Existence of global solutions to multidimensional equations for Bingham fluids

We consider equations describing the multidimensional motion of compressible viscous (non-Newtonian) Bingham-type fluids, i.e., fluids with multivalued function relating the stresses to the tensor of strain rates. We prove the global existence theorem in time and in the initial data for the first initial boundary-value problem corresponding to flows in a bounded domain in the class of “weak” generalized solutions. In this case, we admit an anisotropic relation between the stress and strain rate tensors and study admissible relations of this kind in detail.     

General Solution for the Two-Dimensional System of Static Lame’s Equations with an Asymmetric Elasticity Matrix

We study a two-dimensional system of equations of linear elasticity theory in the case when the symmetric stress and strain tensors are related by an asymmetric matrix of elasticity moduli or elastic compliances. The linear relation between stresses and strains is written in an invariant form which contains three positive eigenmodules in the two-dimensional case. Using a special eigenbasis in the strain space, it is possible to write the constitutive equations with a symmetric matrix, i.e., in the same way as in the case of hyperelasticity. We obtain a representation of the general solution of two-dimensional equations in displacements as a linear combination of the first derivatives of two functions which satisfy two independent harmonic equations. The obtained representation directly implies a generalization of the Kolosov–Muskhelishvili representation of displacements and stresses in terms of two analytic functions of complex variable. We consider all admissible values of elastic parameters, including the case when the system of differential equations may become singular. We provide an example of solving the problem for a plane with a circular hole loaded by constant stresses.     

Two formulations of elastoplastic problems and the theoretical determination of the location of neck formation in samples under tension

Two formulations of elastoplastic problems in the mechanics of deformable solids with finite displacements and deformations are investigated. The first of these is formulated starting from the classical geometrically non-linear equations of the theory of elasticity and plasticity, in which the components of the Cauchy–Green strain tensor, associated with the components of the conditional stress tensor by physically non-linear relations according to flow theory in the simplest version of their representation, are taken as a measure of the deformations. The second formulation is based on the introduction of the true tensile and shear strains which, according to Novoshilov, are associated with the components of the true stresses by physical relations of the above-mentioned form. It is shown that, in the second version of the formulation of the problem, the use of the corresponding equations, complied taking account of the elastoplastic properties of the material with correct modelling of the ends of cylindrical samples and the method of loading (stretching) them, enables the location of the formation of a neck to be determined theoretically and enables the initial stage of its formation to be described without making any assumptions regarding the existence of initial irregularities in the geometry of the samples.     

A variant of the theory of local strains

A variant of the theory of local strains giving the stress tensor as a function of a given strain tensor is formulated. The stresses in an orthogonal coordinate system are established by means of a functional that averages the local stresses expressed by the local stress function. This function is determined by the given strain program. It is shown that in certain practical problems these relations are more convenient than those previously proposed.Mekhanika Polimerov, Vol. 3, No. 5, pp. 800–802, 1967     

A further note on finite-strain force and moment stress elasticity

Earlier considerations [4, 6] of the statics and kinematics of force and moment stresses and strains are complemented by the consideration of constitutive relations for an elastic medium. Known variational theorems for stresses and displacements of finite force stress elasticity, and of infinitesimal force and moment stress elasticity, are generalized to the case of finite force and moment stress elasticity.Based upon work supported by National Science Foundation Grant No. CEE-8213256, and dedicated to my friend Ekkehart Kröner on the occasion of his 65th birthday.     

Anisotropic tensor spaces and functions,the mean values of tensor quantities and the limiting criteria

The concept of an anisotropic vector space with a tensor basis which is invariant under a symmetry transformations of a three-dimensional Euclidean vector space is introduced using the example of symmetric second- and fourth-rank Euclidean tensors. In addition to the traditional operation of summation, the operation of multiplication in a fixed tensor basis is introduced for the elements of this space, that is, the axioms of a ring with an identity element and zero divisors, which enable one to carry out algebraic and functional operations. The possibilities of the proposed mathematical procedure are illustrated using examples of anisotropic tensor functions of a tensor argument, by the general solution of the classical problem of calculating the mean value of the tensor of the moduli of elasticity of a single-phase grain-oriented polycrystalline material and the construction of the strength surfaces of anisotropic composite materials.     

Strictly nonnegative tensors and nonnegative tensor partition

We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.     

Variant of the variable-modulus theory of elasticity

A variant of the variable-modulus theory — a generalization of the ideas of the classical theory of elasticity in which the observed difference in the moduli of elasticity in uniaxial tension and compression and homogeneous shear is taken into account — is considered. Quasilinear expressions are proposed for the stresses in terms of the strains and the strains in terms of the stresses.Tula Polytechnic Institute. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 363–365, March–April, 1969.     

Non-linear relations of anisotropic elasticity and a particular postulate of isotropy

A general approach to the construction of six-dimensional images of strain processes is proposed with the introduction of a vector basis which, in special cases, is identical to the well-known bases of A. A. Il’yushin, V. V. Novozhilov and Ye. I. Shemyakin and S. A. Khristianovich. The analysis of the properties of materials is based on the use of the concept of characteristic elastic states, which was introduced in the papers of J. Rychlewski. In the case of an isotropic material and four types of anisotropic materials belonging to the cubic, hexagonal, trigonal and tetragonal systems, characteristic subspaces, corresponding to the multiple eigenvalues of the elasticity tensor are defined in a six-dimensional space. In accordance with Hooke's law, the components of the stress and strain vectors in these subspaces preserve their axial alignment for any of their orthogonal transformations. The particular postulate of isotropy, formulated by Il’yushin, is therefore satisfied by definition within the framework of isotropic characteristic subspaces for linear elastic materials. An extension of the particular postulate to strain processes in non-linear anisotropic materials is proposed, on the basis of which a general form of constitutive relations is obtained containing a minimum number of experimentally determinable material functions.     

对称张量的分解和它的应用

本文把任一对称张量分解成两个张量的和,其中之一是“应力型”张量,另一个是“应变型”张量.对称张量空间被分解成两个直交子空间的直和.并用几何语言证明了弹性力学的几个基本原理.     

Conditions for strong ellipticity and M-eigenvalues

The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we define M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive definite. The elasticity tensor is rank-one positive definite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive definite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for finding M-eigenvalues are presented.      

Stress and strain-inertia gradient elasticity in free vibration analysis of single walled carbon nanotubes with first order shear deformation shell theory

A gradient-enriched shell formulation is introduced in the present study based on the first order shear deformation shell model and the stress gradient and strain-inertia gradient elasticity theories are used for dynamic analysis of single walled carbon nanotubes. It provides extensions of the first order shear deformation shell formulation with additional higher-order spatial derivatives of strains and stresses. The higher-order terms are introduced in the formulation by using the Laplacian of the corresponding lower-order terms. The proposed shell formulation includes two length scale size parameters related to the strain gradients and inertia gradients. The effects of the transverse shear, aspect ratio, circumferential and half-axial wave numbers and length scale parameters on different vibration modes of the single-walled carbon nanotubes are elucidated. The results are also compared with those obtained from a classical shell theory with Sanders–Koiter strain-displacement relationships.     

The steady axisymmetric flow of ideally plastic materials in a conical channel

The problem of the asymmetric flow of an ideally plastic medium is formulated within the framework of the von Mises model and the total plasticity condition, using the invariant condition of compatibility for the deviator component of the stress tensor. Flow in a converging conical channel, on the boundary of which the shear stresses are specified, is considered. First-order differential equations are obtained, describing the shear-stress distribution in the moving medium, one of which corresponds to the von Mises model, and the other to the total plasticity condition. It is established from an analysis of the solution in the neighbourhood of singular points, that the minus sign in front of the radical in these equations corresponds to positive shear stresses and vice versa. The problem of the shear stresses reaching a maximum value on the specified boundary surface of the channel is investigated. The aperture angle of the channel, beginning from which this value is reached, is determined. It is established that the value of the angle, following from the total plasticity condition, somewhat exceeds its value obtained within the framework of the von Mises model.     

On rational approaches to formulate minimal integrity basis for explicit stress closures

Due to their high efficiency and robustness almost all simulations of complex engineering flows rely on turbulence models with explicit stress closures. Focal point of an explicit stress closure is the stress‐strain‐relationship, that couples the Reynolds‐stresses to the velocity field. Mostly linear eddy‐viscosity models are employed, thus, they are based on the hypothesis and assume Reynolds‐stress proportional to the strain‐rate tensor. Therefore, they are easy to implement, but they fail to predict complex flow fields besides two dimensional shear flow such as secondary flow in non‐circular ducts. In contrast to these models, explicit algebraic stress models (EASM) represent the Reynolds‐stress by the integrity basis of the strain‐rate and vorticity tensors. This leads to a more general stress‐strain‐relationship, however, it increases the implementation and computation effort. In this paper the properties of the integrity basis will be discussed and the derivation of a minimal integrity basis will be proposed.     

The uniqueness of solutions of some variational problems of the theory of phase equilibrium in solid bodies

A variational problem in the theory of phase equilibrium is considered in a geometrically linear statement. A solid body is assumed to have two different phases with the same elasticity tensor. In the case of the Dirichlet condition, it is proved that under some restrictions on the elasticity tensor a solution of the relaxational variational problem is unique provided that the difference of the strain tensors relative to the zero stress state of the corresponding phase is anisotropic. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 220–232.     

On the dependence of the functions of state of a deformable body on the measure of rotation

We show that the measures of strain and initial values of vector and tensor state parameters are divided into subjective nonrotational and objective rotational. Representations of the functions of state are divided in a similar way, and only objective ones do not depend explicitly on the measure of rotation of material axes. We have constructed relations for the reduction of rotation-free differentials of the functions of state to expressions in terms of the tensors of infinitesimal strain and rotation. On this basis, we have obtained objective representations for stress tensor in terms of the derivatives of the state potential of an anisotropic material. The results obtained concern the nonlinear mechanics of initially stressed bodies.     

Stress-strain state in the matrix of stochastically reinforced composites

Conclusion We proposed a method of studying the concentration of stresses and strains in the matrix of composites with a stochastic structure in a three-dimensional formulation. The method is based on the use of tensor operators assigned at the inclusion-matrix interface and results from the theory of effective moduli of stochastically reinforced composites.Advantages of the proposed approach include its relative simplicity and clarity, as well as the fact that it can be used to analyze the three-dimensional stress and strain concentration in the matrix of composites with components having very different properties. However, for high values of the volume concentration of reinforcement c₁ > 0.6, it is necessary to use the results of exact solutions obtained, for example, within the framework of deterministic models. The correction that is introduced here is connected with the average of the stresses and strains over an inclusion. The character of their distribution over the interface remains the same as before.The numerical results obtained here show the significant effect of the relative dimensions of the inclusions on the effective elastic properties and the stress concentration in the matrix. Comparison of theoretical values of the longitudinal elastic modulus with experimental results can serve as grounds for validating the proposed variant of choosing the tensor L₀ in the determination of the corrected characteristics and stress-strain state of the matrix.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 396–402, May–June, 1989     

Shear driven planar Couette and Taylor-like instabilities for a class of compressible isotropic elastic solids

We study the possibility for an isotropic elastic body to support forms of instability induced by shear stress states which are reminiscent of the planar Couette and the Taylor–Couette patterns observed in the flow of viscous fluids. Here, we investigate the emergence of bifurcating periodic deformations for an infinitely long compressible elastic block confined between and attached to parallel plates which are subject to a relative shear displacement. We specialize our analysis by considering a generalized form of the Blatz–Ko strain energy function and show through numerical representative examples that planar Couette modes are always preferred with respect to the twisting Taylor–Couette modes. Finally, we introduce a suitably restricted form of the strong ellipticity condition for the incremental elasticity tensor and discuss its significance in this bifurcation problem.     

Deformation of laminated anisotropic bars in the three-dimensional statement 2. effect of edge boundary layers on the stress-strain properties of the composite

The effect of edge boundary layers on the stress-strain properties of a tensioned multilayer composite bar whose depth-to-width ratio is much less than unity is evaluated. The investigation is carried out by solving a 3D elasticity problem in the Saint-Venant statement. Simple relations for calculating the rigidity of the composite bar and the components of stress tensor are derived. The formulas for stresses are valid almost everywhere in the bar, except in narrow boundary layers close to its ends and longitudinal edges.