A class of universal relations in isotropic elasticity theory

A class of universal relations for isotropic elastic materials is described by the tensor equationTB = BT. This simple rule yields at most three component relations which are the generators of many known universal relations for isotropic elasticity theory, including the well-known universal rule for a simple shear. Universal relations for four families of nonhomogeneous deformations known to be controllable in every incompressible, homogeneous and isotropic elastic material are exhibited. These same universal relations may hold for special compressible materials. New universal relations for a homogeneous controllable shear, a nonhomogeneous shear, and a variable extension are derived. The general universal relation for an arbitrary isotropic tensor function of a symmetric tensor also is noted.     

Local and global universal relations for first-gradient materials

Local universal relations are relations between stress and kinematic variables which hold for all materials of a particular class irrespective of specific material parameters. A method is developed for obtaining local universal relations for most first gradient materials. The currently known local universal relations for isotropic elastic materials have been extended to all isotropic first gradient materials under constant step deformation histories and have also been extended to all isotropic first gradient materials undergoing arbitrary time dependent triaxial extensions along fixed material directions. It has been shown that universal relations exist for some anisotropic materials. A set of pseudo-universal relations has been obtained for anisotropic elastic materials which can be used to decouple the material functions. These pseudo-universal relations contain some, but not all, material functions. A global universal relation has been developed for the extension and torsion of an isotropic cylindrical shaft which holds for all incompressible first gradient materials.     

Towards an acoustoelastic theory for measurement of residual stress

The rudiments of an acoustoelastic theory is developed within the framework of linear elasticity with initial stress. Since no assumption is made about the origin of the initial stress, our acoustoelastic theory will be applicable to evaluation of stress in plastically deformed bodies, provided that the superimposed ultrasonic waves be hyperelastic. New universal relations are deduced. An approach to evaluation of stress which does not use calibration specimens and makes full use of universal relations in our acoustoelastic theory is advocated. Examples are given which illustrate application of our theory to evaluate residual stress in plates. Preliminary corroborations of our theory are provided by the recent experiments of King & Fortunko and Thompson et al.     

Universal Deformations for a Class of Compressible Isotropic Hyperelastic Materials

Differential conditions are derived for a smooth deformation to be universal for a class of isotropic hyperelastic materials that we regard as a compressible variant (a notion we make precise) of Mooney–Rivlin’s class, and that includes the materials studied originally by Tolotti in 1943 and later, independently, by Blatz. The collection of all universal deformations for an incompressible material class is shown to contain, modulo a uniform dilation, all the universal deformations for its compressible variants. As an application of this result, by searching the known families of universal deformations for all incompressible isotropic materials, a nontrivial universal deformation for Tolotti materials is found. This revised version was published online in August 2006 with corrections to the Cover Date.     

Universal relations for instantaneous deformations of viscoelastic fluids

The note considers viscoelastic fluids which undergo an instantaneous homogeneous deformation consisting of shear superposed on triaxial extension. Two relations involving the stress and deformation components are presented, which are valid for all such fluids, and hence are termed “universal relations”. The first contains the Lodge-Meissner relation as a special case; the second arises when a block is deformed by shear traction only. It relates dimensional changes to the amount of shear.     

Three-dimensional Maxwell stress-strain relations in viscoelasticity

Conclusion In this paper three-dimensional Maxwell stress-strain relations were deduced phenomenologically.In the first place we applied the Hamilton's principle to the viscoelastic deformation, and obtained the variational equation with respect to the elastic potential and the dissipation function.Then we assumed that the elastic potential is a function only of the stress, and the dissipation function is a function of stress and rate of stress. By the above variational equation of the virtual stress satisfying the equilibrium equation and the boundary conditions, we obtained the relations to be satisfied by the elastic potential and the dissipation function, and the conditions to be satisfied by the dissipation function.From these relations we obtained the required three-dimensional Maxwell stress-strain relations in viscoelasticity. These relations indicate that the strain is the sum of the internal elastic strain and the internal viscous strain.If a given substance is isotropic with respect to stress, the stress-strain relations are expressed by a linear Maxwell model consisting of Hookian spring in series with a Newtonian dashpot.It is the main result of this paper that the three-dimensional Maxwell stress-strain relations in viscoelasticity are deduced from physically appropriate assumptions.     

PANSYMMETRY AND FIXED PANSYSTEMS THEOREMS OF A CLASS OF PANSYSTEMS RELATIONS

In this paper. some results of pansymmetry and fixed pansystems theorems areextended to the case of a class of binary relations under the framework of pansystemsmethodology It is concretely discussed when a class of binary relations has a common fixedsubset on a finite universal set. Several simple but comprehensive decisive theorems areobtained which can be determined if a class of binary relations has a common fixed subset.Asaresult the main decisive theorem-Markov-Kakutani Theorem-is extended whichis in traditionally fixed point theory and can be determined if a class of mappings has acommon fixed point.     

A Natural Generalization of Hypoelasticity and Eulerian Rate Type Formulation of Hyperelasticity

According to the classical hypoelasticity theory, the hypoelasticity tensor, i.e. the fourth order Eulerian constitutive tensor, characterizing the linear relationship between the stretching and an objective stress rate, is dependent on the current stress and must be isotropic. Although the classical hypoelasticity in this sense includes as a particular case the isotropic elasticity, it fails to incorporate any given type of anisotropic elasticity. This implies that one can formulate the isotropic elasticity as an integrable-exactly classical hypoelastic relation, whereas one can in no way do the same for any given type of anisotropic elasticity. A generalization of classical theory is available, which assumes that the material time derivative of the rotated stress is dependent on the rotated Cauchy stress, the rotated stretching and a Lagrangean spin, linear and of the first degree in the latter two. As compared with the original idea of classical hypoelasticity, perhaps the just-mentioned generalization might be somewhat drastic. In this article, we show that, merely replacing the isotropy property of the aforementioned stress-dependent hypoelasticity tensor with the invariance property of the latter under an R-rotating material symmetry group R⋆ G ₀, one may establish a natural generalization of classical theory, which includes all of elasticity. Here R is the rotation tensor in the polar decomposition of the deformation gradient and G ₀ any given initial material symmetry group. In particular, the classical case is recovered whenever the material symmetry is assumed to be isotropic. With the new generalization it is demonstrated that any two non-integrable hypoelastic relations based on any two objective stress rates predict quite different path-dependent responses in nature and hence can in no sense be equivalent. Thus, the non-integrable hypoelastic relations based on any given objective stress rate constitute an independent constitutive class in its own right which is disjoint with and hence distinguishes itself from any class based on another objective stress rate. Only for elasticity, equivalent hypoelastic formulations based on different stress rates may be established. Moreover, universal integrability conditions are derived for all kinds of objective corotational stress rates and for all types of material symmetry. Explicit, simple, integrable-exactly hypoelastic relations based on the newly discovered logarithmic stress rate are presented to characterize hyperelasticity with any given type of material symmetry. It is shown that, to achieve the latter goal, the logarithmic stress rate is the only choice among all infinitely many objective corotational stress rates. This revised version was published online in August 2006 with corrections to the Cover Date.     

General stress-strain relations in non-linear theory of viscoelasticity for large deformations

Conclusion General phenomenoligical stress-strain relations in non-linear theory of visco-elasticity for large deformations have been presented.In the first place, according to V. V. Novozhilov 1 we express the generalized equilibrium equation for large deformations in the Lagrange representation, and we apply the generalized Hamilton's principle to the equation of energy conservation, which denotes that the sum of the elastic energy and the dissipative energy is equal to the work done by the body force and the surface on the substance; so that we obtain the required general stress-strain relations in comparison with the above two equations.On the condition that the elastic potential is a function only of the strain, and the dissipation function is a function of the rate of strain and of strain; such a substance is reduced to the Voigt material necessarily, and the stresses which act on the substance are given by the sum of elastic- and viscous stresses, and the stress-strain relations are reduced to the so-called Lagrangian form.If elongations, shears and angles of rotation are small and also the strains and rates of strain are sufficiently small, the stress-strain relations are expressed by a linear Voigt model constituting a Hookian spring in parallel with a Newtonian dashpot.Non-linearity in the theory is classified into two groups i. e. the geometrical non-linearity and the physical non-linearity. The former is introduced into the theory through the definition of the generalized strain and of the generalized stress and through the equilibrium equation for large deformation, and the latter through the general stress-strain relations.The main result of this paper is that the general stress-strain relations in viscoelasticity are deduced necessarily from the physically appropriate assumptions.     

Some pansystems studies on panchaos and strange panattractor

Paper [1] discussed the relations among panchaos and strange panattractor and pansystems operators. Paper [2] gave the applications of the fixed-point pansystems theorems to these typical nonlinear problems. In this paper, we firstly present several concepts: increasing relation, maximal panchaos, etc., and discuss the relations among them. We also discuss the problems when two increasing relations have the same panchaos and when panchaos of g is panchaos of g^t as well.     

PROBABILISTIC MODELS FOR LONG FATIGUE CRACK GROWTH RATES OF LZ50 AXLE STEEL

IntroductionStudy on the model of fatigue crack growth rate (da/dN) and measurement of theprobabilistic curves of the model are a basis of the fatigue damage tolerate design and theprobabilistic safety assessment. Some benefit efforts have been made in th…     

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.     

Measurement of stress based upon universal relations in acoustoelasticity

A new approach to ultrasonic measurement of stress was proposed recently by Man and Lu. That approach is based on universal relations that result from material symmetry in an acoustoelastic theory applicable to plastically deformed bodies for which superimposed ultrasonic waves can be taken as elastic. When applicable, the Man-Lu approach circumvents the two main difficulties of the acoustic-birefringence method, namely those of unknown initial texture and plastic deformation. An experiment was performed to verify a new universal relation derived by Man and Lu in their earlier work. Plastic strain was developed during the test. The results were in basic accord with the prediction of the theory.     

Propagating structures in wall-bounded turbulent flows

The Karhunen—Loève procedure is used to analyze two turbulent channel flow simulations. In both instances this reveals the presence of propagating plane wave structures in the turbulent flows. These waves appear to play an essential role in the local production of turbulence via bursting or sweeping events. The envelope of the propagating modes propagates with a speed which is equal to the mean velocity at the locus of maximal average Reynolds stress. Despite marked differences between the two flows similar results are obtained from each simulation. This is suggestive of the existence of universal or near universal features in the turbulent boundary layer. An analogy with critical layer mechanisms of transitional flows is discussed.Dedicated to Professor J.L. Lumley on the occasion of his 60th birthday.We gratefully acknowledge support provided by DARPA-URI under Contract Number N00014-86-K0754. The use of the Pittsburgh Supercomputing Center is also acknowledged.     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.     

Overall dynamic constitutive relations of layered elastic composites

A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity.The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results.     

Universal relations between dynamic and static interfacial stress intensity factors

Universal relations between dynamic and static interfacial stress intensity factors are derived for both in-plane and anti-plane cases under rate independent loading. Explicit and analytical form of the universal function is provided for the simple case of mode III fracture.Sponsored jointly by the National Natural Science Foundation of China and the State Education Commission of China.     

Analysis of the Onsager Relations by Analytic Methods in Kinetic Gas Theory

The well-known Onsager relations L ₁₂=L ₂₁ are verified within the framework of kinetic gas theory with allowance for the mass and heat fluxes localized in the Knudsen layer. On the basis of an analytic solution of the BGK (Bhatnagar-Gross-Krook) equation, it is shown that the Onsager relations are fulfilled correct to at least exponential corrections (–1/Kn) in the Knudsen number Kn.     

Mixed convection about a non-isothermal vertical surface in a porous medium

The problem of mixed convection about non-isothermal vertical surfaces in a saturated porous medium is analysed using boundary layer approximations. The analysis is made assuming that the surface temperature varies as an arbitrary function of the distance from the origin. A perturbation technique has been applied to obtain the solutions. Using the differentials of the wall temperature, which are functions of distance along the surface, as perturbation elements, universal functions are derived for various values of the governing parameter Gr/Re. Both aiding and opposing flows are considered. The universal functions obtained can be used to estimate the heat transfer and fluid velocity inside the boundary layer for any type of wall temperature variation. As a demonstration of the method, heat transfer results have been presented for the case of the wall temperature varying as a power function of the distance from the origin. The results have been studied for various combinations of the parameters Gr/Re and the power index m, taking both aiding and opposing flows into consideration. On comparing these results with those obtained by a similarity analysis, the agreement is found to be good.     

Clarifications of Certain Ambiguities and Failings of?Poisson��s Ratios in Linear Viscoelasticity

Detailed new analytical investigations are presented describing the behavior of Class I, II and III viscoelastic Poisson??s ratios (PR). Their previously demonstrated dependence on stress-time histories, which lead to the inability to consider them as universal viscoelastic material properties and the incapacity to produce a general elastic?Cviscoelastic correspondence principle (EVCP) based, is expanded. A new Class VI PR is analytically derived from the viscoelastic constitutive relations in the Fourier transform (FT) space to achieve the proper FT form of the elastic/viscoelastic correspondence principle, i.e., the elastic-viscoelastic analogy. However, even though this PR Class is a pure universal material property function, it still fails to provide a convenient and useful path to a correspondence principle due to its inopportune constitutive form in real time space vis-à-vis a thermodynamic model with equivalent attributes. Consequently, no general EVCP involving PRs can be formulated. The derived Class VI PRs are equivalent to the defined Class III PRs with 1-D loadings (stresses).