Skip to main content
SpringerLink
Log in

The Multiplicative Decomposition of the Deformation Gradient in Plasticity—Origin and Limitations

  • Chapter
  • First Online:
From Creep Damage Mechanics to Homogenization Methods

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 64))

Abstract

The history of material equations and hence the development of present material theory as a method to describe the behavior of materials is closely related to the development of continuum theory and associated with the beginning of industrialization towards the end of the 19th century. While on the one hand new concepts such as continuum, stresses and strains, deformable body etc. were introduced by Cauchy, Euler, Leibniz and others and mathematical methods were provided to their description, the pressure of industrialization with the need to ever newer, and likewise reliably secure, developments has led to the fact that more appropriate models for the description of elastic-plastic behavior were introduced. Upon this background, this Chapter wants to introduce into the history of plasticity of the sixties and seventies of last century, and likewise highlight the eminent contributions of A.E. Green and P.M. Naghdi, E.H. Lee and J. Mandel to a modern description of finite plasticity theory.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    It should be generally accepted that a strain larger than \(10\,\%\) would violate the requirements of a linear theory.

  2. 2.

    For details we refer to Bruhns (2014a, b). See also the additional works of Prager (1944) and Edelman and Drucker (1951).

  3. 3.

    Although in their paper the plastic strain has been introduced as a primitive variable, and the elastic-like strain is merely defined by the difference of the total strain and this plastic variable, it is common practice to refer to this sum as an additive decomposition of the strain tensor into its elastic and plastic parts.

  4. 4.

    This insures that X and x are in a one-to-one correspondence, which is reasonable for solid bodies. We note, however, that this assumption also has limitations, e.g. in fracture mechanics, crystallographic slip, lattice distortions, etc., which appear on the microscale level during e.g. plastic deformation.

  5. 5.

    Besides this measure is also called Biot strain or Cauchy strain (Ogden 1984; Bertram 2005).

  6. 6.

    This generalization of the strain measure is mainly due to Doyle and Ericksen (1956); Seth (1964); Hill (1968), and includes also non-integral real values of n.

  7. 7.

    This combination with \(m=1\) was introduced by Böck and Holzapfel (2004) as a two-point deformation tensor. The extension to a more general two-parameter form with independent parameters n and m is due to Darijani and Naghdabadi (2010). We here will confine ourselves to the most simple case when \(n=m\).

  8. 8.

    The history of the multiplicative decomposition of the total deformation into elastic and plastic parts seems to be somewhat vague or at least unclear. Different authors mention different origins, and the different disciplines continuum mechanics and materials science involved in this discussion seem to have different sources. In their own paper Lee and Liu (1967) mention: “The concept of an unstressed configuration ...has appeared before in the literature, but does not seem to have been put to satisfactory use.” All in all, it should be almost clear that Eckart (1948) was the first to use local natural configurations in order to separate the elastic from the plastic part of a total deformation. Backman (1964) introduces three continuous configurations, and represents the elastic and plastic components of the total strains in terms of displacement derivatives, which in general cannot be done. It seems that Kröner (1958, 1960) and Bilby et al. (1957) then were the first to use this expression, with respect to sequences of elastic and plastic distortions.

  9. 9.

    The symbol \({(\bullet )}^{\text{ T }}\) herein is used to represent the transpose of the second order tensor.

  10. 10.

    As usual \({\varvec{I}}\) is a second order unit tensor.

  11. 11.

    This idea was exemplified by Macvean (1968), and fully developed by Hill (1968, 1970, 1978). Hill (1978) also introduced the notion of “work conjugacy”, although work-rates or stress powers are discussed (see also the Appendix of Hill 1968).

  12. 12.

    In a review of the development and the usage of internal variables in inelasticity Horstemeyer and Bammann (2010) stated that—with respect to the period between 1870 and 1945: “Over the next 75 years progress (of plasticity) was slow and spotty,...” It seems to the present author that one should be aware that just this period was marked by two big wars that have reduced large amount of Europe and Asia to wrack and ruin.

  13. 13.

    Here for simplicity the notation \((\bullet )' = (\bullet ) - \frac{1}{3}\text {tr}{(\bullet )}{\varvec{I}}\) will be used to mark a deviator of a second order tensor.

  14. 14.

    This amendment was due to Nádai (1931). Nádai having arrived in the USA in 1927 as well as Hencky himself during his time in the former Soviet Union successfully promoted the distribution of this theory.

  15. 15.

    In a generalized manner this is a surface bounding the domain of elastic deformations. In the classical v. Mises stress space formulation a yield function \(\text{ f } = \text{ f }({\varvec{\sigma }}')\) as function of the stress deviator is bound to a so-called yield limit.

  16. 16.

    This function is in general determined by means of the so-called condition of consistency \(\dot{\text{ f }}({\varvec{\sigma }}') = 0\), which during plastic loading forces the stress to stay on the yield surface.

  17. 17.

    To this end scalar-valued and second order tensor-valued internal variables have been introduced to model these phenomena by means of the evolution of these variables.

  18. 18.

    If our notations are used. As objective rate Tokuoka has taken a Jaumann rate—without mentioning this source. Instead he referred to the fundamental work of Truesdell (1955). For the sake of completeness, we add that Truesdell (1952a, b, c, 1955) has introduced the notion of “hypoelasticity” for this specific material, and that in the original forms the tangential stiffness tensor \({\mathbb L}\) has been used in a more general form as function of the stress.

  19. 19.

    Only recently it was demonstrated that this problem was closely related to a second one, namely to prove the integrability of hypoelastic relation (3.51). It turned out that to prove this a new objective time derivative had to be defined which, moreover applied to a yet undetermined Eulerian strain measure, could give the (Eulerian) stretching. It was shown that this (Eulerian) strain had to be the Hencky strain, and the new (objective) time derivative, in turn, was the logarithmic rate. Only with this logarithmic rate applied to \({\varvec{\sigma }}\) Eq. (3.51) could be integrated to give a really elastic relation. We therefore refer to Bruhns (2014b) and the numerous references mentioned therein.

  20. 20.

    Its formulation from a purely mechanical point of view has been developed later by Naghdi and coworkers and others. For details, we refer to Naghdi (1990).

  21. 21.

    Similar internal variables have been first used in line with the amendments of the Prandtl-Reuss relations. We refer e.g. to Bruhns (2014b).

  22. 22.

    Alternatively, the sets \(({\varvec{E}}-{\varvec{E}}_p,{\varvec{E}}_p,{\varvec{\alpha }},\kappa )\) as well as \(({\varvec{S}},{\varvec{E}}_p,{\varvec{\alpha }},\kappa )\) may be useful for certain specific purposes. For instance, the last set is used in a stress space formulation.

  23. 23.

    See, e.g., Naghdi and Trapp (1975a, b, c). For a discussion of the pros and cons of the strain space formulation, we also refer to Naghdi (1990).

  24. 24.

    In a general context, Hill and Rice (1973) have demonstrated that such a relation holds true with \(\hat{\psi }\) being a fully general elastic potential relying on the “prior history of inelastic deformation”, and with \(({\varvec{S}},{\varvec{E}})\) any given work-conjugate pair.

  25. 25.

    In this respect, we refer, e.g., to Lee (1996) and Lee and Germain (1974), and to the discussion of the different strains for the subsequently loaded rod of Sect. 3.2.

  26. 26.

    This was initiated by Lee and Liu (1967) and Lee (1969) and use was made by Fox (1968), Willis (1996), Freund (1970), Rice (1971), albeit it may be traced back to earlier works by Eckart (1948), Eglit (1960), Backman (1964), Sedov (1966); see Clayton and McDowell (2003) and the references therein. We also refer to footnote 8.

  27. 27.

    Sometimes the rotational parts incorporated in \({\varvec{F}}_{\!e}\) and \({\varvec{F}}_{\!p}\) and even in \({\varvec{F}}\) are loosely said to be “superimposed rigid-body rotations”. This expression may produce an impression as if these rotations might be not so essential. However, essential difference exists between each such rotation and any truly rigid body rotation. The latter is constant at all points in a body and should have no effect on both basic equations of motion and constitutive formulations of material behavior, whereas the former varies from point to point and exhibits essential effects on both. In fact, the rotational parts of \({\varvec{F}}\) and \({\varvec{F}}_{\!e}\) and \({\varvec{F}}_{\!p}\) are inseparable parts of deformations and deformation rates, and incorporated in constitutive formulations for both elastic and plastic behavior.

  28. 28.

    We note that in this specific case we also have: \({\varvec{C}}_e = {\varvec{V}}_e^2 = {\varvec{F}}_{\!e}{{\varvec{F}}_{\!e}}^{\text{ T }} ={\varvec{B}}_e\), i.e. the left and right Cauchy-Green tensors are equal. Moreover, for an isotropic material, \({\varvec{V}}_e, {\varvec{B}}_e\) and \(\partial \psi /\partial {\varvec{B}}_e\) have the same principal axes so that the products are commutative.

  29. 29.

    We also refer to the discussions in Nemat-Nasser (1979, 1982); Lee and McMeeking (1980); Lee (1981); Lubarda and Lee (1981); Mandel (1981).

  30. 30.

    Results have been given e.g. by Lubliner (1984, 1986), assuming maximal plastic dissipation principles.

  31. 31.

    For details we refer e.g. to Bruhns (2014b).

  32. 32.

    This discussion was primarily between the two schools and their followers and lasted several years. We therefore refer to Green and Naghdi (1971); Casey and Naghdi (1980, 1981) and Lee (1982) and the instructive discussion therein.

  33. 33.

    This aspect was also discussed in the foregoing references and in Sidoroff (1973); Lee (1981, 1996); Lubarda and Lee (1981); Haupt (1985, 2002); Dashner (1986), and many others.

  34. 34.

    In fact, the assumption (3.68), the separation (3.66) as well as the elastic relation (3.69) become

    $$ {\varvec{F}}_{\!e}^{*T}={\varvec{F}}_{\!e}^*,\quad {\varvec{F}}^*={\varvec{F}}_{\!e}^*{\varvec{F}}_{\!p}^*,\quad {\varvec{\tau }}^*={\varvec{\phi }}({\varvec{F}}_{\!e}^*), $$

    under the change of frame. Here, Eq. (3.69) is written in the form \({\varvec{\tau }}={\varvec{\phi }}({\varvec{F}}_{\!e})\) considering Eq. (3.68). Since \({\varvec{\phi }}\) is isotropic, the last above and the objectivity of \({\varvec{\tau }}\) yield \({\varvec{\phi }}({\varvec{F}}_{\!e}^*)={\varvec{\phi }}({\varvec{Q}}^*{\varvec{F}}_{\!e}{{\varvec{Q}}^{*T}})\). From this follows that \({\varvec{F}}_{\!e}^*={\varvec{Q}}^*{\varvec{F}}_{\!e}{{\varvec{Q}}^{*T}}\) for an invertible \({\varvec{\phi }}\) with a symmetric argument due to Eq. (3.68). Then, \({\varvec{F}}_{\!p}^*={\varvec{Q}}^*{\varvec{F}}_{\!p}\) may be derived from \({\varvec{F}}^*={\varvec{Q}}^*{\varvec{F}}\) and Eq. (3.66), and the second above.

  35. 35.

    See, e.g., Green and Naghdi (1971), Casey and Naghdi (1980), Naghdi (1990), Naghdi and Casey (1992).

  36. 36.

    We also refer to the following works of Mandel (1973a, b, 1974a, b, 1981).

  37. 37.

    There might be some doubt about the physical pertinence of such a non-symmetric flow rule in 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 Mandel (1974a) and the discussion therein.

References

  • Backman M (1964) Form for the relation between stress and finite elastic and plastic strains under impulsive loading. J Appl Phys 35:2524–2533

    Article  Google Scholar 

  • Bertram A (2005) Elasticity and plasticity of large deformations. Springer, Berlin

    MATH  Google Scholar 

  • Böck N, Holzapfel G (2004) A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept. Int J Solids Struct 41:7459–7469

    Article  MATH  Google Scholar 

  • Bilby B, Gardner L, Stroh A (1957) Continuous distributions of dislocations and the theory of plasticity. Extrait des Actes du IX\(^e\) Congrès Intern. de Mécanique Applicqueé, Bruxelles, pp 35–44

    Google Scholar 

  • Bruhns O (2014a) Some remarks on the history of plasticity—Heinrich Hencky, a pioneer of the early years. In: Stein E (ed) The history of theoretical, material and computational mechanics—mathematics meets mechanics and engineering, vol 1. Springer, Heidelberg, pp 133–152

    Chapter  Google Scholar 

  • Bruhns O (2014b) The Prandtl-Reuss equations revisited. Z Angew Math Mech 94:187–202

    Article  MATH  MathSciNet  Google Scholar 

  • Casey J, Naghdi P (1980) A remark on the use of the decomposition \(\mathbf{F}=\mathbf{F}^e\mathbf{F}^p\) in plasticity. Trans ASME J Appl Mech 47:672–675

    Article  MATH  Google Scholar 

  • Casey J, Naghdi P (1981) Discussion of Lubarda and Lee (1981). Trans ASME J Appl Mech 48:983–984

    Article  Google Scholar 

  • Casey J, Naghdi P (1983) On the use of invariance requirements for intermediate configurations associated with the polar decomposition of a deformation gradient. Q Appl Math 41:339–342

    MATH  MathSciNet  Google Scholar 

  • Clayton J, McDowell D (2003) A multiscale multiplicative decomposition for elastoplasticity of polycrystals. Int J Plast 19:1401–1444

    Article  MATH  Google Scholar 

  • Darijani H, Naghdabadi R (2010) Constitutive modeling of solids at finite deformation using a second-order stress-strain relation. Int J Eng Sci 48:223–236

    Article  Google Scholar 

  • Dashner P (1986) Invariance considerations in large strain elasto-plasticity. Trans ASME J Appl Mech 53:55–60

    Article  MATH  Google Scholar 

  • Doyle T, Ericksen J (1956) Nonlinear elasticity. Adv Appl Mech 4:53–115

    MathSciNet  Google Scholar 

  • Drucker D (1949) The significance of the criterion for additional plastic deformation of metals. J Colloid Sci 4:299–311

    Article  Google Scholar 

  • Drucker D (1950) Some implications of work hardening and ideal plasticity. Q Appl Math 7:411–418

    MATH  MathSciNet  Google Scholar 

  • Eckart C (1948) The thermodynamics of irreversible processes. IV: The theory of elasticity and anelasticity. Phys Rev 73:373–382

    Article  MATH  MathSciNet  Google Scholar 

  • Edelman F, Drucker D (1951) Some extensions of elementary plasticity theory. J Franklin Inst 251:581–605

    Article  MathSciNet  Google Scholar 

  • Eglit M (1960) Tensorial characteristics of finite deformations. Prikl Mat Mekh 24:1432–1438

    MATH  Google Scholar 

  • Fox N (1968) On the continuum theories of dislocations and plasticity. Q J Mech Appl Math 21:67–75

    Article  MATH  Google Scholar 

  • Freund L (1970) Constitutive equations for elastic-plastic materials at finite strain. Int J Solids Struct 6:1193–1209

    Article  MATH  Google Scholar 

  • Green A, Naghdi P (1965a) A general theory of an elastic-plastic continuum. Arch Ration Mech Anal 18:251–281

    Google Scholar 

  • Green A, Naghdi P (1965b) Corrigenda. Arch Ration Mech Anal 19:408

    Article  MathSciNet  Google Scholar 

  • Green A, Naghdi P (1971) Some remarks on elastic-plastic deformation at finite strain. Int J Eng Sci 9:1219–1229

    Article  MATH  Google Scholar 

  • Haar A, von Kármán T (1909) Zur Theorie der Spannungszustände in plastischen und sandartigen Medien. Nachr Ges Wiss Göttingen, Math-Phys Kl, pp 204–218

    Google Scholar 

  • Haupt P (1985) On the concept of an intermediate configuration and its application to a representation of viscoelastic-plastic material behavior. Int J Plast 1:303–316

    Article  MATH  MathSciNet  Google Scholar 

  • Haupt P (2002) Continuum mechanics and theory of materials, 2nd edn. Springer, Berlin

    Book  MATH  Google Scholar 

  • Hencky H (1924) Zur Theorie plastischer Deformationen und der hierdurch im Material hervorgerufenen Nachspannungen. Z Angew Math Mech 4:323–334

    Article  Google Scholar 

  • Hencky H (1928) Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Z Tech Phys 9(215–220):457

    Google Scholar 

  • Hill R (1948) A variational principle of maximum plastic work in classical plasticity. Q J Mech Appl Math 1:18–28

    Article  MATH  Google Scholar 

  • Hill R (1950) The mathematical theory of plasticity. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Hill R (1968) On constitutive inequalities for simple materials. J Mech Phys Solids 16(229–242):315–322

    Article  MATH  Google Scholar 

  • Hill R (1970) Constitutive inequalities for isotropic elastic solids under finite strain. Proc R Soc London Ser A 314:457–472

    Article  MATH  Google Scholar 

  • Hill R (1978) Aspects of invariance in solid mechanics. Adv Appl Mech 18:1–75

    MATH  Google Scholar 

  • Hill R, Rice J (1973) Elastic potentials and the structure of inelastic constitutive laws. SIAM J Appl Math 25:448–461

    Article  MATH  MathSciNet  Google Scholar 

  • Horstemeyer M, Bammann D (2010) Historical review of internal state variable theory for inelasticity. Int J Plast 26:1310–1334

    Article  MATH  Google Scholar 

  • Kratochvíl J (1971) Finite-strain theory of crystalline elastic-inelastic materials. J Appl Phys 42:1104–1108

    Article  Google Scholar 

  • Kröner E (1958) Kontinuumstheorie der Versetzungen und Eigenspannungen. Springer, Berlin

    Book  MATH  Google Scholar 

  • Kröner E (1960) Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch Ration Mech Anal 4:273–334

    Article  MATH  Google Scholar 

  • Lee E (1969) Elastic-plastic deformation at finite strains. Trans ASME J Appl Mech 36:1–6

    Article  MATH  Google Scholar 

  • Lee E (1981) Some comments on elastic-plastic analysis. Int J Solids Struct 17:859–872

    Article  MATH  Google Scholar 

  • Lee E (1982) Finite deformation theory with nonlinear kinematics. In: Lee E, Mallett R (eds) Plasticity of metals at finite strain: theory. Computation and experiment. Stanford University and RPI, Stanford

    Google Scholar 

  • Lee E (1996) Some anomalies in the structure of elastic-plastic theory at finite strain. In: Carroll M, Hayes M (eds) Nonlinear effects in fluids and solids. Plenum Press, New York

    Google Scholar 

  • Lee E, Germain P (1974) Elastic-plastic theory at finite strain. In: Sawczuk A (ed) Problems of plasticity. Noordhoff International Publishing, Leyden

    Google Scholar 

  • Lee E, Liu D (1967) Finite-strain elastic-plastic theory with application to plane-wave analysis. J Appl Phys 38:19–27

    Article  Google Scholar 

  • Lee E, McMeeking R (1980) Concerning elastic and plastic components of deformation. Int J Solids Struct 16:715–721

    Article  MATH  MathSciNet  Google Scholar 

  • Lubarda V, Lee E (1981) A correct definition of elastic and plastic deformation and its computational significance. Trans ASME J Appl Mech 48:35–40

    Article  MATH  MathSciNet  Google Scholar 

  • Lubliner J (1984) A maximal-dissipation principle in generalized plasticity. Acta Mechanica 52:225–237

    Article  MATH  MathSciNet  Google Scholar 

  • Lubliner J (1986) Normality rules in large-deformation plasticity. Mech Mater 5:29–34

    Article  Google Scholar 

  • Ludwik P (1909) Elemente der technologischen Mechanik. Springer, Berlin

    Book  MATH  Google Scholar 

  • Macvean D (1968) Die Elementararbeit in einem Kontinuum und die Zuordnung von Spannungs- und Verzerrungstensoren. Z Angew Math Phys (ZAMP) 19:157–185

    Article  MATH  Google Scholar 

  • Mandel J (1972) Plasticité Classique et Viscoplasticité, CISM Courses and Lectures, vol 97. Springer, Wien

    Google Scholar 

  • Mandel J (1973a) Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques. Int J Solids Struct 9:725–740

    Article  MATH  Google Scholar 

  • Mandel J (1973b) Relations de comportement des milieux élastiques-viscoplastiques. Notion de répère directeur. In: Sawczuk A (ed) Foundations of plasticity. Noordhoff International Publishing, Leyden, pp 387–399

    Google Scholar 

  • Mandel J (1974a) Director vectors and constitutive equations for plastic and viscoplastic media. In: Sawczuk A (ed) Problems of plasticity. Noordhoff International Publishing, Leyden, pp 135–143

    Chapter  Google Scholar 

  • Mandel J (1974b) Thermodynamics and plasticity. In: Domingos J, Nina M, Whitelaw J (eds) Foundations of continuum thermodynamics. The MacMillan Press, London, pp 283–304

    Google Scholar 

  • Mandel J (1981) Sur la définition de la vitesse de déformation élastique et sa relation avec la vitesse de contrainte. Int J Solids Struct 17:873–878

    Article  MATH  MathSciNet  Google Scholar 

  • Nádai A (1931) Plasticity, a mechanics of the plastic state of matter. McGraw-Hill, New York

    MATH  Google Scholar 

  • Naghdi P (1990) A critical review of the state of finite plasticity. Z Angew Math Phys (ZAMP) 41:315–394

    Article  MATH  MathSciNet  Google Scholar 

  • Naghdi P, Casey J (1992) A prescription for the identification of finite plastic strain. Int J Eng Sci 30:1257–1278

    Article  MATH  MathSciNet  Google Scholar 

  • Naghdi P, Trapp J (1975a) On the nature of normality of plastic strain rate and convexity of yield surfaces in plasticity. Trans ASME J Appl Mech 42:61–66

    Article  MATH  Google Scholar 

  • Naghdi P, Trapp J (1975b) Restrictions on constitutive equations of finitely deformed elastic-plastic materials. Q J Mech Appl Math 28:25–46

    Article  MATH  MathSciNet  Google Scholar 

  • Naghdi P, Trapp J (1975c) The significance of formulating plasticity theory with reference to loading surfaces in strain space. Int J Eng Sci 13:785–797

    Article  MATH  Google Scholar 

  • Nemat-Nasser S (1979) Decomposition of strain measures and their rates in finite deformation elastoplasticity. Int J Solids Struct 15:155–166

    Article  MATH  Google Scholar 

  • Nemat-Nasser S (1982) On finite deformation elasto-plasticity. Int J Solids Struct 18:857–872

    Article  MATH  Google Scholar 

  • Ogden R (1984) Nonlinear elastic deformations. Ellis Harwood, Chichester

    MATH  Google Scholar 

  • Prager W (1944) Exploring stress-strain relations of isotropic plastic solids. J Appl Phys 15:65–71

    Article  Google Scholar 

  • Prandtl L (1924) Spannungsverteilung in plastischen Körpern. In: Proceedings of 1st international congress on applied mechanics, Delft, pp 43–46

    Google Scholar 

  • Reuss A (1930) Berücksichtigung der elastischen Formänderung in der Plastizitätstheorie. Z Angew Math Mech 10:266–274

    Article  MATH  Google Scholar 

  • Reuss A (1932) Fließpotential oder Gleitebenen? Z Angew Math Mech 12:15–24

    Article  Google Scholar 

  • Rice J (1971) Inelastic constitutive relations for solids: an internal-variable theory and its application to metal plasticity. J Mech Phys Solids 19:433–455

    Article  MATH  Google Scholar 

  • Sedov L (1966) Foundations of the non-linear mechanics of continua. Pergamon Press, Oxford

    MATH  Google Scholar 

  • Seth B (1964) Generalized strain measures with applications to physical problems. In: Reiner M, Abir D (eds) Second-order effects in elasticity. Plasticity and fluid dynamics. Pergamon Press, Oxford

    Google Scholar 

  • Sidoroff F (1973) The geometrical concept of intermediate configuration and elastic-plastic finite strain. Arch Mech 25:299–308

    MATH  MathSciNet  Google Scholar 

  • Tokuoka T (1977) Rate type plastic material with kinematic work-hardening. Acta Mechanica 27:145–154

    Article  Google Scholar 

  • Tokuoka T (1978) Prandtl-Reuss plastic material with scalar and tensor internal variables. Arch Mech 30:801–826

    MATH  MathSciNet  Google Scholar 

  • Truesdell C (1952a) The mechanical foundations of elasticity and fluid dynamics. J Ration Mech Anal 1:125–300

    MATH  MathSciNet  Google Scholar 

  • Truesdell C (1952b) The mechanical foundations of elasticity and fluid dynamics. J Ration Mech Anal 2:595–616

    Google Scholar 

  • Truesdell C (1952c) The mechanical foundations of elasticity and fluid dynamics. J Ration Mech Anal 3:801

    Google Scholar 

  • Truesdell C (1955) Hypo-elasticity. J Ration Mech Anal 4:83–133

    MATH  MathSciNet  Google Scholar 

  • Truesdell C (1964) Second-order effects in the mechanics of materials. In: Reiner M, Abir D (eds) Second-order effects in elasticity. Plasticity and fluid dynamics. Pergamon Press, Oxford

    Google Scholar 

  • Truesdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flügge S (ed) Handbuch der Physik. Springer, Berlin

    Google Scholar 

  • Willis J (1996) Some constitutive equations applicable to problems of large dynamic plastic deformation. J Mech Phys Solids 17:359–369

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mechanics, Ruhr-University Bochum, Universitätsstr. 150, 44780, Bochum, Germany

    Otto T. Bruhns

Authors
  1. Otto T. Bruhns
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Otto T. Bruhns .

Editor information

Editors and Affiliations

  1. Otto-von-Guericke-Universität Magdeburg, Fakultät für Maschinenbau, Institut für Mechanik, Lehrstuhl für Technische Mechanik, Universitätsplatz 2, 39106 Magdeburg, Germany

    Holm Altenbach

  2. Department of Engineering Mechanics and Energy, University of Tsukuba, Tsukuba, Japan

    Tetsuya Matsuda

  3. Department of Mechanical Science and Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan

    Dai Okumura

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bruhns, O.T. (2015). The Multiplicative Decomposition of the Deformation Gradient in Plasticity—Origin and Limitations. In: Altenbach, H., Matsuda, T., Okumura, D. (eds) From Creep Damage Mechanics to Homogenization Methods. Advanced Structured Materials, vol 64. Springer, Cham. https://doi.org/10.1007/978-3-319-19440-0_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-19440-0_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-19439-4

  • Online ISBN: 978-3-319-19440-0

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation