We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma...
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We prove that for nonlinear elastic energies with strong enough energetic control of the outer distortion of admissible deformations, almost everywhere global invertibility as constraint can be obtained in the $\Gamma$-limit of the elastic energy with an added nonlocal self-repulsion term with asymptocially vanishing coefficient. The self-repulsion term considered here formally coincides with a Sobolev-Slobodeckii seminorm of the inverse deformation. Variants near the boundary or on the surface of the domain are also studied.
Liquid crystal elastomers realize a fascinating new form of soft matter that is a composite of a conventional crosslinked polymer gel (rubber) and a liquid crystal. These solid liquid crystal amalgams, quite similarly...
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Liquid crystal elastomers realize a fascinating new form of soft matter that is a composite of a conventional crosslinked polymer gel (rubber) and a liquid crystal. These solid liquid crystal amalgams, quite similarly to their (conventional, fluid) liquid crystal counterparts, can spontaneously partially break translational and/or orientational symmetries, accompanied by novel soft Goldstone modes. As a consequence, these materials can exhibit unconventional elasticity characterized by symmetry-enforced vanishing of some elastic moduli. Thus, a proper description of such solids requires an essential modification of the classical elasticity theory. In this work, we develop a rotationally invariant, nonlinear theory of elasticity for the nematic phase of ideal liquid crystal elastomers. We show that it is characterized by soft modes, corresponding to a combination of long wavelength shear deformations of the solid network and rotations of the nematic director field. We study thermal fluctuations of these soft modes in the presence of network heterogeneities and show that they lead to a large variety of anomalous elastic properties, such as singular length-scale dependent shear elastic moduli, a divergent elastic constant for splay distortion of the nematic director, long-scale incompressibility, universal Poisson ratios and a nonlinear stress-strain relation for arbitrary small strains. These long-scale elastic properties are universal, controlled by a nontrivial zero-temperature fixed point and constitute a qualitative breakdown of the classical elasticity theory in nematic elastomers. Thus, nematic elastomers realize a stable "critical phase", characterized by universal power-law correlations, akin to a critical point of a continuous phase transition, but extending over an entire phase. (C) 2007 Elsevier Inc. All rights reserved.
We propose Mooney-Rivlin (MR) nonlinear elasticity of hyperelastic materials and numerical algorithms for image registration in the presence of landmarks and large deformation. An auxiliary variable is introduced to r...
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ISBN:
(纸本)9783642143656
We propose Mooney-Rivlin (MR) nonlinear elasticity of hyperelastic materials and numerical algorithms for image registration in the presence of landmarks and large deformation. An auxiliary variable is introduced to remove the nonlinearity in the derivatives of Euler-Lagrange equations. Comparing the MR elasticity model with the Saint Venant-Kirchhoff elasticity model (SVK), the results show that the MR. model gives better matching in fewer iterations. To accelerate the slow convergence due to the lack of smoothness of the L-2 gradient, we construct a Sobolev H-1 gradient descent method [13] and take advantage of the smoothing quality of the Soboley operator (Id - Delta)(-1). The MR model with Sobolev H-1 gradient descent (SGMR) improves both matching criterion and computational time substantially. We further apply the L-2 and Sobolev gradient to landmark registration for multi-modal mouse brain data, and observe faster convergence and better landmark matching for the MR model with Sobolev H-1 gradient descent.
nonlinear elasticity has been observed in different materials and at different scales (from laboratory tests to the crust of the Earth). This behavior is evidenced by nonlinear response accompanied by slow dynamics, a...
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nonlinear elasticity has been observed in different materials and at different scales (from laboratory tests to the crust of the Earth). This behavior is evidenced by nonlinear response accompanied by slow dynamics, a process in which elastic properties recover (fully or partially) values previous to the excitation. In this study we show that nonlinear elasticity also occurs at buildings scale, where nonlinear signatures can be observed even for low levels of strain, and slow dynamics is seen at both short and long-term monitoring. Although the physics behind this response remains unknown, it is believed that this fascinating behavior is controlled by the presence of cracks, heterogeneities and dislocations;and therefore, it might be linked to the level of structural damage. By using data from Japan, we analyze the dynamic response of 24 buildings facing strong motions. The objective is to monitor variations of fundamental frequency over time and correlate them with different levels of structural drift and loading. Analyzing nonlinear elastic behavior might become a very useful and valuable way to monitor the time evolution of structural health, and hopefully, a predictive method to estimate deterioration and likely structural damage over time. (C) 2017 The Authors. Published by Elsevier Ltd.
The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin shell is considered,as the thickness h of the shell tends to *** the appropriate scalings of the applied force and of the i...
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The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin shell is considered,as the thickness h of the shell tends to *** the appropriate scalings of the applied force and of the initial data in terms of h,it’s verified that three-dimesional solutions of the nonlinear elastodynamic equations converge to solutions of the time-dependent von Kármán equations or dynamic linear equations for shell of arbitrary geometry.
This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility const...
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This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses.
We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity. Precisely, we show that if a gradient field is L-1-close to a set of the form SO(n)U-1 boolean OR...boolean OR SO(n)U-l and a...
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We derive a quantitative rigidity estimate for a multiwell problem in nonlinear elasticity. Precisely, we show that if a gradient field is L-1-close to a set of the form SO(n)U-1 boolean OR...boolean OR SO(n)U-l and an appropriate bound on the length of the interfaces holds, then the gradient field is actually close to only one of the wells SO(n)U-i. The estimate holds for any connected subdomain and has the optimal scaling.
Separate displacement preconditioners are studied in the context of outer-inner iterations for a model in 3D nonlinear elasticity. Such a preconditioner, already known to be efficient for linear models, arises as the ...
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Separate displacement preconditioners are studied in the context of outer-inner iterations for a model in 3D nonlinear elasticity. Such a preconditioner, already known to be efficient for linear models, arises as the discretization of three independent Laplacian operators. In this paper the resulting condition number is investigated with focus on independence of parameters. Estimates are given which show that the condition number is uniformly bounded w.r.t. both the studied Newton iterate and the chosen discretization. Finally, it is sketched that ill-conditioning caused by nearly incompressible material parameters can be handled by a suitable mixed formulation. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.
Koiter's shell model is derived systematically from nonlinear elasticity theory, and shown to furnish the leading-order model for small thickness when the bending and stretching energies are of the same order of m...
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Koiter's shell model is derived systematically from nonlinear elasticity theory, and shown to furnish the leading-order model for small thickness when the bending and stretching energies are of the same order of magnitude. An extension of Koiter's model to finite midsurface strain emerges when stretching effects are dominant.
In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and ...
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In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.
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