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Quasi-static evolution for fatigue debonding

Alessandro Ferriero (2008)

ESAIM: Control, Optimisation and Calculus of Variations

The propagation of fractures in a solid undergoing cyclic loadings is known as the fatigue phenomenon. In this paper, we present a time continuous model for fatigue, in the special situation of the debonding of thin layers, coming from a time discretized version recently proposed by Jaubert and Marigo [C. R. Mecanique333 (2005) 550–556]. Under very general assumptions on the surface energy density and on the applied displacement, we discuss the well-posedness of our problem and we give the main...

Quasistatic frictional problems for elastic and viscoelastic materials

Oanh Chau, Dumitru Motreanu, Mircea Sofonea (2002)

Applications of Mathematics

We consider two quasistatic problems which describe the frictional contact between a deformable body and an obstacle, the so-called foundation. In the first problem the body is assumed to have a viscoelastic behavior, while in the other it is assumed to be elastic. The frictional contact is modeled by a general velocity dependent dissipation functional. We derive weak formulations for the models and prove existence and uniqueness results. The proofs are based on the theory of evolution variational...

Quasistatic problems in contact mechanics

Meir Shillor (2001)

International Journal of Applied Mathematics and Computer Science

We describe some of our recent results concerning the modeling and analysis of quasistatic contact problems between a deformable body and a foundation. We concentrate mainly on frictional contact, and in some of the problems thermal effects and the wear of the contacting surfaces are also taken into account. We describe the physical processes involved, the mathematical models, their variational formulation and then present statements of our results. We conclude with a description of some unresolved...

Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Matteo Negri (2014)

ESAIM: Control, Optimisation and Calculus of Variations

We characterize quasi-static rate-independent evolutions, by means of their graph parametrization, in terms of a couple of equations: the first gives stationarity while the second provides the energy balance. An abstract existence result is given for functionals ℱ of class C1 in reflexive separable Banach spaces. We provide a couple of constructive proofs of existence which share common features with the theory of minimizing movements for gradient flows. Moreover, considering a sequence of functionals...

Rank 1 convex hulls of isotropic functions in dimension 2 by 2

Miroslav Šilhavý (2001)

Mathematica Bohemica

Let f be a rotationally invariant (with respect to the proper orthogonal group) function defined on the set M 2 × 2 of all 2 by 2 matrices. Based on conditions for the rank 1 convexity of f in terms of signed invariants of 𝔸 (to be defined below), an iterative procedure is given for calculating the rank 1 convex hull of a rotationally invariant function. A special case in which the procedure terminates after the second step is determined and examples of the actual calculations are given.

Rate independent Kurzweil processes

Pavel Krejčí, Matthias Liero (2009)

Applications of Mathematics

The Kurzweil integral technique is applied to a class of rate independent processes with convex energy and discontinuous inputs. We prove existence, uniqueness, and continuous data dependence of solutions in B V spaces. It is shown that in the context of elastoplasticity, the Kurzweil solutions coincide with natural limits of viscous regularizations when the viscosity coefficient tends to zero. The discontinuities produce an additional positive dissipation term, which is not homogeneous of degree...

RBF Based Meshless Method for Large Scale Shallow Water Simulations: Experimental Validation

Y. Alhuri, A. Naji, D. Ouazar, A. Taik (2010)

Mathematical Modelling of Natural Phenomena

2D shallow water equations with depth-averaged k−ε model is considered. A meshless method based on multiquadric radial basis functions is described. This methods is based on the collocation formulation and does not require the generation of a grid and any integral evaluation. The application of this method to a flow in horizontal channel, taken as an experimental device, is presented. The results of computations are compared with experimental data...

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