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- ItemThe von Mises model vor one-dimensional elastoplastic beams and Prandtl-Ishlinskii hysteresis operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Krejčí, Pavel; Sprekels, JürgenIn this paper, the one-dimensional equation for the transversal vibrations of an elastoplastic beam is derived from a general three-dimensional system. The plastic behavior is modeled using the classical three-dimensional von Mises plasticity model. It turns out that this single-yield model without hardening leads after a dimensional reduction to a multi-yield one-dimensional hysteresis model with kinematic hardening, given by a hysteresis operator of Prandtl-Ishlinskii type whose density function can be determined explicitly. This result indicates that the use of Prandtl-Ishlinskii hysteresis operators in the modeling of elastoplasticity is not just a questionable phenomenological approach, but in fact quite natural. In addition to the derivation of the model, it is shown that the resulting partial differential equation with hysteresis can be transformed into an equivalent system for which the existence and uniqueness of a strong solution is proved. The proof employs techniques from the mathematical theory of hysteresis operators.
- ItemElastoplastic Timoshenko beams(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Krejčí, Pavel; Sprekels, Jürgen; Wu, HaoA Timoshenko type elastoplastic beam equation is derived by dimensional reduction from a general 3D system with von Mises plasticity law. It consists of two second-order hyperbolic equations with an anisotropic vectorial Prandtl--Ishlinskii hysteresis operator. Existence and uniqueness of a strong solution for an initial-boundary value problem is proven via standard energy and monotonicity methods.
- ItemRate independent Kurzweil processes(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Krejčí, Pavel; Liero, MatthiasThe 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 $BV$ 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 one.
- ItemSmall strain oscillations of an elastoplastic Kirchhoff plate(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Guenther, Ronald B.; Krejčí, Pavel; Sprekels, JürgenThe two dimensional equation for transversal vibrations of an elastoplastic plate is derived from a general three dimensional system with a single yield tensorial von Mises plasticity model in the five dimensional deviatoric space. It leads after dimensional reduction to a multiyield three dimensional Prandtl-Ishlinskii hysteresis model whose weight function is explicitly given. The resulting partial differential equation with hysteresis is solved by means of viscous approximations and a monotonicity argument.
- ItemStability results for a soil model with singular hysteretic hydrology(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Krejčí, Pavel; O'Kane, J. Philip; Pokrovskii, Alexei; Rachinskii, DmitriiWe consider a differential equation describing the mass balance in a soil hydrology model with noninvertible Preisach-type hysteresis. We approximate the singular Preisach operator by regular ones and show, as main result, that the solutions of the regularized problem converge to a solution of the original one as the regularization parameter tends to zero. For monotone right hand sides, we prove that the solution is unique. If in addition the external water sources are time periodic, then we find sufficient conditions for the existence, uniqueness, and asymptotic stability of periodic solutions.
- ItemA nonlocal phase-field model with nonconstant specific heat(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Krejčí, Pavel; Rocca, Elisabetta; Sprekels, JürgenWe prove the existence, uniqueness, thermodynamic consistency, global boundedness from both above and below, and continuous data dependence for a strong solution to an integrodifferential model for nonisothermal phase transitions under nonhomogeneous mixed boundary conditions. The specific heat is allowed to depend on the order parameter, and the convex component of the free energy may or may not be singular.
- ItemMagnetohydrodynamic flow with hysteresis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Eleuteri, Michela; Kopfová, Jana; Krejčí, PavelWe consider a model system describing the 2D flow of a conducting fluid surrounded by a ferromagnetic solid under the influence of the hysteretic response of the surrounding medium. We assume that this influence can be represented by the Preisach hysteresis operator. Existence and uniqueness of solutions for the resulting system of PDEs with hysteresis nonlinearities is established in the convexity domain of the Preisach operator.
- ItemAsymptotic behavior of a Neumann parabolic problem with hysteresis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Eleuteri, Michela; Krejčí, PavelA parabolic equation in two or three space variables with a Preisach hysteresis operator and with homogeneous Neumann boundary conditions is shown to admit a unique global regular solution. A detailed investigation of the Preisach memory dynamics shows that the system converges to an equilibrium in the state space of all admissible Preisach memory configurations as time tends to infinity.
- ItemRegularity and uniqueness in quasilinear parabolic systems(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Krejčí, Pavel; Panizzi, LuciaInspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma.
- ItemAsymptotic convergence results for a system of partial differential equations with hysteresis(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Eleuteri, Michela; Krejčí, PavelA partial differential equation motivated by electromagnetic field equations in ferromagnetic media is considered with a relaxed rate dependent constitutive relation. It is shown that the solutions converge to the unique solution of the limit parabolic problem with a rate independent Preisach hysteresis constitutive operator as the relaxation parameter tends to zero.