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Classical and Nonclassical Treatment of Problems in Elastic-Plastic and Creep Deformation for Rotating Discs
Published in Satya Bir Singh, Alexander V. Vakhrushev, A. K. Haghi, Materials Physics and Chemistry, 2020
A. Temesgen, S. B. Singh, Pankaj Thakur
Equations characterizing the individual material and its response to applied loads are called constitutive equations. The macroscopic behavior is described by these equations resulting from the internal constitution of the material. Materials in the solid state behave in such a complex way that when entire range of possible temperatures and deformations is considered, it is not feasible to write down one equation or set of equations to describe accurately a real material over its entire range of behavior. Instead, separate equations are formulated to describe the various kinds of ideal materials response, each of which is a mathematical formulation designed to approximate physical observations of a real material’s response over a suitably restricted range. The classical equations were introduced separately to meet specific needs and made as simple as possible to describe many physical situations. Some of the ideas involved in formulating simple equations for such ideal material are illustrated below.
Fundamental Laws and Equations
Published in G. Thomas Mase, Ronald E. Smelser, Jenn Stroud Rossmann, Continuum Mechanics for Engineers, 2020
G. Thomas Mase, Ronald E. Smelser, Jenn Stroud Rossmann
The global balance laws and resulting field equations developed earlier in this chapter are applicable to all continuous media, but say nothing about the response of specific materials to force or temperature loadings. To fill this need, we introduce constitutive equations which specify the mechanical and thermal properties of particular materials based upon their internal constitution. Mathematically, these constitutive equations are needed to describe the relationships among the kinematic, mechanical, and thermal field equations and to permit the formulations of well-posed problems in continuum mechanics. The constitutive equations define various idealized materials which serve as models for the behavior of real materials. However, it is not possible to write down one equation which is capable of representing a given material over its entire range of application, since many materials behave quite differently under changing levels of loading, such as, elastic-plastic response due to increasing stress. And so in this sense it is perhaps better to think of constitutive equations as being representative of a particular behavior rather than of a particular material.
On the necessity of stochastic material descriptions in the computational analysis of soils
Published in G.N. Pande, S. Pietruszczak, H.F. Schweiger, Numerical Models in Geomechanics, 2020
The relation between the (discretised) random field and the response in quasi-static conditions is, neglecting body forces, governed by the boundary value problem {∇·σ=0inΩ;u=u¯(t)in∂Ω1;σ·n=σ¯(t)on∂Ω2 at each instant t, where σ is the stress tensor, u is the displacement field, ∂Ω1 ⋃ ∂Ω2 = Ω, η is the outward normal vector to ∂Ω. and and are the prescribed boundary displacements and the loading, respectively. The relation between the stress, displacement and material properties is set up by the constitutive equation, which is usually defined in a rate form for plastic materials and in a total form for quasi-brittle (damaging) materials.
An analytical approach for dynamic response of viscoelastic annular sector plates
Published in Mechanics of Advanced Materials and Structures, 2022
Seyed Hashem Alavi, Hamidreza Eipakchi
The constitutive equation for a viscoelastic material represents the stress-strain relation in which the stress and strain are functions of time. Having achieved the constituting equations, it is obligatory to select a rheological model. In general, three classical models called the Maxwell, Kelvin-Voigt, and SLS models are used. In the viscoelastic analysis, it is usual to separate the deviatoric and dilatational parts of the stress components. For the deviatoric part, we have P1τij=Q1γij. P1, Q1 are the viscoelastic operators, τij, γij denote the shear stress and strain. In the elastic case, the shear stress-strain relation is τij=2Gεij, so G = Q1/2P1. In the presented work, we assume that the viscoelastic material obeys the SLS model in shear and elastic in bulk i.e. K = K0 where K0 is a constant (elastic bulk modulus) and it is introduced in Eqs. (23). The viscoelastic operators are expressed as the following [26] (Figure 2): where τ is the relaxation time, D is the time derivative operator. By substituting G, K into Eqs. (7) and applying the time derivative operator on the equations, the governing equations of motion for a viscoelastic sector plate are derived in the general following form:
Effect of cutting edge radius when milling hardened steels: a finite element analysis and surface integrity investigation
Published in Machining Science and Technology, 2022
Mohamd Imad, Hossam A. Kishawy, Nima Z. Yussefian, Ali Hosseini
An accurate representation of the behavior of workpiece material during development of FEA models greatly impacts the accuracy of the results. Material models are presented in terms of constitutive equations. These equations are able to represent the instantaneous flow stress of the workpiece materials during FEA simulations. Due to the complex nature of machining process, material flow stress depends on the strains, strains rates and temperatures experienced by the workpiece during machining. In the 1980s, Gordon R. Johnson and William H. Cook proposed a constitutive material model, which is widely known as Johnson-Cook (J-C) strength model. J-C model is one of the most commonly utilized constitutive equations in machining simulations and it has the ability to represent the behavior of a material subjected to high strains, high strain rates, and high temperatures (Johnson, 1983; Johnson and Cook, 1985). It expresses the behavior of a material during its deformation process by calculating its von Mises flow stress (as shown in equation (1).
Evaluation of in-situ stress state along the shotcrete lined high-pressure headrace tunnel at a complex Himalayan geological condition
Published in Geosystem Engineering, 2021
Chhatra Bahadur Basnet, Krishna Kanta Panthi
In FLAC3D software, the differential equations that describe the mechanical behavior of the rock mass are solved with explicit finite difference method (ITASCA, 2017). The major calculation steps in FLAC3D involve as; (a) nodal forces are calculated from the loading such as, stresses, body forces etc., (b) the equations of motion are used to derive new nodal velocities and displacements, (c) new strain rates are derived from nodal velocities, (d) constitutive equations are used to calculate new stresses from the strain rates and stresses at the previous time, (e) the equations of motion are again used to derive new nodal velocities and displacements from stresses and forces. These calculation steps are repeated at each cycle until the maximum out-of-balance force approach to zero indicating that the system is reaching to an equilibrium state. It is highlighted here that the geometry, material properties, boundary and initial conditions should be well defined before the execution of these calculation steps. The Tamakoshi project area consists of single rock type; i.e. foliated gneiss. For simplicity, the rock mass is considered as a homogeneous and isotropic material even though the rock mass possess some degree of anisotropic behavior because of the developed schistosity. The constitutive equations derived for a linearly elastic model is used where the material is expected to exhibit linear stress-strain behavior. Principally this is correct for the in-situ stress state evaluation of a large area as of UTKHP. Hence, it is believed that the assumptions are representative enough to find the in-situ stress state for a large area.