zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids. (English) Zbl 0783.73024
Summary: Key qualitative features of solutions exhibiting strong discontinuities in rate-independent inelastic solids are identified and exploited in the design of a new class of finite element approximations. The analysis shows that the softening law must be re-interpreted in a distributional sense for the continuum solutions to make mathematical sense and provides a precise physical interpretation to the softening modulus. These results are verified by numerical simulations employing a regularized discontinuous finite element method which circumvent the strong mesh- dependence exhibited by conventional methods, without resorting to viscosity or introducing additional ad-hoc parameters. The analysis is extended to a new class of anisotropic rate-independent damage models for brittle materials.
MSC:
74C99Plastic materials, etc.
74S05Finite element methods in solid mechanics
74R99Fracture and damage
References:
[1]Anzellotti, G.; Giaquinta, M. (1982): On the existence of the fields of stresses and displacements for an elasto-perfectly plastic body in static equilibrium. J. Math. Pures et Appliqu?es 61, 219-244
[2]Bazant, Z. P. (1986): Mechanics of distributed cracking. Appl. Mech. Res. 5, 675-705 · doi:10.1115/1.3143724
[3]Bazant, Z. P.; Belytschko, T. (1985): Wave propagation in a strain-softening bar: exact solution. J. Eng. Mech. Div. ASCE. 111, 381-389 · doi:10.1061/(ASCE)0733-9399(1985)111:3(381)
[4]Belytschko, T.; Fish, J.; Engelmann, B. E. (1988): A finite element with embedded localization zones. Comput. Meth. Appl. Mech. Eng. 70, 59-89 · Zbl 0653.73032 · doi:10.1016/0045-7825(88)90180-6
[5]Crisfield, M. A.; Wills, J. (1988): Solution strategies and softening materials. Comput. Meth. Appl. Mech. Eng. 66, 267-289 · Zbl 0618.73035 · doi:10.1016/0045-7825(88)90002-3
[6]De Borst, R. (1987): Computation of post-bifurcation and post-failure behavior of strain-softening solids. Comput. Struct. 2, 211-224
[7]Demengel, F. (1989): Compactness theorems of spaces of functions with bounded derivatives and applications to limit analysis problems in plasticity. Arch. Rat. Mech. Anal. 105, 123-161 · Zbl 0669.73030 · doi:10.1007/BF00250834
[8]Duvaut, G.; Lions, J. L. (1976): Inequalities in mechanics and physics. Berlin, Heidelberg, New York: Springer
[9]Hadamard, J. (1903): Lecons sur la Propagation des Ondes et les Equations de l’Hydrodynamique, Chap. VI. Paros: Hermann
[10]Hill, R. (1950): The mathematical theory of plasticity. Oxford University Press
[11]Hill, R. (1962): Acceleration waves in solids. J. Mech. Phys. Solids 6, 1-10 · Zbl 0080.18204 · doi:10.1016/0022-5096(57)90040-6
[12]Hillerborg, A. (1985): Numerical methods to simulate softening and fracture of concrete. In: Sih, G. C.; DiTommaso, A. (eds): Fracture mechanics of contrete: structural application and numerical calculation, 141-170
[13]Hillerborg, A.; Modeer, M.; Petersson, P. (1976): Analysis of crack formation and crack growth in concrete by means fo fracture mechanics and finite elements. Cement and concrete research 6, 773-782 · doi:10.1016/0008-8846(76)90007-7
[14]Johnson, C. (1976): Existence theorems for plasticity problems. J. Math. Pures et Appliqu?es 55, 431-444
[15]Johnson, C. (1978): On plasticity with hardening. J. Math. Anal. Appl. 62, 325-336 · Zbl 0373.73049 · doi:10.1016/0022-247X(78)90129-4
[16]Johnson, C.; Hansbo, P. (1992): Adaptive finite element methods for small strain elastoplasticity. In: Proceedings IUTAM conference. Springer Verlag (in press)
[17]Johnson, W. (1987): Henri tresca as the originator of adiabatic heat lines. Intern. J. Mech. Sci. 29, 301-310
[18]Knowles, J. (1979): On the dissipation associated with weak shocks in finite elasticity. J. Elasticity 9, 131-158 · Zbl 0407.73037 · doi:10.1007/BF00041322
[19]Katchanov, L. M. (1958): Time of the rupture process under creep conditions. IVZ Akad. Nauk S.S.R., Otd Tech Nauk 8, 26-31
[20]Koiter, W. T. (1966): General theorems for elastic-plastic solids. In: Progress in Solid Mechanics, Chapter IV, vol. 1, 67-221
[21]Kohn, R.; Temam, R. (1983): Dual spaces of stresses and strains, with applications to Hencky plasticity. Appl. Math. Optim. 10, 1-35. · Zbl 0532.73039 · doi:10.1007/BF01448377
[22]Lesaint, P. (1979): Sur les Syst?mes Hyperboliques, Th?se d’Etat
[23]LeVeque, R. J. (1990): Numerical methods of conservation laws. Basel, Birkhauser Verlag
[24]Mandel, J. (1966): Conditions de Stabilit? et Postulat de Drucker. In: Kravtchenko, J.; Sirieys, P. M. (eds): Rheology and soil mechanics, IUTAM Symposium Grenoble 1964, 58-68
[25]Matthies, H.; Strang, G.; Christiansen, E. (1979): The saddle point of a differential program. In: Glowinski, Robin, Zienkiewicz (eds). Energy methods in finite element analysis, 309-318. London: J. Wiley and Sons
[26]Matthies, H. (1979): Existence theorems in thermo-plasticity. J. Mec. 18, 695-712
[27]Needleman, A. (1987): Material rate dependent and mesh sensitivity in localization problems. Comp. Appl. Mech. Eng. 67, 68-85.
[28]Needleman, A.; Tvergard, V. (1992): Analysis of plastic localization in metals. Appl. Mech. Rev. 3-18
[29]Oliver, J. (1989): A consistent characteristic length for smeared cracking models. Int. J. Num. Meth. Eng. 28, 461-474. · Zbl 0676.73066 · doi:10.1002/nme.1620280214
[30]Ortiz, M.; Leroy, Y.; Needleman, A. (1987): A finite element method for localization failure analysis. Comp. Meth. Appl. Mech. Eng. 61, 189-214 · Zbl 0597.73105 · doi:10.1016/0045-7825(87)90004-1
[31]Ortiz, M. (1985): A constitutive theory for the inelastic behavior of concrete. Mech. Mat. 4, 67-93. · doi:10.1016/0167-6636(85)90007-9
[32]Ortiz, M.; Quigley, J. J. (1991): Adaptive mesh refinement in strain localization. Com. Meth. Appl. Mech. Eng. 90, 781-804 · doi:10.1016/0045-7825(91)90184-8
[33]Ottosen, N. (1985): Thermodynamic consequences of Strain-softening in tension. J. Eng. Mech. Div. ASCE, 112, 1152-1164 · doi:10.1061/(ASCE)0733-9399(1986)112:11(1152)
[34]Pietrszczak, St.; Mr?z (1981): Finite element analysis of deformation of strain-softening materials. Int. J. Num. Meth. Eng. 17, 327-334 · Zbl 0461.73063 · doi:10.1002/nme.1620170303
[35]Pironneau, O. A. (1989): Finite element methods for fluids. John Wiley and Sons
[36]Prandtl, L. (1920): Ueber di Haerte plastischer Koerper. Goett. Nach., 74-84
[37]Prandtl, L. (1921): Ueber die Eindringungsfestigkeit plastischer Baustoffe und die Festigkeit von scneiden. Z. angew. Math. Mech. 1, 15-20 · doi:10.1002/zamm.19210010102
[38]Read, H. E.; Hegemier, G. A. (1984): Strain softening of rock, soil and concrete?a review article. Mech. Mat. 3, 271-294 · doi:10.1016/0167-6636(84)90028-0
[39]Reddy and Simo, J. C. (1992): Stability and convergence analysis of a class of assumed strain mixed methods. Num. Math. 3, 271-294.
[40]Simo, J. C.; Ju, J. W. (1984): A continuum strain based damage model. Part I: Formulation and Part II: Numerical algorithms. Int. J. Solids Struct. 23, 821-840 and 841-870 · Zbl 0634.73106 · doi:10.1016/0020-7683(87)90083-7
[41]Simo, J. C.; Rifai, M. S. (1990): A class of mixed assumed strain methods and the method of incompatible modes. Int. J. Num. Meth. Eng. 29, 1595-1638 · Zbl 0724.73222 · doi:10.1002/nme.1620290802
[42]Simo, J. C.; Hughes, T. J. R. (1991): Plasticity, viscoplasticity and viscoelasticity: formulation and numerical analysis Aspects. Berlin, Heidelberg, New York: Springer
[43]Strang, G.; Fix, G. (1972): An analysis of the finite element method. In: Energy methods in finite element analysis. Englewood Cliffs, New Jersey: Prentice-Hall, pp. 1595-1638
[44]Suquet, P. M. (1981): Sur les ?quations de la plasticit?: existence et regularit? des solutions. J. Mech. 20, 5-39
[45]Suquet, P. M. (1978): Existence and regularity of solutions for plasticity problems. In: Nemat-Nasser (ed): Variational methods in the mechanics of solids, 304-309
[46]Temam, R.; Strang, G. (1980): Functions of bounded deformation. Arch. Rational Mech. Anal. 75, 7-21 · Zbl 0472.73031 · doi:10.1007/BF00284617
[47]Temam, R. (1986): A generalized Norton-Hoff model and the Prandtl-Reuss law of plasticity. Arch. Rational. Mech. Anal. 95, 137-183 · Zbl 0615.73035 · doi:10.1007/BF00281085
[48]Thomas, T. Y. (1961): Plastic flow and fracture in solids, Academic Press
[49]Willam, K.; Sobh, N.; Sture (1987): Elastic-plastic tangent operators: localization study on the constitutive and finite element levels. In: Nakazawa, S.; William, K.; Robelo, N. (eds): Advances in inelastic analysis, AMD-Vol. 28, 107-126