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2013 | OriginalPaper | Chapter

Material Strain Tensor

Authors : Pavel A. Zhilin, Holm Altenbach, Elena A. Ivanova, Anton Krivtsov

Published in: Generalized Continua as Models for Materials

Publisher: Springer Berlin Heidelberg

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Abstract

The problem of description of large inelastic deformations of solids is considered. On a simple discrete model it is shown that the classical concept of deformations used in continuum mechanics can exhibit serious difficulties due to reorganizations of the internal structure of materials. The way of construction of constitutive equations in continuum mechanics aimed to avoid these problems is proposed. A method of introduction of material strain tensor for the inelastic continuum is suggested. The paper is based on the report: P. A. Zhilin, A. Krivtsov: Point mass simulation of inelastic extension process. It was prepared for the ICIAM 95 (Third International Congress on Industrial and Applied Mathematics, Hamburg, Germany, July 3–7, 1995), but not accepted for publication.

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previous chapter A Theory of Disclination and Dislocation Fields for Grain Boundary Plasticity

Among such theories probably the best results in explanation of experimental phenomena are given by the so-called “deformation theory” of H. Hencky, sometimes much better than the rate theory can do [13]. As it can be seen from below, there are serious reasons for that.

This model was proposed by P. A. Zhilin and analyzed by A. Krivtsov.

Here it is used: \(\displaystyle \dot{\varvec{Q}}\varvec{\!}\cdot \!\varvec{Q}^T\)—antisymmetric tensor, identity \(\displaystyle {\varvec{A}\varvec{\cdot \cdot }\varvec{B}\varvec{\!}\cdot \!\varvec{C} = \varvec{A}\varvec{\!}\cdot \!\varvec{B}\varvec{\cdot \cdot }\varvec{C}}\) and statement: \(\displaystyle {\varvec{A}\varvec{\cdot \cdot }\varvec{B}=\mathrm{0},\quad \forall \varvec{A}:\ \varvec{A}^T=-\varvec{A}\quad \Rightarrow \quad \varvec{B}^T = \varvec{B}}\).

This statement becomes more evident if we consider the linear theory. Indeed, in the linear theory the elasticity relations have the form \(\displaystyle \varvec{\tau }= \mathbf C \varvec{\cdot \cdot }\varvec{\varepsilon }\), where \(\displaystyle \mathbf C \) is the stiffness tensor and \(\displaystyle \varvec{\varepsilon }\) is the linear strain tensor, which has pure geometrical definition. In the case of an anisotropic material the principal axis of the tensors \(\displaystyle \varvec{\varepsilon }\) and \(\displaystyle \mathbf C \varvec{\cdot \cdot }\varvec{\varepsilon }\) have different orientations. In our case we have to introduce an alternative strain tensor \(\displaystyle \varvec{\mathcal{{E}}}\) in such way, that it should be coaxial to the tensor \(\displaystyle \mathbf C \varvec{\cdot \cdot }\varvec{\mathcal{{E}}}\). It is clear, that such a strain tensor should by some means take into account the anisotropy of the material.

This proof is suggested by A. Krivtsov, the original proof by P.A. Zhilin unfortunately is lost.

This result was obtained by P. A. Zhilin and it was explained in private communications to his pupils before 1995, however it was not officially published. In 1995 a short paper with this result was submitted to ICIAM 95 proceedings, however it was rejected. In 1997 a paper by other authors was published in Acta Mechanica [9], where the same result is presented as obtained for the first time.

Proof of these statements by P. A. Zhilin unfortunately is not preserved.

Frequently an alternative form of the corotational rate is used, where the difference is in the sign of \(\displaystyle \varvec{\Omega }\). This is because the definition of the gradient of a vector can be as in this chapter and [7] or in the transposed form. As a consequence the sign of the spin tensor can differ.

This formula for logarithmic rate differs from the one in [9] by the sign of \(\displaystyle \varvec{\Omega }^{log}\) (see the previous footnote).

For some particular strain fields (e.g. when all the tensors \(\displaystyle \varvec{H}\) are coaxial) the tensor \(\displaystyle \varvec{\Omega }^{\log }\) is reduced to the vorticity tensor \(\displaystyle (\varvec{\nabla }\varvec{v})^A\) and logarithmic rate became Jaumann’s rate. However in general case the representation for \(\displaystyle \varvec{\Omega }^{\log }\) is much more complex, which is connected with existence of two independent rotations—rotation of media and rotation of the main axis of the strain tensor.

Personal communication by O.T. Bruhns

Truesdell, C.: First Course in Rational Continuum Mechanics: General Concepts (2nd edn.), vol. 1. Academic Press, San Diego (1991)

Palmov, V.A.: Vibrations of Elasto-Plastic Bodies. Springer, Berlin (1998)MATHCrossRef

Altenbach, H.: Kontinuumsmechanik - Eine elementare Einführung in die materialunabhängigen und materialabhängigen Gleichungen. Springer-Vieweg, Heidelberg (2012)

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Krivtsov, A.: Deformation and Fracture of Solids with Microstructure (in Russ.). Fizmatlit, Moscow (2007)

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Lurie, A.I.: Theory of Elasticity. Springer, Berlin (2005)CrossRef

Lebedev, L.P., Cloud, M.J., Eremeyev, V.A.: Tensor Analysis with Applications to Mechanics. World Scientific, Singapore (2010)CrossRef

Xiao, H., Bruhns, O.T., Meyers, A.: Logarithmic strain, logarithmic spin and logarithmic rate. Acta. Mech. 124, 89–105 (1997)MathSciNetMATHCrossRef

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Palmov, V.A.: Fundamental Laws of Nature in Nonlinear Thermomechanics of Deformoble Bodies (in Russ.). SPbSTU Publishing, St. Petersburg (2008)

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Golovanov, A.I.: Kinematics of finite deformations 3D isoparametrical finite elements for shells (in Russ.). Probl. Strength. Plast. 70, 109–122 (2008)

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Xiao, H., Bruhns, O.T., Meyers, A.: Existence and uniqueness of the integrable-exactly hypoelastic equation \({{\vec{\tau }} }^{\circ *} = \lambda ({\rm {tr}}{\bf {D}}){\bf {I}}+2\upmu {\bf {D}}\) and its significance to finite inelasticity. Acta. Mech. 138, 31–50 (1999)

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Xiao, H., Bruhns, O.T., Meyers, A.: Elastoplasticity beyond small deformations. Acta.Mech. 182, 31–111 (2006)MATHCrossRef

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Korobeynikov, S.N.: Families of continuous spin tensors and applications in continuum mechanics. Acta. Mech. 216, 301–332 (2011)MATHCrossRef

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Xiao, H., Bruhns, O.T., Meyers, A.: A new aspect in the kinematics of large deformations. In: Gupta, N.K. (ed.) Plasticity and Impact Mechanics, pp. 100–109. New Age International Ltd Publishing, New Delhi (1997)

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Bruhns, O.T., Meyers, A., Xiao, H.: On non-corotational rates of Oldroyd’s type and relevant issues in rate constitutive formulations. Proc. Royal Soc. Lond. A. 460, 909–928 (2004)MathSciNetMATHCrossRef

Title: Material Strain Tensor
Authors: Pavel A. Zhilin
Holm Altenbach
Elena A. Ivanova
Anton Krivtsov
Publisher: Springer Berlin Heidelberg
Book: Generalized Continua as Models for Materials
Print ISBN: 978-3-642-36393-1

Electronic ISBN: 978-3-642-36394-8

Copyright Year: 2013
DOI: https://doi.org/10.1007/978-3-642-36394-8_19

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