Deformation of a Peridynamic Bar

The deformation of an infinite bar subjected to a self-equilibrated load distribution is investigated using the peridynamic formulation of elasticity theory. The peridynamic theory differs from the classical theory and other nonlocal theories in that it does not involve spatial derivatives of the displacement field. The bar problem is formulated as a linear Fredholm integral equation and solved using Fourier transform methods. The solution is shown to exhibit, in general, features that are not found in the classical result. Among these are decaying oscillations in the displacement field and progressively weakening discontinuities that propagate outside of the loading region. These features, when present, are guaranteed to decay provided that the wave speeds are real. This leads to a one-dimensional version of St. Venant's principle for peridynamic materials that ensures the increasing smoothness of the displacement field remotely from the loading region. The peridynamic result converges to the classical result in the limit of short-range forces. An example gives the solution to the concentrated load problem, and hence provides the Green's function for general loading problems.     

Physically-Based Approach to the Mechanics of Strong Non-Local Linear Elasticity Theory

In this paper the physically-based approach to non-local elasticity theory is introduced. It is formulated by reverting the continuum to an ensemble of interacting volume elements. Interactions between adjacent elements are classical contact forces while long-range interactions between non-adjacent elements are modelled as distance-decaying central body forces. The latter are proportional to the relative displacements rather than to the strain field as in the Eringen model and subsequent developments. At the limit the displacement field is found to be governed by an integro-differential equation, solved by a simple discretization procedure suggested by the underlying mechanical model itself, with corresponding static boundary conditions enforced in a quite simple form. It is then shown that the constitutive law of the proposed model coalesces with the Eringen constitutive law for an unbounded domain under suitable assumptions, whereas it remains substantially different for a bounded domain. Thermodynamic consistency of the model also has been investigated in detail and some numerical applications are presented for different parameters and different functional forms for the decay of the long range forces. For simplicity, the problem is formulated for a 1D continuum while the general formulation for a 3D elastic solid has been reported in the appendix.     

Forced axisymmetric motions of viscoelastic cylindrical shells

The isothermal response of a viscoelastic cylindrical shell, of finite length, to arbitary axisymmetric surface forces, initial conditions, and boundary conditions is considered within the linear theory of thin shells. The problem is formulated with the effects of shear deformation and rotatory inertia included; the viscoelastic properties are assumed to be isotropic and homogeneous. The response is first found formally in terms of a causal Green's function. It is then shown that when Poisson's ratio is constant, the causal Green's function can be expanded in a series of orthonormal spatial eigenfunctions of an associated elastic shell eigenvalue problem. The resulting solution for the general problem is an eigenfunction series with Laplace transformed time-dependent coefficients. The general solution is applied to predicting the motion of a uniform, simply-supported cylindrical shell, initially quiescent, which is subjected to a step pressure moving with constant velocity. For this example, the relaxation function of the shell material in uniaxial extension is taken to be that of a standard linear solid. The motions predicted by simpler shell models, namely, shells with bending only and without bending, are also considered for comparison. Here, the absolute values of the Fourier coefficients in the shell displacement series go to zero faster than the inverse of the first or second power of positive integers when bending is excluded or included, respectively. Numerical results are presented for a moderately long and relatively thick, nearly elastic, cylindrical shell.     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅰ)──MICROPOLAR THERMOELASTICITY

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Motion of gas bubbles in a liquid under the influence of a temperature gradient

This study deals with the motion of a gas bubble developing under the influence of surface-tension forces in an imponderable viscous liquid with a temperature gradient. A theory of steady-state motion of a bubble in a field with constant temperature gradient is given for the case of small Reynolds numbers. Experimental results that show qualitative agreement with the theory are presented.The authors wish to thank M. A. Lavrent'ev for formulating the problem and giving constant attention to their work.     

The self-similarly expanding Eshelby ellipsoidal inclusion: I. Field solution

The solution of a self-similarly (subsonically) dynamically expanding ellipsoidal inclusion with general spatially uniform transformation strain temporally constant is obtained by the use of the Radon transform and the satisfaction of the zero initial conditions and the radiation condition at infinity. It constitutes the self-similar evolution of the inclusion singularity (jump discontinuity at the inclusion boundary) starting from zero dimension. The field solutions for the displacement gradient and particle velocity are presented. Due to the fact that for a self-similarly expanding subsonic motion the hyperbolic system of the partial differential equations of motion becomes elliptic (as proved in Ni and Markenscoff, 2015), it is shown here explicitly that the solution for the displacement gradient in the interior domain of the expanding ellipsoid is constant, thus extending the Eshelby property to the self-similarly expanding ellipsoids as pointed out by Burridge and Willis (1969). Also, the particle velocity is shown to be zero in the interior domain (lacuna) as the waves emitted by the self-similarly expanding inclusion cancel each other due to the symmetries of geometry and motion.     

Restudy of coupled field theories for micropolar continua (I)—Micropolar thermoelasticity

Problems of micropolar thermoelasticity have been presented and discussed by some authors in the traditional framework of micropolar continuum field theory. In this paper the theory of micropolar thermoelasticity is restudied. The reason why it was restricted to a linear one is analyzed. The rather general principle of virtual work and the new formulation for the virtual work of internal forces as well as the rather complete Hamilton principle in micropolar thermoelasticity are established. From this new Hamilton principle not only the equations of motion, the balance equation of entropy, the boundary conditions of stress, couple stress and heat, but also the boundary conditions of displacement, microrotation and temperature are simultaneously derived. Contributed by DAI Tian-min Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235) and the DFG (51520001); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)     

Two-Dimensional Electroelastic Problem for a Multiply Connected Piezoelectric Body

In this paper, we present the basic relationships for the complex potentials of a two-dimensional electroelastic problem, their general representations for a multiply connected domain, expressions for stress, displacement, electrostatic field intensity and induction, and potential. A closed solution is found for a body with one elliptic cavity or one elliptic crack under the action at infinity of a constant electroelastic field or concentrated forces and charges     

Axisymmetric contact problem of cubic quasicrystalline materials

The axisymmetric elasticity theory of cubic quasicrystal was developed in Ref. [1]. The axisymmetric elasticity problem of cubic quasicrystal is reduced to a single higher-order partial differential equation by introducing a displacement function, based on which, the exact analytic solutions for the elastic field of an axisymmetric contact problem of cubic quasicrystalline materials are obtained for universal contact stress or contact displacement. The result shows that if the contact stress has order −1/2 singularity on the edge of the contact domain, the contact displacement is a constant in the contact domain. Conversely, if the contact displacement is a constant, the contact stress must have order −1/2 singularity on the edge of the contact domain. Project supported by the National Natural Science Foundation of China (No. 19972011).     

Non-linear viscoelastodynamic equations of three-dimensional rotating structures in finite displacement and finite element discretization

This paper deals with the non-linear viscoelastodynamics of three-dimensional rotating structure undergoing finite displacement. In addition, the non-linear dynamics is studied with respect to geometrical and mechanical perturbations. On part of the boundary of the structure, a rigid body displacement field is applied which moves the structure in a rotation motion. A time-dependent Dirichlet condition is applied to another part of the boundary. For instance, this corresponds to the cycle step of a helicopter rotor blade. A surface force field is applied to the third part of the boundary and depends on the time history of the structural displacement field. For example, this might corresponds to general unsteady aerodynamics forces applied to the structure. The objective of this paper is to model the non-linear dynamic behavior of such a rotating viscoelastic structure undergoing finite displacements, and to allow small geometrical and mechanical (mass, constitutive equations) perturbations analysis to be performed. The model is constructed by the introduction of a reference configuration which is deduced from the non-linear steady boundary value problem. A constitutive equation deduced from the Coleman and Noll theory concerning the viscoelasticity in finite displacement is used. Thereafter, the weak formulation of the boundary value problem is constructed and discretized using the finite element method. In order to simplify the mathematical study of the equations, multilinear forms are introduced in the algebraic calculation and their mathematical properties are presented.