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1.
The existence of conservation laws in linear elasticity based upon divergence transformations of the Lagrangian density function is investigated. It is found that there exist a set of conservation laws which correspond to infinitesimal homogeneous perturbations of the strain and velocity fields. These conservation laws have a unique feature not shared by other conservation laws in linear elasticity in that they contain an arbitrary free parameter.  相似文献   

2.
The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) and the dual configuration force (dual energy–momentum tensor) ,
This and other results derived in this paper may lead to a better understanding of configurational mechanics and therefore of mechanics of defects.  相似文献   

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

4.
Conservation laws in linear viscoelastodynamics   总被引:1,自引:0,他引:1  
Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain two conservation laws associated with linear viscoelastodynamics. These laws represent viscoelastic generalizations of two conservation laws in elasticity.  相似文献   

5.
Conservation laws in linear elastodynamics   总被引:6,自引:0,他引:6  
Noether's theorem on variational principles invariant under a group of infinitesimal transformations is used to obtain a class of conservation laws associated with linear elastodynamics. These laws represent dynamical generalizations of certain path-independent integrals in elastostatics which have been of considerable recent interest. It is shown that the conservation laws obtained here are the only ones obtainable by Noether's theorem from invariance under a reasonably general group of infinitesimal transformations.  相似文献   

6.
The existing various couple stress theories have been carefully restudied.The purpose is to propose a coupled Noether’s theorem and to reestablish rather complete conservation laws and balance equations for couple stress elastodynamics. The new concrete forms of various conservation laws of couple stress elasticity are derived. The precise nature of these conservation laws which result from the given invariance requirements are established. Various special cases are reduced and the results of micropolar continua may be naturally transited from the results presented in this paper.  相似文献   

7.
The first order conservation laws for an arbitrary homogeneous linear planar elastic material are completely classified. In all cases, both isotropic and anisotropic, besides the standard Betti reciprocity laws, there are two infinite-dimensional families of conservation laws, each depending on an arbitrary analytic function of two complex variables.  相似文献   

8.
严格而言,流体力学中所有守恒定律均是针对物质体系的(或称流体系统),如质量、动量、动量矩和能量等守恒定律。如果跟随物质体系描述和表征流体质点系的运动行为,即为Lagrange描述方法;如果把物质体系的运动和守恒定律转换到空间坐标系中,即为人们常说的Euler描述方法。因此,对于具体考察(跟随的)的流体物质系统而言,各守恒定律存在由物质体系表征到空间体系表征的转换,这个转换关系就是著名的Reynolds输运方程。本文从动边界微积分关系式出发,系统推导了在不同运动速度控制体上的雷诺输运方程,并通过讨论进一步阐明各种不同形式输运方程的物理意义。  相似文献   

9.
皮道华 《力学学报》1990,22(4):490-494
本文用无穷小变换群使作用量不变的思想证明了广义Noether定理,且得到一类守恒律,对线性均匀微孔弹性材料阐明了尺度变换下守恒律的可能性,且给出了完备性定理的证明。  相似文献   

10.
The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether’s theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity.   相似文献   

11.
It is the purpose of this work to derive the balance laws (in the Günther–Knowles–Sternberg sense) pertaining to dipolar gradient elasticity. The theory of dipolar gradient (or grade 2) elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain–energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the gradient of both strain and rotation (additional terms). The balance laws are derived here through a more straightforward procedure than the one usually employed in classical elasticity (i.e. Noether’s theorem). Indeed, the pertinent balance laws are obtained through the action of the standard operators of vector calculus (grad, curl and div) on appropriate forms of the Hamiltonian of the system under consideration. These laws are directly related to the energy release rates in the processes of crack translation, rotation and self-similar expansion. Under certain conditions, they are identified with conservation laws and path-independent integrals are obtained.  相似文献   

12.
Motivated by Benney’s general theory, we propose new models for short wave–long wave interactions when the long waves are described by nonlinear systems of conservation laws. We prove the strong convergence of the solutions of the vanishing viscosity and short wave–long wave interactions systems by using compactness results from compensated compactness theory and new energy estimates obtained for the coupled systems. We analyze several of the representative examples, such as scalar conservation laws, general symmetric systems, nonlinear elasticity and nonlinear electromagnetism.  相似文献   

13.
薄板理论的正交关系及其变分原理   总被引:6,自引:2,他引:4  
利用平面弹性与板弯曲的相似性理论,将弹性力学新正交关系中构造对偶向量的思路推广到 各向同性薄板弹性弯曲问题,由混合变量求解法直接得到对偶微分方程并推导了对应的变分 原理. 所导出的对偶微分矩阵具有主对角子矩阵为零矩阵的特点. 发现了两个独立的、对称 的正交关系,利用薄板弹性弯曲理论的积分形式证明了这种正交关系的成立. 在恰当选择对 偶向量后,弹性力学的新正交关系可以推广到各向同性薄板弹性弯曲理论.  相似文献   

14.
I. INTRODUCTION It is well known there are close relationships between the symmetries and conservation laws inmechanical systems. The symmetric principles are among the key issues in mechanics. Two e?ectivemethods of studying the symmetries and conservation laws of mechanical are Noether’s method[1] andLie’s method. The approach to Lie symmetries was reported in the 19th century, but no applicationin mechanics appeared until 1979[2]. In recent years, studies of Lie’s method have be…  相似文献   

15.
This work aims at obtaining a covariant representation of the elasticity tensor of a hyperelastic material when the elastic strain energy potential is written employing the volumetric–distortional decomposition of the deformation. This requires the careful definition of some fundamental fourth-order tensors: the identity, the spherical operator, and the deviatoric operator, which appear in the material and spatial expressions of the elasticity tensor. These operators can be defined in the spatial or the material setting and admit pulled-back and pushed-forward forms, respectively. These forms are intimately related to the pulled-back and pushed-forward metric tensors, and are somewhat awkward to derive in Cartesian coordinates, because of the loss of the distinction between a vector space and its dual, and therefore between objects having contravariant and covariant components, which obey to different transformation laws. The relationship between the deformation and the various forms of the identity, spherical, and deviatoric operators can be entirely clarified only within a covariant theory, where the central role played by the spatial and material metric tensors, and their pulled-back and pushed-forward counterparts, which are deformation tensors, can be emphasised.  相似文献   

16.
. We analyze a class of vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of fields. Then we apply this theory to analyze entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class of systems with affine characteristic hypersurfaces. The analysis in also extends to . (Accepted July 16, 1998)  相似文献   

17.
The (static) energy-momentum tensor, angular momentum tensor and scaling flux vector of micropolar elasticity are derived within the framework of Noether’s theorem on variational principles. Certain balance (or broken conservation) laws of broken translational, rotational and dilatational symmetries are found including inhomogeneities, elastic anisotropy, body forces, body couples and dislocations and disclinations present. The non-conserved J-, L- and M-integrals of micropolar elasticity are derived and discussed. We gave explicit formulae for the configurational forces, moments and work terms.  相似文献   

18.
In this work the continuum theory of defects has been revised through the development of kinematic defect potentials. These defect potentials and their corresponding variational principles provide a basis for constructing a new class of conservation laws associated with the compatibility conditions of continua. These conservation laws represent configurational compatibility conditions which are independent of the constitutive behavior of the continuum. They lead to the development of a new concept termed configurational compatibility, dual to the concept of configurational force. The contour integral of the corresponding conserved quantity is path-independent, if the domain encompassed by the integral is defect-free. It is shown that the Peach-Koehler force can be recovered as one of these invariant integrals. Based on the proposed defect potentials and their corresponding defect energies, two-field multiscale mixed variational principles can be employed to construct multiscale energy momentum tensors. An application is outlined in the form of a mode III elasto-plastic crack problem for which the new configurational quantities are calculated.  相似文献   

19.
We study 2×2 systems of hyperbolic conservation laws near an umbilic point. These systems have Undercompressive shock wave solutions, i.e., solutions whose viscous profiles are represented by saddle connections in an associated family of planar vector fields. Previous studies near umbilic points have assumed that the flux function is a quadratic polynomial, in which case saddle connections lie on invariant lines. We drop this assumption and study saddle connections using Golubitsky-Schaeffer equilibrium bifurcation theory and the Melnikov integral, which detects the breaking of heteroclinic orbits. The resulting information is used to construct solutions of Riemann problems.  相似文献   

20.
The steady flow of incompressible elasto-viscoplastic liquids through a planar expansion–contraction is investigated. A novel constitutive model is employed to describe the mechanical behavior of the flowing liquids. Numerical solutions of the constitutive and conservation equations were obtained via a finite element method to investigate the role of elasticity, yield stress, and inertia. The fields of velocity, stress, elastic strain, and rate of strain were obtained for different combinations of the governing parameters. It was observed that these fields, as well as the shape and position of the yield surface, are all strong functions of elasticity, yield stress, and inertia. The trends observed agree well with previous numerical and visualization results available in the literature. The present work offers a detailed study on the effects of elasticity, presenting, in particular, the fields of elastic strain.  相似文献   

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