Analytical stress solutions of a closed deformation path with stretching and shearing using the hypoelastic formulations

Lin, R.C.; Schomburg, U.; Kletschkowski, Thomas

doi:10.1016/s0997-7538(03)00031-7

Cited by 24 publications

(13 citation statements)

References 16 publications

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“…We will come back to that point shortly. 43 A result which already has been demonstrated by Jaumann in 1911. where Q * is a proper orthogonal tensor. In an Ω * -frame relative to a fixed background frame, this pair becomes (Q * sQ * T , Q * eQ * T ).…”

Section: The Logarithmic Rate and Related Propertiesmentioning

confidence: 61%

“…• would be a very good approximation to D for sufficiently small deformations 43 . Inspired by their works and using an explicit formula for the gradient of the general strain measure e with respect to the stretch tensor V (see [8] and [96]) Xiao et al [98] could prove that an objective corotational rate of the logarithmic strain measure ln V could be identical with the stretching tensor D, and furthermore that in all possible strain measures only ln V would enjoy this property, i.e.…”

Section: The Logarithmic Rate and Related Propertiesmentioning

confidence: 98%

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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 Logarithmic Rate and Related Propertiesmentioning

confidence: 61%

Section: The Logarithmic Rate and Related Propertiesmentioning

confidence: 98%

The Prandtl‐Reuss equations revisited

Bruhns

2013

Z Angew Math Mech

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“…For this formulation the stress-strain relation (24) and almost all evolution equations must be applied. By application of the chain rule, the stress-strain relation (24) and the evolution equations included in (32) and after a lengthy calculation we get…”

Section: Algorithm Aspectsmentioning

confidence: 99%

“…By this approach, one can also reformulate the Chaboche laws for finite strain cases. But, it has been shown that the hypoelastic equations related to the conventional stress rates are not consistent with elasticity and cause spurious constitutive responses in deformation processes with finite rotations [23,24]. The reason for these spurious responses is that the types of these conventional stress rates are different from the type of the spatial deformation rate, which is a logarithmic corotational rate of the spatial logarithmic strain.…”

Section: Introductionmentioning

confidence: 99%

Modeling of finite strain viscoplasticity based on the logarithmic corotational description

Lin

Betten²,

Brocks

2006

Arch Appl Mech

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It has been proven that the deformation rate is identical to the logarithmic corotational rate of the spatial logarithmic strain. Based on the logarithmic corotational description we present an extension of a Chaboche's infinitesimal viscoplastic law for finite strain cases. An additive decomposition of the logarithmic corotational rate of the logarithmic strain, implicitly included in an internal dissipation inequality, is applied for the extension. Functionally, this additive decomposition corresponds to the additive decomposition of the material derivative of the logarithmic strain used in infinitesimal inelastic problems. Using the additive decomposition and replacing the material time derivatives of all second-order tensors in the infinitesimal viscoplastic law with the corresponding logarithmic corotational rates we arrive at a new finite viscoplastic law with nonlinear isotropic and kinematic hardening. The numerical algorithms suitable for the finite element implementation of this law are formulated and several numerical examples are presented. These numerical examples prove that the finite viscoplastic law and the algorithms are effective and reliable.

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“…Moreover, special care had to be undertaken in the selection of the rate elastic constitutive equation (2) for the approach to be compatible with the notion of hyperelasticity (see, e.g., Simo and Pister [1984]) which requires the existence of a stored energy function. Elasticity without a stored energy function is difficult to motivate physically, since it may result in aberrant elastic behavior which may be manifested by hysteretic energy dissipation (see, e.g., Bernstein [1960]) and/or residual stresses after a closed strain path (see, e.g., Lin [2002]; Lin et al [2003], Meyers et al [2003]). In this work, the large strain generalized plasticity theory is revisited and further extended by introducing the concept of the logarithmic rate that treats the aforementioned inconsistency.…”

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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Analytical stress solutions of a closed deformation path with stretching and shearing using the hypoelastic formulations

Cited by 24 publications

References 16 publications

The Prandtl‐Reuss equations revisited

The Prandtl‐Reuss equations revisited

Modeling of finite strain viscoplasticity based on the logarithmic corotational description

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

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