Edoardo Mainini, Danilo Percivale
We obtain linear elasticity as $Γ$-limit of finite elasticity under incompressibility assumption and Dirichlet boundary conditions. The result is shown for a large class of energy densities for rubber-like materials.
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We consider pure traction problems and we show that incompressible linearized elasticity can be obtained as variational limit of incompressible finite elasticity under suitable conditions on external loads.
We consider the topic of linearization of finite elasticity for pure traction problems. We characterize the variational limit for the approximating sequence of rescaled nonlinear elastic energies. We show that the limiting minimal value can be strictly lower than the minimal value of the standard linear elastic energy if a strict compatibility condition for external loads does not hold. The results are provided for both the compressible and the incompressible case.
We consider Riesz' fractional gradient and a truncated version of it. The equations of nonlocal nonlinear elasticity based on those gradients are known. We perform a formal linearization and arrive at the equations of linear elasticity based on those nonlocal operators. We prove the existence of solutions of the linear equations, notably, by a nonlocal version of Korn's inequality. Finally, we show that the linearizations obtained are particular cases of Eringen's model with singular kernels.
Linearization of finite elasticity with surface tension
针对含表面能的非简单材料非线性弹性问题，提出考虑边界活动表面载荷的模型，并通过 $\Gamma$-收敛建立其线性化模型与原模型的联系。
A new energy functional for pure traction problems in elasticity has been deduced in [23] as the variational limit of nonlinear elastic energy functional for a material body subject to an equilibrated force field: a sort of Gamma limit with respect to the weak convergence of strains when a suitable small parameter tends to zero. This functional exhibits a gap that makes it different from the classical linear elasticity functional. Nevertheless a suitable compatibility condition on the force field ensures coincidence of related minima and minimizers. Here we show some relevant properties of the…
We study non-convex elastic energy functionals associated to (spatially) periodic, frame indifferent energy densities with a single non-degenerate energy well at SO(n). Under the assumption that the energy density admits a quadratic Taylor expansion at identity, we prove that the Gamma-limits associated to homogenization and linearization commute. Moreover, we show that the homogenized energy density, which is determined by a multi-cell homogenization formula, has a quadratic Taylor expansion with a quadratic term that is given by the homogenization of the quadratic term associated to the line…
Linearization of finite-strain poro-visco-elasticity with degenerate mobility
本文在拉格朗日框架下研究有限应变多孔粘弹性非线性模型，利用 Moser 迭代证明浓度有界性，并在小载荷假设下严格证明非线性有限应变系统解收敛于线性小应变系统解。
In this paper, we systematically study the regularity theory of the linear system of nearly incompressible elasticity. In the setting of stochastic homogenization, we develop new techniques to establish the large-scale estimates of displacement and pressure, which are uniform in both the scale parameter and the incompressibility parameter. In particular, we obtain the boundary estimates in a new class of Lipschitz domains whose boundaries are smooth at large scales and bumpy at small scales.
Linearization and Homogenization of nonlinear elasticity close to stress-free joints
本文研究具有周期性微结构和接近无应力接头预应力的超弹性复合材料，证明线性化与均匀化的连续极限和同时极限收敛到同一含均匀化预应力的线性弹性模型，并揭示材料均匀化与预应力均匀化的耦合关系。
We derive geometrically linearized theories for incompressible materials from nonlinear elasticity theory in the small displacement regime. Our nonlinear stored energy densities may vary on the same (small) length scale as the typical displacements. This allows for applications to multiwell energies as, e.g., encountered in martensitic phases of shape memory alloys and models for nematic elastomers. Under natural assumptions on the asymptotic behavior of such densities we prove Gamma-convergence of the properly rescaled nonlinear energy functionals to the relaxation of an effective model. The …
Upper bounds for the homogenization problem in nonlinear elasticity: the incompressible case
该研究针对带不可压缩约束的超弹性模型周期均匀化问题，证明常规均匀化积分泛函给出了周期尺度趋于零时$\text{Γ}$-极限的上界。
A Mixed Finite Element Method for Multi-Cavity Computation in Incompressible Nonlinear Elasticity
本文提出一种结合Q2-P1和P2+-P1参数化元的混合有限元方法，用于不可压缩非线性弹性体多腔计算，证明其锁刚性和收敛性，并通过数值实验验证其准确性和效率，发现多腔生长中的新分岔现象。
We consider a general class of parametrized displacement boundary value problems in incompressible nonlinear elasticity. We prove the existence of an unbounded solution branch of classical injective solutions emanating from the unforced stress-free reference configuration, which constitutes a sharp global implicit function theorem. The ramifications of this are: There exists at least one solution for any given applied loading, and or there exists solutions of arbitrarily large norm for some finite loading. We refer to this as solutions in the large. The nonlinear constraint equation enforcing …
A Locking-free DP-Q2-P1 MFEM for Incompressible Nonlinear Elasticity Problems
本文提出了一种DP-Q2-P1混合有限元方法，用于求解具有大变形梯度的不可压缩非线性弹性问题，该方法通过双参数分段双二次和线性有限元近似变形和压力，采用阻尼牛顿法求解离散问题，并证明了该方法无锁且稳定，通过数值实验验证了其准确性和效率。
Linearization of quasistatic fracture evolution in brittle materials
本文证明非线性弹性准静态断裂演化在刚度趋于无穷时收敛到线性弹性准静态裂纹扩展解，无需裂纹几何先验假设，推广了静态线性化结果至演化情形。
Linearization in incompatible elasticity for general ambient spaces
本研究通过Γ-收敛，针对一般环境空间中的弹性体序列，建立了线性化极限模型，并利用曲率差异验证了弹性能量与空间曲率关系的长期猜想，同时借助Sobolev截断和平行移动解决了位移概念的技术难题。
Homogenization, dimension reduction and linearization of thin elastic plate
本文研究复合薄板在外部载荷下的均匀化、降维与线性化，证明不同极限顺序下极限能量不变，并用Γ-收敛证明极限问题解的唯一性，推广至穿孔板。
Discrete-to-continuum linearization in atomistic dynamics
本文将原子到连续与线性化极限的 Γ-收敛分析从静态推广到动态，证明原子运动方程解收敛到线性化弹性连续体动量方程解，并获得能量的逐时收敛。
Vector and matrix monotonicity characterisation of the "load forces" in nonlinear elasticity: compressible versus incompressible response
本研究针对可压缩与不可压缩超弹性材料，探究不同单调性概念的关系，在正定对称张量流形及对应谱流形上刻画单调性，明确Biot应力等载荷张量的单调性性质。
We study compressible and incompressible nonlinear elasticity variational problems in a general context. Our main result gives a sufficient condition for an equilibrium to be a global energy minimizer, in terms of convexity properties of the pressure in the deformed configuration. We also provide a convex relaxation of the problem together with its dual formulation, based on measure-valued mappings, which coincides with the original problem under our condition.
The paper is devoted to the derivation, by linearization, of simplified (fully homogenized) homogenized models of an immiscible incompressible two-phase flow in double porosity media in the case of thin fissures. In a simplified dual porosity model derived previously by the authors the matrix-fracture source term is approximated by a convolution type source term. This approach enables to exclude the cell problem, in form of the imbibition equation, from the global double porosity model. In this paper we propose a new linear version of the imbibition equation which leads to a new simplified dua…
The rigorous derivation of linear elasticity from finite elasticity by means of Gamma-convergence is a well-known result, which has been extended to different models also beyond the elastic regime. However, in these results the applied forces are usually assumed to be dead loads, that is, their density in the reference configuration is independent of the actual deformation. In this paper we begin a study of the variational derivation of linear elasticity in the presence of live loads. We consider a pure traction problem for a nonlinearly elastic body subject to a pressure live load and we comp…
Under a suitable notion of equivalence of integral densities we prove a $Γ$-closure theorem for integral functionals: The limit of a sequence of $Γ$-convergent families of such functionals is again a $Γ$-convergent family. Its $Γ$-limit is the limit of the $Γ$-limits of the original problems. This result not only provides a common basic principle for a number of linearization and homogenization results in elasticity theory. It also allows for new applications as we exemplify by proving that geometric linearization and homogenization of multi-well energy functionals commute.
In this paper, we prove the local well-posedness of the free boundary problem in incompressible elastodynamics under a natural stability condition, which ensures that the evolution equation describing the free boundary is strictly hyperbolic. Our result gives a rigorous confirmation that the elasticity has a stabilizing effect on the Rayleigh-Taylor instability.
For calculating large deformations in finite elasticity, we have proposed a method of successive linear approximation, by considering the relative descriptional formulation. In this article we briefly describe this method and we prove the existence and uniqueness of weak solutions for boundary value problems for nearly incompressible Mooney-Rivlin materials, that arise in each step of the method.
We study the Boltzmann equation with external forces, not necessarily deriving from a potential, in the incompressible Navier-Stokes perturbative regime. On the torus, we establish local-in-time, for any time, Cauchy theories that are independent of the Knudsen number in Sobolev spaces. The existence is proved around a time-dependent Maxwellian that behaves like the global equilibrium both as time grows and as the Knudsen number decreases. We combine hypocoercive properties of linearized Boltzmann operators with linearization around a time-dependent Maxwellian that catches the fluctuations of …
A general theory of nonlocal elasticity based on nonlocal gradients and connections with Eringen's model
本文提出基于非局部梯度的非局部弹性理论，证明其包含Eringen模型，并在两种情况下展示非局部解向经典局部解收敛。
We consider a class of elasticity equations in ${\mathbb R}^d$ whose elastic moduli depend on $n$ separated microscopic scales, are random and expressed as a linear expansion of a countable sequence of random variables which are independently and identically uniformly distributed in a compact interval. The multiscale Hellinger-Reissner problem that allows for computing the stress directly, and the multiscale mixed problem for nearly incompressible isotropic materials are considered. The stochastic problems are studied via deterministic problems that depend on a countable number of real paramet…
In this paper, we consider multiscale methods for nonlinear elasticity. In particular, we investigate the Generalized Multiscale Finite Element Method (GMsFEM) for a strain-limiting elasticity problem. Being a special case of the naturally implicit constitutive theory of nonlinear elasticity, strain-limiting relation has presented an interesting class of material bodies, for which strains remain bounded (even infinitesimal) while stresses can become arbitrarily large. The nonlinearity and material heterogeneities can create multiscale features in the solution, and multiscale methods are theref…
Diffeological Tangent Spaces and Distributional Linearization for Lifted Euler--Reynolds Limits
该研究为不可压缩欧拉方程的欧拉-雷诺次解构建微分拓扑框架，通过提升型公式构造极限空间，建立一阶理论，引入有限混合模型实现切方向并得到相计数界与极小性结果。
This paper studies the inviscid limit of the two-dimensional incompressible viscoelasticity, which is a system coupling a Navier-Stokes equation with a transport equation for the deformation tensor. The existence of global smooth solutions near the equilibrium with a fixed positive viscosity was known since the work of F. H. Lin, C. Liu, and P. Zhang in "On hydrodynamics of viscoelastic fluids". The inviscid case was solved recently by the second author Z. Lei. in "Global well-posedness of incompressible elastodynamics in two dimensions". While the latter was solely based on the techniques fro…
Asymptotic variational analysis of incompressible elastic strings
本文从不可压缩三维弹性出发，通过Γ-收敛分析推导弦的一维极限模型，揭示体积守恒对能量密度或变形类的影响，并利用内部扰动构造恢复序列以处理非线性约束。
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