The limit equilibrium of an anisotropic medium under the general plasticity condition

A theory of the limit equilibrium of an anisotropic medium under the general plasticity condition in the plane strain state is developed. The proposed yield criterion (the limit equilibrium condition) is obtained by combining the von Mises–Hill yield criterion of an ideally plastic anisotropic material and Prandtl's limit equilibrium condition for a medium under the general plasticity law. It is shown that the problem is statically determinate, i.e., if the boundary conditions are specified in stresses, the stress state in plastic region can only be obtained using equilibrium equations. It is established that the equations describing the stress state are hyperbolic and have two families of characteristic curves that intersect at variable angles. In deriving the equations describing the velocity field, the material is assumed to be rigid plastic, and the associated law of flow is applied. It is shown that the equations for the velocities are also hyperbolic, and their characteristic curves are identical with those of the equations for stresses. However, the directions of the principal values of the stress and strain rate tensors are different due to the anisotropy of the material. The characteristic directions differ from the isotropic case in that the normal and tangential components of the stress tensor do not satisfy the limit conditions. It is established that the equations obtained allow of partial solutions, and in this case, at least one family of characteristic curves consists of straight lines. The conditions along the lines of discontinuity of the velocity are investigated, and it is shown that, as in the isotropic case, these are characteristic curves of the system of governing equations. In the anisotropic formulation, the well-known Rankine problem of the limit state of a ponderable layer is solved. From an analysis of the velocity field it is shown that plastic flow of the entire layer is possible only for a slope angle equal to the angle of internal friction. For slope angles less than the angle of internal friction, the solutions obtained are solutions of problems of the pressure of the medium on the retaining walls. The change in this pressure as a function of the parameters of anisotropy is investigated, and turns out to be significant.     

Simulation of thick adhesive layers at finite strains utilizing an associated flow rule in a thermodynamically consistent framework

The application of elastoplastic material models is commonly used for the modelling of adhesive layers with high strength adhesives as realized with polyurethane or epoxy resin. To fulfill thermodynamic consistency often restrictions on the choice of material parameters are requested. One of them is the introduction of a non-associated flow rule, which always ensures positive dissipation. Nevertheless, this assumption is a non-essential criterion, which will be addressed in this work. Continuing along this argumentation, the constitutive relations for the material is modified based on an associated flow rule. The applied model for the simulation of the adhesives is based on a small strain theory. A yield surface including two stress invariants, the hydrostatic pressure as well as the deviator stress state, set the elastic limit of the material response. Linear as well as exponential hardening is incorporated and material softening that arises subsequently is also included by substituting effective invariants in the yield function. This material model as proposed from literature was extended to finite strain application with the concept of generalized stress-strain-measure, which was realized in a previous work. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

双I—型裂纹断裂动力学问题的非局部理论解

研究了非局部理论双中I－型裂纹弹性波散射的力学问题,并利用富里叶变换使本问题的求解转换为三重积分方程的求解,进而采用新方法和利用一维非局部积分核代替二维非局部积分核来确定裂纹尖端的应力状态,这种方法就是Schmidt方法,所得结是比艾林根研究断裂静力学问题的结果准确和更加合理,克服了艾林根研究断裂静力学问题时遇到的数学困难,与经典弹性解相比,裂纹尖端不再出现物理意义下不合理的应力奇异性,并能够解释宏观裂纹与微观裂纹的力学问题。     

Nonlinear analysis and elastic-plastic behavior of anisotropic structures

An elastic-plastic analysis of anisotropic work-hardening materials based on a quadratic approximation of the Tsai-Wu criterion is presented. General expressions for the anisotropic parameters in the yield condition are derived for initial and subsequent yielding. Particularly, the plastic constitutive relations are expressed by means of both the flow theory as well as the deformation theory extended to anisotropic plasticity. The numerical algorithms are based upon the notion of a return mapping procedure and a consistent tangent operator valid for anisotropic elastic-plastic materials including work-hardening effects is developed. The solution equations are evaluated by consistent linearization of a nonlinear variational principle and a Newton-Raphson scheme is adopted for the iterative solution of the nonlinear problems. Numerical examples exhibit the reliable performance of the proposed algorithm in some practical calculations. The effects of anisotropy and the differences between flow and deformation theories in the obtained solutions are discussed and compared with available numerical results.     

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.     

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.     

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.     

A method of differential iteration for boundary value problems in extended thermodynamics

The classical well-posed boundary conditions in Navier–Stokes–Fourier (NSF) theory are usually insufficient for the corresponding problems in extended theories of thermodynamics. Some additional boundary data may be needed for the uniqueness of solutions. Owing to the specific structure of systems of balance equations in extended thermodynamics, no such data will be needed in the proposed iterative method by decoupling the system into two subsystems and solving them alternatively with an iterative procedure. One of them can be solved uniquely with the classical boundary conditions, and the other determines the remaining non-equilibrium field variables by direct evaluation. The method does not rely on any criterion for uniqueness, in contrast to various physical criteria proposed for such problems recently. In Part I, the shearing flow with heat conduction is considered as an illustrative numerical example for the proposed method. The condition for convergence based on the estimated error that can easily be checked in numerical iterations is proved in Part II. Furthermore, stability and uniqueness of the numerical iterative solution are also considered. Additional examples are given to substantiate the main results on convergence, stability and uniqueness.     

On the Convergence of Formal Solutions of a System of Partial Differential Equations

We study a version of the classical problem on the convergence of formal solutions of systems of partial differential equations. A necessary and sufficient condition for the convergence of a given formal solution (found by any method) is proved. This convergence criterion applies to systems of partial differential equations (possibly, nonlinear) solved for the highest-order derivatives or, which is most important, “almost solved for the highest-order derivatives.”     

Variational formulations for discontinuous displacement fields in problems of the deformation theory of plasticity without hardening

The problem of the construction and use of extended variational formulations which enable an explicit analysis to be made of discontinuous displacement fields for a wide class of problems of the deformation theory of plasticity is discussed. Three-dimensional, as well as plane problems with the Mises and Schleicher-Moreau criteria are investigated. In the case of a piecewise-continuous discontinuity line it is shown that the existence of a saddle point of an extended Lagrangian results in an integral inequality, which imposes certain conditions on the trace of the stress tensor on the line of discontinuity. Different arguments were used in [1–3] to obtain different versions of this condition for a number of problems of the theory of plasticity. When sufficient regularity of the stresses is assumed, then from the condition in question a simple algebraic relation follows connecting, at the line of discontinuity, the value of the stress tensor with the parameters determining the magnitude and direction of the discontinuity. Examples are given, which show that, generally speaking, only some of the stress states lying on the yield surface correspond to discontinuous solutions.     

EXPONENTIAL STABILITY FOR NONLINEAR HYBRID STOCHASTIC PANTOGRAPH EQUATIONS AND NUMERICAL APPROXIMATION

The paper develops exponential stability of the analytic solution and convergence in probability of the numerical method for highly nonlinear hybrid stochastic pantograph equation. The classical linear growth condition is replaced by polynomial growth conditions, under which there exists a unique global solution and the solution is almost surely exponentially stable. On the basis of a series of lemmas, the paper establishes a new criterion on convergence in probability of the Euler-Maruyama approximate solution. The criterion is very general so that many highly nonlinear stochastic pantograph equations can obey these conditions. A highly nonlinear example is provided to illustrate the main theory.     

Control of a hyperbolic problem with pointwise stress constraints

Control of a hyperbolic equation with a quadratic criterion in the presence of a pointwise constraint on the stress is considered. Existence and uniqueness of the solution is given. Approximating problems by penalized regularization are studied, and convergence and regularity results of the solutions are given.     

FEM solutions for plane stress mode-I and mode-II cracks in strain gradient plasticity

The strain gradient plasticity theory is used to investigate the crack-tip field in a power law hardening material. Numerical solutions are presented for plane-stress mode I and mode II cracks under small scale yielding conditions. A comparison is made with the existing asymptotic fields. It is found that the size of the dominance zone for the near-tip asymptotic field, recently obtained by Chen et al., is on the order 5% of the intrinsic material lengthI. Remote from the dominance zone, the computed stress field tends to be the classical HRR field. Within the plastic zone only force-stress dominated solution is found for either mode I or mode II crack.     

A continuous model for investigation of physically nonlinear deformation of orthotropic sandwich plates

A phenomenological yield condition for quasi-brittle and plastic orthotropic materials with initial stresses is suggested. All components of the yield tensor are determined from experiments on uniaxial loading. The reliability estimates of the criterion suggested is discussed. For a plastic material without initial stresses, the given condition transforms into the Marin—Hu criterion. The defining equations of the deformation theory of plasticity with isotropic and “anisotropic” hardening, associated with the yield condition suggested, are obtained. These equations are used as the basis for a highly accurate nonclassical continuous model for nonlinear deformation of thick sandwich plates. The approximations with respect to the transverse coordinate take into account the flexural and nonflexural deformations in transverse shear and compression. The high-order approximations allow us to model the occurrence of layer delamination cracks by introducing thin nonrigid interlayers without violating the continuity concept of the theory. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. pp. 329–340, May–June, 2000.     

Adaptive Strategy for the Damping Parameters in an Iteratively Regularized Gauss–Newton Method

In this paper, a convergence analysis of an adaptive choice of the sequence of damping parameters in the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed operator equations is presented. The selection criterion is motivated from the damping parameter choice criteria, which are used for the efficient solution of nonlinear least-square problems. The performance of this selection criterion is tested for the solution of nonlinear ill-posed model problems.     

Explicit expressions for the plastic normality-flow rule associated to the Tresca yield criterion

The Tresca yield criterion is classical and important to the theory of plasticity. It is usually formulated in terms of the difference between the maximum and minimum principal stresses. Difficulties have been encountered in attempts to explicitly express the plastic normality-flow rule associated to it, because the difference between the maximum and minimum principal stresses is generally not differentiable but subdifferentiable with respect to the stress tensor. In this work, the corresponding subdifferential is determined and specified for all possible cases; the explicit mathematical expressions are obtained for the plastic normality-flow rule relative to the Tresca yield criterion.Received: November 30, 2004     

Stress Integration of the Hoek-Brown Material Model Using Incremental Plasticity Theory

This paper presents the formulation of the stress integration procedure for the Hoek-Brown (HB) material model with non-associative yielding condition by using the incremental plasticity method. Main idea of this method is to reach the solution by calculating the plastic matrix according to the method of incremental plasticity (used for elastic constitutive matrix corrections), and with the use of the total strain increment. Computational procedure is implemented within the PAK program package. Results of this procedure were compared with the solutions obtained by other program that contains this material model. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-step-prox-regularization method for solving convex variation problems

The present paper deals with a special scheme of iterative prox-regularization applied to approximation of ill-posed convex variational problems. In distinction to the standard iterative regularization, here for each approximate problem the number of steps of the prox-method is determined within the iteration method by means of a distance criterion between two succeeding iterates. Convergence is proved under conditions which do not contradict the usual organization of discretization methods. Apriori bounds for the distance between the current solutions of the approximate problems and a solution of the original problem are described. That permits to control the number of steps of the pro x-method with the goal to use rough approximations more effectively.Rate of convergence of the minimizing sequence is estimated under the condition that the choice of controlling parameters is suitably regulated during the iteration method. For special classes of ill-posed variational problems a linear rate of convergence W.r.t. the objective functional values and the arguments is established.     

Characterizations of set order relations and constrained set optimization problems via oriented distance function

Set-valued optimization problems are important and fascinating field of optimization theory and widely applied to image processing, viability theory, optimal control and mathematical economics. There are two types of criteria of solutions for the set-valued optimization problems: the vector criterion and the set criterion. In this paper, we adopt the set criterion to study the optimality conditions of constrained set-valued optimization problems. We first present some characterizations of various set order relations using the classical oriented distance function without involving the nonempty interior assumption on the ordered cones. Then using the characterizations of set order relations, necessary and sufficient conditions are derived for four types of optimal solutions of constrained set optimization problem with respect to the set order relations. Finally, the image space analysis is employed to study the c-optimal solution of constrained set optimization problems, and then optimality conditions and an alternative result for the constrained set optimization problem are established by the classical oriented distance function.     

An exact finite deformation elasto-plastic solution for the outside-in free eversion problem of a tube of elastic linear-hardening material

An exact solution is obtained in this paper for the elasto-plasticoutside-in free eversion problem of a tube of elastic linear-hardeningmaterial using a tensorial formulation. The solution is basedon a finite-strain version of Hencky's deformation theory, thevon Mises yield criterion, and the assumptions of volume incompressibilityand axial length constancy. All expressions for the stress,strain distributions and the eversion load are derived in anexplicit form. In addition, with both the linear-elastic andstrain-hardening-plastic responses of the material being includedand with the thickness effect of the tube being incorporated,this solution provides a rigorous and complete theoretical analysisof the elasto-plastic eversion problem, unlike existing solutions.Two specific solutions are also presented as limiting casesof the solution. Also provided are some numerical results andthe related observations to show quantitatively applicationsof the solution.