Moritz Doll, André Froehly, René Schulz
We discuss the determination of the Lamé parameters of an elastic material by the means of boundary measurements. We will combine previous results of Eskin-Ralston and Isakov to prove inverse results in the case of bounded domains with partial data. Moreover, we generalise these results to infinite cylinders.
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We rigorously derive linear elasticity as a low energy limit of pure traction nonlinear elasticity. Unlike previous results, we do not impose any restrictive assumptions on the forces, and obtain a full $Γ$-convergence result. The analysis relies on identifying the correct reference configuration to linearize about, and studying its relation to the rotations preferred by the forces (optimal rotations). The $Γ$-limit is the standard linear elasticity model, plus a term that penalizes for fluctuations of the reference configurations from the optimal rotations. However, on minimizers this additio…
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.
On Some Models in Linear Thermo-Elasticity with Rational Material Laws
我们将在一个共同的结构框架内考虑线性热弹性力学中的一些常见模型。由于结构视角的灵活性，我们将获得一大类广义模型的适定性结果，允许更一般的材料属性，如各向异性，不均匀性等。
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.
Damage as Gamma-limit of microfractures in anti-plane linearized elasticity
均匀化的结果给出了一种材料具有脆性夹杂物安排在一个周期性的结构。根据软度参数与微结构尺寸之间的关系，利用伽玛收敛法推导出三种不同的极限模型。特别地，损伤作为周期性分布的微裂纹的极限而获得。
In this paper, we establish a novel unique continuation property for two-dimensional anisotropic elasticity systems with partial information. More precisely, given a homogeneous elasticity system in a domain, we investigate the unique continuation by assuming only the vanishing of one component of the solution in a subdomain. Using the corresponding Riemann function, we prove that the solution vanishes in the whole domain provided that the other component vanishes at one point up to its second derivatives. Further, we construct several examples showing the possibility of further reducing the a…
Rotational elasticity and couplings to linear elasticity
本文的目的是提出一个新的观点，在非线性设置使用正交矩阵的旋转弹性。在线性近似下，该模型可与已知的静态线弹性平衡方程相比较。确定了一个合适的动能，并提出了一个转动弹性动力学模型。研究了弹性波在这种介质中的传播，发现了两类波：横向旋转波和纵向旋转波，它们都是非线性偏微分方程的解。对于某些参数选择，横波速度可以大于纵波速度。此外，参数范围被确定的模型描述拉胀材料。然而，在所有情况下，势能泛函是正定的。最后，我们将旋转波耦合到线弹性波，研究耦合系统的行为。我们找到了耦合方程的波动解，并可以借助适当的图形将结果可视化。
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.
Stress solution of static linear elasticity with mixed boundary conditions via adjoint linear operators
本文从几何与泛函分析视角研究混合边界条件下静态线性弹性应力问题，通过构造一对无界伴随算子，将应力解表示为基本子空间仿射平移的交集，并利用位移边界条件问题的算子性质进行分析。
We provide an approximation result for the pure traction problem of linearized elasticity in terms of local minimizers of finite elasticity, under the constraint of vanishing average curl for admissible deformation maps. When suitable rotations are included in the constraint, the limit is shown to be the linear elastic equilibrium associated to rotated loads.
Inverse Density Problem for Linear Elasticity: Uniqueness from Local Measurements on a Partially Accessible Boundary
研究弹性系统中的逆边界值问题，证明从局部柯西数据可唯一确定边界处的密度函数及其导数，若密度解析则可进一步确定内部埋藏物、未知边界及边界条件。
Spectral asymptotics for linear elasticity: the case of mixed boundary conditions
本文研究了弹性力学算子在混合边界条件下的谱渐进行为，通过二维和三维显式例子验证了分析和数值结果。
We construct the microlocal solutions of Rayleigh and Stoneley waves in linear elasticity with variable coefficients and a curved boundary (interface). We compute the direction of the polarization and show the retrograde elliptical motion of both of them.
A local-effective relation in planar linear elasticity
我们给出了一个例子的弹性结构的局部和有效属性之间的关系，几何常数。所考虑的模型是平面线弹性中局部弹性张量各向同性且齐次的周期结构。相应的物理模型是具有孔的平面二维体，例如穿孔板。
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。
Consider a three dimensional piecewise homogeneous anisotropic elastic medium $Ω$ which is a bounded domain consisting of a finite number of bounded subdomains $D_α$, with each $D_α$ a homogeneous elastic medium. One typical example is a finite element model with elements with curvilinear interfaces for an ansiotropic elastic medium. Assuming the $D_α$ are known and Lipschitz, we are concerned with the uniqueness in the inverse boundary value problem of identifying the anisotropic elasticity tensor on $Ω$ from a localized Dirichlet to Neumann map given on a part of the boundary $\partial D_{α_…
For homogeneous higher gradient elasticity models we discuss frame-indifference and isotropy requirements. To this end, we introduce the notions of local versus global SO(3)-invariance and identify frame-indifference (traditionally) with global left SO(3)-invariance and isotropy with global right SO(3)-invariance. For specific restricted representations, the energy may also be local left SO(3)-invariant as well as local right SO(3)-invariant. Then we turn to linear models and consider a consequence of frame-indifference together with isotropy in nonlinear elasticity and apply this joint invari…
We consider a linearized inverse boundary value problem for the elasticity system. From the linearized Dirichlet-to-Neumann map at zero frequency, we show that a transversely isotropic perturbation of a homogeneous isotropic elastic tensor can be uniquely determined. From the linearized Dirichlet-to-Neumann map at two distinct positive frequencies, we show that a transversely isotropic perturbation of a homogeneous isotropic density can be identified at the same time.
Uniform W^{1,p} Estimates for Systems of Linear Elasticity in a Periodic Medium
设$\mathcal{L}_ε$是一族具有快速振荡周期系数的线性弹性椭圆方程组。我们得到了狄利克雷问题解在Lipschitz域上的一致$W^{1,p}$估计，其中$(2n/(n+1)) -δ<p<(2n/(n-1))+δ$ .对于$n=2$或3，$p$的范围是尖锐的。
We prove existence and uniqueness for solutions to equilibrium problems for free-standing, traction-free, non homogeneous crystals in the presence of plastic slips. Moreover we prove that this class of problems is closed under G-convergence of the operators. In particular the homogenization procedure, valid for elliptic systems in linear elasticity, depicts the macroscopic features of a composite material in the presence of plastic deformation.
A limit elastic energy for pure traction problem is derived from re-scaled nonlinear energy of an hyperelastic material body subject to an equilibrated force field. We show that the strains of minimizing sequences associated to re-scaled non linear energies weakly converge, up to subsequences, to the strains of minimizers of a limit energy, provided an additional compatibility condition is fulfilled by the force field. The limit energy is different from classical energy of linear elasticity; nevertheless the compatibility condition entails the coincidence of related minima and minimizers. A st…
Mathematical analysis of successive linear approximation for Mooney-Rivlin material model in finite elasticity
对于有限弹性力学中的大变形问题，我们提出了一种考虑相对论列式的逐次线性逼近方法。在这篇文章中，我们简要地介绍了这种方法，我们证明了存在性和唯一性的弱解的边值问题几乎不可压缩的Mooney-Rivlin材料，出现在每一个步骤的方法。
We show how to derive (variants of) Michell truss theory in two and three dimensions rigorously as the vanishing weight limit of optimal design problems in linear elasticity in the sense of $Γ$-convergence. We improve our previous results in that our treatment here includes the three dimensional case and that we allow for more general boundary conditions and applied forces.
Reconstruct Lame parameters of linear isotropic elasticity system
瞬态弹性成像能够检测和表征组织异常。在本文中，我们假设位移是由线性各向同性弹性系统建模，并通过混合方法的第一步得到组织位移。然后，我们重建该系统的Lamé参数从组织位移的知识。我们表明，对于足够大的弹性系统的解决方案和一个开集精心选择的边界条件，$(λ,μ)$可以唯一和稳定地重建。一组精心挑选的边界条件的特点是在适当的复杂的几何光学解决方案。
Asymptotic Expansions for High-Contrast Linear Elasticity
本文研究高对比系数下的线性弹性问题，通过最近引入的渐近极限推导渐近展开，并在H¹范数下分析其收敛性。
It is well-known that classical linear elasticity equations are not form-invariant under local transformations. This is intrinsically related to the inhomogeneity of elastic media. However, the reported new linear elasticity equations for inhomogeneous media may appear in different forms. This paper tries to clarify this issue by investigating the form-invariance of the Lagrangian under local temporal or spatial gauge transformations. In this way, these new equations in different forms can be easily understood as the results from different choices of gauge fixing schemes. It recommends to choo…
From polynomial integrals of Hamiltonian flows to a model of non-linear elasticity
本文证明了Ericksen提出的一个非线性弹性模型中的拟线性系的光滑解的不存在性。该系统是椭圆-双曲混合型的。我们还讨论了这样一个系统与经典哈密顿系统的多项式积分的关系。
Consider the commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ such that %satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. %$\mathbb{B}$ is unique. Let $D$ be a domain in $xOy$, $D_ζ:=\{xe_1+ye_2:(x,y) \in D\}\subset \mathbb{B}$. We say that $\mathbb{B}$-valued function $Φ\colon D_ζ \longrightarrow \mathbb{B}$, $Φ(ζ)=U_{1}\,e_1+U_{2}\,ie_1+ U_{3}\,e_2+U_{4}\,ie_2$, $ζ=xe_1+ye_2$, $U_{k}=U_{k}(x,y)\colon D\longrightarrow \mathbb{R}$, $k=\bar{1,4}$, is {\em monogenic} in $D_ζ$ iff $Φ$ has the classic derivative in every point in $D_…
We investigate and clarify the mathematical properties of linear poro-elastic systems in the presence of classical (linear, Kelvin-Voigt) visco-elasticity. In particular, we quantify the time-regularizing and dissipative effects of visco-elasticity in the context of the quasi-static Biot equations. The full, coupled pressure-displacement presentation of the system is utilized, as well as the framework of implicit, degenerate evolution equations, to demonstrate such effects and characterize linear poro-visco-elastic systems. We consider a simple presentation of the dynamics (with convenient bou…
We first formulate an inverse problem for a linear fractional Lamé system. We determine the Lamé parameters from exterior partial measurements of the Dirichlet-to-Neumann map. We further study an inverse obstacle problem as well as an inverse problem for a nonlinear fractional Lamé system. Our arguments are based on the unique continuation property for the fractional operator as well as the associated Runge approximation property.
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…
In this paper, we deal with the inverse problem of the shape reconstruction of cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement's measurements. For, we consider a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term that penalizes the perimeter of the cavity or inclusion to be identified. Then using a phase-field approach we derive a robust algorithm for the reconstruction of elastic inclusions and of cavities modeled as inclusions with a very small elasticity tensor.
Surface penalization of self-interpenetration in linear and nonlinear elasticity
本文分析了表面自相交惩罚项作为软约束以近似Ciarlet-Nečas条件，证明了线性弹性下受惩罚泛函的收敛性，并扩展至非线性非简单材料模型，数值实验验证了3d自接触问题的结果。
Strong convergence of the solutions of the linear elasticity and uniformity of asymptotic expansions in the presence of small inclusions
本文考虑线弹性的Lamé系统，当夹杂具有极端弹性常数时。我们证明了当剪切模量趋于无穷大（另一个模量，压缩模量是固定的），以及当体积模量和剪切模量趋于零时，Lamé系统的解收敛于适当的$H^1$ -范数.利用这个结果，我们证明了在小夹杂存在时位移向量的渐近展开关于Lamé参数是一致的。
We study the numerical approximation of the inverse scattering problem in the two-dimensional homogeneous isotropic linear elasticity with an unknown linear load given by a square matrix. For both backscattering data and fixed-angle scattering data, we show how to obtain numerical approximations of the so-called Born approximations and propose new iterative algorithms that provide sequences of approximations to the unknown load. Numerical evidences of the convergence for not too large loads are also given.
Embedded corrector problems for homogenization in linear elasticity
本文将嵌入式校正问题从标量扩散方程扩展到线弹性均匀化，通过将有限尺寸非均匀弹性区域嵌入均匀无限介质来构造均匀化张量的有效近似，并证明其收敛性，特别关注各向同性材料。
We consider a singularly-perturbed two-well problem in the context of planar geometrically linear elasticity to model a rectangular martensitic nucleus in an austenitic matrix. We derive the scaling regimes for the minimal energy in terms of the problem parameters, which represent the {shape} of the nucleus, the quotient of the elastic moduli of the two phases, the surface energy constant, and the volume fraction of the two martensitic variants. We identify several different scaling regimes, which are distinguished either by the exponents in the parameters, or by logarithmic corrections, for w…
Since the seminal contribution of Geymonat, Müller, and Triantafyllidis, it is known that strong ellipticity is not necessarily conserved through periodic homogenization in linear elasticity. This phenomenon is related to microscopic buckling of composite materials. Consider a mixture of two isotropic phases which leads to loss of strong ellipticity when arranged in a laminate manner, as considered by Gutiérrez and by Briane and Francfort. In this contribution we prove that the laminate structure is essentially the only microstructure which leads to such a loss of strong ellipticity. We perfor…
This paper concerns with a family of elliptic systems of linear elasticity with rapidly oscillating periodic coefficients, arising in the theory of homogenization. We establish uniform optimal regularity estimates for solutions of Neumann problems in a bounded Lipschitz domain with $L^2$ boundary data. The proof relies on a boundary Korn inequality for solutions of systems of linear elasticity and uses a large-scale Rellich estimate obtained in \cite{Shen-2016}.
Many biological and engineering materials have nonperiodic microstructures for which classical periodic homogenization results do not apply. Certain nonperiodic microstructures may be approximated by locally periodic microstructures for which homogenization techniques are available. Motivated by the consideration that such materials are often anisotropic and can posses residual stresses, a broad class of locally periodic microstructures is considered and the resulting effective macroscopic equations are derived. The effective residual stress and effective elasticity tensor are determined by so…
For the Generalized Plane Stress (GPS) problem in linear elasticity, we obtain an optimal stability estimate of logarithmic type for the inverse problem of determining smooth cavities inside a thin isotropic cylinder {}from a single boundary measurement of traction and displacement. The result is obtained by reformulating the GPS problem as a Kirchhoff-Love plate-like problem in terms of the Airy's function, and by using the strong unique continuation at the boundary for a Kirchhoff-Love plate operator under homogeneous Dirichlet conditions, which has been recently obtained in \cite{l:arv}.
In this article we deduce necessary and sufficient conditions for the presence of `Conti-type', highly symmetric, exactly-stress free constructions in the geometrically non-linear, planar $n$-well problem, generalising results of [CKZ17]. Passing to the limit $n\rightarrow \infty$, this allows us to treat solid crystals and nematic elastomer differential inclusions simultaneously. In particular, we recover and generalise (non-linear) planar tripole star type deformations which were experimentally observed in [MA80,MA80a,KK91]. Further we discuss the corresponding geometrically linearised probl…
This article develops duality principles applicable to the Ginzburg-Landau system in superconductivity. The main results are obtained through standard tools of convex analysis, functional analysis, calculus of variations and duality theory. In the second section, we present the general result for the case including a magnetic field and the respective magnetic potential in a local extremal context. Finally, in the last section we develop some global existence results for a model in elasticity.
Uniform rectifiability of brittle fractures in linear elasticity
本文研究了线弹性中脆性断裂的均匀可 rectifiability，通过证明裂缝具有大量大投影，提供了一种新的方法来克服现有方法在Griffith设定中的限制。
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…
In this paper we study Steklov eigenvalues for the Lamé operator which arise in the theory of linear elasticity. In this eigenproblem the spectral parameter appears in a Robin boundary condition, linking the traction and the displacement. To establish the existence of a countable spectrum for this problem, we present an extension of Korn's inequality. We also show that a proposed conforming Galerkin scheme provides convergent approximations to the true eigenvalues. A standard finite element method is used to conduct numerical experiments on 2D and 3D domains to support our theoretical findings…
We study an inverse problem for fractional elasticity. In analogy to the classical problem of linear elasticity, we consider the unique recovery of the Lamé parameters associated to a linear, isotropic fractional elasticity operator from fractional Dirichlet-to-Neumann data. In our analysis we make use of a fractional matrix Schrödinger equation via a generalization of the so-called Liouville reduction, a technique classically used in the study of the scalar conductivity equation. We conclude that unique recovery is possible if the Lamé parameters agree and are constant in the exterior, and th…
Homogenized moderately wrinkled shell theory from 3D Koiter's linear elasticity
本文通过双重尺度收敛方法，基于三维线弹性推导出周期性皱褶壳体模型，采用Koiter模型描述应变能。
A Rigidity Result for a Reduced Model of a Cubic-to-Orthorhombic Phase Transition in the Geometrically Linear Theory of Elasticity
本文研究了形状记忆合金中的一个二维相变简化模型。在低温状态下，线性弹性理论预测了马氏体排列的各种可能模式：除了众所周知的层压结构外，这种相变还显示了涉及四种马氏体变体的额外结构-所谓的交叉孪晶。引入一个包括表面能的变分模型，我们证明了这些结构在小的能量扰动下是刚性的。结合一个上界构造，给出了非相容微结构的最优标度行为。这些结果与卡佩拉和奥托的论文以及多尔兹曼和穆勒的论文有关。
Convergence Rates in Periodic Homogenization of Systems of Elasticity
本文研究具有快速振荡周期系数的线弹性力学系统的均匀化问题。对于可测系数有界的混合边值问题，我们建立了$L^2$的快速收敛速度.
The energy scaling behaviour of singular perturbation models of staircase type in linearized elasticity for higher order laminates
该研究在几何线性化弹性理论中探究了含高阶层合数据的阶梯型奇异摄动模型的能量标度行为，揭示了其下限标度界依赖于边界数据层合阶数和（非）退化对称秩一方向数量这两个参数，并验证了下限估计的尖锐性。
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。
It is shown that in core-radius cutoff regularized simplified elasticity (where the elastic energy depends quadratically on the full displacement gradient rather than its symmetrized version), the force on a dislocation curve by the negative gradient of the elastic energy asymptotically approaches the mean curvature of the curve as the cutoff radius converges to zero. Rigorous error bounds in Hölder spaces are provided. As an application, convergence of dislocations moving by the gradient flow of the elastic energy to dislocations moving by the gradient flow of the arclength functional, when t…
Thin-walled beams with a cross-section of arbitrary geometry: derivation of linear theories starting from 3D nonlinear elasticity
本文严格推导了截面由平滑曲线的薄壁管邻域给出的非线性弹性薄壁梁的低维模型。用h和δ_h分别表示截面的直径和厚度，我们分析了弹性能的标度因子为ε_h^2的情况，其中ε_h/δ_h^2 \rightarrow l \in [0，+ \infty）.根据δ h相对于h的相对数量级，推导出不同的线性化模型。
Heat kernel for the elliptic system of linear elasticity with boundary conditions
本文考虑在第二个Korn不等式成立的区域中具有有界可测系数的椭圆型线性弹性力学系统。在假设椭圆型方程组的弱解在内部是Hölder连续的条件下，构造了Dirichlet、Neumann或混合边界条件下的热核.此外，我们证明了如果混合问题的弱解在边界上是Hölder连续的，那么相应的热核有Gauss界.特别是，如果该域是一个二维Lipschitz域满足螺旋形或非切向的可达性条件的集合上，我们指定Dirichlet边界条件，那么我们证明了热核有一个高斯界。作为应用，我们构造了椭圆型混合问题的格林函数。
A new formulation of boundary value problems in gradient elasticity is presented in this work. The main outcome is the construction of partial differential systems of second order, which are typically equivalent with the well known fourth order equation of gradient elasticity. Two alternative methodologies are developed and presented in the present work. The first approach is based on the framework of the pseudo-differential calculus and exploits the special characteristics of this approach to deal with the non local behavior of gradient elasticity. The second implementation is purely differen…
Two-term spectral asymptotics in linear elasticity
该研究在任意维光滑紧黎曼流形上建立线性弹性边值问题的两项谱渐近结论，给出示例与主题历史回顾，并纠正了已发表的相关错误结果。
Spectral asymptotics for linear elasticity with mixed boundary conditions
本文指出Capoferri和Mann给出的线性弹性算子两项谱渐近实为前人已知结果，并纠正了相关文献中一般公式与实例计算的错误。
We reconsider the minimization of the compliance of a two dimensional elastic body with traction boundary conditions for a given weight. It is well known how to rewrite this optimal design problem as a nonlinear variational problem. We take the limit of vanishing weight by sending a suitable Lagrange multiplier to infinity in the variational formulation. We show that the limit, in the sense of $Γ$-convergence, is a certain Michell truss problem. This proves a conjecture by Kohn and Allaire.
Non-local approximation of free-discontinuity problems in linear elasticity and application to stochastic homogenisation
本文分析了广义非局部卷积型泛函的Γ-收敛性，证明其极限为局部自由间断泛函，并通过渐近胞元公式刻画体积项，进而推导出随机设定下的均匀化结果。
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