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

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.Keywords Plasticity · Large deformation · Additive decomposition · Multiplicative decomposition · Intermediate configuration · Green-Naghdi theory · Lee-Liu theory · Isoclinic configuration · Director triads IntroductionThe present article continues a series of contributions intended to highlight the history of plasticity. The first was devoted to the life and works of Heinrich Hencky and the rapid development of plasticity at the beginning of last century (Bruhns 2014a). At that time the main focus of the different considerations was on the mathematical description of the basics of plasticity within the general frame of continuum mechanics, and the combination of the solid-like elastic behavior of most O.T. Bruhns (B)

Computational anisotropic hardening multiplicative elastoplasticity based on the corrector elastic logarithmic strain rate

“…The consistency requirement A = 0 over Eq. (50) when y > 0 yields the consistency requirement at any time t as 1 / AJ e ", :…”

“…A recent publication considering the numerics of the multiplicative decomposition, including combined isotropic-kinematic hardening models, can be found in [48]. However, the most controversial aspect of the theory is arguably associated with the derivation of continuum evolution equations for the plastic flow [49] and with their numerical integration [50], i.e. the "rate issue" as coined by Simo [11].…”

A nonlinear thermomechanical formulation for anisotropic volume and surface continua

“…For two consecutive deformation gradients F 1 and F 2 , such that F = F 2 F 1 , E ( n ) = E 1 ( n ) + E 2 ( n ) is true if the logarithmic strain n = 0 is used. Here, E i ( n ) is defined as It means that the logarithmic strain can be additively decomposed for finite deformations [166]. For n ≠ 0 , E ( n ) can be written as …”

“…It means that the logarithmic strain can be additively decomposed for finite deformations [166]. For n ≠ 0 , E ( n ) can be written as…”

A convenient form of the multiplicative decomposition of the deformation gradient

“…The multiplicative decomposition of the deformation gradient F into two factors, "F e " and "F p " (say), has been fruitfully exploited in diverse areas of continuum mechanics, particularly in finite plasticity [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20], thermoelasticity [21][22][23], and growth mechanics [24,25]. 1 In contrast to the purely kinematical right and left polar decompositions of F, in which the rotation tensor and each of the stretch tensors are uniquely defined, it is well-known that both F e and F p have an essential rotational non-uniqueness.…”