XVII.—Generalized Hypo-Elasticity

Green, A. E.; McInnis, B.

doi:10.1017/s0080454100008074

Cited by 22 publications

(22 citation statements)

References 5 publications

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“…17 Here ε p stands for the irreversible plastic part of the strain ε. 18 We should mention here that none of the cited works has used the notion of "internal variable". This term has been introduced much later.…”

Section: The Original Work Of Prandtl and Reussmentioning

confidence: 99%

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The Prandtl‐Reuss equations revisited

Bruhns

2013

Z Angew Math Mech

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At the beginning of the last century two different types of constitutive relations to describe the complex behavior of elastoplastic material were presented. These were the deformation theory originally developed by Hencky and the Prandtl-Reuss theory. Whereas the former provides a direct solid-like relation of stress as function of strain, the latter has been based on an additive composition of elastic and plastic parts of the increments of strains. These in turn were taken as a solid-and fluid-like combination of the de Saint-Venant/Lévy theory with an incremental form of Hookes law.Even nowadays this Prandtl-Reuss theory is still accepted within the restriction of small elastic deformations, i.e. it is generally stated in most textbooks on plasticity that this theory due to a number of defects can not be applied to large deformations. In the present article it is shown that this restrictive statement may be no longer true. Introducing a specific objective time derivative it could be shown that these defects disappear.

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Section: The Original Work Of Prandtl and Reussmentioning

confidence: 99%

“…Dienes with his "current theory" used the Green-Naghdi rate or "polar rate" (refer to [17,18]) where the material time derivative is applied to a rotated stress tensor, thus replacing the vorticity W of the Jaumann rate by a skew-symmetric rate of rotation Ω R with…”

Section: Wwwzamm-journalorgmentioning

confidence: 99%

The Prandtl‐Reuss equations revisited

Bruhns

2013

Z Angew Math Mech

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“…To determine such a derivative we first observe that instead of U rather a combination of the two-point tensors f and r should be applied. From However, B = 0 leads to a Zaremba-Jaumann type of derivative and such case was introduced by Green and Mclnnis [7] in the 3-D context. 6.…”

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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“…As a result any tensorial quantity defined in this configuration will be material frame-indifferent such that the homonym principle is trivially satisfied. Next, motivated by the material rotated description of elasticity (see Green and McInnis [1967]; Simo and Marsden [1984a,b]; see also [Simo and Hughes, 1998, pp. 271-275] for the elastic-plastic case), this configuration is identified as a reference one for the development of the theory.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

Soldatos

Triantafyllou

2016

Int. J. Appl. Mechanics

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In this work we present a new rate type formulation of large deformation generalized plasticity which is based on the consistent use of the logarithmic rate concept. For this purpose, the basic constitutive equations are initially established in a local rotationally neutralized configuration which is defined by the logarithmic spin. These are then rephrased in their spatial form, by employing some standard concepts from the tensor analysis on manifolds. Such an approach, besides being compatible with the notion of (hyper)elasticity, offers three basic advantages, namely:(i) The principle of material frame-indifference is trivially satisfied.(ii) The structure of the infinitesimal theory remains essentially unaltered.(iii) The formulation does not preclude anisotropic response.A general integration scheme for the computational implementation of generalized plasticity models which are based on the logarithmic rate is also discussed. The performance of the scheme is tested by two representative numerical examples.

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XVII.—Generalized Hypo-Elasticity

Abstract: SynopsisThe theory of Hypo-Elasticity is generalized so as to include the theory of (anisotropic) elasticity as a special case, and to include thermal effects. Explicit restrictions on the constitutive equations, arising from thermodynamics, are obtained.

Cited by 22 publications

References 5 publications

The Prandtl‐Reuss equations revisited

The Prandtl‐Reuss equations revisited

On objective surface rates

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

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