On the thermomechanics of materials that have multiple natural configurations Part I: Viscoelasticity and classical plasticity

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasicity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

On the thermomechanics of materials that have multiple natural configurations

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasticity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.Received: February 17, 2004     

On the thermomechanics of materials that have multiple natural configurations Part I: Viscoelasticity and classical plasticity

Many bodies, both solid and fluid, are capable of being stress-free in numerous configurations that are not related to each other through a rigid body motion. Moreover, it is possible that these bodies could have different material symmetries in these different stress-free natural configurations. In order to describe the response of such bodies, it is necessary to know the manner in which these natural configurations evolve as well as a class of response functions for the stress that are determined by kinematical quantities that are measured from these evolving natural configurations. In this review article, we provide a framework to describe the mechanics of such bodies whose natural configurations evolve during a thermodynamic process. The framework is capable of describing a variety of responses and has been used to describe traditional metal plasticity, twinning, traditional viscoelasticity of both solids and fluids, solid-to-solid phase transitions, polymer crystallization, response of multi-network polymers, and anisotropic liquids. The classical theories of elastic solids and viscous fluids are included as special cases of the framework. After a review of the salient features of the framework, we briefly discuss the status of viscoelasicity, traditional plasticity, twinning and solid to solid phase transitions within the context of the framework.     

Spectral synthesis for Euler type operators

A=(a _ij) _i ^j=1 — k-o ,a _ij . :

Necessary and sufficient conditions for interpolation with functions having monotoner-th derivatives

,

Global properties of solutions to 1D-viscous compressible barotropic fluid equations with density dependent viscosity

The Navier-Stokes equations for compressible barotropic fluid in 1D with the mass force under zero velocity boundary conditions are studied. We prove the uniform upper and lower bounds for the density as well as the uniform in time L ²()-estimates for _x and u _x (u is the velocity). Moreover, a collection of the decay rate estimates for - (with being the stationary density) and u in ²()-norm and H ¹()-norm as time t are established. The results are given for general state function p() (but mainly monotone) and viscosity coefficient µ() of arbitrarily fast (or slow) growth as well as for the large data.     

Some remarks on Turán's inequality. III: The completion

. 0pq, 1–1/p+1/p0. f(x) — n, [–1,1],

On the Pólya conjecture concerning the maximum and minimum of the modulus of an entire function of finite order given by a lacunary power series

, (fz) , ,

Sharpening of Stečkin's theorem to strong approximation

. . , , : f — ,

On the class of an elliptic projective surface

Research done within the framework of the 40% project Algebraic Geometry of the Italian M. U. R. S. T.     

The Number of Linearly Independent Binary Vectors with Applications to the Construction of Hypercubes and Orthogonal Arrays, Pseudo ( t, m, s)-Nets and Linear Codes

We study formulae to count the number of binary vectors of length n that are linearly independent k at a time where n and k are given positive integers with 1kn. Applications are given to the design of hypercubes and orthogonal arrays, pseudo (t, m, s)-nets and linear codes.     

On the divergence of linear summation methods of Fourier series

[0, 1],fL(0,2),

New covering numbers and spectral properties of operators in Banach spaces

, , , . ., . — s- , . , s- (,s- ) ; _n (T), ¦ _l (T)¦¦ ₂ (T)¦...0, T:EE .

Projection minima of a symmetric convex body

R ⁿ n- , : RⁿPRⁿ/ o - . —

Blocks of orthogonal random variables and the strong law of large numbers

(X _k ),k=1,2,... — _k ² >1; (X _k ) , E(X _k X _t )=0 p k<>(p+1) (p,k,l=1, 2, ...) , , ,

A test of convergence of Fourier series with respect to multiplicative systems,analogous to the Jordan test

, {p _n} _n=0 (p₀=1, _n2 n2). : x f(t) V(G)

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition of a natural number n, the partition h() of n is defined so as to obtain a certain set of zeros in the table of characters for S_n. Namely, h() is the greatest (under the lexicographic ordering ) partition among P(n) such that (g) 0. Here, is an irreducible character of S_n, indexed by a partition , and g is a conjugacy class of elements in S_n, indexed by a partition . We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition P(n), the partition f() of n is defined so that f() is greatest among the partitions of n which are opposite in sign to h() and are such that (g) 0 (Thm. 1). Also, for any self-associated partition of n > 1, we construct a partition () P(n) such that () is greatest among the partitions of n which are distinct from h() and are such that (g) 0 (Thm. 2).Supported by RFBR grant No. 04-01-00463 and by RFBR-BRFBR grant No. 04-01-81001.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 24–43, January–February, 2005.