Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.      

A bifurcation-based decohesion model for simulating the transition from localization to decohesion with the MPM

Based on recent experimental observations on the formation of localization before delamination, a bifurcation-based decohesion approach is proposed in this paper to simulate the transition from localization to decohesion involved in the delamination process of compressed films. The onset and orientation of discontinuous failure are identified from the discontinuous bifurcation analysis. A discrete constitutive model is then formulated based on the bifurcation analysis to predict the evolution of material failure as decohesion or separation of continuum. The Material Point Method, that does not employ fixed mesh-connectivity, is developed as a robust spatial discretization method to accommodate the multi-scale discontinuities involved in the film delamination. To demonstrate the potential of the proposed approach, a parametric study is conducted to explore the effects of aspect ratio and failure mode on the evolution of failure patterns under different boundary conditions, which provides a better understanding on the physics behind the film delamination process.Received: January 16, 2003; revised: August 11, 2003     

Strain gradient plasticity solution for an internally pressurized thick-walled cylinder of an elastic linear-hardening material

An elastic-plastic solution is presented for an internally pressurized thick-walled plane strain cylinder of an elastic linear-hardening plastic material. The solution is derived in a closed form using a strain gradient plasticity theory. The inner radius of the cylinder enters the solution not only in non-dimensional forms but also with its own dimensional identity, which differs from that in classical plasticity based solutions and makes it possible to capture the size effect at the micron scale. The classical plasticity solution of the same problem is recovered as a special case of the current solution. To further illustrate the newly derived solution, formulas and numerical results for the plastic limit pressure are provided. These results reveal that the load-carrying capacity of the cylinder increases with decreasing inner radius at the micron scale. It is also seen that the macroscopic behavior of the pressurized cylinder can be well described by using classical plasticity based solutions.     

Concentrators of deformations and fracture of plastic bodies

An approach to the investigation of shape discontinuity regions as strain concentrators is proposed. The near-concentrator strain fields are determined on the basis of the theory of ideal rigid-plastic body; under the condition of plane deformation, their determination is reduced to integration of ordinary differential equations. The deformation as a function of the location of the plastic region and its shape evolution in the process of plastic flow is studied. The plastic flow is demonstrated to be not unique (within the framework of solution completeness). A deformation criterion for the choice of the preferred plastic flow is suggested. The fracture of a V-notched strip is considered. On the basis of the solutions obtained, an approach to the investigation of the fracture processes for more complicated models is formulated.     

Procedures for calculating the nonconvexity measures of a plane set

The geometry of nonconvex sets is analyzed. The measure of nonconvexity of a closed set that has the sense of an angle is considered. Characteristic manifolds of nonconvex sets are constructed. Procedures for calculating the measure of nonconvexity are proposed for a class of plane sets.     

A condensed approach to model the constitutive behavior of multiphase ferroelectric and multiferroic materials

Ferroelectric as well as ferromagnetic materials are widely used in smart structures and devices as actuators, sensors etc. Most of the developed models, describing the nonlinear behavior, are implemented within the framework of the Finite Element Method. Most investigations, however, are restricted to simple boundary value problems under uniaxial or biaxial loading and their goal is the calculation of hysteresis loops or to determine e.g. electromechanical coupling coefficients. Regarding these circumstances, the so-called condensed method (CM) is introduced to investigate the macroscopic polycrystalline ferroelectric material behavior at a macroscopic material point without any kind of discretization scheme. In the presented paper, the CM is extended towards multiphase ferroelectric material behavior. Moreover, first numerical results of a multiphase ferroelectric material at the morphotropic phase boundary are presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermomechanics of twinning from a parent to a twin with mirror image symmetry

In the paper we develop a modeling with multiple configurations and mirror image of parent crystal in the twinned structure, to describe the behavior of partially twinned structure. In the constitutive framework we take into account that: (1) the untwinned and twinned material have distinct natural configurations by virtue of their miscrostructure being different, (2) the material symmetry groups of the untwinned and twinned structures characterize the peculiar feature that the presence of the mirror image structure is related to the untwinned structure, but it can exist only as a counterpart of the previous one. The partially twinned structure is described by the evolution equations for the growth of twins, characterized by a pair of a deformation like tensorial variable and a scalar field with meaning of the volume fraction for the twins. The capability of the material to twin and untwin at a constant rate of strain in uniaxial compression has been analyzed and the oscillatory behavior predicted by the model reveals qualitative agreement with experimental evidences.      

Weak solutions of the interior boundary value problems of plane Cosserat elasticity

In this paper we formulate interior Dirichlet and Neumann boundary value problems of plane Cosserat elasticity in Sobolev spaces, show that these problems are well-posed and find the corresponding weak solutions in the form of integral potentials. Received: April 7, 2005     

An evolutionary game based particle swarm optimization algorithm

Particle swarm optimization (PSO) is an evolutionary algorithm used extensively. This paper presented a new particle swarm optimizer based on evolutionary game (EGPSO). We map particles’ finding optimal solution in PSO algorithm to players’ pursuing maximum utility by choosing strategies in evolutionary games, using replicator dynamics to model the behavior of particles. And in order to overcome premature convergence a multi-start technique was introduced. Experimental results show that EGPSO can overcome premature convergence and has great performance of convergence property over traditional PSO.     

A dual-parametric finite element method for cavitation in nonlinear elasticity

A dual-parametric finite element method is introduced in this paper for the computation of singular minimizers in the 2D cavitation problem in nonlinear elasticity. The method overcomes the difficulties, such as the mesh entanglement and material interpenetration, generally encountered in the finite element approximation of problems with extremely large expansionary deformation. Numerical experiments show that the method is highly efficient in the computation of cavitation problems. Numerical experiments are also conducted on discrete problems without the radial symmetry to show the validity of the method to more general settings and the potential of its application to the study of mechanism of cavity nucleation in nonlinear elastic materials.     

Derivation of the material momentum equation from the energy balance

Based on a recent paper of Yavari et al, we show how the equation of material momentum can be obtained from invariance of the energy balance under translations in the material space. This approach further highlights the duality between the physical and material space.      

Variational formulation of a modified couple stress theory and its application to a simple shear problem

A variational formulation is provided for the modified couple stress theory of Yang et al. by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the boundary conditions, thereby complementing the original work of Yang et al. where the boundary conditions were not derived. Also, the displacement form of the modified couple stress theory, which is desired for solving many problems, is obtained to supplement the existing stress-based formulation. All equations/expressions are presented in tensorial forms that are coordinate-invariant. As a direct application of the newly obtained displacement form of the theory, a simple shear problem is analytically solved. The solution contains a material length scale parameter and can capture the boundary layer effect, which differs from that based on classical elasticity. The numerical results reveal that the length scale parameter (related to material microstructures) can have a significant effect on material responses.      

Analysis of a dynamic unilateral contact problem for a cracked viscoelastic body

This work deals with the mathematical analysis of a dynamic unilateral contact problem with friction for a cracked viscoelastic body. We consider here a Kelvin-Voigt viscoelastic material and a nonlocal friction law. To prove the existence of a solution to the unilateral problem with friction, an auxiliary penalized problem is studied. Several estimates on the penalized solutions are given, which enable us to pass to the limit by using compactness results. Received: February 16, 2005     

Boundary conditions in intrinsic nonlinear elasticity

The intrinsic formulation of the displacement-traction problem of nonlinear elasticity is a system of partial differential equations and boundary conditions whose unknown is the Cauchy–Green strain tensor field instead of the deformation as is customary. We explicitly identify here the boundary conditions satisfied by the Cauchy–Green strain tensor field appearing in such intrinsic formulations.     

Analysis of a dynamic contact problem for electro-viscoelastic cylinders

We consider a mathematical model which describes the antiplane shear deformations of a piezoelectric cylinder in frictional contact with a foundation. The process is mechanically dynamic and electrically static, the material behavior is described with a linearly electro-viscoelastic constitutive law, the contact is frictional and the foundation is assumed to be electrically conductive. Both the friction and the electrical conductivity condition on the contact surface are described with subdifferential boundary conditions. We derive a variational formulation of the problem which is of the form of a system coupling a second order hemivariational inequality for the displacement field with a time-dependent hemivariational inequality for the electric potential field. Then we prove the existence of a unique weak solution to the model. The proof is based on abstract results for second order evolutionary inclusions in Banach spaces. Finally, we present concrete examples of friction laws and electrical conductivity conditions for which our result is valid.     

Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity. Received: June 1, 2005     

Solvability of dynamic antiplane frictional contact problems for viscoelastic cylinders

We study a mathematical model which describes the antiplane shear deformations of a cylinder in frictional contact with a rigid foundation. The process is dynamic, the material behavior is described with a linearly viscoelastic constitutive law and friction is modeled with a general subdifferential boundary condition. We derive a variational formulation of the model which is in a form of an evolutionary hemivariational inequality for the displacement field. Then we prove the existence of a weak solution to the model. The proof is based on an abstract result for second order evolutionary inclusions in Banach spaces. Also, we prove that, under additional assumptions, the weak solution to the model is unique. We complete our results with concrete examples of friction laws for which our results are valid.     

Geometry and asymptotics of wavefronts

Methods of convex analysis and differential geometry are applied to the study of properties of nonconvex sets in the plane. Constructions of the theory of α-sets are used as a tool for investigation of problems of the control theory and the theory of differential games. The notions of the bisector and of a pseudovertex of a set introduced in the paper, which allow ones to study the geometry of sets and compute their measure of nonconvexity, are of independent interest. These notions are also useful in studies of evolution of sets of attainability of controllable systems and in constructing of wavefronts. In this paper, we develop a numerically-analytical approach to finding pseudovertices of a curve, computation of the measure of nonconvexity of a plane set, and constructing front sets on the basis these data.In the paper, we give the results of numerical constructing of bisectors and wavefronts for plane sets. We demonstrate the relation between nonsmoothness of wavefronts and singularity of the geometry of the initial set. We also single out a class of sets whose bisectors have an asymptote.     

General decay rate estimates for viscoelastic dissipative systems

The linear viscoelastic equation is considered. We prove uniform decay rates of the energy by assuming a nonlinear feedback acting on the boundary, without imposing any restrictive growth assumption on the damping term and strongly weakening the usual assumptions on the relaxation function. Our estimate depends both on the behavior of the damping term near zero and on the behavior of the relaxation function at infinity.