An elementary discussion of definitions of stress rate

Prager, William

doi:10.1090/qam/116567

Cited by 139 publications

(67 citation statements)

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“…Nevertheless, numerous theories based on different concepts of a relaxed, intermediate state are still subjects of actual research. Recently, within the framework of finite elastoplasticity, the irreversible objective time derivative has been proposed by Stumpf [29], In turn, Zaremba's solution was developed and applied by Jaumann [12] first and later revindicated and fully justified within the framework of Rational Mechanics in Noll's dissertation [21], This "corotational" derivative, but in Jaumann's version, has been proposed for plasticity also by Prager [24], Independent of a controversial discussion about the Zaremba-Jaumann derivative, a quite different approach to constructing the Maxwell model was considered by Truesdell [31, § §56, 81; 32]. Overcoming the original Natanson line towards a finite viscoelasticity, Truesdell as a starting point uses an assumption about a linear superposition of elastic and viscotic agendae ("superposition theories").…”

Section: Geometry Of Deformationmentioning

confidence: 99%

On objective surface rates

Stumpf¹,

Badur²

1993

Quart. Appl. Math.

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Abstract. In this paper we derive objective, in the sense of surface, rates of tensors and we give a correct formulation of a two-dimensional continuum. Furthermore we present objective rates of tensors on the boundary line enclosing the surface under consideration. It can be considered as a first step to derive objective rates for generalized continua as Cosserat continua and Kirchhoff-Love type nonlinear shell theories.1. Introduction. Objective time derivatives were introduced into the constitutive equations primarily to describe properly the time change of stress and strain tensors. Starting from the pioneering papers of Zaremba [36] and Jaumann [12], the concept of objective rates has been the subject of several works. Among them the fruitful concept of "convective derivatives" was given by Oldroyd [22], It can be considered as a method of time differentiation of any tensor field attached to the deformed body, but performed after some mapping into a place where a comparison of different tensors can be made. Since Oldroyd has used the gradient of deformation for the mapping operation, all objective derivatives based on the deformation gradient or its parts are called "convective rates." Special cases are presently known as ZarembaJaumann, Cotter-Rivlin [4] or Rivlin-Ericksen derivatives related to strain rates, and the Truesdell derivative [31] related to stress rates. Additional arguments in favor of the convective derivatives were given by Sedov [25] and Masur [17]. The latter has shown how the Zaremba-Jaumann derivative is connected with that of Oldroyd.Recently, great efforts have been made for a correct formulation of two-dimensional thermomechanics of a curved surface. For many phenomena, like dynamics of a phase interfaces layer [34], viscoelastic 2-dimensional flow of the products of friction and wear [37], capillary flow [9], bubble dynamics [11], dynamics of a crack [27], elastoviscoplasticity of thin shells, etc., it is required to define useful constitutive equations. Frequently, the constitutive equations of differential or rate type are used, since they are easier to implement in computer codes than equations of integral type. These phenomenological relations should contain information about the surface geometry as well as objective surface rates both of stresses and strains.The aim of this paper is to consider a correct formulation of the two-dimensional continuum and to derive objective, but in the sense of surface, rates of tensors.

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Section: Geometry Of Deformationmentioning

confidence: 99%

On objective surface rates

Stumpf¹,

Badur²

1993

Quart. Appl. Math.

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“…the stress rate, has aroused much interest recently and has been discussed and investigated by Truesdell [1,2], Noll [3], Oldroyd [4], Thomas [5], Cotter and Rivlin [6], and Prager [7]; the last, with the limitation to rectangular Cartesian coordinates, contains an account of the various definitions for the stress rate proposed in [1,3,4,5,6], as well as in others, including the earlier work of Jaumann [8]1. Although the expressions for the stress rate deduced in [1][2][3][4][5][6] differ from one another, their underlying objective is the requirement that the constitutive equations of the medium involving the stress rate must remain invariant under rigid motions.…”

Section: Introductionmentioning

confidence: 99%

“…Although the expressions for the stress rate deduced in [1][2][3][4][5][6] differ from one another, their underlying objective is the requirement that the constitutive equations of the medium involving the stress rate must remain invariant under rigid motions. In addition, Prager [7] has pointed out the desirability for a definition of the stress rate, the vanishing of which will render the invariants of the stress tensor stationary. Also, mention should be made of a very recent paper by Sedov [9], dealing with the time derivative of tensors, which will be referred to again.…”

Section: Introductionmentioning

confidence: 99%

“…requirement is equivalent to that stipulated by Prager [7] when the coordinate system is rectangular Cartesian and the time derivative is taken in the sense of Jaumann.…”

Section: Introductionmentioning

confidence: 99%

“…(»« 1/ + I.) (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) for the rate of strain tensor. Although the application of (3.13) to the stress tensor will be postponed until the next section, it should be emphasized that the time derivative operator defined by (3.13) is applicable to any tensor, whether relative or absolute, provided the transformation law for the tensor in question is known a priori.…”

Section: Introductionmentioning

confidence: 99%

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