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The von Mises model vor one-dimensional elastoplastic beams and Prandtl-Ishlinskii hysteresis operators
In 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.
Small strain oscillations of an elastoplastic Kirchhoff plate
The 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.
Magnetohydrodynamic flow with hysteresis
We 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.
Asymptotic behavior of a Neumann parabolic problem with hysteresis
A 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.
Elastoplastic Timoshenko beams
A 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.
Stability results for a soil model with singular hysteretic hydrology
We 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.
Analysis of a tumor model as a multicomponent deformable porous medium
We propose a diffuse interface model to describe tumor as a multicomponent deformable porous medium. We include mechanical effects in the model by coupling the mass balance equations for the tumor species and the nutrient dynamics to a mechanical equilibrium equation with phase-dependent elasticity coefficients. The resulting PDE system couples two Cahn--Hilliard type equations for the tumor phase and the healthy phase with a PDE linking the evolution of the interstitial fluid to the pressure of the system, a reaction-diffusion type equation for the nutrient proportion, and a quasistatic momentum balance. We prove here that the corresponding initial-boundary value problem has a solution in appropriate function spaces.
Rate independent Kurzweil processes
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 $BV$ 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.
A nonlocal phase-field model with nonconstant specific heat
We 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.
A variational inequality for the derivative of the scalar play operator
We show that the directional derivative of the scalar play operator is the unique solution of a certain variational inequality. Due to the nature of the discontinuities involved, the variational inequality has an integral form based on the Kurzweil-Stieltjes integral.