Stress functions for continua with couple stresses

In this paper new representations of stresses and couple stresses in terms of stress functions are obtained for three- and two-dimensional Cosserat continua using the motor analysis, and the particular cases of these representations are compared with known results. As an application of the introduced stress functions we consider several examples of determining the stress and couple stress fields due to discrete dislocations and disclinations.     

The effects of couple stresses on dislocation strain energy

The correspondence theorem which relates the solutions of displacement boundary value problems in classical and couple stress elasticity is formulated and applied to derive the solutions for edge and screw dislocations in an infinite medium. The effects of couple stresses on the dislocation strain energy is evaluated for both types of dislocations. It is shown that within a radius of influence of each dislocation in a metallic crystal with dislocation density of 10¹⁰ cm⁻², the strain energy contribution from couple stresses (excluding the core energy) is about 15% in the case of an edge dislocation, and about 11% in the case of a screw dislocation. It is then shown that couple stresses make large effect on the total work of tractions acting on the dislocation core surface.     

From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses

In the present contribution, a transition from the dynamics of single particles to a Cosserat continuum is discussed. Based on the definition of volume averages, expressions for the macroscopic stress tensors and for the couple stress tensors are derived. It is found that an ensemble of particles allows for a non-symmetric macroscopic stress tensor and, thus, for the existence of couple stresses, even if the single particles are considered as standard continua. Discrete element method simulations of a biaxial box are used for the validation of the proposed homogenization technique.     

Continuum theory of dislocations and disclinations in nonlinearly elastic micropolar media

A nonlinear theory of continuously distributed dislocation and disclination type defects in elastic media with intrinsic rotational degrees of freedom and couple stresses is proposed. The mediumstrains are assumed to be finite. The solving equations of the continuum theory of defects are obtained by passing to the limit from a discrete set of isolated dislocations and disclinations to their continuous distribution. The notions of dislocation and disclination densities in a micropolar body under large deformations are introduced. Incompatibility equations are obtained and a boundaryvalue problem of equilibriumis posed for an elastic micropolar body with a given density of distributed defects. A nonlinear problem of determining the intrinsic stresses in a hollow circular cylinder due to a given distribution of disclinations is solved.     

New Principles of Power and Energy Rate of Incremental Rate Type for Generalized Continuum Field Theories

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Determining the effective thermoelastic characteristics of regularly laminated composite in the asymmetric theory of elasticity

The asymmetric theory of elasticity is used to model a hybrid laminated composite of regular structure with all phases isotropic. The effective thermoelastic characteristics of the composite are determined. It is shown that the equations derived can be used to determine stress–strain state in all the phases of the composite using the average components of the tensors of force stresses, couple stresses, strains, and wryness in a layered material, which is of fundamental importance for the design of composites based on structural theories of failure     

The effect of couple stresses on the tip of a crack

The effect of couple stresses at a crack tip is investigated by considering two particular problems. A formally exact solution is obtained (for couple-stress and micropolar elasticity) for the case of a semi-infinite crack with a prescribed internal stress. Secondly, the problem of a finite crack in an infinite medium (with couple stresses) under uniform tension at infinity, is solved by matched expansions when the couple stress parameter is small compared with the crack length. In each case it is shown that the energy release rate from a crack tip tends to the classical elastic value as the couple stress (or micropolar) parameter tends to zero.     

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

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

Defects in gradient micropolar elasticity—I: screw dislocation

A gradient micropolar elasticity is proposed based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium. This theory is an extension of the theory of micropolar elasticity with couple stresses together with gradient elasticity in a way that in addition to hyper stresses, hyper couple stresses also appear. In particular, the strain energy, besides its dependence upon the distortion and bend-twist terms of a micropolar medium (Cosserat continuum), depends also on distortion and bend-twist gradients. Using a simplified but rigorous version of this gradient theory, we can connect it to Eringen's nonlocal micropolar elasticity. In addition, it is used to study a screw dislocation in gradient micropolar elasticity. One important result is that we obtained nonsingular expressions for the force and couple stresses. The components of the force stress have maximum values near the dislocation line and those of the couple stress have maximum values at the dislocation line.     

Numerical simulation of interaction of deformable gas-permeable packets of grids with shock waves

A mathematical model that describes a one-dimensional approximation of interrelated processes of unsteady deformation of plane, cylindrical, and spherical packets of metallic woven grids and wave processes in the gas contained in pores is presented. Nonlinear equations of dynamics of two interpenetrating continua are solved numerically with the use of a modified Godunov scheme. Solutions of problems of the shock wave impact on plane and cylindrical packets of grids are obtained. Numerical results are found to agree with available experimental data. The character of explosive waves passing through a cylindrical packet of grids is studied.     

Generalization of Saint-Venant's principle to micropolar continua

The mathematical formulation and proof of Saint-Venant's principle as given by Toupin for non-polar solids is generalized to the case of micropolar elasticity. On one end of a micropolar cylinder of arbitrary length and cross-section we apply a system of self-equilibrated stresses and couple stresses. We first prove that the norms of the stress and couple stress tensors are bounded by the energy density. By means of Rayleigh's principle for the lowest natural eigenfrequency for a slice of the cylinder we then prove that the energy, stored in the cylinder beyond a certain distance from the loaded end, has an exponential decrease with this distance, thus establishing Saint-Venant's principle for the system.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

Propagation of a straight crack in dipolar elastic bodies

Our study is concerned with some results involving the energy release rate for a sharp and straight crack in an elastic dipolar solid. The effect of the dipolar couple stresses and microinertia on the energy release rate is also investigated.     

Defects in gradient micropolar elasticity—II: edge dislocation and wedge disclination

A theory of gradient micropolar elasticity based on first gradients of distortion and bend-twist tensors for an isotropic micropolar medium has been proposed in Part I of this paper. Gradient micropolar elasticity is an extension of micropolar elasticity such that in addition to double stresses double couple stresses also appear. The strain energy depends on the micropolar distortion and bend-twist terms as well as on distortion and bend-twist gradients. We use a version of this gradient theory which can be connected to Eringen's nonlocal micropolar elasticity. The theory is used to study a straight-edge dislocation and a straight-wedge disclination. As one important result, we obtained nonsingular expressions for the force and couple stresses. For the edge dislocation the components of the force stress have extremum values near the dislocation line and those of the couple stress have extremum values at the dislocation line and for the wedge disclination the components of the force stress have extremum values at the disclination line and those of the couple stress have extremum values near the disclination line.     

基于介观结构的饱和与非饱和多孔介质有效应力

基于描述含液颗粒材料介观结构的Voronoi 胞元模型和离散颗粒集合体与多孔连续体间的介-宏观均匀化过程, 定义饱和与非饱和多孔介质有效应力. 导出了计及孔隙液压引起之颗粒体积变形的饱和多孔介质广义有效应力. 用以定义广义有效应力的Biot 系数不仅依赖于颗粒材料的多孔连续体固体骨架及单个固体颗粒的体积模量(材料参数),同时与固体骨架当前平均广义有效应力及单个固体颗粒的体积应变(状态量) 有关. 提出了描述非饱和多孔介质中非混和固体颗粒、孔隙液体和气体等三相相互作用的具介观结构的Voronoi 胞元模型.具体考虑在低饱和度下双联(binary bond) 模式的摆动(pendular) 液桥系统介观结构. 导出了基于介观水力-力学模型的非饱和多孔介质的各向异性有效应力张量与有效压力张量. 考虑非饱和多孔介质Voronoi 胞元模型介观结构的各向同性情况,得到了与非饱和多孔连续体理论中唯象地假定的标量有效压力相同的有效压力形式.但本文定义的与确定非饱和多孔介质有效应力和有效压力相关联的Bishop 参数由基于三相介观水力-力学模型, 作为饱和度、孔隙度和介观结构参数的函数导出,而非唯象假定.      

横观各向同性饱和层状地基的三维稳态动力响应

首先引入状态向量,将直角坐标系下横观各向同性饱和土的Biot波动方程转化为一组状态方程,然后基于双重Fourier变换,求解了状态方程,得到传递矩阵.进而利用传递矩阵,并结合饱和地基的边界条件、排水条件及层间接触和连续条件,求解了横观各向同性饱和层状地基的稳态动力响应问题.     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

RENEWAL OF BASIC LAWS AND PRINCIPLES FOR POLAR CONTINUUM THEORIES (Ⅲ)—NOETHER''''S THEOREM

IntroductionThispaperisadirectcontinuationofthepreviouspapers [1 ,2 ] .ThefieldequationsoflinearcouplestresselasticityweredevelopedbyMindlinandTiersten[3].AgeneralcouplestresstheorywaspresentedbyToupin[4 ].Eringen[5 ,6 ]haspointedoutthatthecouplestresstheoriesofelasticmediafluidsmaybededucedfromthemicropolarelasticityandfluidmechanicswhenthemicrorotationsiandψiareconstrainedbytherotationvectorωiofclassicalelasticityandvorticityvectorrioffluidmechanics,respectively .AnexpressionoftheJ_int…     

Continuum Dynamics on Manifolds: Application to Elasticity of Residually-Stressed Bodies

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of the equations of motion, viewing motion as a curve in an infinite-dimensional Banach manifold of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies, where the residual stresses result from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two-dimensional curved annulus embedded in a sphere.