Uniqueness and minimum theorems for a multifield model of brittle solids

Some minimum theorems potentially useful to construct numerical schemes related to quasi-static evolution of damage in brittle elastic solids are proposed. The approach is that of multifield theories, with a second-order damage tensor describing the microcrack density. The use of damage entropy flux and damage pseudo-potential are both investigated.     

Uniqueness in linear coupled thermoelasticity

It is shown that the boundary-initial-value problem of linear coupled thermoelasticity has at most one regular solution on unbounded domains in a very large function class.     

Use of comparison theorems to calculate flow in media with “double porosity”

A method is described for constructing estimates of the solution to a system of nonlinear differential equations of parabolic type. The obtained estimates can also be used to determine the intervals of applicability of approximate methods and to determine the accuracy of these methods.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 147–151, May–June, 1981.     

Uniqueness theorems for finite elastodynamics

Described in this paper is a study of the uniqueness of solutions to the boundary-initial value problems of nonlinear dynamical elasticity. Particular consideration is given to the uniqueness implications of certain well-known a priori restrictions on the material response.     

Monotonicity theorems for two-phase solids

Certain alloys, such as gold-copper, have two solid phases. We establish a general mathematical framework in which we show that the fraction in one phase and the compositions within each phase are in some sense decreasing in the overall composition. The tools used include useful new lemmas on minima of functions of several variables and parameters.     

Damping associated with porosity and crack in solids

When a material is subjected to alternating stresses, there are temperature fluctuations indicative of damping. Temperature effects give rise to entropy production. An analysis is made to obtain the entropy produced for a vibration cycle. This corresponds to the reciprocity of temperature rise and strain yielded that alter the material damping factor (MDF) as a function of shape and magnitude of material porosity or existing cracks. A homogeneous, isotropic, elastic bar is considered. It consists of uniformly distributed cavities or a single-edge crack subjected to alternating axial stresses. Dynamic characteristics of the porous or cracked medium are determined to evaluate the damping factor of the bar and/or of the material. The experimental data correlate well with the analytical results. The calculated damping factor in this work can be used as an indicator of structural integrity.     

On the dual-phase-lag thermoelasticity theory

The uniqueness and reciprocal theorems are proved without the use of Laplace Transforms for the Dual-Phase-Lag thermoelasticity theory. Variational principle is established for a linear anisotropic and inhomogeneous thermoelastic solid. The dissipative inequality is used to obtain a continuous dependence result for isotropic solid.     

Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics

Uniquencess theorems are proved for the fundamental boundary value problems of linear elastostatics in bodies of arbitrary shape. The displacement fields are required to have finite strain energy in bounded portions of the bodies and satisfy the principle of virtual work. For bounded bodies, the total strain energy is finite and uniquencess is proved without additional hypotheses. In particular, no restrictions other than the energy condition are placed on the field singularities that may occur at sharp edges and corners. For unbounded bodies, uniqueness can be proved as in the bounded case if the total strain energy is finite. Sufficient conditions for this are shown to be the finiteness of the strain energy in bounded portions of the body together with the growth restriction % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWdraqaaiaabwhadaWg% aaWcbaGaaeyAaaqabaGccaGGOaGaaeiEaiaacMcacaqG1bWaaSbaaS% qaaiaabMgaaeqaaOGaaiikaiaabIhacaGGPaGaaeizaiaabIhacaqG% 9aGaaGimaiaacIcacaqGYbGaaiykaiaacYcacaqGYbGaeyOKH4Qaey% OhIukaleaacqGHPoWvdaWgaaadbaGaaeOCaiaacYcacqaH0oazaeqa% aaWcbeqdcqGHRiI8aaaa!5E73!\[\int_{\Omega _{{\text{r}},\delta } } {{\text{u}}_{\text{i}} ({\text{x}}){\text{u}}_{\text{i}} ({\text{x}}){\text{dx = }}0({\text{r}}),{\text{r}} \to \infty } \] on the displacement fieldu _i , where _r, is the portion of the body that lies between concentric spheres with radiir andr+ and >0.This research was supported by the Air Force Office of Scientific Research. Reproduction in whole or part is permitted for any purpose of the United States Government.Prepared under Contract No. F 49620-77-C-0053 for Air Force Office of Scientific Research.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

Uniqueness theorems for the entropy in any differential body of complexity one

By using in an essential way a certain condition of mutual physical equivalence between admissible response functions for the heat flux, in a previous paper uniqueness theorems were proved for the response functions of the internal energy and of the equilibrium stress, in connection with differential bodies of complexity 1. It was then pointed out that the equality expressing the vanishing of the static internal dissipation uniquely determines the rate of entropy variation in terms of the rate of the internal energy variation and of the equilibrium stress. This paper shows, in a threefold manner, that the last result also holds if one does not impose the condition of physical equivalence. The first proof uses the assumption that the response functions are Euclidean invariant. The second proof uses (i) the weaker assumption of Galilean invariance and (ii) a greater degree of smoothness of the response function for the internal energy. Both of these proofs use an axiom postulating the possibility of putting the body in contact with a vacuum. The third proof of the uniqueness property for the entropy is independent of the isolation axiom and uses the assumptions of the second proof. Whereas any of the first two proofs is a consequence of the uniqueness theorem for the internal energy-proved here by using the afore-mentioned axiom-the third proof does not depend on this theorem. Rather, disregarding the above isolation axiom, it implies that uniqueness of the entropy is compatible with non-uniqueness of both the stress and internal energy.

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Sommario In un precedente articolo, utilizzando in maniera essenziale una certa condizione di mutua equivalenza fisica tra ammissibili funzioni costitutive per il flusso di calore, si dimostrano teoremi di unicità per le funzioni costitutive di energia interna e di stress statico in corpi di tipo differenziale e complessità 1. Inoltre si osserva che, di conseguenza, la uguaglianza esprimente l'annullarsi della dissipazione interna statica permette di determinare univocamente la velocità di variazione dell'entropia in termini della velocità di variazione dell'energia interna e dello stress statico. Nel presente lavoro si dimostra, in triplice maniera, che l'ultimo risultato vale anche se non si impone la accennata condizione di equivalenza fisica. Nella prima dimostrazione si usa l'ipotesi di invarianza Euclidea per le funzioni costitutive. Nella seconda si usa (i) la più debole ipotesi di invarianza Galileiana e (ii) un maggiore grado di regolarità per la funzione costitutiva dell'energia interna. In entrambe le dimostrazioni si usa un assioma che postula la possibilità di porre in contatto il corpo con il vuoto. La terza dimostrazione della proprietà di unicità per l'entropia, che usa le medesime ipotesi della seconda, è indipendente da tale assioma. Mentre nelle prime due dimostrazioni si richiama il teorema il teorema di unicità per l'energia interna-qui dimostrato con l'assioma di isolamento-la terza dimostrazione non dipende da tale teorema. Invece, se non si assume l'assioma di isolamento con il vuoto, l'unicità dell'entropia risulta compatibile con certe non-unicità per stress ed energia interna.

     

Uniqueness theorems for the entropy in any differential body of complexity one

Uniqueness theorems for the entropy in any differential body of complexity one     

Fractional order theory of thermoelasticity

In this work, a new theory of thermoelasticity is derived using the methodology of fractional calculus. The theories of coupled thermoelasticity and of generalized thermoelasticity with one relaxation time follow as limit cases. A uniqueness theorem for this model is proved. A variational principle and a reciprocity theorem are derived.     

Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids

The present paper concerns with the linear theory of micropolar thermoelasticity for materials with voids. Some basic properties of wave numbers of the longitudinal and transverse plane harmonic waves are treated. The existence theorems of non-trivial solutions and eigenfrequencies of the interior homogeneous boundary value problems of steady vibrations are proved. The connection between plane harmonic waves and eigenfrequencies of the aforementioned problems is established.     

非线性随动强化条件下的安定定理

基于经典的安定理论与随动强化模型的一般性质,将结构在强化过程中的背应力计入Von Mises屈服准则,建立了随动强化条件下结构的静力安定定理;将背应力与对应的塑性应变率的点积在一个载荷循环内的积分计入塑性耗散功,建立了随动强化条件下结构的机动安定定理,扩展了经典安定理论的应用范围。针对两种定理的存在格式进行了理论证明,并以推论形式给出了结构在随动强化条件下静力安定和机动安定另外两种存在格式。结果表明,随动强化材料的安定状态和安定极限不受强化过程的影响,只取决于材料的初始屈服应力和最终屈服应力。     

General solution technique for transient thermoelasticity of transversely isotropic solids in Cartesian coordinates

Summary In the present paper we propose a new general solution technique for transient thermoelastic problems of transversely isotropic solids in Cartesian coordinates. The solution technique consists of five fundamental solutions. By considering the relations among the material constants of transverse isotropy, the solution technique is classified into five groups. One among those corresponds to Goodier's thermoelastic potential function as well as the generalized Boussinesq solutions and the Michell function. For an application of the solution technique, an inverse problem of transient thermoelasticity in a transversely isotropic semi-infinite solid is analyzed.

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Ein allgemeines Lösungsverfahren in kartesischen Koordinaten für transiente Thermoelastizität transversal-isotroper Körper

Übersicht Vorgeschlagen wird ein neues, allgemeines Verfahren zur Lösung transienter thermoelastischer Probleme transversal-isotroper Körper in kartesischen Koordinaten. Das Lösungsverfahren besteht aus fünf Fundamentallösungen und kann unter Berücksichtigung der Beziehungen zwischen den Materialkonstanten der Transversal-Isotropie in fünf Gruppen eingeteilt werden Eine Gruppe entspricht dem Goodierschen thermoelastischen Potential bzw. der verallgemeinerten Boussinesq-Lösung und der Mitchell-Funktion. Als Anwendung dieses Lösungsverfahrens wird ein inverses transient thermoelastisches Problem des transversal-isotropen, unendlichen Halbraumes untersucht.