Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions

Herzog, Roland; Meyer, Christian; Wachsmuth, Gerd

doi:10.1016/j.jmaa.2011.04.074

Cited by 49 publications

(63 citation statements)

References 12 publications

(10 reference statements)

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“…A typical differential operator occurring in the theory of Cosserat models is given by the following weak form for

(u, R), (v, Q) \in {double-struckW}_{D}^{1, 2} (normalΩ)

〈 A (\begin{matrix} u \\ R \end{matrix}), (\begin{matrix} v \\ Q \end{matrix}) 〉 = \int_{Ω} 2 μe (u) : \overset{true¯}{e (v)} + λ div u \overset{true¯}{div v} + 2 μ_{c} skew (\nabla u - R) : \overset{true¯}{skew (\nabla v - Q)} + γ \nabla axl R : \overset{true¯}{\nabla axl Q} d x .

If in addition to (A1), the domain is a Lipschitz domain and if for the Lamé‐constants λ , μ , the Cosserat couple modulus μ c and the parameter γ , it holds μ > 0,2 μ + 3 λ > 0, μ c ≥0 and γ > 0, then condition (A2) is satisfied , where also more general situations are discussed. Hence, Theorem 6.2 is applicable. Remark We finally remark that on the basis of the previous example, the results from for nonlinear elasticity models can be extended to the situation discussed here by repeating the arguments in , Section 3].…”

Section: Elliptic Regularity For Systemsmentioning

confidence: 63%

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Haller-Dintelmann

Jonsson

Knees

et al. 2015

Math Methods in App Sciences

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“…A typical differential operator occurring in the theory of Cosserat models is given by the following weak form for

(u, R), (v, Q) \in {double-struckW}_{D}^{1, 2} (normalΩ)

〈 A (\begin{matrix} u \\ R \end{matrix}), (\begin{matrix} v \\ Q \end{matrix}) 〉 = \int_{Ω} 2 μe (u) : \overset{true¯}{e (v)} + λ div u \overset{true¯}{div v} + 2 μ_{c} skew (\nabla u - R) : \overset{true¯}{skew (\nabla v - Q)} + γ \nabla axl R : \overset{true¯}{\nabla axl Q} d x .

Section: Elliptic Regularity For Systemsmentioning

confidence: 63%

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Haller-Dintelmann

Jonsson

Knees

et al. 2015

Math Methods in App Sciences

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“…The regularity of u is given by theorem 1.1 of [25]. The Fréchet differentiability of f γ is done in [26].…”

Section: Differentiability Of the Regularized Formulationmentioning

confidence: 99%

Elasto-plastic Shape Optimization Using the Level Set Method

Maury¹,

Allaire²,

Jouve³

2018

SIAM J. Control Optim.

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This article is concerned with shape optimization of structures made of a material obeying Hencky's laws of plasticity, with the stress bound expressed by the von Mises effective stress. The ill-posedness of the model is circumvented by using two regularized versions of the mechanical problem. The first one is the classical Perzyna formulation which is regularized, the second one is a new regularized formulation proposed for the von Mises criterion. Shape gradients are calculated thanks to the adjoint method. The optimal shape is numerically computed by using the level set method. To illustrate the validity of the method, 2D examples are performed.

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“…Remark 2.6 The critical assumption is Assumption 2.5.1. If N = 2, then this condition is automatically fulfilled, see [18,Lemma 3.2] and [12]. The situation changes however if one turns to N = 3.…”

Section: Notation and Standing Assumptionsmentioning

confidence: 99%

Optimal Control of a Viscous Two‐Field Gradient Damage Model

Susu¹

2018

GAMM-Mitteilungen

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The paper deals with a class of optimal control problems governed by a viscous damage model including two damage variables which are coupled through a penalty term. The existence and directional differentiability of the associated control‐to‐state operator is established. Moreover, necessary optimality conditions are derived. It is shown that these are equivalent to an optimality system, provided that a strict complementarity condition is satisfied.

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Integrability of displacement and stresses in linear and nonlinear elasticity with mixed boundary conditions

Cited by 49 publications

References 12 publications

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

Elasto-plastic Shape Optimization Using the Level Set Method

Optimal Control of a Viscous Two‐Field Gradient Damage Model

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