A Comprehensive Energy Formulation for General Nonlinear Material Continua

1997 ◽  
Vol 64 (2) ◽  
pp. 353-360 ◽  
Author(s):  
A. Carini ◽  
O. De Donato
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Small Strain ◽  
Constitutive Law ◽  
Nonlinear Material ◽  
Total Potential ◽  
Initial Boundary Value ◽  
Energy Formulation ◽  
The Continuum

By specialization to the continuum problem of a general formulation of the initial/boundary value problem for every nonpotential operator (Tonti, 1984) and by virtue of a suitable choice of the “integrating operator,” a comprehensive energy formulation is established. Referring to the small strain and displacement case in the presence of any inelastic generally nonlinear constitutive law, provided that it is differentiable, this formulation allows us to derive extensions of well-known principles of elasticity (Hu-Washizu, Hellinger-Reissner, total potential energy, and complementary energy). An illustrative example is given. Peculiar properties of the formulation are the energy characterization of the functional and the use of Green functions of the same problem in the elastic range for every inelastic, generally nonlinear material considered.

Computational Modeling of Material Failure

1994 ◽  
Vol 47 (6S) ◽  
pp. S34-S42 ◽  
Author(s):  
A. Needleman
Keyword(s):  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Problem Formulation ◽  
Material Failure ◽  
Structural Metals ◽  
Natural Outcome ◽  
Initial Boundary Value ◽  
The Continuum

Analyses of fracture are discussed where the initial-boundary value problem formulation allows for the possibility of a complete loss of stress carrying capacity, with the associated creation of new free surface. No additional failure criterion is employed so that fracture arises as a natural outcome of the deformation process. Two types of analyses are reviewed. In one case, the material’s constitutive description incorporates a model of the failure mechanism; the nucleation, growth and coalescence of microvoids for ductile fracture in structural metals. In some analyses this is augmented with a simple characterization of failure by cleavage to analyze ductile-brittle transitions. The other class of problems involves specifying separation relations for one or more cohesive surfaces present in the continuum. The emphasis is on reviewing recent work on dynamic failure phenomena and the discussion centers around issues of length scales, size effects and the convergence of numerical solutions.

ON BOUNDARY CONDITIONS FOR FIRST-ORDER SYMMETRIC HYPERBOLIC SYSTEMS WITH CONSTRAINTS

2013 ◽  
Vol 10 (04) ◽  
pp. 725-734 ◽  
Author(s):  
NICOLAE TARFULEA
Keyword(s):  
Boundary Conditions ◽  
Wave Equations ◽  
Hyperbolic Systems ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
First Order ◽  
The Cauchy Problem ◽  
Initial Boundary Value ◽  
Well Posed

The Cauchy problem for many first-order symmetric hyperbolic (FOSH) systems is constraint preserving, i.e. the solution satisfies certain spatial differential constraints whenever the initial data does. Frequently, artificial space cut-offs are performed for such evolution systems, usually out of the necessity for finite computational domains. However, it may easily happen that boundary conditions at the artificial boundary for such a system lead to an initial boundary value problem which, while well-posed, does not preserve the constraints. Here we consider the problem of finding constraint-preserving boundary conditions for constrained FOSH systems in the well-posed class of maximal non-negative boundary conditions. Based on a characterization of maximal non-negative boundary conditions, we discuss a systematic technique for finding such boundary conditions that preserve the constraints, pending that the constraints satisfy a FOSH system themselves. We exemplify this technique by analyzing a system of wave equations in a first-order formulation subject to divergence constraints.

JUSTIFICATION OF HOMOGENIZATION IN VISCOPLASTICITY: FROM CONVERGENCE ON TWO SCALES TO AN ASYMPTOTIC SOLUTION IN L2(Ω)

2009 ◽  
Vol 01 (02) ◽  
pp. 223-244 ◽  
Author(s):  
HANS-DIETER ALBER ◽  
SERGIY NESENENKO
Keyword(s):  
Boundary Value Problem ◽  
Asymptotic Solution ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Material Model ◽  
Small Strain ◽  
Internal Variables ◽  
The Difference ◽  
Initial Boundary Value ◽  
Strong Norm

A homogenized material model can be used effectively for simulation, if the difference of the solutions of this model and the microscopic model converges to zero in a strong norm when the microstructure is scaled. The second author recently showed for the quasistatic initial-boundary value problem with internal variables modeling an inelastic solid body Ω at small strain that this convergence holds in an averaged sense over phase shifts of the microstructure. Based on this result, we construct an asymptotic solution, which converges to the solution of the microscopic problem in the L2(Ω)-norm, thus avoiding the averaging.

Internal Variable Formulations of Problems in Elastoplasticity: Constitutive and Algorithmic Aspects

1994 ◽  
Vol 47 (9) ◽  
pp. 429-456 ◽  
Author(s):  
B. D. Reddy ◽  
J. B. Martin
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Internal Variable ◽  
Solution Algorithm ◽  
Strain Increment ◽  
Constitutive Law ◽  
Discrete Problem ◽  
Initial Boundary Value ◽  
Continuous Problem ◽  
Entire Treatment

This work surveys a broad range of related issues in quasistatic elastoplasticity, beginning with a development of an internal variable constitutive theory. The initial-boundary value problem is then considered, and the remainder of the work is concerned with the properties of the time-discrete problem. It is shown how this discrete problem has associated with it a holonomic constitutive law (that is, one relating stress to strain or strain increment), and this holonomic law in turn forms the basis of a solution algorithm. Conditions for the convergence of the algorithm are discussed. The entire treatment applies to the spatially continuous problem.

Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

1990 ◽  
Vol 115 (1-2) ◽  
pp. 39-59 ◽  
Author(s):  
John A. Nohel ◽  
Robert L. Pego ◽  
Athanasios E. Tzavaras
Keyword(s):  
Shear Stress ◽  
Strain Rate ◽  
Newtonian Fluid ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Steady States ◽  
Constitutive Law ◽  
Shear Strain Rate ◽  
Initial Boundary Value

SynopsisWe study the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain-rate that results in steady states having, in general, discontinuities in the strain rate. We show that every solution tends to a steady state as t → ∞, and we identify steady states that are stable.

scholarly journals On the existence of weak solutions for the initial-boundary value problem in the Jeffreys model of motion of a viscoelastic medium

2004 ◽  
Vol 2004 (10) ◽  
pp. 815-829 ◽  
Author(s):  
D. A. Vorotnikov ◽  
V. G. Zvyagin
Keyword(s):  
Boundary Value Problem ◽  
Weak Solutions ◽  
Viscoelastic Medium ◽  
Dimensional Space ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Constitutive Law ◽  
Existence Of Weak Solutions ◽  
Initial Boundary Value

This paper deals with the initial-boundary value problem for the system of motion equations of an incompressible viscoelastic medium with Jeffreys constitutive law in an arbitrary domain of two-dimensional or three-dimensional space. The existence of weak solutions of this problem is obtained.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

RESEARCH OF THE IMPACT DEVICE MODEL OF THE ROD TYPE BY DIFFERENCE METHOD

2020 ◽  
pp. 73-80
Author(s):  
А.М. Слиденко ◽  
В.М. Слиденко
Keyword(s):  
Boundary Value Problem ◽  
Difference Method ◽  
Model Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Cross Sectional ◽  
Impact Device ◽  
Initial Boundary Value ◽  
The Impact

Приводится анализ механических колебаний элементов ударного устройства с помощью модели стержневого типа. Ударник и инструмент связаны упругими и диссипативными элементами, которые имитируют их взаимодействие. Аналогично моделируется взаимодействие инструмента с рабочей средой. Сформулирована начально-краевая задача для системы двух волновых уравнений с учетом переменных поперечных сечений стержней. Площади поперечных сечений определяются параметрическими формулами при сохранении объемов стержней. Параметрические формулы позволяют получать различного вида зависимости площади поперечного сечения стержня от его длины. Начальные условия отражают физическую картину взаимодействия инструмента с ударником и рабочей средой. Краевые условия описывают контактные взаимодействия ударника с инструментом и последнего с рабочей средой. В качестве модельной задачи рассматривается соударение ударника и инструмента через элемент большой жесткости. Начально-краевая задача исследуется разностным методом. Проводится сравнение решений задачи, полученных с помощью двухслойной и трехслойной разностных схем. Такие схемы реализованы в общей компьютерной программе в системе Mathcad. Показано, что при вычислениях распределения нормальных напряжений по длине стержня лучшими свойствами относительно устойчивости обладает двухслойная схема The article gives the analysis of mechanical vibrations of the impact device elements using the model of the rod type. The hammer and the tool are connected by elastic and dissipative elements that simulate their interaction. The interaction of the tool with the processing medium is simulated in a similar way. An initial boundary-value problem is formulated for a system of two wave equations taking into account the variable cross sections of the rods. Cross-sectional areas are determined by parametric formulas maintaining the volume of the rods. Parametric formulas allow one to obtain various dependence types of the cross-sectional area of the rod on its length. The initial and boundary conditions reflect the physical phenomenon of the tool interaction with the processing medium, and also describe the contact interactions of the hammer with the tool. The impacting of the hammer and the tool through an element of high rigidity is considered as a model problem. To control the limiting values, the solution of the model problem by the Fourier method is used. The initial-boundary-value problem is investigated by the difference method. A comparison of solutions obtained for the two-layer and three-layer difference schemes is given. Such schemes are realized in a common computer program in the Mathcad. It is shown that the two-layer scheme has the best properties in relation to stability while calculating the distribution of normal voltage along the length of the rod

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