Abstract
In this Chapter nonsmooth computational mechanics problems are formulated and studied by using the quasidifferentiable modelling techniques. Discrete variational inequality problems, hemivariational inequality problems and systems of variational inequalities are formulated based on the results of Chapters 3 and 4. All problems treated here are pilot applications which can be followed to develop nonsmooth modelling techniques for other branches in engineering.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Argyris J.H. (1966), Continua and discontinua. Proc. 1st Conf. Matrix Meth. Struct. MechWright Petterson Air Force Base, Dayton, Ohio 1965, AFFDL TR 66–80.
Auchmuty G. (1989), Duality algorithms for nonconvex variational principles. Num. Functional Analysis and Optimization, 10, 211–264.
Bazant Z.P. and Cedolin L. (1991), Stability of structures. Elastic, inelastic, fracture and damage theories, Oxford University Press, New York, Oxford.
Comi C., Maier G. and Perego U. (1992), Generalized variable finite element modelling and extremum theorems in stepwise holonomic elastoplasticity with internal variables, Computer Method in Applied Mechanics and Engineering 96, 213–237.
Demyanov V.F. and Vasiliev L.N. (1985), Nondifferentiable Optimization, Optimization Software, New York.
Demyanov V.F and Rubinov A.M. (1995), Introduction to Constructive Nonsmooth Analysis, Peter Lang Verlag, Frankfurt a.M. — Bern — New York, 414 p.
Duvaut G. and Lions J.L. (1972), Les inequations en mechanique et en physique, Dunod, Paris.
Eve R.A., Reddy B.D. and Rockafellar R.T. (1990), An internal variable theory of plasticity based on the maximum plastic work inequality, Quarterly of Applied Mathematics 68 (1), 59–83.
Glowinski R., Lions J.-L. and Tremolieres R. (1981), Numerical analysis of variational inequalities, Studies in Mathematics and its Applications Vol. 8, North-Holland, Elsevier, Amsterdam, New York.
Glowinski R. and Le Tallec P. (1989), Augmented lagrangian and operator-splitting methods for nonlinear mechanics. Philadelphia, SIAM.
Green A.E. and Naghdi P.M. (1965), A general theory of an elastic-plastic continuum. Arch. Rat. Mech. Anal., 18, 251–281.
Hiriart-Urruty J.-B. (1985), Generalized differentiability, duality and optimization for problems dealing with differences of convex functions. In: Convexity and duality in optimization, Ed. J. Ponstein, 37–50, Lect. Notes in Economics and Mathematical Systems Vol. 256, Springer.
Hiriart-Urruty J.-B. and Lemarechal C. (1993), Convex analysis and minimization algorithms I, Springer-Verlag, Berlin, Heidelberg.
Horst, R. and Tuy, H. (1990), Global Optimization, Springer Verlag, Berlin — Heidelberg.
Kim S.J. and Oden J.T. (1985), Generalized flow potentials in finite elastoplasticity. II. Examples. Int. J. Engng, Sci., 23 (5), 515–530.
LeTallec P. (1990), Numerical analysis of viscoelastic problems, Masson, Paris and Springer Verlag, Berlin.
Lubliner L. (1990), Plasticity Theory, Macmillan Publ., New York and Collier Macmillan Publ., London.
Malvern L.E. (1969), Introduction to the mechanics of a continuous medium, Prentice-Hall Inc., Englewood Cliffs, N.Jersey.
Martin J.B. and Caddemi S. (1994), Sufficient conditions for convergence of the Newton-Raphson iterative algorithms in incremental elastic-plastic analysis, European Journal of Mechanics, A/Solids, 13 (3), 351–365.
Moreau, J.J. (1968), La notion de sur-potentiel et les liaisons unilaterales en elasto-statique, C.R. Acad. Sc. Paris 267A, 954–957.
Nadai A. (1963), Theory of flow and fracture of solids. Vol. II, McGraw-Hill, New York.
Panagiotopoulos P.D. (1976), Convex analysis and unilateral static problems. Ing. Archiv, 45, 55–68.
Panagiotopoulos P.D. (1985), Inequality problems in mechanics and applications. Convex and nonconvex energy functions, Birkhauser Verlag, Basel — Boston — Stuttgart, (russian translation, MIR Publ., Moscow 1988 ).
Panagiotopoulos P.D. (1988), Nonconvex superpotentials and hemivariational inequalities. Quasidiferentiability in mechanics, in Moreau, J.J. and Panagiotopoulos, P.D. (eds.), Nonsmooth Mechanics and Applications, CISM Lect. Nr. 302, Springer Verlag, Wien — New York.
Panagiotopoulos P.D. and Stavroulakis G.E. (1992), New types of variational principles based on the notion of quasidifferentiability, Acta Mechanica 94, 171–194.
Panagiotopoulos P.D. (1993), Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer Verlag, Berlin — Heidelberg — New York.
Polyakova L.N. (1986), On minimizing the sum of a convex function and a concave function, Mathematical Programming Study 29, 69–73.
Reddy B.D. (1991), Algorithms for the solution of internal variable problems in plasticity. Computer Methods in Applied Mechanics and Engineering, 93, 253–273.
Rockafellar R.T. (1970), Convex Analysis, Princeton University Press, Princeton.
Romano G., Rosati L. and Marotti de Sciarra F. (1993), A variational theory for finite-step elasto-plastic problems, Int. J. Solids Structures, 30 (17), 2317–2334.
Romano G., Rosati L. and Marotti de Sciarra F. (1993), Variational principles for a class of finite step elastoplastic problems with non-linear mixed hardening, Computer Methods in Applied Mechanics and Engineering, 109, 293–314.
Rubinov A.M. and Yagubov A.A. (1986), The space of star-shaped sets and its apli-cations in nonsmooth optimization, Mathematical Programming Study 29, 176–202.
Salencon J. and Tristan-Lopez A. (1980), Analyse de la stailite des talus en sols coherents anisotropes. C.R. Acad. Sci. Paris, 290B, 493–496.
Salencon J. (1983), Calcul a la Rupture et analyse limite, Paris, Presses de l’ecole nationale des Ponts et chaussees.
Simo J.C., Kennedy J.G. and Govindjee S. (1988), Nonsmooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical analysis, Int. J. Num. Meth. Engng., 26, 2161–2185.
Simo J.C. (1993), Recent developments in the numerical analysis of plasticity, in Progress in computational analysis of inelastic structures, E. Stein ed., CISM Courses and Lectures No. 321, Springer Verlag, Wien, New York, 115–174.
Stavroulakis G.E. (1991), Analysis of structures with interfaces. Formulation and study of variational-hemivariational inequality problems, Ph.D. Dissertation, Aristotle University, Thessaloniki.
Stavroulakis G.E. (1993), Convex decomposition for nonconvex energy problems in elastostatics and applications, European Journal of Mechanics A/Solids 12 (1), 1–20.
Stavroulakis G.E. and Panagiotopoulos P.D. (1993), Convex multilevel decomposition algorithms for non-monotone problems, Int. J. Num. Meth. Engng. 36, 1945–1966.
Stavroulakis G.E. and Panagiotopoulos P.D. (1994), A new class of multilevel decomposition algorithms for nonmonotone problems based on the quasidifferentiability concept, Comp. Meth. Appl. Mech. Engng. 117, 289–307.
Stavroulakis G.E., Demyanov V.F. and Polyakova L.N. (1995), Quasidifferentiability in Mechanics, Journal of Global Optimization, 6 (4), 327–345.
Stavroulakis G.E. (1995), Variational problems for nonconvex elastoplasticity based on the quasidifferentiability concept. In: Proc. 4th Greek Nat. Congress on Mechanics, Eds. P.S. Theocaris, E.E. Gdoutos, Vol. I, 527–534.
Stavroulakis G.E. (1995), Quasidifferentiability and star-shaped sets. Application in nonconvex, finite dimensional elastoplasticity. Communications on Applied Nonlinear Analysis, 2 (3), 23–46.
Stavroulakis G.E. and Mistakidis E.S. (1995), Numerical treatment of hemivariational inequalities. Computational Mechanics, 16, 406–416.
Stuart C.A. and Toland J.F. (1980), A variational method for boundary value problems with discontinuous nonlinearities. J. London Math. Soc. (2), 21, 319–328.
Tao P.D. and Souad F.B. (1986), Algorithms for solving a class of nonconvex optimization problems. Method of subgradients. In: FERMAT Days 85: Mathematics for Optimization, Ed. J.-B. Hiriart-Urruty, 249–271.
Temam R. (1983), Problemes mathematiques en plasticite, Gauthier-Villars, Paris.
Toland J.F. (1979), A duality principle for nonconvex optimization and the calculus of variations. Arch. Rat. Mech. Analysis, 71, 41–61.
Tzaferopoulos M.Ap. (1991), Numerical analysis of structures with monotone and nonmonotone, nonsmooth material laws and boundary conditions. Algorithms and applications, Doctoral Dissertation, Aristotle University, Department of Civil Engineering, Thessaloniki.
Tzaferopoulos M.Ap. and Panagiotopoulos P.D. (1993), Delamination of composites as a substationarity problem: Numerical approximation and algorithms. Computer Methods in Applied Mechanics and Engineering, 110 (1–2), 63–86.
Tzaferopoulos M.Ap. and Panagiotopoulos P.D. (1994), A numerical method for a class of hemivariational inequalities. Computational Mechanics, 15, 233–248.
Panagiotopoulos P.D. and Tzaferopoulos M.Ap. (1995), On the numerical treatment of nonconvex energy problems. Multilevel decomposition methods for hemivariational inequalities. Computer Methods in Applied Mechanics and Engineering, 123 (1–4), 81–94.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 1996 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Dem’yanov, V.F., Stavroulakis, G.E., Polyakova, L.N., Panagiotopoulos, P.D. (1996). Nonsmooth Computational Mechanics. In: Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics. Nonconvex Optimization and Its Applications, vol 10. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-4113-4_7
Download citation
DOI: https://doi.org/10.1007/978-1-4615-4113-4_7
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-6844-1
Online ISBN: 978-1-4615-4113-4
eBook Packages: Springer Book Archive