anisotropic interfacial energy density over sharp interfaces given by the jumpset of $c.$
