NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)——RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NONLOCAL THERMOELASTIC BODIES

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Some Primitive Concepts in Continuum Mechanics Regarded in Terms of Objective Space-Time Molecular Averaging: The Key Rôle Played by Inertial Observers

In Continuum Mechanics the notions of body, material point, and motion, are primitive. Here these concepts are derived for any (possibly time-dependent) material system via mass and momentum densities whose values are local spacetime averages of molecular quantities. The averaging procedure necessary to ensure molecular-based densities can be agreed upon by all observers (that is, are objective) has implications for constitutive relations. Specifically, such relations should first be expressed in terms of Galilean-invariant functions of the motion relative to an inertial frame. Thereafter such relations can be re-phrased for general observers, thereby yielding general-frame constitutive relations compatible with material frame-indifference. Two postulates concerning observer agreement (which together constitute a statement of material frame-indifference) are shown to imply that any stress response function which is assumed to depend upon the motion in an inertial (general) frame must be Galilean-invariant (invariant under superposed rigid body motions). Accordingly, invariance under superposed rigid body motions is not a fundamental tenet of continuum physics, but rather a consequence of material frame-indifference whenever constitutive dependence upon motion in a general observer frame is postulated.     

NEW POINTS OF VIEW ON THE NONLOCAL FIELD THEORY AND THEIR APPLICATIONS TO THE FRACTURE MECHANICS(Ⅱ)--RE-DISCUSS NONLINEAR CONSTITUTIVE EQUATIONS OF NON-LOCAL THERMOELASTIC BODIES

In this paper, nonlinear constitutive equations are deduced strictly according to the constitutive axioms of rational continuum mechanics. The existing judgments are modified and improved. The results show that the constitutive responses of nonlocal thermoelastic body are related to the curvature and higher order gradient of its material space, and there exists an antisymmetric stress whose average value in the domain occupied by thermoelastic body is equal to zero. The expressions of the antisymmetric stress and the nonlocal residuals are given. The conclusion that the directions of thermal conduction and temperature gradient are consistent is reached. In addition, the objectivity about the nonlocal residuals and the energy conservation law of nonlocal field is discussed briefly, and a formula for calculating the nonlocal residuals of energy changing with rigid motion of the spatial frame of reference is derived. Foundation item: the Natural Science Foundation of Province Jiangshu (BK97063)     

On Objectivity and Material Symmetry for Simple Elastic Solids

Restrictions imposed on stress response functions by objectivity and material symmetry are usually obtained by appeal to both a reference configuration and a response function common to all observers. Here these restrictions are obtained by regarding the reference configuration to be observer-dependent in a natural manner, and without the need to assume the existence of a common response function. Remarks are made on the distinction which should be drawn between the notions of observer transformation and rigid deformation, the consequences of this distinction for the correct interpretation of ‘invariance under superposed rigid motions’, and on the domains of response functions. This revised version was published online in July 2006 with corrections to the Cover Date.     

Convergence of Peridynamics to Classical Elasticity Theory

The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.      

关于时空标架变换和物质标架无关性原理

建立了欧几里德标架无关性的一般表述与某一参考标架\phi下作相对刚体运动的一般曲线坐标、分量表述之间的关系,从而使欧几里德标架无关性的张量表述有相应的分量表述形式.建立这种联系的基础是, 参考标架\phi^\ast 下描述的运动在参考标架\phi 中某坐标系下的分量结果, 对应于\phi下另一相对该坐标系作相同刚体运动的坐标系下描述\phi观测的运动的分量结果. 通过例子说明了Truesdell和Noll用等价动力过程,关于物质标架无关性原理的陈述,比Svendsen和Bertram及Liu的本构关系在参考标架变换下的形式不变性更合理且方便,尤其是对微分方程形式的本构关系更是如此.      

Active disturbance rejection adaptive control of uncertain nonlinear systems: theory and application

This paper proposes an active disturbance rejection adaptive controller for tracking control of a class of uncertain nonlinear systems with consideration of both parametric uncertainties and uncertain nonlinearities by effectively integrating adaptive control with extended state observer via backstepping method. Parametric uncertainties are handled by the synthesized adaptive law and the remaining uncertainties are estimated by extended state observer and then compensated in a feedforward way. Moreover, both matched uncertainties and unmatched uncertainties can be estimated by constructing an extended state observer for each channel of the considered nonlinear plant. Since parametric uncertainties can be reduced by parameter adaptation, the learning burden of extended state observer is much reduced. Consequently, high-gain feedback is avoided and improved tracking performance can be expected. The proposed controller theoretically guarantees a prescribed transient tracking performance and final tracking accuracy in general while achieving asymptotic tracking when the uncertain nonlinearities are not time-variant. The motion control of a motor-driven robot manipulator is investigated as an application example with some suitable modifications and improvements, and comparative simulation results are obtained to verify the high tracking performance nature of the proposed control strategy.     

Realizing the Willis equations with pre-stresses

This paper proves that the linear elastic behavior of the material with inhomogeneous pre-stresses can be described by the Willis equations. In this case, the additional terms in the Willis equations, compared with the classical linear elastic equations for homogeneous media, are related to the gradient of pre-stresses. In this way, the material length scale is naturally incorporated in the framework of continuum mechanics. All these findings also coincide with the results of transformation elastodynamics, so that they can meet the requirement of the principle of material objectivity and the principle of general invariance.     

Microscopic symmetric bifurcation condition of cellular solids based on a homogenization theory of finite deformation

In this paper, we establish a homogenization framework to analyze the microscopic symmetric bifurcation buckling of cellular solids subjected to macroscopically uniform compression. To this end, describing the principle of virtual work for infinite periodic materials in the updated Lagrangian form, we build a homogenization theory of finite deformation, which satisfies the principle of material objectivity. Then, we state a postulate that at the onset of microscopic symmetric bifurcation, microscopic velocity becomes spontaneous, yet changing the sign of such spontaneous velocity has no influence on the variation in macroscopic states. By applying this postulate to the homogenization theory, we derive the conditions to be satisfied at the onset of microscopic symmetric bifurcation. The resulting conditions are verified by analyzing numerically the in-plane biaxial buckling of an elastic hexagonal honeycomb. It is thus shown that three kinds of experimentally observed buckling modes of honeycombs i.e., uniaxial, biaxial and flower-like modes, are attained and classified as microscopic symmetric bifurcation. It is also shown that the multiplicity of bifurcation gives rise to the complex cell-patterns in the biaxial and flower-like modes.     

Passive-based adaptive control with the full-order observer for induction motor without speed sensor

The conventional linear control methods are difficult to meet the control requirements of high-performance speed regulation of asynchronous motor due to the nonlinear and multi-variable problems of induction motor. A passive-based control method of induction motor with the full-order state observer is proposed with the Euler–Lagrange equation of motion of the induction motor. Based on the relationship between passivity and stability of induction motor, the state feedback is used for torque and speed tracking. The full-order state observer is adopted with rotor current and rotor flux as state variables, and the adaptive speed controller is designed to realize the passive-based control. The experimental results show that the errors between the estimated value based on the proposed full-order observer and the actual value of rotor current, speed and flux are small; the speed with the proposed adaptive control can reach the expected value quickly. The proposed control method can effectively meet the high-performance requirements of induction motor.

     

连续介质力学中变形梯度张量客观性表述的分歧

张量的客观性是连续介质力学中一个重要的概念,但现有文献对张量客观性的定义不一致导致有关变形梯度张量客观性的表述存在分歧.本文主要基于张量的逆及功共轭角度分析了不同客观性定义的差别,旨在加深对张量客观性,特别是对变形梯度等两点张量客观性的认识.     

Further remarks on Euclidean objectivity and the principle of material frame-indifference

Murdoch [1] claimed that purely objective considerations imply the restrictions imposed upon response functions by the principle of invariance under superposed rigid body motions. In response to the criticism raised by Liu [2], Murdoch failed to recognize the mathematical implication of the condition of objectivity, as pointed out by Liu, that if a response function for an observer is given, the corresponding one for any other observer can be defined uniquely. Consequently his criticism against Lius counter-example, based on the assumption of observer agreement concerning response functions of different observers, cannot be taken as a valid argument. Furthermore, regarding objective considerations as playing the central role in continuum physics, besides our disagreement, especially on the assumption that it is possible to contemplate arbitrary relative motions between two given observers, it is impractical, because, it cannot be stated unambiguously in mathematical terms, and some physical intuitions need to be purposely conceived for occasional cases. Therefore, it runs against the rational spirit deeply cherished in Modern Continuum Mechanics.Received: 21 April 2004, Accepted: 18 August 2004, Published online: 24 March 2005PACS: 83.10Ff     

Acoustic radiation induced by bubble motion in compressible fluid

Based on the theory of compressible fluid, a three-dimension boundary element method is utilized to research the motion of bubble. The far-field noise radiation during the growth and contraction is calculated by the Kirchhoff formula and the Ffowcs Williams-Hawkings （FW-H） formula with a fixed radiation surface being arranged at the near-field of bubble as a new acoustic source. The results show that the amplitude of the sound pressure induced by non-spherical bubble is lower than that of spherical bubble in the contraction phase. The retardance effect is more obvious when the observer is farther away from the bubble. In the anaphase of contraction, the observer with the maximum amplitude of sound pressure moves up with the obvious jet. Larger buoyance parameters will generate lower sound pressure amplitudes in the anaphase, while larger intensive parameters will cause higher sound pressure amplitudes in the whole procedure of bubble motion.     

Traction problem in electro-elasticity and bifurcation theory

This paper is the first of a series of two. It will deal with the problem of static traction problem with minor deformations for a material which is governed by the electrostriction phenomenon. Two approaches to this problem will be described. We can consider either the equilibrium equations which are naturally non-linear, or the equations after linearization. The linearization of equations must be done near a natural state. Locally, under some conditions, we can establish the existence and the uniqueness of the solutions. We use the local theorem of implicit functions. The problem can be approached more globally. If we consider the non-linear equations, we can use a natural principle of these equations: the independence of the choice of the observer, also known as objectivity property. This property makes it possible for us to take into account an action of the rotations group of the Euclidean space, and consequently to take into account all the trivial solutions. It is then possible to prove within the space of all configurations the existence of the non-linear equations solutions and to find their number.This work presents a thorough and detailed approach to a non-linear theory, the geometric arguments of which make it possible for us to prove the existence of all the solutions and to study their stability in the aggregate; this last aspect will be developed in the second paper. Not only can this theory anticipate the eventual existence of a stable solution, it can also anticipate that an unstable solution in terms of the elasticity can, thanks to the effect of an electric field, become stable in terms of the electro-elasticity.     

Fourier analysis of the signal scattered by three-dimensional objects in translational motion — I

The signal scattered by objects in translational motion is evaluated. The incident wave is a time harmonic plane wave. The scattered fields are expressed in terms of the precursor position of the scatterer. To determine the Doppler shift experienced by a stationary observer, the Fourier spectrum of the received signal is calculated. The spectrum turns out to be band limited and to evidence two (infinite) peaks.     

Model of a thermo-elastic-inelastic process with large deformations and structural changes in material

The dependence of a scalar measure of the structural changes occurring in a material under plastic deformation on a plastic strain measure and the dependence of a free energy measure on a structural change measure are constructed using experimental data that allow the expended plastic work to be divided into a latent part and a thermal part. The obtained dependences, kinematic relations, a constitutive equation, and a heat-conduction equation that satisfy the principles of thermodynamics and objectivity are used to construct a model of thermo-elastic-inelastic processes in the presence of finite deformations and structural changes in the material. The model is tested on the problem of temperature changes in the process of adiabatic elastic-plastic compression, which has experimental support.     

An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.     

Coupled shear-torsional motion of a rubber support system

The finite amplitude, coupled shear-torsional motion of a circular disk supported between identical rubber spring cylinders is studied. The material of the springs is assumed to be an incompressible elastic material. The oscillatory motion oscillatory of the disk is studied for two different cases. In the first case, the material of the spring is assumed to be an incompressible elastic material whose response functions are constants. Typical examples include the Mooney-Rivlin model. The motion of the disk in this case is governed by two independent equations whose closed form solutions are noted. For the second case, the material of the spring is assumed to be an incompressible quadratic material. The motion oscillatory of the disk in this case is governed by two coupled nonlinear differential equations. The stability properties of small shearing oscillation superimposed on finite torsion and small torsional oscillation superimposed on finite shearing are studied.