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On the Decomposition of the Deformation Gradient in Plasticity

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Abstract

Starting from the premise that the distances between points are the only measurable quantities, plasticity is placed into the more general context of the continua with a two-scale representation of the deformation. The Kröner-Lee multiplicative decomposition of the deformation gradient comes out to be incompatible with the geometry of such continua, while the Clifton multiplicative decomposition is compatible but geometrically irrelevant. On the contrary, an approximation theorem taken from the theory of structured deformations provides a measure-theoretic justification for the additive decomposition. It also leads to a decomposition of the strain energy into the sum of two parts, one for each term of the decomposition of the deformation.

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Notes

  1. I found no trace of this condition in the literature. As shown in the next sections, ignoring this condition was the origin of the many conflicting opinions about the indifference requirements.

  2. See e.g. [5, 24, 28].

  3. Del Piero and Owen [10, 11].

  4. Noll, [29] Sect. 3 and [30] Sect. 7. Before Noll, the term configuration was used in a vague, informal way. To him is due the distinction between extrinsic configurations, which he called placements, and intrinsic configurations, which he identified with distance functions. In spite of Noll’s precise definitions, the term configuration is still being used in a vague and informal way.

  5. Typically, the “fixed stars”, or the walls of the laboratory, see [40], Sect. 17.

  6. For example, a deviation may be a change of the crystal lattice directions in a crystalline solid.

  7. Though each \(\nabla f_{X_{\emptyset }}\) is the gradient of the corresponding \(f_{X_{\emptyset }}\), \(F\) is not in general a gradient, because \(\nabla f_{X_{\emptyset }}\) is the gradient of different functions \(f_{X_{\emptyset }}\) at different points.

  8. See Del Piero [9].

  9. Kröner [18], Lee and Liu [21]. According to the historical outline by Sadik and Yavari [35], the idea of this decomposition goes back to Eckart in 1948, and was developed in the 1950’s by Kondo, Bilby, and Kröner.

  10. “Ein gedachter Zustand” [18], “a thought experiment” [20].

  11. In gradient plasticity, which is based on the same two-scale geometry, there are further stress measures, partly energetic and partly dissipative. They are assumed to depend on \(F^{e}\) and \(\nabla F^{e}\) and on \(F^{p}\) and \(\nabla F^{p}\), respectively. See, e.g., Gurtin et al. [15], Sect. 90.

  12. This describes the phenomena of elastic unloading and elastic reloading, typical of plasticity.

  13. Nevertheless, once the constitutive equation has been constructed, the pair \((\nabla f,F^{e})\) can be taken as a purely geometric pair of independent variables. Indeed, once this pair is known, \(F^{p}\) and \(T\) follow from (13) and (14), respectively.

  14. See Truesdell and Noll [40], Sect. 47.

  15. In [3], Casey and Naghdi pointed out that the symmetry of \(F^{e}\) is not preserved under all arbitrary distance preserving transformations, since the transformed \(F^{e}\) is given by equation (17). This led them to conclude that Lee’s proposal was “unduly restrictive on physical grounds”.

  16. An attempt in this direction was made by Lubarda [23].

  17. Here the term configuration is used in the sense of classical continuum mechanics, that is, according to the definition given in Sect. 2.

  18. Sidoroff [36] and, independently, Green and Naghdi [13]. Both disregarded the fact that, if only the preservation of distances is required, there is no reason for assuming \(\hat{Q}\) constant over the body.

  19. See Truesdell and Noll [40], Sect. 43.

  20. This conclusion suggested to consider condition (16)2 not as an indifference requirement, but as a constitutive restriction on materials without preferred material directions, like amorphous materials (Anand and Gurtin [1]) or isotropic-viscoplastic materials (Gurtin and Anand [14]).

  21. See Truesdell and Noll [40], Sect. 28.

  22. See Truesdell and Noll [40], Sect. 43.

  23. See [26] Sect. 8.2, and the papers cited therein.

  24. “Observers view only the deformed configuration” [14]. That is, they view only the total deformation \(\nabla f\).

  25. To my knowledge, the first explicit statement of the rule \(F^{p*}=F^{p}\) is due to Šilhavý, who deduced it from general properties of deformation histories for materials with elastic range [37]. It must be said, however, that he did not consider this rule as “the only one possible”. On the contrary, he explicitly mentioned the rule \(F^{p*}=QF^{p}\) as a possible alternative. Relatively recent positions pro and against the full invariance rules can be found in [16] and [15], respectively.

  26. In [21], [36], [22], [33], [15] Sect. 91.2, respectively.

  27. Indeed, the full invariance rules (16) state the indifference under all distance-preserving changes of placement, while the no invariance rules (21) state the indifference only under changes of placement within the same configuration. Neither involves the state of the material, which is an array of independent variables, not necessarily geometric (see, e.g., [29]). For an example of different interpretations of the “intermediate configuration”, compare those of Gurtin et al. [15] Sect. 91.2, and of Simo and Hughes [39], Sect. 9.1.2.

  28. Surprisingly enough, similar conclusions have been reached in the recent paper [34], starting from a very different approach.

  29. This has been recognized by some authors. See, e.g., Lee [20] and Nemat-Nasser [28]. To consider \(F^{p}\) as an independent geometric variable may lead to misconceptions, as confirmed by the dispute about the indifference of the transformation rules.

  30. Clifton [5]. See also Nemat-Nasser [28], Owen [31], Del Piero and Owen [10], Lubarda [24].

  31. See [31] and [10].

  32. Both decompositions “are formal, and their practical usefulness remains to be established” [28].

  33. For a more general setting in the space \(SBV\) of simple functions with bounded variation see Choksi and Fonseca [4].

  34. Additive decompositions in finite plasticity were proposed by Green and Naghdi [12] and by Nemat-Nasser [28]. Their formal equivalence to the multiplicative decompositions of Kröner-Lee and of Clifton was shown in [10].

  35. Therefore, (32) is the Radon-Nikodým decomposition of this measure into an absolutely continuous and a singular part, with the particularity that the singular part also has a volume density. For further details see [4], and for some examples see [8], Sect. 4.3.

  36. Green and Naghdi [12].

  37. Choksi and Fonseca [4].

  38. In [4], the \(SBV\) functional setting and a notion of convergence weaker than (27) are assumed. In this broader functional setting, the energies of the approximating sequences to a given structured deformation need not converge to the same limit. This is the reason for taking the relaxed energy (34) as the energy of a structured deformation.

  39. In the proof, the functions \(f\) and \(F\) were supposed to be piecewise continuously differentiable and piecewise continuous, respectively, as assumed in the original paper [10]. The energy densities \(\varphi \) and \(\theta \) were supposed to be continuous, and \(\theta \) was supposed to have a nonnegative right derivative \(\theta '(0^{+})\) at zero. In [7], the proof was extended to pairs \((f,F)\) in \(SBV\times L^{1}\) and to pairs \((\varphi ,\theta )\) of non-negative functions, which in the decomposition (36) are replaced by their convex and lower semicontinuous envelope and by their subadditive envelope, respectively.

  40. Baìa et al. [2]. A finer characterization of the dependence of \(\varphi \) and \(\theta \) upon \(F\) and \(M\) was given in the recent paper [38] by Šilhavý.

  41. However, \(\theta \) may also have an energetic part, related to the back stress of Prager’s kinematic hardening model. See, e.g., Anand and Gurtin [1].

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Acknowledgements

I thank the anonymous reviewers for their pertinent and constructive comments. In particular, the one of them who detected a substantial inaccuracy in the first version of the paper.

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  1. Dipartimento di Ingegneria, Università di Ferrara, 44100, Ferrara, Italy

    Gianpietro Del Piero

  2. International Research Center M&MoCS, Cisterna di Latina, Italy

    Gianpietro Del Piero

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Del Piero, G. On the Decomposition of the Deformation Gradient in Plasticity. J Elast 131, 111–124 (2018). https://doi.org/10.1007/s10659-017-9648-z

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