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Large deformation plasticity

From basic relations to finite deformation

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The theory of plasticity as a special field of continuum mechanics deals with the irreversible, i.e. permanent, deformation of solids. Under the action of given loads or deformations, the state of the stresses and strains or the strain rates in these bodies is described. In this way, it complements the theory of elasticity for the reversible behavior of solids. In practice, it has been observed that many materials behave elastically up to a certain load (yield point), beyond that load, however, increasingly plastic or liquid-like. The combination of these two material properties is known as elastoplasticity. The classical elastoplastic material behavior is assumed to be time-independent or rate-independent. In contrast, we call a time- or rate-dependent behavior visco-elastoplastic and visco-plastic—if the elastic part of the deformation is neglected. In plasticity theory, because of the given loads the states of the state variables stress, strain and temperature as well as their changes are described. For this purpose, the observed phenomena are introduced and put into mathematical relationships. The constitutive relations describing the specific material behavior are finally embedded in the fundamental relations of continuum theory and physics. Historically, the theory of plasticity was introduced in order to better estimate the strength of constructions. An analysis based purely on elastic codes is not in a position to do this, and can occasionally even lead to incorrect interpretations. On the other hand, the entire field of forming techniques requires a theory for the description of plastic behavior. Starting from the classical description of plastic behavior with small deformations, the present review is intended to provide an insight into the state of the art when taking into account finite deformations.

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  1. 1.

    A somewhat more detailed outline of the history of plasticity can be found in Refs. [1,2,3,4]. Parts of these works were included in the present review article.

  2. 2.

    Here and in what follows, a prime will mark the deviator of a second order tensor.

  3. 3.

    In the English literature, these models are also referred to as incremental and total theories.

  4. 4.

    We note that during loading \(\varrho \) takes the value 1.

  5. 5.

    This calculation was first carried out in Ref. [7] for the specific case of an isotropic material with hardening.

  6. 6.

    In several of these works, the progress of the plastic processes is described by accumulated plastic strains \({\varvec{\varepsilon }}_{{\mathrm{p}}}\) rather than by the accumulated plastic work \(w^{{\mathrm{p}}}\). This, however, does not change the results significantly.

  7. 7.

    The interested reader may in particular follow the discussion in Sects. 3 and 4 of this critical review.

  8. 8.

    In passing, we note that for an adiabatic process, i.e. for \(- \nabla {\varvec{\cdot \,}}{\varvec{ q}}+r \equiv 0\), due to the different sources of dissipation the remaining thermodynamic process is not isentropic.

  9. 9.

    The proof of this limit is given with Ref. [4].

  10. 10.

    Unfortunately, the original work of Hencky contained a small error. Instead of the above spin tensor, he used an alternative definition, which differs by a minus sign. Due to this deviation, however, his derivative (52) loses its objectivity.

  11. 11.

    We note that in the original literature instead of the small circle a wavy line was used to designate an objective time derivative. In some more recent literature, the raising and lowering of the indices c is in analogy to the accidentals of music notation designated by a sharp (\(\sharp \)) or a flat (\(\flat \)).

  12. 12.

    Noll called this a general invariance requirement principle of isotropy of space, whereas Thomas used the term absolute time derivative.

  13. 13.

    Unfortunately, this paper was written in German and submitted for publication to a Romanian journal in 1968. Due to severe production problems in those days, the article appeared not until 1972. These circumstances may explain why this paper was widely ignored.

  14. 14.

    It seems that a direct precursor of Eq. (65) is relation (2.13) with Eqs. (2.11) and (2.15) in Ref. [115], which says that the Jaumann rate of \({\varvec{ h}}\) should exactly give \({\varvec{ D}}\) in some cases; see also Refs. [110, 111].

  15. 15.

    Almost at the same time several groups were seeking a solution for this problem, namely to express the stretching \({\varvec{ D}}\) as an objective rate of an Eulerian strain. It is reported that P.A. Zhilin 1995 starting from a quite different idea came to a conclusion comparable to Eq. (68). Unfortunately, however, his result remained unpublished then (refer to Ref. [118]). Later, Eq. (68) was also discovered by Reinhardt and Dubey [119, 120]. This relationship was derived in the general sense of studying Eq. (65) independently, and its intrinsic uniqueness property was thus revealed for the first time in Refs. [78, 114, 121, 122] from different contexts. Regrettably, Profs. Lehmann, Guo, and Zhilin were not able to realize the publication of their works and their seminal ideas.

  16. 16.

    For details, we refer to Ref. [13].

  17. 17.

    For a discussion of the pros and cons of the strain-space formulation, we refer to Ref. [13].

  18. 18.

    Here, \({\varvec{ R}}_{{\mathrm{e}}} = {\varvec{ I}}\) was used. An alternative approximation has been introduced in Ref. [157] with \({\varvec{ W}}_{{\mathrm{p}}} = \mathbf{0}\).

  19. 19.

    This discussion was primarily between the two schools and their followers and lasted several years. We therefore refer to Refs. [166,167,168,169] and the instructive discussion therein.

  20. 20.

    There might, however, be some doubt about the physical pertinence of such non-symmetric flow rule in a 9-dimensional space. It would imply that nine, instead of six, rate equations governing plastic flow should be needed even in the case of infinitesimal deformation, except for some particular cases. We also refer to Ref. [172] and the discussion therein.


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Bruhns, O.T. Large deformation plasticity. Acta Mech. Sin. (2020).

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  • Finite deformation
  • Elasto-plasticity
  • Constitutive relations
  • Thermodynamics
  • History of plasticity