zbMATH — the first resource for mathematics

On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects. (English) Zbl 0588.73082
A fully three-dimensional finite-strain viscoelastic model is developed, characterized by: (i) general anisotropic response, (ii) uncoupled bulk and deviatoric response over any range of deformations, (iii) general relaxation functions, and (iv) recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin). Continuum damage mechanics is employed to develop a simple isotropic damage mechanism, which incorporates softening behavior under deformation, and leads to progressive degradation of the storage in a cyclic test with increasing amplitude (Mullins’ effect). A numerical integration procedure is proposed which trivially satisfies objectivity and bypasses the use of midpoint configurations. The resulting algorithm can be exactly linearized in closed form, and leads to symmetric tangent moduli. Quasi-incompressible response is accounted for within the context of a three-field variational formulation of the Hu-Washizu type.

74D10 Nonlinear constitutive equations for materials with memory
74S30 Other numerical methods in solid mechanics (MSC2010)
74D05 Linear constitutive equations for materials with memory
Full Text: DOI
[1] Blatz, P.J.; Ko, W.L., Application of finite elastic theory to the deformation of rubbery materials, Trans. soc. rheology, 6, 223-251, (1968)
[2] Bernstein, B.; Kearsley, E.A.; Zapas, L.J., Trans. soc. rheology, 7, 391-410, (1963)
[3] Bernstein, B.; Kearsley, E.A.; Zapas, L.J., J. res. nat. bur. standards, 68B, 103, (1964)
[4] Browning, R.V.; Gurtin, M.E.; Williams, W.O., A viscoplastic constitutive theory for filled polymers, Internat. J. solids and structures, 20, 11/12, 921-934, (1983) · Zbl 0567.73005
[5] Chaboche, J.L., Continuous damage mechanics—A tool to describe phenomena before crack initiation, Nucl. engrg. design, 64, 233-247, (1981)
[6] Chaboche, J.L., Le concept de contrainte effective appliqué à l’élasticité et á la viscoplasticité en présence d’un endommagement anisotrope, (), 737-760 · Zbl 0516.73014
[7] Christensen, R.M., A nonlinear theory of viscoelasticity for application to elastomers, J. appl. mech., 47, 762-768, (1980) · Zbl 0456.73021
[8] Christensen, R.M., Theory of viscoelasticity: an introduction, (1971), Academic Press New York
[9] Coleman, B.D.; Gurtin, M., Thermodynamics with internal variables, J. chem. phys., 47, 597-613, (1967)
[10] Farris, R.J., The stress-strain behavior of mechanically degradable polymers, ()
[11] Ferry, J.D., Viscoelastic properties of polymers, (1970), Wiley New York
[12] Flory, R.J., Thermodynamic relations for high elastic materials, Trans. Faraday soc., 57, 829-838, (1961)
[13] Fortin, M.; Glowinski, R., Méthodes de lagrangien augmenté. application á la résolution numérique de problèmes aux limites, (1982), Dunod-Bordas Paris · Zbl 0491.65036
[14] Gent, A.N., The mechanics of fracture, (), 55
[15] Green, M.S.; Tobolsky, A.V., A new approach to the theory of relaxing polymeric media, J. phys. chem., 14, 80-92, (1946)
[16] Glowinski, R., Numerical methods for nonlinear variational problems, (1984), Springer Berlin · Zbl 0575.65123
[17] Gurtin, M.E.; Francis, E.C., Simple rate-independent model for damage, J. spacecraft, 18, 3, 285-286, (1981)
[18] Hallquist, J.O., NIKE 2D: an implicit, finite deformation, finite element code for analyzing the static and dynamic response of two-dimensional solids, () · Zbl 0319.70015
[19] Herrmann, L.R.; Peterson, F.E., A numerical procedure for viscoelastic stress analysis, ()
[20] Hughes, T.J.R., Generalization of selective integration procedures to anisotropic and nonlinear media, Internat. J. numer. meths. engrg., 15, 9, 1413-1418, (1980) · Zbl 0437.73053
[21] Hughes, T.J.R.; Winget, J., Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, Internat. J. numer. meths. engrg., 15, 9, 1413-1418, (1980)
[22] Kachanov, L.M., Time of the rupture process under creep conditions, Izv. akad. nauks SSR otd. tech. nauk, 26-31, (1958)
[23] Kelly, J.M.; Celebi, M., Verification testing of prototype bearings for based isolated building, ()
[24] Key, S.W.; Krieg, R.D., On the numerical implementation of inelastic time dependent and time independent, finite strain constitutive equations in structural mechanics, Comput. meths. appl. mech. engrg., 33, 439-452, (1982) · Zbl 0504.73054
[25] Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity, J. appl. mech., 51, 2, 299-307, (1984) · Zbl 0544.73052
[26] Kramer, E.J., Microscopic and molecular fundamentals of crazing, Adv. polymer sci., 52/53, (1983)
[27] Lemaitre, J., How to use damage mechanics, Nucl. engrg. design, 80, 233-245, (1984)
[28] Lemaitre, J., A continuous damage mechanics model for ductile fracture, J. engrg. mater. tech., 107, 83-89, (1985)
[29] Leaderman, H., Elastic properties of filamentous materials, (1943), The Textile Foundation Washington, D.C
[30] Lubliner, J., A model of rubber viscoelasticity, Mech. res. comm., 12, 93-99, (1985)
[31] Naghdi, P.M.; Trapp, J.A., The significance of formulating plasticity theory with reference to loading surfaces in strain space, Internat. J. engrg. sci., 13, 785-797, (1975) · Zbl 0315.73050
[32] Nagtegaal, J.C.; Parks, D.M.; Rice, J.R., On numerically accurate finite element solutions in the fully plastic range, Comput. meths. appl. mech. engrg., 4, 153-177, (1974) · Zbl 0284.73048
[33] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1983), Prentice-Hall Englewood-Cliffs, NJ · Zbl 0545.73031
[34] Mullins, L., Softening of rubber by deformation, Rubber chem. technol., 42, 339-362, (1969)
[35] Oden, J.T.; Carey, G., Finite elements vol. IV. mathematical aspects, (1983), Prentice-Hall Englewood-Cliffs, NJ
[36] Ogden, R.W., Elastic deformations in rubberlike solids, (), 499-537 · Zbl 0491.73045
[37] Ogden, R.W., Nonlinear elastic deformations, (1984), Ellis Horwood Chichester, U.K · Zbl 0541.73044
[38] Pinsky, P.M.; Ortiz, M.; Pister, K.S., Numerical integration of rate constitutive equations in finite deformation analysis, Comput. meths. appl. mech. engrg., 40, 137-158, (1983) · Zbl 0504.73057
[39] Pipkin, A.C.; Rogers, T.G., A non-linear integral representation for viscoelastic behavior, J. mech. phys. solids, 16, 59-72, (1968) · Zbl 0158.43601
[40] Rabotnov, Yu.N., Elements of hereditary solid mechanics, (1980), Mir Moscow · Zbl 0515.73026
[41] Schapery, R.A., A theory of non-linear viscoelasticity based on irreversible thermodynamics, (), 511-530
[42] Sidoroff, F., Un modele viscoelastique non lineaire avec configuration intermediaire, J. Méc., 13, 679-713, (1974) · Zbl 0321.73029
[43] Simo, J.C.; Taylor, R.L., A simple three dimensional model accounting for damage effects, () · Zbl 0585.73059
[44] Simo, J.C.; Taylor, R.L., A three dimensional finite deformation viscoelastic model accounting for damage effects, () · Zbl 0585.73059
[45] Simo, J.C.; Taylor, R.L.; Pister, K.S., Variational and projection methods for the volume constraint in finite deformation elastoplasticity, Comput. meths. appl. mech. engrg., 51, 177-208, (1985) · Zbl 0554.73036
[46] C. Truesdell and W. Noll, The Nonlinear Field Theories, Handbuch der Physik, Band III/3 Springer, Berlin. · Zbl 1068.74002
[47] Mullins, L.; Tobin, N.R., Stress softenings in rubber vulcanizates, J. appl. polymer sci., 9, 2993-3010, (1965), Part 1
[48] Harwood, J.A.C.; Mullins, L.; Payne, A.R., Stress softenings in natural rubber vulcanizates, J. appl. polymer sci., 9, 3011-3021, (1965), Part II
[49] Payne, A.R., Dynamic properties of heat-treated butyl vulcanizates, J. appl. polymer sci., 7, 873, (1963)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.