Unilateral dynamic contact of von Kármán plates with singular memory

Bock, Igor; Jarušek, Jiřı́

doi:10.1007/s10492-007-0030-5

Cited by 11 publications

(26 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(cf. [1]) hence it satisfies the corresponding requirements and the quadratic forms generated by such defined A , B, E ·, · Q are weakly lower semicontinuous and we are done. We remark that M(u) = b(e 1 m(u) + e 0 m(u)), where m(u) = △u + (1 − ν) 2n 1 n 2 ∂ 12 u − n 2 1 ∂ 22 u − n 2 2 ∂ 11 u .…”

Section: Introduction and Notationmentioning

confidence: 99%

“…And, as well, it is realistic to assume that such a bound cannot be reached. u = Au + Bu − Eu + p(u + g) + f in X on I, D(u) = 0 ∈ Y, u(0) = u 0 ,u(0) = u 1 (1) Here D is a general differential operator of a Dirichlet or somewhat combined type. If X = H 2 (Ω), the space of square integrable functions having the (possibly generalized) first and the second derivatives square integrable as well and A, B are differential operators of the fourth order then D(u) ≡ {D 1 (u), D 2 (u)}, D 1 (u) = u−u 0 for both cases, D 2 (u) = ∂ñ(u−u 0 ) (the outer co-)normal derivative) or D 2 (u) = M(u) a Neumann-type operator, which ensures that after the integration by parts in the space variable in the variational formulation of the problem no additional boundary term occurs.…”

Section: Introduction and Notationmentioning

confidence: 99%

“…Let us denote by ·, · Ω the duality pairing of X and X * derived from the L 2 (Ω) scalar product and by ·, · Q the duality pairing of X and X * derived from the L 2 (Q) scalar product. Let X 0 be a subspace of elements of X satisfying the appropriate homogeneous Dirichlet boundary condition in (1)…”

Section: Introduction and Notationmentioning

confidence: 99%

See 2 more Smart Citations

Solvability of a dynamic rational contact with limited interpenetration for viscoelastic plates

Jarušek

2020

Appl.Math.

Self Cite

View full text Add to dashboard Cite

The solvability of the rational contact with limited interpenetration of different kind of viscolastic plates is proved. The biharmonic plates, von Kármán plates, Reissner-Mindlin plates and full von Kármán systems are treated. The viscoelasticity can have the classical ("short memory") form or the form of a certain singular memory. For all models some convergence of the solutions to the solutions of the Signorini contact is proved provided the thickness of the interpenetration tends to zero.These are the premises of the rational contact model which was introduced by [7] and [8], where the solvability of its static version has been proved. The first dynamic (frictionless) rational contact has been investigated in [9]. It concerns a boundary contact of a body with a foundation.Since 2006 a series of papers about the solvability of dynamic Signorini contact problems for different models of plates [1]-[4] was published. The purpose of this paper is to extend these results to the rational contact with limited interpenetration. Unlike [9] we face here a domain contact.2 Abstract formulation of the problem for the clamped or simply supported viscoelastic plate and the scheme of its solutionLet Ω ⊂ R 2 be a bounded domain with a sufficiently smooth boundary Γ . Let X be a Sobolev-type Hilbert space defined on Ω, let Y be the space of traces of elements from X on Γ . Let A, B : X → X * be two linear symmetric strongly elliptic operators in the form D * aD, D * bD, respectively, where D is a differential operator and a, b are positively definite matrices or tensors of time constant but possibly space-dependent elements. Let I ≡ [0, T ] be a time interval. Here the dual space X * is defined via the suitable generalization of the L 2 (Ω) scalar product. Let X ≡ L 2 (I; X). We introduce the bilinear forms A :We shall denote the elements of v ∈ X or v : I → X such that v ∈ X as displacements, and their first time derivatives (denoted by dots) as velocities. Let γ be a negative real number. Let p : R →R ≡ R ∪ {+∞} be a nonincreasing function such that p(x) = 0 for x ≥ 0, p(x) ∈ R for x > γ, and lim xցγ p(x) = +∞, where γ ∈ R is a given bound of the interpenetration. Our problem is to find u ∈ X such thatu ∈ X for which the following set of relations holds

show abstract

Section: Introduction and Notationmentioning

confidence: 99%

Section: Introduction and Notationmentioning

confidence: 99%

See 1 more Smart Citation

Solvability of a dynamic rational contact with limited interpenetration for viscoelastic plates

Jarušek

2020

Appl.Math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The inner dynamic obstacle problem for a viscoelastic plate with moderately large deflections has been solved in [2]. We deal here with an optimal design problem for a viscoelastic cantilever beam in a dynamic contact on one part of the boundary.…”

Section: Introductionmentioning

confidence: 99%

An optimal design with respect to a variable thickness of a viscoelastic beam in a dynamic boundary contact

Bock¹,

Kečkemétyová²

2011

Tatra Mountains Mathematical Publications

Self Cite

View full text Add to dashboard Cite

show abstract

“…[5], for bodies [7], and for the linear models of plates [6]. The inner obstacle problems for viscoelastic von Kármán plates with a short and a singular long memory have been solved in [1] and [2], respectively. The aim of the present paper is to extend these results to the dynamic full von Kármán system for short memory plates with Signorini conditions for plane displacements on the boundary.…”

Section: Introduction and Notationmentioning

confidence: 99%

Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system

Bock

Jarušek

2010

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

The existence of solutions is proved for a full system of dynamic von Kármán equations expressing vibrations of geometrically nonlinear viscoelastic plate, the viscosity of which has the character of a short memory. The system models the behaviour of a bridge. The in-plane acceleration terms are taken into account. The boundary contact conditions for plane displacements and possibly the contact with the rigid support are considered. (2000). 35L85 · 74D10 · 74K20. Mathematics Subject Classification

show abstract

Unilateral dynamic contact of von Kármán plates with singular memory

Cited by 11 publications

References 7 publications

Solvability of a dynamic rational contact with limited interpenetration for viscoelastic plates

Solvability of a dynamic rational contact with limited interpenetration for viscoelastic plates

An optimal design with respect to a variable thickness of a viscoelastic beam in a dynamic boundary contact

Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system

Contact Info

Product

Resources

About