RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE

IntroductionTherelationsbetweenvariousstresstensors,theequationofmomentumandtheboundaryconditionsofincrementalratetypeforclassicalcontinuummechanicshavebeensystematicallyderivedbyKuanginRef.[1 ] .InRef.[2 ]wehavederivedtherelationsbetweenvariouscouplestresstensorsandtheirratesandpresentedtheequationsofmotionandtheboundaryconditionsofincrementalratetypeofCauchyform ,PiolaformandKirchhoffformforpolarcontinua .InRef.[3 ]wehavepresentednewprinciplesofpowerandenergyrateofincrementalratetypeformi…     

On the Use of the Continuum Mechanics Method for Describing Interactions in Discrete Systems with Rotational Degrees of Freedom

Elastic interactions in a system of two body-points possessing both translational and rotational degrees of freedom are studied for the most general case of motion in 3D space. The continuum mechanics method is used as a theoretical foundation for describing the interactions. A definition of strain measures for the discrete system is given by analogy with that in continuum mechanics. Constitutive equations for force and moment vectors are derived based on the energy balance equation. Several new interaction potentials are suggested.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES(Ⅶ)—INCREMENTAL RATE TYPE

The purpose is to establish the rather complete equations of motion, boundary conditions and equation of energy rate of incremental rate type for micropolar continua. To this end the rather complete definitions for rates of deformation gradient and its inverse are made. The new relations between various stress and couple stress rate tensors are derived. Finally, the coupled equations of motion, boundary conditions and equation of energy rate of incremental rate type for continuum mechanics are obtained as a special case. Contributed by Dai Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: Dai Tian-min (1931≈)     

Modeling of properties and motions of an inhomogeneous one-dimensional continuum of a complicated Cosserat-type microstructure

We consider an approach to modeling the properties of the one-dimensional Cosserat continuum [1] by using the mechanical modeling method proposed by Il’yushin in [2] and applied in [3]. In this method, elements (blocks, cells) of special form are used to develop a discrete model of the structure so that the average properties of the model reproduced the properties of the continuum under study. The rigged rod model, which is an elastic structure in the form of a thin rod with massive inclusions (pulleys) fixed by elastic hinges on its elastic line and connected by elastic belt transmissions, is taken to be the original discrete model of the Cosserat continuum. The complete system of equations describing the mechanical properties and the dynamical equilibrium of the rigged rod in arbitrary plane motions is derived. These equations are averaged in the case of a sufficiently smooth variation in the parameters of motion along the rod (the long-wave approximation). It was found that the average equations exactly coincide with the equations for the one-dimensional Cosserat medium [1] and, in some specific cases, with the classical equations of motion of an elastic rod [4–6]. We study the plane motions of the one-dimensional continuum model thus constructed. The equations characterizing the continuum properties and motions are linearized by using several assumptions that the kinematic parameters are small. We solve the problem of natural vibrations with homogeneous boundary conditions and establish that each value of the parameter distinguishing the natural vibration modes is associated with exactly two distinct vibration mode shapes (in the same mode), each of which has its own frequency value.     

Gradient elasticity and nonstandard boundary conditions

Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain energy is shown to include a bulk and surface strain energy. A minimum total potential energy principle is provided for the related structural boundary-value problem. The Toupin–Mindlin polar-type strain gradient material model is also addressed and compared with the above one, their substantial differences are pointed out, particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations. A gradient one-dimensional bar sample in tension is considered for a few applications of the proposed theory.     

A gradient elasticity theory for second-grade materials and higher order inertia

Second-grade elastic materials featured by a free energy depending on the strain and the strain gradient, and a kinetic energy depending on the velocity and the velocity gradient, are addressed. An inertial energy balance principle and a virtual work principle for inertial actions are envisioned to enrich the set of traditional theoretical tools of thermodynamics and continuum mechanics. The state variables include the body momentum and the surface momentum, related to the velocity in a nonstandard way, as well as the concomitant mass-accelerations and inertial forces, which do intervene into the motion equations and into the force boundary conditions. The boundary traction is the sum of two parts, i.e. the Cauchy traction and the Gurtin–Murdoch traction, whereas the traction boundary condition exhibits the typical format of the equilibrium equation of a material surface (as known from the principles of surface mechanics) whereby the Gurtin–Murdoch traction (incorporating the inertial surface force) plays the role of applied surfacial force density. The body’s boundary surface constitutes a thin boundary layer which is in global equilibrium under all the external forces applied on it, a feature that makes it possible to exploit the traction Cauchy theorem within second-grade materials. This means that a second-grade material is formed up by two sub-systems, that is, the bulk material operating as a classical Cauchy continuum, and the thin boundary layer operating as a Gurtin–Murdoch material surface. The classical linear and angular momentum theorems are suitably extended for higher order inertia, from which the local motion equations and the moment equilibrium equations (stress symmetry) can be derived. For an isotropic material featured by four constants, i.e. the Lamé constants and two length scale parameters (Aifantis model), the dynamic evolution problem is characterized by a Hamilton-type variational principle and a solution uniqueness theorem. Closed-form solutions of the wave dispersion analysis problem for beam models are presented and compared with known results from the literature. The paper indicates a correct thermodynamically consistent way to take into account higher order inertia effects within continuum mechanics.     

A couple-stress theory for gridwork-reinforced media

A two-dimensional continuum model with couple stress for a gridwork-reinforced composite is developed. The derivation is based on the calculation of equivalent potential and kinetic energies stored in the representative medium. Hamilton's principle is used to derive the equations of motion and the boundary conditions. Deformation variables are defined and the constitutive relations are subsequently derived. The in-plane transverse vibration problem is investigated as an evaluation example for which both the continuum approach and the discrete element method are employed to compute the frequencies. Numerical results show that solutions according to both methods agree reasonably well.     

Analytical solutions for buckling of size-dependent Timoshenko beams

The inconsistences of the higher-order shear resultant expressed in terms of displacement(s) and the complete boundary value problems of structures modeled by the nonlocal strain gradient theory have not been well addressed. This paper develops a size-dependent Timoshenko beam model that considers both the nonlocal effect and strain gradient effect. The variationally consistent boundary conditions corresponding to the equations of motion of Timoshenko beams are reformulated with the aid of the weighted residual method. The complete boundary value problems of nonlocal strain gradient Timoshenko beams undergoing buckling are solved in closed forms. All the possible higher-order boundary conditions induced by the strain gradient are selectively suggested based on the fact that the buckling loads increase with the increasing aspect ratios of beams from the conventional mechanics point of view. Then, motivated by the expression for beams with simply-supported(SS) boundary conditions, some semiempirical formulae are obtained by curve fitting procedures.     

New Principles of Work and Energy as Well as Power and Energy Rate for Continuum Field Theories

Ｉｎｔｒｏｄｕｃｔｉｏｎ1 999ｉｓｔｈｅ 90ｔｈａｎｎｉｖｅｒｓａｒｙｏｆｔｈｅｐｕｂｌｉｃａｔｉｏｎｏｆＥ .Ｆ .Ｃｏｓｓｅｒａｔ’ｓｂｏｏｋ“ＴｈｅｏｒｉｅｄｅｓＣｏｒｐｓＤｅｆｏｒｍａｂｌｅ”[1],ｗｈｉｃｈｗａｓｔｈｅｆｏｕｎｄａｔｉｏｎｓｔｏｎｅａｎｄｎｏｗｉｓｓｔｉｌｌａｇｕｉｄｉｎｇｍｏｎｏｇｒａｐｈｉｎｔｈｅｓｔｕｄｙｏｆｇｅｎｅｒａｌｉｚｅｄｃｏｎｔｉｎｕｕｍｆｉｅｌｄｔｈｅｏｒｉｅｓ.Ｈｏｗｅｖｅｒ,ｉｔｗａｓｎｏｔｔｉｌｌｔｈｅｐｕｂｌｉｃａｔｉｏｎｏｆｐａｐｅｒｓｏｆＥｒｉｃｋｓ…     

On virtual work and stress in granular media

A granular medium can be treated as an equivalent continuum. Appropriate representative stresses can be derived from the virtual work principle. However, the expression of virtual work is not unique and therefore may lead to different results of stress expressions in terms of discrete quantities—contact forces, contact moments, and branch vectors. In this paper, we introduced a generalized expression of virtual work that includes the restriction of boundary conditions. To show the advantages of the current expression, the virtual work expression is applied to derive expressions for stress, couple stress, a higher-order stress, and the stress moment. A distinction is made between the average stress within a granular volume and the representative stress that is conjugate with the representative strain of the volume. The current work is compared with that of [International Journal of Solids and Structures 38 (2) (2001) 353–367], and the current stress expressions are shown to satisfy three essential conditions of a stress measure.     

The equilibrium equations and constitutive equations of the growing deformable body in the framework of continuum theory

In this paper, a general framework of continuum theory for a growing deformable body is established. Firstly, the so-called material accretion derivative is defined. Based on this definition, a general form of the equilibrium equation and its growing boundary condition describing motion of the growing deformable body are deduced in detail. From the process of deduction, the concept of coupling function of growth is derived, which reflects the influence of the accretive boundary surface. Then, the equilibrium equations, including the equation of mass, momentum, moment of momentum and energy, are discussed. Also, the entropy inequality is given according to the assumption of local equilibrium of non-equilibrium thermodynamics. In the meantime, the related constitutive equations are deduced. All these equations constitute a group of closed equations characterizing the growth and motion of the body.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅱ)-THERMOPIEZOELECTRICITY AND MAGNETOTHERMOELASTICITY

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｏｒｙｏｆｍｉｃｒｏｐｏｌｏａｒｔｈｅｒｍｏｅｌａｓｔｉｃｉｔｙｐｒｅｓｅｎｔｅｄｂｙＷ .Ｎｏｗａｃｋｉｉｓｒｅｓｔｕｄｉｅｄｉｎｏｕｒｐａｐｅｒ[1].Ｔｈｉｓｐａｐｅｒｉｓａｄｉｒｅｃｔｃｏｎｔｉｎｕａｔｉｏｎｏｆｒｅｆｅｒｅｎｃｅ [1 ] .Ｔｈｅｐｒｏｂｌｅｍｓｏｃｃｕｒｒｉｎｇｉｎｔｈｅｔｈｅｏｒｉｅｓｏｆｔｈｅｒｍｏｐｉｅｚｏｅｌｅｃｔｒｉｃｉｔｙａｎｄｍａｇｎｅｔｏｔｈｅｒｍｏｅｌａｓｔｉｃｉｔｙｆｏｒｍｉｃｒｏｐｏｌａｒｃｏｎｔｉｎｕａａｒｅｓｉｍｉｌａｒｔ…     

One-dimensional dynamically consistent gradient elasticity models derived from a discrete microstructure: Part 1: Generic formulation

This paper is the first in a series of two that focus on gradient elasticity models derived from a discrete microstructure. In this first paper, a new continualization method is proposed in which each higher-order stiffness term is accompanied by a higher-order inertia term. As such, the resulting models are dynamically consistent. A new parameter is introduced that accounts for the nonlocal interaction between variables of the discrete model and of the continuous model. When this parameter is set to proper values, physically realistic behavior is obtained in statics as well as in dynamics. In this sense, the proposed methodology is superior to earlier approaches to derive gradient elasticity models, in which anomalies in the dynamic behavior have been found. A generic formulation of field equations and boundary conditions is given based on Hamilton's principle. In the second paper, analytical and numerical results of static and dynamic response of the second-order model and the fourth-order model will be treated.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅴ)--POLAR THERMOMECHANICAL CONTINUA

The purpose is to reestablish rather complete basic balance equations and boundary conditions for polar thermomechanical continua based on the restudy of the traditional theories of micropolar thermoelasticity and thermopiezoelectricity . The equations of motion and the local balance equation of energy rate for micropolar thermoelasticity are derived from the rather complete principle of virtual power. The equations of motion, the balance equation of entropy and all boundary conditions are derived from the rather complete Hamilton principle . The new balance equations of momentum and energy rate which are essentially different from the existing results are presented. The corresponding results of micromorphic thermoelasticity and couple stress elastodynamics may be naturally obtained by the transition and the reduction from the micropolar case , respectively . Finally , the results of micropolar thermopiezoelectricity are directly given .     

Equations of motion and boundary conditions of incremental rate type for polar continua

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ[1]ｔｈｅｒｅｌａｔｉｏｎｓｏｆｖａｒｉｏｕｓｓｔｒｅｓｓｔｅｎｓｏｒｓ,ｔｈｅｅｑｕａｔｉｏｎｓｏｆｍｏｍｅｎｔｕｍａｎｄｔｈｅｃｏｒｒｅｓｐｏｎｄｉｎｇｂｏｕｎｄａｒｙｃｏｎｄｉｔｉｏｎｓｏｆｖａｒｉｏｕｓｆｏｒｍｓｆｏｒｃｌａｓｓｉｃａｌｃｏｎｔｉｎｕｕｍｍｅｃｈａｎｉｃｓａｒｅｓｙｓｔｅｍａｔｉｃａｌｌｙｄｅｒｉｖｅｄ.Ｉｎ[2]ｔｈｅｇｅｎｅｒａｌｉｚｅｄｃｏｎｔｉｎｕｕｍｆｉｅｌｄｔｈｅｏｒｉｅｓａｒｅｃｏｍｐｒｅｈｅｎｓｉｖｅｌｙｒｅｖｉｅｗｅｄａｎｄｃｌａｒ…     

Plane viscoplastic flow in narrow channels with deformable walls

The equations of motion of a continuum in a thin layer are derived for a given functional dependence of the stress tensor on the strain rate tensor. The general problem of viscoplastic flow is considered in the thin-layer approximation for boundary surface material points travelling in the lateral direction in a predetermined fashion.The projections of the continuum point velocity, pressure, flow rate through a cross-section of the channel, and the power of external forces are expressed as functions of the boundary deformation law. The problem of determining the channel boundary deformation law is formulated for a given boundary pressure distribution. The expressions for the continuum flow rate and pressure and the power of external forces written as functionals of the channel width allow formulation of the problems of controlling viscoplastic flows in thin layers and optimizing the processes.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 2, pp. 23–31, March–April, 1996.     

RESTUDY OF THEORIES FOR ELASTIC SOLIDS WITH MICROSTRUCTURE

ＩｎｔｒｏｄｕｃｔｉｏｎＵｐｔｏｎｏｗｔｈｅｒｅｈａｓｂｅｅｎｖｅｒｙｍｕｃｈｗｒｉｔｔｅｎｗｏｒｋｏｎｔｈｅｓｕｂｊｅｃｔｓｏｆｃｏｎｔｉｎｕｕｍｔｈｅｏｒｉｅｓｉｎｗｈｉｃｈｔｈｅｄｅｆｏｒｍａｔｉｏｎｉｓｄｅｓｃｒｉｂｅｄｎｏｔｏｎｌｙｂｙｔｈｅｕｓｕａｌｖｅｃｔｏｒｄｉｓｐｌａｃｅｍｅｎｔｆｉｅｌｄ ,ｂｕｔｂｙｏｔｈｅｒｖｅｃｔｏｒｏｒｔｅｎｓｏｒｆｉｅｌｄｓａｓｗｅｌｌ.Ｉｎａｆａｍｏｕｓｍｏｎｏｇｒａｐｈ ,Ｅ .ＣｏｓｓｅｒａｔａｎｄＦ .Ｃｏｓｓｅｒａｔ[1]ｇａｖｅａｓｙｓｔｅｍａｔｉｃ…     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅴ)——POLAR THERMOMECHANICAL CONTINUA

The purpose is to reestablish rather complete basic balance equations and boundary conditions for polar thermomechanical continua based on the restudy of the traditional theories of micropolar thermoelasticity and thermopiezoelectricity. The equations of motion and the local balance equation of energy rate for micropolar thermoelasticity are derived from the rather complete principle of virtual power. The equations of motion, the balance equation of entropy and all boundary conditions are derived from the rather complete Hamilton principle. The new balance equations of momentum and energy rate which are essentially different from the existing results are presented. The corresponding results of micromorphic thermoelasticity and couple stress elastodynamics may be naturally obtained by the transition and the reduction from the micropolar case, respectively. Finally, the results of micropolar thermopiezoelectricity are directly given. Contributed by DAI Tian-min, Original Member of Editorial Committee, AMM Foundation items: the National Natural Science Foundation of China (10072024); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

用于弹性蠕变损伤问题的参变量变分原理

参变量变分原理是近年来发展的用于处理数学物理问题中边界待定边值问题的一种有效方法,本文建立起用于蠕变损伤问题结构分析的参变量变分原理,该原理将原问题化为求解带约束条件的泛函极值,其约束条件就是由蠕变损伤本构关系推导出的系统状态方程组;该原理物理意义明确、表达式简单并且规范,容易为计算机实现。本文给出原理的证明,并就2.25Cr-1Mo钢在550℃下的蠕变问题给出实例。