Applications of mathematical programming concepts to incremental elastic-plastic analysis

Martin, J. B.; Reddy, B. D.; Griffin, T. B.; Bird, W.W.

doi:10.1016/0141-0296(87)90012-5

Cited by 44 publications

(9 citation statements)

References 4 publications

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“…In addition, internal variable formulations of elastoplasticity has provided a sound basis for the development of the mathematical theory of plasticity and the link with mathematical programming and optimization techniques (Germain, 1973, Maier, 1970a, Halphen and Son, 1975, Martin, 1975, Martin et al, 1987, Martin and Nappi, 1990. The technique of introducing internal (or hidden) variables in a mechanical model, which count for the (from the past time or loading history) nonlinear response, can be traced back to Ziegler, 1963.…”

Section: Convex Energy and Dissipation Problems In Standard Generalizmentioning

confidence: 99%

Nonconvex Optimization in Mechanics

Mistakidis

Stavroulakis

1998

Nonconvex Optimization and Its Applications

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This book is dedicated to our teacher, Professor P. D. Pa nagiotopoulos 206 xiii XV xvi NONCONVEX OPTIMIZATION IN MECHANICSThe book is divided into four parts. The first part presents a number of engineering problems involving nonmonotone stress-strain or reaction-displacement laws, leading to nonconvex energy functions. The difficulties introduced by the nonconvexity and nonsmoothness is demonstrated by very simple examples involving one or two variables.The second part gives the mathematical background of applied convex and nonconvex optimization. After a short review of the literature, new algorithms inspired from engineering applications are proposed for the solution of the mathematical problem.The third part deals with the mathematical formulation of convex and nonconvex, smooth and nonsmooth problems in mechanics and engineering. Chapter 3 deals with convex problems, while in Chapter 4 nonconvex problems are presented. The presentation of the problems follows the same sequence in both Chapters in order to facilitate the comparison of nonconvex problems with respect to the corresponding convex problems. Starting from the basic relations of mechanics, a detailed formulation is presented for a large class of engineering problems. First, interface problems are treated and the corresponding variational relations are obtained. Then the treatment is extended in order to include material nonlinearities.In the last Chapter of this part {Chapter 5), some problems of optimal design of structures are briefly treated. After the theoretical formulation of the problem, certain algorithms are presented for its numerical treatment.The fourth part presents algorithms specialized for the treatment of a large class of nonconvex, nonsmooth optimization problems appearing in mechanics and engineering. The algorithms are presented in Chapter 6 and described in all possible detail; certain hints on their computer implementation are given. In Chapter 7 a large number of engineering applications is presented. After the detailed examination of the behaviour of the algorithms on very simple examples, realistic engineering applications are presented with large number of unknowns, exhibiting the potential of the optimization approach.The authors are deeply greatful to Professor P.O. Panagiotopoulos {Aristotle University of Thessaloniki, Greece and RWTH Aachen, Germany) for his encouragement to publish this book, for his helpful suggestions and his continuous support. We would also like to thank Professor K.T. Thomopoulos and Professor C.C. Baniotopoulos for the helpful discussions and their advices concerning the engineering oriented topics and Dr. O.K. Panagouli for her help on the topics concerning the application of fractal geometry to mechanics.

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Section: Convex Energy and Dissipation Problems In Standard Generalizmentioning

confidence: 99%

Nonconvex Optimization in Mechanics

Mistakidis

Stavroulakis

1998

Nonconvex Optimization and Its Applications

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“…The second class is furnished by holonomic approximations constructed from 0029-598 1/89/08 1839-1 5$07.50 0 1989 by John Wiley & Sons, Ltd. incrementally extremal paths. 3 In the latter case, the rate equations of plasticity are replaced by a deformation theory based on the incremental strains. The existence of an incremental strain energy potential insures symmetry of the consistent tangents.…”

Section: Introductionmentioning

confidence: 99%

Symmetry‐preserving return mapping algorithms and incrementally extremal paths: A unification of concepts

Ortiz

Martin

1989

Numerical Meth Engineering

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in this work we seek to characterize the conditions under which an elastic-plastic stress update algorithm preserves the symmetries inherent to the material response. From a numerical standpoint, the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality. For the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving. For hardcning plasticity, symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations. We show that these restrictions amount to the existence of a pseudo-internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli. In view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and, hence, in symmetric tangents, we address the question of whether any connections exist between the two approaches. We show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence.

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“…Now the relevant quantities are those appearing in the dissipation inequality (21). Of these, g and y are scalars, and hence insensitive to the particular description, while T and D7' are spatial quantities.…”

Section: S(fv/a) = S(qfv/a)mentioning

confidence: 99%

“…This may be seen in work presented in [25] and [26]; the former contains an analysis of a problem which has an identical structure to that which is under discussion, while the latter treats a broad class of problems with unilateral constraints, so that the problem being discussed here is a special case. Secondly, it has been demonstrated in [19,20,21] that this same variational framework, when time-discretised in a rational way, gives rise to a problem which in discrete form is amenable to solution by the use of simple algorithms. Naturally it is of great interest to ascertain the extent to which these advantages carry over to the finite-strain case.…”

Section: Fp = Ifmentioning

confidence: 99%

“…(33) or (36)). It has been demonstrated [19,20,21], however, that there are considerable advantages to using an evolution equation in the form (34) involving the dissipation function, particularly when numerical solutions to the corresponding initial-boundary value problems are sought. The properties of the dissipation function, particularly its convexity and positive homogeneity, allow one to construct rational approximations which in turn place in perspective the well-known schemes for integration of constitutive equations (see [20] for a review).…”

Section: Fp = Ifmentioning

confidence: 99%

See 1 more Smart Citation

An internal variable finite-strain theory of plasticity within the framework of convex analysis

Eve¹,

Gültop²,

Reddy³

1990

Quart. Appl. Math.

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Abstract. An internal variable constitutive theory for elastic-plastic materials undergoing finite strains is presented. The theory is based on a corresponding study in the context of small strains [6], and has the following features: first, with a view to embracing the classical notions of convex yield surfaces and the normality law, the evolution law is developed within the framework of nonsmooth convex analysis, which proves to be a powerful unifying tool; secondly, the special case of elastic materials is recovered from the theory in a natural manner. After presentation of the theory a concrete example is discussed in detail.1. Introduction. The theory of plasticity in its classical small-strain form is wellestablished, especially that form which the theory takes in its application to metals. The finite-strain theory, on the other hand, while showing indications that it is on the way to becoming an established branch of mechanics, is still nevertheless the subject of considerable effort and debate, and certain of its aspects remain unsettled. The literature on the subject has in the meantime acquired voluminous proportions; without attempting a comprehensive survey we mention as important contributions the early work of Green and Naghdi [7] and of Lee [17] in which were addressed, inter alia, the question of the decomposition of deformation into elastic and plastic parts, the former proposing an additive decomposition of strain and the latter a multiplicative decomposition of deformation gradient. The multiplicative decomposition has proved more popular, and forms the basis for a number of alternative theories; as examples we mention the works of Mandel [18], Halphen and Nguyen [8], Simo and

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Applications of mathematical programming concepts to incremental elastic-plastic analysis

Cited by 44 publications

References 4 publications

Nonconvex Optimization in Mechanics

Nonconvex Optimization in Mechanics

Symmetry‐preserving return mapping algorithms and incrementally extremal paths: A unification of concepts

An internal variable finite-strain theory of plasticity within the framework of convex analysis

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