## Analyses of large quasistatic deformations of inelastic bodies by a new hybrid-stress finite element algorithm: Applications.(English)Zbl 0532.73070

This paper deals with the implementation of a hybrid stress FE algorithm which the authors have presented before [ibid. 39, 245-295 (1983; Zbl 0505.73045)]. First, the energy principle underlying the method is discussed. Then, the FE equations are outlined. Examples dealing with plane strain problems serve for testing the algorithm’s performance. The first example treats the necking phenomenon occuring in tension tests of elastic-plastic specimens. Bifurcation analysis is performed, and results are compared with other works. The same example is used for testing the sensitivity of the results to variations in the number, shape, and type of elements. The last example deals with plane extension of an elastic- viscous material which contains a periodic array of cylindrical voids. An appendix provides the specific shape functions used in this study.
Reviewer: Sh.Ginsburg

### MSC:

 74S05 Finite element methods applied to problems in solid mechanics 74-04 Software, source code, etc. for problems pertaining to mechanics of deformable solids 74B20 Nonlinear elasticity 74S99 Numerical and other methods in solid mechanics

Zbl 0505.73045
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### References:

 [1] Reed, K.W.; Atluri, S.N., Analyses of large quasistatic deformations of inelastic bodies by a new hybrid-stress finite element algorithm, Comput. meths. appl. mech. engrg., 39, 245-295, (1983) · Zbl 0505.73045 [2] Pian, T.H.H., Derivation of element stiffness matrices by assumed stress distributions, Aiaa j., 2, 7, 1333-1336, (1964) [3] Herrmann, L.R., Elasticity equation for incompressible materials by a variational theorem, Aiaa j., 3, 10, 1896-1900, (1965) [4] Key, S.W., A variational principle for incompressible and nearly-incompressible anisotropic elasticity, Internat. J. solids structures, 5, 951-964, (1969) · Zbl 0175.22101 [5] Tong, P., An assumed stress hybrid finite element method for an incompressible and near-incompressible material, Internat. J. solids structures, 5, 455-461, (1969) · Zbl 0167.52803 [6] Washizu, K., Variational methods in elasticity and plasticity, (1975), Pergamon London · Zbl 0164.26001 [7] de Veubeke, B.F., A new variational principle for finite elastic deformations, Internat. J. engrg. sci., 10, 745-763, (1972) · Zbl 0245.73031 [8] Atluri, S.N., On some new general and complementary energy theorems for the rate problems of finite strain, classical elasto-plasticity, J. structural mech., 8, 1, 61-92, (1980) [9] Truesdell, C.; Noll, W., The nonlinear field theories of mechanics, () · Zbl 0779.73004 [10] Hill, R., Aspects of invariance in solid mechanics, Adv. appl. mech., 18, 1-75, (1978) · Zbl 0475.73026 [11] Prager, W., Introduction to the mechanics of continua, (1961), Ginn New York · Zbl 0094.18602 [12] Fung, Y.C., Foundations of solid mechanics, (1965), Prentice-Hall Englewood Cliffs, NJ [13] Dill, E.H., The complementary energy principle in nonlinear elasticity, Lett. appl. engrg. sci., 5, 95-106, (1977) [14] Ergatoudis, I.; Irons, B.M.; Zienkiewicz, O.C., Curved, isoparametric, quadrilateral elements for finite element analysis, Internat. J. solids structures, 4, 31-42, (1968) · Zbl 0152.42802 [15] Reed, K.W., Analysis of large quasistatic deformations of inelastic solids by a new stress based finite element method, () · Zbl 0505.73045 [16] Tong, P.; Rossettos, J.N., Finite element method: basic technique and implementation, (1977), MIT Cambridge, MA · Zbl 0293.73037 [17] Bathe, K.-J., Finite element procedures in engineering analysis, (1982), Prentice-Hall Englewood Cliffs, NJ [18] Tong, P.; Pian, T.H.H., A variational principle and the convergence of a finite element method based on assumed stress distribution, Internat. J. solids structures, 5, 463-472, (1969) · Zbl 0167.52805 [19] Conte, S.D.; de Boor, C., Elementary numerical analysis, (1972), McGraw-Hill New York · Zbl 0257.65002 [20] Cormeau, I., Numerical stability in quasistatic elasto-visco-plasticity, Internat. J. numer. meths. engrg., 9, 109-127, (1975) · Zbl 0293.73022 [21] Hughes, T.J.R.; Taylor, R.L., Unconditionally stable algorithms for quasi-static elasto-visco-plastic finite element analysis, Comput. & structures, 8, 169-173, (1978) · Zbl 0365.73029 [22] Argyris, J.H.; Vaz, L.E.; Willam, K.J., Improved solution methods for inelastic rate problems, Comput. meths. appl. mech. engrg., 16, 231-277, (1978) · Zbl 0405.73068 [23] Kanchi, M.B.; Zienkiewicz, O.C.; Owen, D.R.J., The visco-plastic approach to problems of plasticity and creep involving geometric nonlinear effects, Internat. J. numer. meths. engrg., 12, 169-181, (1978) · Zbl 0366.73032 [24] Hutchinson, J.W.; Miles, J.P., Bifurcation analysis of the onset of necking in an elastic/plastic cylinder under uniaxial tension, J. mech. phys. solids, 22, 61-71, (1974) · Zbl 0271.73042 [25] Miles, J.P., The initiation of necking in rectangular elastic/plastic specimens under uniaxial and biaxial tension, J. mech. phys. solids, 23, 197-213, (1975) · Zbl 0324.73035 [26] Hill, R.; Hutchinson, J.W., Bifurcation phenomena in the plane tension test, J. mech. phys. solids, 23, 239-264, (1975) · Zbl 0331.73048 [27] Burke, M.A.; Nix, W.D., A numerical study of necking in the plane tension test, Internat. J. solids structures, 15, 379-393, (1979) · Zbl 0404.73045 [28] Osias, J.R., Finite deformation of elastic-plastic solids: the example of necking in flat tensile bars, () [29] McMeeking, R.M.; Rice, J.R., Finite element formulations for problems of large elastic-plastic deformation, Internat. J. solids structures, 11, 601-616, (1975) · Zbl 0303.73062 [30] Hill, R., A general theory of uniqueness and stability in elastic-plastic solids, J. mech. phys. solids, 6, 236-249, (1958) · Zbl 0091.40301 [31] Hill, R., Some basic principles in the mechanics of solids without a natural time, J. mech. phys. solids, 7, 209-225, (1959) · Zbl 0086.17301 [32] Cowper, G.R.; Onat, E.T., The initiation of necking and buckling in plane plastic flow, () [33] de Veubeke, B.F.; Millard, A., Discretization of stress field in the finite element method, J. franklin inst., J. franklin inst., 302, 6, 389-412, (1976) · Zbl 0352.73061 [34] Burke, M.A.; Nix, W.D., A numerical analysis of void growth in tension creep, Internat. J. solids structures, 15, 55-71, (1979) · Zbl 0392.73031 [35] Bird, R.B.; Armstrong, R.C.; Hassager, O., () [36] Timoshenko, S.P.; Goodier, J.N., Theory of elasticity, (1970), McGraw-Hill New York · Zbl 0266.73008 [37] Argyris, J.H.; Doltsinis, J.St.; Pimenta, P.M.; Wüstenberg, H., Thermomechanical response of solids at high strains natural approach, Comput. meths. appl. mech. engrg., 32, 3-57, (1982) · Zbl 0505.73062 [38] Argyris, J.H.; Doltsinis, J.St., On the large strain analysis in natural formulation—part I. quasistatic problems, Comput. meths. appl. mech. engrg., 20, 213-252, (1980) · Zbl 0437.73065 [39] Argyris, J.H.; Doltsinis, J.St., On the large strain analysis in natural formulation—part II. dynamic problems, Comput. meths. appl. mech. engrg., 21, 91-128, (1980) · Zbl 0437.73067
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