Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Knees, Dorothee; Schröder, Jörg

doi:10.1002/mma.2598

Cited by 37 publications

(19 citation statements)

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“…The study of the existence and uniqueness of a solution for frictionless contact and its regularity were performed, for instance, in [15], [9], more recently in [4], [5] and, thanks to the use of pseudo-differential operators, in [54]. Similar results on frictionless auto-contact and auto-contact with Tresca friction can also be found in [40]. However, as soon as a more realistic friction model is taken into consideration, results on existence and uniqueness become harder to obtain [16], [38].…”

Section: Introductionmentioning

confidence: 93%

Shape optimisation with the level set method for contact problems in linearised elasticity

Maury¹,

Allaire²,

Jouve³

2017

The SMAI Journal of Computational Mathematics

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Abstract. This article is devoted to shape optimisation of contact problems in linearised elasticity, thanks to the level set method. We circumvent the shape non-differentiability, due to the contact boundary conditions, by using penalised and regularised versions of the mechanical problem. This approach is applied to five different contact models: the frictionless model, the Tresca model, the Coulomb model, the normal compliance model and the Norton-Hoff model. We consider two types of optimisation problems in our applications: first, we minimise volume under a compliance constraint, second, we optimise the normal force, with a volume constraint, which is useful to design compliant mechanisms. To illustrate the validity of the method, 2D and 3D examples are performed, the 3D examples being computed with an industrial software.Math. classification. 74P05, 75P10, 74P15, 74M10, 74M15, 49Q10, 49Q12, 35J85.

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Section: Introductionmentioning

confidence: 93%

Shape optimisation with the level set method for contact problems in linearised elasticity

Maury¹,

Allaire²,

Jouve³

2017

The SMAI Journal of Computational Mathematics

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“…Чтобы завершить доказательство, остается доказать сильную сходимость ξ k → ξ t * в H(Ω γ ). Подстановкой χ = 2ξ t и χ = 0 в вариационные неравенства (4) для t ∈ (0, T ], находим (13) ξ t ∈ K t , B(ξ t , ξ t ) = В силу эквивалентности норм (см. замечание 1), легко видеть, что ξ k → ξ t * сильно H(Ω γ ) при k → ∞.…”

Section: рис 1 срединная плоскость пластиныunclassified

“…Начиная с 1990-х годов, начали активно разрабатываться задачи теории трещин с условиями непроникания противоположных берегов трещины [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Используя вариационный подход, успешно исследован широкий круг задач о деформировании композитных тел, содержащих жесткие включения см., например, [18,19,20,21,22,23,24].…”

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Optimal control of a thin rigid stiffener for a model describing

Lazarev¹,

Das²,

Grigoryev³

2018

Sib. Elektron. Mat. Izv.

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All Russian mathematical portal N. P. Lazarev, S. Das, M. P. Grigoryev, Optimal control of a thin rigid stiffener for a model describing equilibrium of a Timoshenko plate with a crack, Sib.Èlektron.Abstract. We consider a family of variational problems describing equilibrium of plates containing a crack and rigid thin stiffener on the outer boundary. Nonlinear conditions of the Signorini type on the crack faces are imposed. For this family of problems, we formulate an optimal problem with a control parameter determining the length of the thin rigid stiffener. Meanwhile, a cost functional is specified with the help of an arbitrary continuous functional in the solution space. The existence of the solution to the optimal control problem is proved. We state the continuous dependence of the solutions with respect to the stiffener's size parameter.

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“…(vi) The numerical simulations that we provide in Section 8 for the cohesive fracture evolutions agree with physically relevant requirements, such as the crack initiation criterion (see [6,Theorem 4.6]), which states that a crack appears only when the maximum sustainable stress of the material is reached. We also mention that numerical simulations, obtained instead with the vanishing viscosity approach, have already appeared in [24].…”

Section: 2mentioning

confidence: 99%

Linearly constrained evolutions of critical points and an application to cohesive fractures

Artina

Cagnetti

Fornasier

et al. 2017

Math. Models Methods Appl. Sci.

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We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.Moreover, we say that a measurable function v : [0, T ] → E is an approximable quasistatic evolution with initial condition v 0 and constraint f , if for every t ∈ [0, T ] there exists a sequence δ k → 0 + and a sequence (v δ k ) k∈N of discrete quasistatic evolutions with time step δ k , initial condition v 0 , and constraint f , such thatWe are now ready to state the main results of the paper (see Theorem 2.15 and Theorem 2.16).Dissipative systems. (0)). Under suitable assumptions on J, ψ, A, and f (see Theorem 2.15) we prove that thereis an approximable quasistatic evolution with initial condition v 0 and constraint f ;(C) Energy inequality: the function s → q(s),ḟ (s) F belongs to L 1 (0, T ) andis defined by (2.9) and ·, · F denotes the duality product in F.Any function v ∈ BV ([0, T ]; E) satisfying (A), (B), and (C) is a rate independent evolution. We also remark that, as a consequence of the local stability (B), the energy inequality becomes an equality in all the nontrivial intervals where the solution v(t) is absolutely continuous (see Theorem 2.18, where also the nondissipative case is treated). In addition, in such intervals the doubly nonlinear inclusion A * q(t) ∈ ∂J(v(t)) + ∂ψ(v(t)) is satisfied. It is however a well-known fact that, due to nonconvexity, our solutions can in general develop time discontinuities, where additional dissipation appears (see [27]). Non dissipative systems. When there is no dissipation we can still prove an existence result, although the evolution obtained is in general expected to be less regular in time. This is due to the fact that the absence of dissipation causes loss of compactness, since simple estimates of the total variation of the approximating solutions are now missing. This is undoubtedly a point of great interest in the analysis of the degenerate case (see also [2] for an abstract approach in the unconstrained setting). To compensate the loss of compactness, we need ...

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Global spatial regularity for elasticity models with cracks, contact and other nonsmooth constraints

Cited by 37 publications

References 35 publications

Shape optimisation with the level set method for contact problems in linearised elasticity

Shape optimisation with the level set method for contact problems in linearised elasticity

Optimal control of a thin rigid stiffener for a model describing

Linearly constrained evolutions of critical points and an application to cohesive fractures

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