Abstract
A novel thermodynamic framework for the continuum mechanical response of nonlinear solids is described. Large deformations, nonlinear hyperelasticity, viscoelasticity, and property changes due to evolution of damage in the material are all encompassed by the general theory. The deformation gradient is decomposed in Gram–Schmidt fashion into the product of an orthogonal matrix and an upper triangular matrix, where the latter can be populated by six independent strain attributes. Strain attributes, in turn, are used as fundamental independent variables in the thermodynamic potentials, rather than the usual scalar invariants of deformation tensors as invoked in more conventional approaches. A complementary set of internal variables also enters the thermodynamic potentials to enable history and rate dependence, i.e., viscoelasticity, and irreversible stiffness degradation, i.e., damage. Governing equations and thermodynamic restrictions imposed by the entropy production inequality are derived. Mechanical, thermodynamic, and kinetic relations are presented for material symmetries that reduce to cubic or isotropic thermoelasticity in the small strain limit, restricted to isotropic damage. Representative models and example problems demonstrate utility and flexibility of this theory for depicting nonlinear hyperelasticity, viscoelasticity, and/or damage from cracks or voids, with physically measurable parameters.
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References
Fung, Y.-C.: Biomechanics: Motion, Flow, Stress, and Growth. Springer, New York (1990)
Fung, Y.-C.: Biomechanics: Mechanical Properties of Living Tissues, 2nd edn. Springer, New York (1993)
Humphrey, J.D.: Continuum biomechanics of soft biological tissues. Proc. R. Soc. Lond. A 459, 3–46 (2003)
Rodríguez, J.F., Cacho, F., Bea, J.A., Doblaré, M.: A stochastic-structurally based three dimensional finite-strain damage model for fibrous soft tissue. J. Mech. Phys. Solids 54, 864–886 (2006)
Balzani, D., Brinkhues, S., Holzapfel, G.A.: Constitutive framework for the modeling of damage in collagenous soft tissues with application to arterial walls. Comput. Methods Appl. Mech. Eng. 213, 139–151 (2012)
Regueiro, R.A., Zhang, B., Wozniak, S.L.: Large deformation dynamic three-dimensional coupled finite element analysis of soft biological tissues treated as biphasic porous media. Comput. Model. Eng. Sci. (CMES) 98, 1–39 (2014)
Fankell, D.P., Regueiro, R.A., Kramer, E.A., Ferguson, V.L., Rentschler, M.E.: A small deformation thermoporomechanics finite element model and its application to arterial tissue fusion. J. Biomech. Eng. 140, 031007 (2018)
Simo, J.C.: On a fully three-dimensional finite-strain viscoelastic damage model: formulation and computational aspects. Comput. Methods Appl. Mech. Eng. 60, 153–173 (1987)
Holzapfel, G.A., Simo, J.C.: A new viscoelastic constitutive model for continuous media at finite thermomechanical changes. Int. J. Solids Struct. 33, 3019–3034 (1996)
Holzapfel, G.A.: On large strain viscoelasticity: continuum formulation and finite element applications to elastomeric structures. Int. J. Numer. Meth. Eng. 39, 3903–3926 (1996)
Liu, H., Holzapfel, G.A., Skallerud, B.H., Prot, V.: Anisotropic finite strain viscoelasticity: constitutive modeling and finite element implementation. J. Mech. Phys. Solids 124, 172–188 (2019)
Reese, S., Govindjee, S.: A theory of finite viscoelasticity and numerical aspects. Int. J. Solids Struct. 35, 3455–3482 (1998)
Clayton, J.D., McDowell, D.L., Bammann, D.J.: A multiscale gradient theory for elastoviscoplasticity of single crystals. Int. J. Eng. Sci. 42, 427–457 (2004)
Clayton, J.D.: A continuum description of nonlinear elasticity, slip and twinning, with application to sapphire. Proc. R. Soc. Lond. A 465, 307–334 (2009)
Clayton, J.D.: Nonlinear Mechanics of Crystals. Springer, Dordrecht (2011)
Ghosh, P., Srinivasa, A.R.: Development of a finite strain two-network model for shape memory polymers using QR decomposition. Int. J. Eng. Sci. 81, 177–191 (2014)
Flory, P.J.: Thermodynamic relations for high elastic materials. Trans. Faraday Soc. 57, 829–838 (1961)
Freed, A.D., Zamani, S.: On the use of convected coordinate systems in the mechanics of continuous media derived from a QR factorization of F. Int. J. Eng. Sci. 127, 145–161 (2018)
Freed, A.D., Zamani, S.: Elastic Kelvin-Poisson-Poynting solids described through scalar conjugate stress/strain pairs derived from a QR factorization of F. J. Mech. Phys. Solids 129, 278–293 (2019)
Freed, A.D., Graverend, J.-B., Rajagopal, K.R.: A decomposition of Laplace stretch with applications in inelasticity. Acta Mech. 230, 3423–3429 (2019)
Criscione, J.C.: A constitutive framework for tubular structures that enables a semi-inverse solution to extension and inflation. J. Elast. 77, 57–81 (2004)
McLellan, A.G.: Finite strain coordinates and the stability of solid phases. J. Phys. C: Solid State Phys. 9, 4083–4094 (1976)
McLellan, A.G.: The classical thermodynamics of deformable materials. Cambridge University Press, Cambridge (1980)
Srinivasa, A.R.: On the use of the upper triangular (or QR) decomposition for developing constitutive equations for Green-elastic materials. Int. J. Eng. Sci. 60, 1–12 (2012)
Criscione, J.C., Humphrey, J.D., Douglas, A.S., Hunter, W.C.: An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity. J. Mech. Phys. Solids 48, 2445–2465 (2000)
Criscione, J.C., Hunter, W.C.: Kinematics and elasticity framework for materials with two fiber families. Contin. Mech. Thermodyn. 15, 613–628 (2003)
Freed, A.D., Erel, V., Moreno, M.: Conjugate stress/strain base pairs for planar analysis of biological tissues. J. Mech. Mater. Struct. 12, 219–247 (2017)
Freed, A.D.: A note on stress/strain conjugate pairs: explicit and implicit theories of thermoelasticity for anisotropic materials. Int. J. Eng. Sci. 120, 155–171 (2017)
Rajagopal, K.R., Srinivasa, A.R.: On the response of non-dissipative solids. Proc. R. Soc. A 463, 357–367 (2006)
Freed, A.D., Einstein, D.R.: An implicit elastic theory for lung parenchyma. Int. J. Eng. Sci. 62, 31–47 (2013)
Freed, A.D., Rajagopal, K.R.: A promising approach for modeling biological fibers. Acta Mech. 227, 1609–1619 (2016)
Freed, A.D., Rajagopal, K.R.: A viscoelastic model for describing the response of biological fibers. Acta Mech. 227, 3367–3380 (2016)
Freed, A.D.: Soft Solids. Birkhauser, Cham (2016)
Mackenzie, J.K.: The elastic constants of a solid containing spherical holes. Proc. Phys. Soc. B 63, 2–11 (1950)
Bristow, J.R.: Microcracks, and the static and dynamic elastic constants of annealed and heavily cold-worked metals. Br. J. Appl. Phys. 11, 81–85 (1960)
Budiansky, B., O’Connell, R.J.: Elastic moduli of a cracked solid. Int. J. Solids Struct. 12, 81–97 (1976)
Rice, J.R.: Continuum mechanics and thermodynamics of plasticity in relation to microscale deformation mechanisms. In: Argon, A.S. (ed.) Constitutive Equations in Plasticity, pp. 23–79. Massachusetts Institute of Technology Press, Cambridge (1975)
Clayton, J.D.: Differential Geometry and Kinematics of Continua. World Scientific, Singapore (2014)
Clayton, J.D.: On anholonomic deformation, geometry, and differentiation. Math. Mech. Solids 17, 702–735 (2012)
Voyiadjis, G.Z., Kattan, P.I.: Advances in Damage Mechanics: Metals and Metal Matrix Composites. Elsevier, Amsterdam (1999)
Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)
Jog, C.S.: Continuum Mechanics: Volume 1: Foundations and Applications of Mechanics, 3rd edn. Cambridge University Press, Delhi (2015)
Smith, G.F., Rivlin, R.S.: The strain-energy function for anisotropic elastic materials. Trans. Am. Math. Soc. 88, 175–193 (1958)
Truesdell, C.A., Noll, W.: The Non-linear Field Theories of Mechanics. Springer, Berlin (1965)
Clayton, J.D.: Analysis of shock compression of strong single crystals with logarithmic thermoelastic-plastic theory. Int. J. Eng. Sci. 79, 1–20 (2014)
Guinan, M.W., Steinberg, D.J.: Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements. J. Phys. Chem. Solids 35, 1501–1512 (1974)
Poirier, J.-P., Tarantola, A.: A logarithmic equation of state. Phys. Earth Planet. Inter. 109, 1–8 (1998)
Clayton, J.D., Tonge, A.: A nonlinear anisotropic elastic-inelastic constitutive model for polycrystalline ceramics and minerals with application to boron carbide. Int. J. Solids Struct. 64–65, 191–207 (2015)
Clayton, J.D.: Finite strain analysis of shock compression of brittle solids applied to titanium diboride. Int. J. Impact Eng. 73, 56–65 (2014)
Anand, L.: On H. Hencky’s approximate strain-energy function for moderate deformations. J. Appl. Mech. 46, 78–82 (1979)
Fung, Y.-C.: Elasticity of soft tissues in simple elongation. Am. J. Physiol. 213, 1532–1544 (1967)
Vawter, D.L., Fung, Y.-C., West, J.B.: Constitutive equation of lung tissue elasticity. J. Biomech. Eng. 101, 38–45 (1979)
Clayton, J.D.: Crystal thermoelasticity at extreme loading rates and pressures: analysis of higher-order energy potentials. Extrem. Mech. Lett. 3, 113–122 (2015)
Clayton, J.D.: Nonlinear Elastic and Inelastic Models for Shock Compression of Crystalline Solids. Springer, Cham (2019)
Simo, J.C., Pister, K.S.: Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput. Methods Appl. Mech. Eng. 46, 201–215 (1984)
Fung, Y.-C., Patitucci, P., Tong, P.: Stress and strain in the lung. ASCE J. Eng. Mech. 104, 201–223 (1978)
Lee, G.C., Frankus, A.: Elasticity properties of lung parenchyma derived from experimental distortion data. Biophys. J. 15, 481–493 (1975)
Clayton, J.D., Banton, R.J., Freed, A.D.: A nonlinear thermoelastic-viscoelastic continuum model of lung mechanics for shock wave analysis. In: Lane, J.M.D. (ed.) Shock Compression of Condensed Matter, volume in press. AIP Conference Proceedings (2019)
Clayton, J.D., Freed, A.D.: A continuum mechanical model of the lung. Technical Report ARL-TR-8859, CCDC Army Research Laboratory, Aberdeen Proving Ground (MD) (2019)
Valanis, K.C.: Irreversible Thermodynamics of Continuous Media: Internal Variable Theory. Springer, Wien (1972)
Wineman, A.: Nonlinear viscoelastic solids—a review. Math. Mech. Solids 14, 300–366 (2009)
Hughes, R., May, A.J., Widdicombe, J.G.: Stress relaxation in rabbits’ lungs. J. Physiol. 146, 85–97 (1959)
Zeng, Y.J., Yager, D., Fung, Y.C.: Measurement of the mechanical properties of the human lung tissue. J. Biomech. Eng. 109, 169–174 (1987)
Malvern, L.E.: Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, Englewood Cliffs, NJ (1969)
Krajcinovic, D.: Damage Mechanics. North-Holland, Amsterdam (1996)
Bourdin, B., Francfort, G.A., Marigo, J.-J.: The variational approach to fracture. J. Elast. 91, 5–148 (2008)
Clayton, J.D., Knap, J.: A geometrically nonlinear phase field theory of brittle fracture. Int. J. Fract. 189, 139–148 (2014)
Clayton, J.D., Knap, J.: Phase field modeling of coupled fracture and twinning in single crystals and polycrystals. Comput. Methods Appl. Mech. Eng. 312, 447–467 (2016)
Clayton, J.D.: Finsler geometry of nonlinear elastic solids with internal structure. J. Geom. Phys. 112, 118–146 (2017)
Clayton, J.D.: Generalized Finsler geometric continuum physics with applications in fracture and phase transformations. Zeitschrift für Angewandte Mathematik und Physik (ZAMP) 68, 9 (2017)
Clayton, J.D., Knap, J.: Continuum modeling of twinning, amorphization, and fracture: theory and numerical simulations. Cont. Mech. Thermodyn. 30, 421–455 (2018)
Marshall, J.S., Naghdi, P.M., Srinivasa, A.R.: A macroscopic theory of microcrack growth in brittle materials. Philos. Trans. R. Soc. Lond. A 335, 455–485 (1991)
Kachanov, M.: Effective elastic properties of cracked solids: critical review of some basic concepts. Appl. Mech. Rev. 45, 304–335 (1992)
Rajendran, A.M.: Modeling the impact behavior of AD85 ceramic under multiaxial loading. Int. J. Impact Eng. 15, 749–768 (1994)
Espinosa, H.D., Zavattieri, P.D., Dwivedi, S.K.: A finite deformation continuum-discrete model for the description of fragmentation and damage in brittle materials. J. Mech. Phys. Solids 46, 1909–1942 (1998)
Clayton, J.D.: A model for deformation and fragmentation in crushable brittle solids. Int. J. Impact Eng. 35, 269–289 (2008)
Clayton, J.D.: Deformation, fracture, and fragmentation in brittle geologic solids. Int. J. Fract. 173, 151–172 (2010)
Kachanov, M., Tsukrov, I., Shafiro, B.: Effective moduli of solids with cavities of various shapes. Appl. Mech. Rev. 47, S151–S174 (1994)
Fond, C.: Cavitation criterion for rubber materials: a review of void-growth models. J. Polym. Sci. B 39, 2081–2096 (2001)
Hang-Sheng, H., Abeyaratne, R.: Cavitation in elastic and elastic-plastic solids. J. Mech. Phys. Solids 40, 571–592 (1992)
Steenbrink, A.C., Van Der Giessen, E., Wu, P.D.: Void growth in glassy polymers. J. Mech. Phys. Solids 45, 405–437 (1997)
Volokh, K.Y.: Cavitation instability in rubber. Int. J. Appl. Mech. 3, 299–311 (2011)
Voyiadjis, G.Z., Shojaei, A., Li, G.: A generalized coupled viscoplastic-viscodamage-viscohealing theory for glassy polymers. Int. J. Plast 28, 21–45 (2012)
Xue, L.: Constitutive modeling of void shearing effect in ductile fracture of porous materials. Eng. Fract. Mech. 75, 3343–3366 (2008)
Clayton, J.D., Knap, J.: Phase field modeling of directional fracture in anisotropic polycrystals. Comput. Mater. Sci. 98, 158–169 (2015)
Clayton, J.D., Knap, J.: Nonlinear phase field theory for fracture and twinning with analysis of simple shear. Phil. Mag. 95, 2661–2696 (2015)
Rajagopal, K.R., Srinivasa, A.R.: On a class of non-dissipative materials that are not hyperelastic. Proc. R. Soc. A 465, 493–500 (2008)
Stamenovic, D.: Micromechanical foundations of pulmonary elasticity. Physiol. Rev. 70, 1117–1134 (1990)
Denny, E., Schroter, R.C.: A model of non-uniform lung parenchyma distortion. J. Biomech. 39, 652–663 (2006)
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J.D.C. acknowledges support of the CCDC Army Research Laboratory. A.D.F. acknowledges support of a Joint Faculty Appointment with the CCDC Army Research Laboratory.
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Clayton, J.D., Freed, A.D. A constitutive framework for finite viscoelasticity and damage based on the Gram–Schmidt decomposition. Acta Mech 231, 3319–3362 (2020). https://doi.org/10.1007/s00707-020-02689-5
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DOI: https://doi.org/10.1007/s00707-020-02689-5