Top

Journal of Elasticity

Published in:

15-11-2021

A Discussion of Multiplicative Decompositions and Strain Measures

Author: James Casey

Published in: Journal of Elasticity | Issue 1/2022

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

In the context of a purely mechanical development for materials that possess some degree of elasticity, two-factor and three-factor multiplicative decompositions of the deformation gradient, as well as related strain measures, are discussed in detail. Different factors in the decompositions (and their polar subfactors) have different degrees of rotational non-uniqueness. Moreover, different deformation measures generally behave differently under superposed rigid motions, which has important implications for the roles that they are allowed to play in constitutive equations. A certain unique right stretch tensor emerges, which yields strain measures suitable for describing anisotropic elastic responses of solids that have evolving stress-free local configurations.

previous article Cosserat Elasticity of Lattice Solids

next article On the Inflation of Residually Stressed Spherical Shells

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

The rotation tensor is proper orthogonal and both stretch tensors are symmetric positive definite.

Latin indices will take on values $1,2,3$ and Einstein’s summation convention will be adopted.

This can be seen at once by choosing rectangular Cartesian coordinates $X_{A}$ on the reference configuration $\boldsymbol {\kappa }_{0}$ and convecting them; here, $x_{i}$ are the Cartesian coordinates of $\mathbf {x}$.

Other pull-backs, involving contravariant and mixed components, are also discussed in [21].

In connection with (13)₁ and (18), we may regard $\mathbf {C}$ as a tensor defined on one manifold, the reference configuration $\boldsymbol {\kappa }_{0}$, and describing the metric tensor for another manifold, the configuration of the body at time $t$.

An example will suffice to illustrate this. Consider a family of homogeneous deformations, parametrized by $\mathbf {X}_{0}$:

$$ \boldsymbol {\chi }\strut_{\mathbf {X}_{0}} (\mathbf {X})=\mathbf {A}(\mathbf {X}_{0})(\mathbf {X}-\mathbf {X}_{0}). $$

For each $\mathbf {X}_{0} \in \boldsymbol {\kappa }_{0}(\mathscr {B})$, the deformation $\boldsymbol {\chi }\strut _{\mathbf {X}_{0}}$ leaves $\mathbf {X}_{0}$ in its reference position. Being homogeneous, each of the deformations satisfies compatibility conditions. Clearly,

$$ \text{GRAD }\boldsymbol {\chi }\strut_{\mathbf {X}_{0}}(\mathbf {X})=\mathbf {A}(\mathbf {X}_{0}),\qquad\text{GRAD }\boldsymbol {\chi }\strut_{\mathbf {X}_{0}}(\mathbf {X}_{0})=\mathbf {A}(\mathbf {X}_{0}). $$

The field $\mathbf {A}(\mathbf {X}_{0})$ may be taken to be continuously differentiable, but need not satisfy any further restrictions. In particular, it is not required to satisfy compatibility conditions anywhere.

By the Nash imbedding theorem of differential geometry, it is possible to imbed ℬ isometrically in some higher dimensional Euclidean space.

However, there is an important distinction between the tensors $\mathbf {E}_{e}$ and $\mathbf {E}-\mathbf {E}_{p}$: as will be seen in Sect. 3.3, these two strain measures behave differently from one another under superposed rigid motions.

There was much controversy in the literature regarding invariance requirements associated with multiplicative decompositions (see especially [3, 6‐10, 12, 13]). In light of the results in (87), (89), (90), and (91), it should be clear now that even though full invariance requirements are being adopted, it is still possible that certain factors in some decompositions transform by only one rotation $\mathbf {Q}$, and that some other factors are not altered at all. Thus, for the decomposition (56), both $\mathbf {U}_{p}$ and $\mathbf {U}_{*}$ are unaltered when the rotations $\mathbf {Q}$ and $\overline{\mathbf {Q}}$ occur, and $\mathbf {R}_{*}$ is transformed into $\mathbf {Q}\mathbf {R}_{*}$. Likewise, in Lee’s decomposition (58), $\mathbf {V}_{e}$ is transformed into $\mathbf {Q}\mathbf {V}_{e}\mathbf {Q}^{T}$, and $\mathbf {F}_{p*}$ into $\mathbf {Q}\mathbf {F}_{p*}$. This sheds new light on the controversy just mentioned.

Clearly, the form (111) allows $\psi$ to be expressed as a different function of $\left(\mathbf {R}_{p}^{T}\mathbf {C}_{e}\mathbf {R}_{p},\mathbf {C}_{p}\right)$.

Equation (62) indicates how $\mathbf {V}_{e}$ is related to $\mathbf {U}_{*}$. A function $\psi(\mathbf {V}_{e})$, of $\mathbf {V}_{e}$ alone, would have to be isotropic.

Lee, E.H.: Elastic-plastic deformation at finite strains. J. Appl. Mech. 36, 1–6 (1969) CrossRef

Sidoroff, F.: Quelques réflexions sur le principe d’indifférence matérielle pour un milieu ayant un état relâché. C. R. Acad. Sci., Paris 271, 1026–1029 (1970) MATH

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

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

Naghdi, P.M., Trapp, J.A.: On finite elastic-plastic deformation of metals. J. Appl. Mech. 41, 254–260 (1974) CrossRef

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

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

Casey, J., Naghdi, P.M.: Discussion of [7]. J. Appl. Mech. 48, 983–984 (1981) CrossRef

Lubarda, V.A., Lee, E.H.: Authors’ closure to [8]. J. Appl. Mech. 48, 984–985 (1981) CrossRef

10.

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

11.

Bammann, D.J., Johnson, G.C.: On the kinematics of finite deformation plasticity. Acta Mech. 70, 1–13 (1987) CrossRef

12.

Naghdi, P.M.: A critical review of the state of finite plasticity. J. Appl. Math. Phys. 41, 315–394 (1990) MathSciNetMATH

13.

Lee, E.H.: Some anomalies in the structure of elastic-plastic theory at finite strain. In: Carroll, M.M., Hayes, M.A. (eds.) Nonlinear Effects in Fluids and Solids, pp. 227–249. Plenum Press, New York and London (1996) CrossRef

14.

Lubarda, V.A.: Constitutive theories based on the multiplicative decomposition of deformation gradient: thermoelasticity, elastoplasticity, biomechanics. Appl. Mech. Rev. 57, 95–108 (2004) CrossRef

15.

Stojanović, R., Djurić, S., Vujošević, L.: On finite thermal deformations. Arch. Mech. Stosow. 16, 103–108 (1964) MathSciNet

16.

Imam, A., Johnson, G.C.: Decomposition of the deformation gradient in thermoelasticity. J. Appl. Mech. 65, 362–366 (1998) MathSciNetCrossRef

17.

Vujošević, L., Lubarda, V.A.: Finite strain thermoelasticity based on multiplicative decomposition of deformation gradient. Theor. Appl. Mech. 28, 379–399 (2002) MathSciNetCrossRef

18.

Rodriguez, E.K., Hoger, A., McCulloch, A.D.: Stress-dependent finite growth in soft elastic tissues. J. Biomech. 27, 455–467 (1994) CrossRef

19.

Lubarda, V.A., Hoger, A.: On the mechanics of solids with a growing mass. Int. J. Solids Struct. 39, 4627–4664 (2002) CrossRef

20.

Casey, J.: A convenient form of the multiplicative decomposition of the deformation gradient. Math. Mech. Solids 22, 528–537 (2016) MathSciNetCrossRef

21.

Casey, J., Papadopoulos, P.: Material transport of sets and fields. Math. Mech. Solids 7, 647–676 (2002) MathSciNetCrossRef

Title: A Discussion of Multiplicative Decompositions and Strain Measures
Author: James Casey
Publication date: 15-11-2021
Publisher: Springer Netherlands
Published in: Journal of Elasticity / Issue 1/2022
Print ISSN: 0374-3535
Electronic ISSN: 1573-2681
DOI: https://doi.org/10.1007/s10659-021-09867-z

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Other articles of this Issue 1/2022

The Generalised Mooney Space for Modelling the Response of Rubber-Like Materials

Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

Cosserat Elasticity of Lattice Solids

Isometries Between Given Surfaces and the Isometric Deformation of a Single Unstretchable Material Surface

Preface

Analysis of Stiction in Nanoelectromechanical Systems Using Molecular Dynamics Simulations and Continuum Theory

Premium Partners