Constitutive theories based on the multiplicative decomposition of deformation gradient: Thermoelasticity, elastoplasticity, and biomechanics

[+] Author and Article Information
Vlado A Lubarda

Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411vlubarda@ucsd.edu

Appl. Mech. Rev 57(2), 95-108 (Apr 26, 2004) (14 pages) doi:10.1115/1.1591000 History: Online April 26, 2004
Copyright © 2004 by ASME
Your Session has timed out. Please sign back in to continue.


Eckart  C (1948), The thermodynamics of irreversible processes, IV: The theory of elasticity and anelasticity, Phys. Rev. 73, 373–380.
Kröner  E (1960), Allgemeine Kontinuumstheorie der Versetzungen und Eigensspannungen, Arch. Ration. Mech. Anal. 4, 273–334.
Sedov LI (1966), Foundations of the Non-Linear Mechanics of Continua, Pergamon Press, Oxford.
Stojanović  R, Djurić  S, and Vujošević  L (1964), On finite thermal deformations, Arch. Mech. Stosow. 16, 103–108.
Lee  EH (1969), Elastic-plastic deformation at finite strains, ASME J. Appl. Mech. 36, 1–6.
Asaro  RJ and Rice  JR (1977), Strain localization in ductile single crystals, J. Mech. Phys. Solids 25, 309–338.
Hill  R and Havner  KS (1982), Perspectives in the mechanics of elastoplastic crystals, J. Mech. Phys. Solids 30, 5–22.
Asaro  RJ (1983), Crystal plasticity, ASME J. Appl. Mech. 50, 921–934.
Asaro  RJ (1983), Micromechanics of crystals and polycrystals, Adv. Appl. Mech. 23, 1–115.
Havner KS (1992), Finite Plastic Deformation of Crystalline Solids, Cambridge Univ Press, Cambridge.
Rodrigez  EK, Hoger  A, and McCulloch  AD (1994), Stress-dependent finite growth in soft elastic tissues, J. Biomech. 27, 455–467.
Taber  LA and Eggers  DW (1996), Theoretical study of stress-modulated growth in the aorta, J. Theor. Biol. 180, 343–357.
Chen  Y-C and Hoger  A (2000), Constitutive functions of elastic materials in finite growth and deformation, J. Elast. 59, 175–193.
Klisch  SM and Van Dyke  J (2001), A theory of volumetric growth for compressible elastic biological materials, Math. Mech. Solids 6, 551–575.
Lubarda  VA and Hoger  A (2002), On the mechanics of solids with a growing mass, Int. J. Solids Struct. 39, 4627–4664.
Stojanović  R, Vujošević  L, and Blagojević  D (1970), Couple stresses in thermoelasticity, Rev. Roum. Sci. Techn.-Méc. Appl. 15, 517–537.
Miehe  C (1995), Entropic thermoelasticity at finite strains. Aspects of the formulation and numerical implementation, Comput. Methods Appl. Mech. Eng. 120, 243–269.
Holzapfel  GA and Simo  JC (1996), Entropy elasticity of isotropic rubber-like solids at finite strains, Comput. Methods Appl. Mech. Eng. 132, 17–44.
Imam  A and Johnson  GC (1998), Decomposition of deformation gradient in thermoelasticity, ASME J. Appl. Mech. 65, 362–366.
Vujošević  L and Lubarda  VA (2002), Finite-strain thermoelasticity based on multiplicative decomposition of deformation gradient, Theor Appl. Mech. 28–29, 379–399.
Backman  ME (1964), From the relation between stress and finite elastic and plastic strains under impulsive loading, J. Appl. Phys. 35, 2524–2533.
Lee  EH and Liu  DT (1967), Finite-strain elastic-plastic theory particularly for plane wave analysis, J. Appl. Phys. 38, 19–27.
Fox  N (1968), On the continuum theories of dislocations and plasticity, Q. J. Mech. Appl. Math. 21, 67–75.
Willis  JR (1969), Some constitutive equations applicable to problems of large dynamic plastic deformation, J. Mech. Phys. Solids 17, 359–369.
Mandel J (1971), Plasticité classique et viscoplasticité, Courses and Lectures, No 97, Int Center for Mechanical Sciences, Udine, Springer, New York.
Mandel  J (1973), Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques, Int. J. Solids Struct. 9, 725–740.
Kröner E and Teodosiu C (1973), Lattice defect approach to plasticity and viscoplasticity, Problems of Plasticity, A Sawczuk (ed), Noordhoff, Leyden, 45–88.
Freund  LB (1970), Constitutive equations for elastic-plastic materials at finite strain, Int. J. Solids Struct. 6, 1193–1209.
Sidoroff  F (1975), On the formulation of plasticity and viscoplasticity with internal variables, Arch. Mech. 27, 807–819.
Kleiber  M (1975), Kinematics of deformation processes in materials subjected to finite elastic-plastic strains, Int. J. Eng. Sci. 13, 513–525.
Nemat-Nasser  S (1979), Decomposition of strain measures and their rates in finite deformation elastoplasticity, Int. J. Solids Struct. 15, 155–166.
Nemat-Nasser  S (1982), On finite deformation elasto-plasticity, Int. J. Solids Struct. 18, 857–872.
Lubarda  VA and Lee  EH (1981), A correct definition of elastic and plastic deformation and its computational significance, ASME J. Appl. Mech. 48, 35–40.
Johnson  GC and Bammann  DJ (1984), A discussion of stress rates in finite deformation problems, Int. J. Solids Struct. 20, 725–737.
Simo  JC and Ortiz  M (1985), A unified approach to finite deformation elasto-plastic analysis based on the use of hyperelastic constitutive equations, Comput. Methods Appl. Mech. Eng. 49, 221–245.
Needleman  A (1985), On finite element formulations for large elastic-plastic deformations, Comput. Struct. 20, 247–257.
Dashner  PA (1986), Invariance considerations in large strain elasto-plasticity, ASME J. Appl. Mech. 53, 55–60.
Dafalias  YF (1987), Issues in constitutive formulation at large elastoplastic deformation, Part I: Kinematics, Acta Mech. 69, 119–138.
Dafalias  YF (1988), Issues in constitutive formulation at large elastoplastic deformation, Part II: Kinetics, Acta Mech. 73, 121–146.
Agah-Tehrani  A, Lee  EH, Mallett  RL, and Onat  ET (1987), The theory of elastic-plastic deformation at finite strain with induced anisotropy modeled as combined isotropic-kinemetic hardening, J. Mech. Phys. Solids 35, 519–539.
Van der Giessen  E (1989), Continuum models of large deformation plasticity, Parts I and II, Eur. J. Mech. A/Solids 8, 15–34 and 89–108.
Moran  B, Ortiz  M, and Shih  CF (1990), Formulation of implicit finite element methods for multiplicative finite deformation plasticity, Int. J. Numer. Methods Eng. 29, 483–514.
Naghdi  PM (1990), A critical review of the state of finite plasticity, Z. Angew. Math. Phys. 41, 315–394.
Aravas  N (1992), Finite elastoplastic transformations of transversely isotropic metals, Int. J. Solids Struct. 29, 2137–2157.
Lubarda  VA and Shih  CF (1994), Plastic spin and related issues in phenomenological plasticity, ASME J. Appl. Mech. 61, 524–529.
Xiao  H, Bruhns  OT, and Meyers  A (2000), A consistent finite elastoplasticity theory combining additive and multiplicative decomposition of the stretching and the deformation gradient, Int. J. Plast. 16, 143–177.
Lubarda  VA and Benson  DJ (2001), On the partitioning of the rate of deformation gradient in phenomenological plasticity, Int. J. Solids Struct. 38, 6805–6817.
Aravas  N and Aifantis  EC (1991), On the geometry of slip and spin in finite plasticity deformation, Int. J. Plast. 7, 141–160.
Bassani  JL (1993), Plastic flow of crystals, Adv. Appl. Mech. 30, 191–258.
Lubarda  VA (1999), On the partition of rate of deformation in crystal plasticity, Int. J. Plast. 15, 721–736.
Gurtin  ME (2000), On the plasticity of single crystals: free energy, microforces, plastic-strain gradients, J. Mech. Phys. Solids 48, 989–1036.
Taber  LA and Perucchio  R (2000), Modeling heart development, J. Elast. 61, 165–197.
Hoger A, Van Dyke TJ, and Lubarda VA (2002), Symmetrization of the growth deformation and velocity gradients in residually stressed biomaterials, Z. Angew. Math. Phys. (submitted ).
Stojanović R (1972), Nonlinear thermoelasticity, CISM Lecture Notes, Udine.
Mićunović  M (1974), A geometrical treatment of thermoelasticity of simple inhomogeneous bodies: I and II, Bull. Acad. Polon. Sci., Ser. Sci. Techn. 22, 579–588, and 633–641.
Lu  SCH and Pister  KS (1975), Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids, Int. J. Solids Struct. 11, 927–934.
Lubarda VA (2002), Multiplicative decomposition of deformation gradient in continuum mechanics: thermoelasticity, elastoplasticity and biomechanics, Proc of Montenegrin Acad of Sci and Arts14 , 53–86.
Carlson DE (1972), Linear thermoelasticity, Handbuch der Physik, Band VIa/2, S Flügge (ed), Springer-Verlag, Berlin, 297–346.
Nowacki W (1986), Thermoelasticity (2nd ed), Pergamon Press, Oxford; PWN—Polish Sci Publ, Warszawa.
Green  AE and Naghdi  PM (1971), Some remarks on elastic-plastic deformation at finite strain, Int. J. Eng. Sci. 9, 1219–1229.
Casey  J and Naghdi  PM (1980), Remarks on the use of the decomposition F=FeFp in plasticity, ASME J. Appl. Mech. 47, 672–675.
Kleiber M and Raniecki B (1985), Elastic-plastic materials at finite strains, Plasticity Today, A Sawczuk and G Bianchi (eds), Elsevier Applied Science, UK, 3–46.
Casey  J (1987), Discussion of “Invariance considerations in large strain elasto-plasticity,” ASME J. Appl. Mech. 54, 247–248.
Lubarda  VA (1991), Constitutive analysis of large elasto-plastic deformation based on the multiplicative decomposition of deformation gradient, Int. J. Solids Struct. 27, 885–895.
Lubarda VA (2002), Elastoplasticity Theory, CRC Press, Boca Raton FL.
Simo  JC and Ju  JW (1987), Strain- and stress-based continuum damage models, I: Formulation, Int. J. Solids Struct. 23, 821–840.
Lubarda  VA (1994), An analysis of large-strain damage elastoplasticity, Int. J. Solids Struct. 31, 2951–2964.
Lubarda  VA and Krajcinovic  D (1995), Some fundamental issues in rate theory of damage-elastoplasticity, Int. J. Plast. 11, 763–797.
Ilyushin  AA (1961), On the postulate of plasticity, Prikl. Mat. Mekh. 25, 503–507.
Hill  R and Rice  JR (1973), Elastic potentials and the structure of inelastic constitutive laws, SIAM J. Appl. Math. 25, 448–461.
Hill  R (1978), Aspects of invariance in solid mechanics, Adv. Appl. Mech. 18, 1–75.
Khan AS and Huang S (1995), Continuum Theory of Plasticity, John Wiley and Sons, New York.
Simo JC and Hughes TJR (1998), Computational Plasticity, Springer-Verlag, New York.
Casey  J and Naghdi  PM (1983), On the nonequivalence of the stress and strain space formulations of plasticity theory, ASME J. Appl. Mech. 50, 350–354.
Lubarda  VA (1994), Elastoplastic constitutive analysis with the yield surface in strain space, J. Mech. Phys. Solids 42, 931–952.
Lubarda VA (2001), Continuum Mechanics of Materials, Encyclopedia of Materials: Science and Technology, Elsevier, Amsterdam, 5295–5307.
Kratochvil  H (1973), On a finite strain theory of elastic-inelastic materials, Acta Mech. 16, 127–142.
Lubarda  VA (1999), Duality in constitutive formulation of finite-strain elastoplasticity based on F=FeFp and F=FpFe decompositions, Int. J. Plast. 15, 1277–1290.
Lubarda  VA (1991), Some aspects of elasto-plastic constitutive analysis of elastically anisotropic materials, Int. J. Plast. 7, 625–636.
Steinmann  P, Miehe  C, and Stein  E (1996), Fast transient dynamic plane stress analysis of orthotropic Hill-type solids at finite elastoplastic strain, Int. J. Solids Struct. 33, 1543–1562.
Lee  EH, Mallett  RL, and Wertheimer  TB (1983), Stress analysis for anisotropic hardening in finite-deformation plasticity, ASME J. Appl. Mech. 50, 554–560.
Loret  B (1983), On the effects of plastic rotation in the finite deformation of anisotropic elastoplastic materials, Mech. Mater. 2, 287–304.
Dafalias  YF (1983), Corotational rates for kinematic hardening at large plastic deformations, ASME J. Appl. Mech. 50, 561–565.
Dafalias  YF (1985), The plastic spin, ASME J. Appl. Mech. 52, 865–871.
Zbib  HM and Aifantis  EC (1988), On the concept of relative and plastic spins and its implications to large deformation theories, Part II: Anisotropic hardening, Acta Mech. 75, 35–56.
Van der Giessen  E (1991), Micromechanical and thermodynamic aspects of the plastic spin, Int. J. Plast. 7, 365–386.
Nemat-Nasser  S (1992), Phenomenological theories of elastoplasticity and strain localization at high strain rates, Appl. Mech. Rev. 45, S19–S45.
Dafalias  YF (1998), Plastic spin: Necessity or redundancy, Int. J. Plast. 14, 909–931.
Taylor  GI (1938), Plastic strain in metals, J. Inst. Met. 62, 307–324.
Hill  R and Rice  JR (1972), Constitutive analysis of elastic-plastic crystals at arbitrary strain, J. Mech. Phys. Solids 20, 401–413.
Mandel J (1974), Thermodynamics and plasticity, Foundations of Continuum Thermodynamics, JJD Domingos, MNR Nina and JH Whitelaw (eds), McMillan Publishers, London, 283–311.
Lubarda  VA (1999), On the partition of the rate of deformation in crystal plasticity, Int. J. Plast. 15, 721–736.
Hsu  F (1968), The influences of mechanical loads on the form of a growing elastic body, Biomechanics 1, 303–311.
Cowin  SC and Hegedus  DH (1976), Bone remodeling I: Theory of adaptive elasticity, J. Elast. 6, 313–326.
Skalak  R, Dasgupta  G, Moss  M, Otten  E, Dullemeijer  P, and Vilmann  H (1982), Analytical description of growth, J. Theor. Biol. 94, 555–577.
Taber  LA (1995), Biomechanics of growth, remodeling, and morphogenesis, Appl. Mech. Rev. 48(8), 487–545.
Humphrey  JD (1995), Mechanics of the arterial wall: Review and directions, Crit. Rev. Biomed. Eng. 23, 1–162.
Holzapfel  GA, Gasser  TC, and Ogden  RW (2000), A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elast. 61, 1–48.
Sacks  MS (2000), Biaxial mechanical evaluation of planar biological materials, J. Elast. 61, 199–246.
Fung  Y-C (1973), Biorheology of soft tissues, Biorheology 9, 139–155.
Fung  Y-C (1995), Stress, strain, growth, and remodeling of living organisms, Z. Angew. Math. Phys. 46, S469–S482.
Humphrey  JD (2003), Continuum thermomechanics and the clinical treatment of disease and injury, Appl. Mech. Rev. 56, 231–260.
Fung Y-C (1990), Biomechanics: Motion, Flow, Stress, and Growth, Springer, New York.
Liu  SQ and Fung  Y-C (1988), Zero-stress states of arteries, ASME J. Biomech. Eng. 110, 82–84.
Liu  SQ and Fung  Y-C (1989), Relationship between hypertension, hypertrophy, and opening angle of zero-stress state of arteries following aortic constriction, ASME J. Biomech. Eng. 111, 325–335.
Truesdell C and Noll N (1965), The nonlinear field theories of mechanics, Handbuch der Physik, Band III/3, S Flügge (ed), Springer-Verlag, Berlin.
Bruhns  OT, Xiao  H, and Meyers  A (2001), A self-consistent Eulerian rate type model for finite deformation elastoplasticity with isotropic damage, Int. J. Solids Struct. 38, 657–683.
Boyce  MC, Parks  DM, and Argon  AS (1988), Large inelastic deformation of glassy polymers, Part I: Rate-dependent constitutive model, Mech. Mater. 7, 15–33.
Boyce  MC, Weber  GG, and Parks  DM (1989), On the kinematics of finite strain plasticity, J. Mech. Phys. Solids 37, 647–665.
Arruda  EM and Boyce  MC (1993), A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, J. Mech. Phys. Solids 41, 389–412.
Wu  PD and Van der Giessen  E (1993), On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers, J. Mech. Phys. Solids 41, 427–456.
Lion  A (1997), A physically based method to represent the thermomechanical behavior of elastomers, Acta Mech. 123, 1–25.


Grahic Jump Location
The intermediate configuration Bθ at a nonuniform temperature θ is obtained from the deformed configuration B by isothermal destressing to zero stress. The deformation gradient from initial to deformed configuration F is decomposed into elastic part Fe and thermal part Fθ, such that F=Fe⋅Fθ.
Grahic Jump Location
The intermediate configuration Bp is obtained from the deformed configuration B by destressing to zero stress. The elastoplastic deformation gradient is decomposed into its elastic and plastic part, such that F=Fe⋅Fp.
Grahic Jump Location
Kinematic model of elastoplastic deformation of a single crystal. The material flows through the crystalline lattice by crystallographic slip, which gives rise to deformation gradient Fp. Subsequently, the material with embedded lattice deforms elastically from the intermediate to current configuration. The corresponding deformation gradient is F*.
Grahic Jump Location
Schematic representation of the multiplicative decomposition of deformation gradient into its elastic and growth parts. The mass of an infinitesimal volume element in the initial configuration Bo is dmo. The corresponding mass in the configurations Bg and B is dm.




Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In