Formulation of boundary integral equations for three-dimensional elasto-plastic flow

A general theory of elasto-plastic flow is formulated for work-hardening materials that may be both anisotropic and compressible. Because the theory is quasi-linear, it may be cast in terms of integral equations and the result is an extended form of Somigliana's identity. When these relations are evaluated on the boundary of a solid, their dimensionality is reduced. Previous experience with simpler materials shows that arbitrary problems may be solved in a direct manner.     

Nonlinear elasto-plastic model for dense granular flow

This work proposes a model for granular deformation that predicts the stress and velocity profiles in well-developed dense granular flows. Recent models for granular elasticity [Jiang, Y., Liu, M., 2003. Granular elasticity without the Coulomb condition. Phys. Rev. Lett. 91, 144301] and rate-sensitive fluid-like flow [Jop, P., Forterre, Y., Pouliquen, O., 2006. A constitutive law for dense granular flows. Nature 441, 727] are reformulated and combined into one universal elasto-plastic law, capable of predicting flowing regions and stagnant zones simultaneously in any arbitrary 3D flow geometry. The unification is performed by justifying and implementing a Kröner–Lee decomposition, with care taken to ensure certain continuum physical principles are necessarily upheld. The model is then numerically implemented in multiple geometries and results are compared to experiments and discrete simulations.     

A CONDENSED METHOD FOR LINEAR COMPLEMENTARY EQUATIONS OF ELASTO－PLASTIC PROBLEMS

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On the various forms of the conservation equations in two-phase flow

The conservation equations for one-dimensional two-phase flow are derived from first principles. The effects of the radial distributions of velocities, enthalpies, and void fraction are taken into account through the use of correlation coefficients. Several simplified separated-flow model formulations that have appeared in the literature are derived from these equations by specializing the values of the correlation coefficients. The equivalence of these formulations under certain assumptions is demonstrated. Finally, new Lagrangian forms of the conservation equations, written in terms of the velocities of the center of mass, momentum, and energy are presented.     

On numerical solutions of the time-dependent Euler equations for incompressible flow

This paper presents finite element methods for the non-stationary Euler equations of a two dimensional inviscid and incompressible flow. For the time discretization, we compare numerical results obtained by the use of a leap-frog scheme and a semi-implicit scheme of order two.     

On the evaluation of elasto-plastic strains measured with strain gages

The analysis of plane-stress conditions using strain gages is feasible even when the deformations of the materials are in the elasto-plastic range. The calculation of stresses from the measured strain values is based on an analysis which requires knowledge of the stress-strain curve of the material as obtained from a uniaxial tensile test. In this paper, the practical procedure of such an analysis is described together with the application of a nomograph to allow a simplified evaluation to be made of the results of measurements. A numerical example is also given.     

On the existence of indeterminate solutions to the equations of motion under non-associated flow

Under certain conditions, an indeterminate solution exists to the equations of motion for dynamic elastic–plastic deformation of materials using constitutive laws based on non-associated flow that suggests that an initially unbounded dynamic perturbation in the stress can develop from a quiescent state on the yield surface. The existence of this indeterminate solution has been alleged to discourage use of non-associated flow rules for both dynamic and quasi-static analysis theoretically. It is shown in this paper that the indeterminate solution that may solve the equations of motion is intrinsically dynamic, and it determinately goes to zero in the quasi-static limit regardless of other indeterminate parameters. Consequently, the existence of this unstable dynamic solution has no impact on stability and use of non-associated flow rules for analysis of the quasi-static problem. More importantly, for dynamic applications, it is also shown that the indeterminate solution solves the equations of motion only if critical restrictions are applied to the constitutive equations such that the effective modulus during loading is constant and the direction of the perturbation is unidirectional over a finite time interval. It is shown that common components of the constitutive laws used in metal forming and deformation analysis are inconsistent with these restrictions. So, these common models can be generalized to include non-associated flow for analysis of the dynamic problem without concern that the solution will become indeterminate.     

Extraction of characteristic roots on Bessel-Neumann's mixed equations

This paper discusses the problem of the extraction of characteristic roots {λ_t}(λ₀<λ₁<λ₂…) on Bessel-Neumann’s mixed equations. It gives the expressions and the evaluation of the minimum root. The advantage of the method has no use for the table of the multi-figure number Bessel function and it does not need computer but can calculate all the characteristic roots {λ_t}. The precision of these roots is still high.     

On the structure of characteristic surfaces related to partial differential equations of first and higher orders. II

The generalized method of characteristics is developed within the framework of the geometric Monge picture. Hopf-Lax-type extremality solutions are obtained for a broad class of Cauchy problems for nonlinear partial differential equations of the first and higher orders. A special Hamilton-Jacobi-type case is analyzed separately. An exact extremality Hopf-Lax-type solution of the Cauchy problem for the nonlinear Burgers equation is obtained, and its linearization to the Hopf-Cole expression and to the corresponding Airy-type linear partial differential equation is found and discussed. Published in Neliniini Kolyvannya, Vol. 8, No. 4, pp. 529–543, October–December, 2005.     

Vibrations of the elasto-plastic pendulum

A numerical strategy for vibrations of elasto-plastic beams with rigid-body degrees-of-freedom is presented. Beams vibrating in the small-strain regime are considered. Special emphasis is laid upon the development of plastic zones. An elasto-plastic beam performing plane rotatory motions about a fixed hinged end is used as example problem. Emphasis is laid upon the coupling between the vibrations and the rigid body rotation of the pendulum. Plastic strains are treated as eigenstrains acting in the elastic background structure. The formulation leads to a non-linear system of differential algebraic equations which is solved by means of the Runge-Kutta midpoint rule. A low dimension of this system is obtained by splitting the flexural vibrations into a quasi-static and a dynamic part. Plastic strains are computed by means of an iterative procedure tailored for the Runge-Kutta midpoint rule. The numerical results demonstrate the decay of the vibration amplitude due to plasticity and the development of plastic zones. The pendulum approaches a state of plastic shake-down after sufficient time.     

On the existence of self-similar solutions of the equations governing unsteady flow through a porous medium

In this paper the conditions for the existence of self-similar solutions of the equations governing unsteady flows through a porous medium are presented and discussed. The first two sections deal with the case of non-Newtonian fluids of power-law behavior; the third section analyzes the case of non-Darcy gas flows. The boundary and initial conditions occuring currently in a large class of fluid mechanics problems, of practical interest in engineering, are considered.     

关于 App ell方程1）--分析力学札记之二十八

Appell 方程是非完整系统独具特色的一类方程,凡涉及非完整力学的著作几乎都会介绍Appell方程. 本札记介绍Appell方程的形成和发展,包括Appell的原述,诸多名家对Appell方程的理解和表述,方程的称谓以及方程的后续发展.     

关于Appell方程——分析力学札记之二十八

Appell 方程是非完整系统独具特色的一类方程,凡涉及非完整力学的著作几乎都会介绍Appell方程. 本札记介绍Appell方程的形成和发展,包括Appell的原述,诸多名家对Appell方程的理解和表述,方程的称谓以及方程的后续发展.     

A framework for numerical integration of crystal elasto-plastic constitutive equations compatible with explicit finite element codes

In this paper we summarize the elements of a numerical integration scheme for elasto-plastic response of single crystals. This is intended to be compatible with large-scale explicit finite element codes and therefore can be used for problems involving multiple crystals and also overall behavior of polycrystalline materials. The steps described here are general for anisotropic elastic and plastic response of crystals. The crystallographic axes of the lattice are explicitly stored and updated at each time step. A plastic predictor–elastic corrector scheme is used to calculate the plastic strain rates on all active slip systems based on a rate-dependent physics-based constitutive model without the need of further auxiliary assumptions. Finally we present the results of numerous calculations using a physics-based rate- and temperature-dependent model of copper and the effect of elastic unloading, elastic crystal anisotropy, and deformation-induced lattice rotation are emphasized.