A $\Gamma$-Convergence Result for the Upper Bound Limit Analysis of Plates

Bleyer, Jérémy; Carlier, Guillaume; Duval, Vincent; Mirebeau, Jean-Marie; Peyré, Gabriel

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 215-235.

Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the $\Gamma$-convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials.

Classification : 73E20, 73K10, 73V05
Mots clés : Bounded Hessian functions, Finite element method, Γ-convergence

R.A. Adams, Sobolev Spaces. Academic Press New York (1975). | MR | Zbl

L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr. Oxford University Press (2000). | MR | Zbl

H. Ben Dhia and T. Hadhri, Existence result and discontinuous finite element discretization for a plane stresses hencky problem. Math. Meth. Appl. Sci. 11 (1989) 169–184. | DOI | MR | Zbl

J. Bleyer and P. De Buhan, On the performance of non-conforming finite elements for the upper bound limit analysis of plates. Int. J. Numer. Methods Eng. 94 (2013) 308–330. | DOI | MR | Zbl

M.W. Bræstrup, Yield-line Theory and Limit Analysis of Plates and Slabs. Danmarks Tekniske Højskole, Afdelingen for Bærende Konstruktioner (1971).

A. Braides, Gamma-convergence for Beginners. Oxford University Press, USA (2002). | MR | Zbl

S.C. Brenner and R. Scott, The Mathematical Theory of Finite Element Methods. Texts Appl. Math. Springer (2008). | MR | Zbl

A. Capsoni and L. Corradi, Limit analysis of plates- a finite element formulation. Struct. Eng. Mech. 8 (1999) 325–341. | DOI

P. Ciarlet and P. Destuynder, Justification of the 2-dimensional linear plate model. J. Mécanique 18 (1979) 315–344. | MR | Zbl

G. Dal Maso, An Introduction to $\Gamma$-Convergence. Vol. 8 of Progr. Nonlin. Differ. Eq. Appl. Birkhäuser, Boston, MA (1993). | MR | Zbl

F. Demengel, Problèmes variationnels en plasticité parfaite des plaques. Numer. Funct. Anal. Optim. 6 (1983) 73–119. | DOI | MR | Zbl

F. Demengel, Fonctions à hessien borné. Ann. Inst. Fourier 34 (1984) 155–190. | DOI | Numdam | MR | Zbl

F. Demengel, Compactness theorems for spaces of functions with bounded derivatives and applications to limit analysis problems in plasticity. Archive for Rational Mechanics and Anal. 105 (1989) 123–161. | DOI | MR | Zbl

F. Demengel and R. Temam, Convex function of a measure and its applications. Indiana Univ. Math. J. 33 (1984) 673–709. | DOI | MR | Zbl

E.N. Fox, Limit analysis for plates: the exact solution for a clamped square plate of isotropic homogeneous material obeying the square yield criterion and loaded by uniform pressure. Philos. Trans. Roy. Soc. London. Ser. A, Math. Phys. Sci. 277 (1974) 121–155. | Zbl

T. Hadhri, Fonction convexe d’une mesure. C.R. Acad. Sci. Paris 301 (1985) 687–690. | MR | Zbl

T. Hadhri, Étude dans $HB\times BD$ d’un modèle de plaques élastoplastiques comportant une non-linéarité géométrique. ESAIM: M2AN 19 (1985) 235–283. | DOI | Numdam | MR | Zbl

T. Hadhri, Présentation et analyse mathématique de quelques modèles pour des structures élastoplastiques homogènes ou hétérogènes. Thèse d’état, Université Pierre et Marie Curie, Paris (1986).

T. Hadhri, Prise en compte d’une force linéique de frontière dans un modèle de plaques de Hencky comportant une non-linéarité géométrique. ESAIM: M2AN 22 (1988) 457–468. | DOI | Numdam | MR | Zbl

P.G. Hodge Jr and T. Belytschko, Numerical methods for the limit analysis of plates. J. Appl. Mech. 35 (1968) 796. | DOI | Zbl

A. Jennings, On the identification of yield-line collapse mechanisms. Eng. Struct. 18 (1996) 332–337. | DOI

D. Johnson. Mechanism determination by automated yield-line analysis. Struct. Eng. 72 (1994) 323–323.

C.V. Le, M. Gilbert and H. Askes, Limit analysis of plates using the EFG method and second-order cone programming. Int. J. Numer. Methods Eng. 78 (2009) 1532–1552. | DOI | MR | Zbl

C.V. Le, H. Nguyen-Xuan and H. Nguyen-Dang, Upper and lower bound limit analysis of plates using FEM and second-order cone programming. Comput. Struct. 88 (2010) 65–73. | DOI

Mosek, The Mosek optimization toolbox for MATLAB manual (2008).

J. Munro and A.M.A. Da Fonseca, Yield line method by finite elements and linear programming. Struct. Eng. 56 (1978) 37–44.

C. Ortner and D. Praetorius, On the convergence of adaptive nonconforming finite element methods for a class of convex variational problems. SIAM J. Numer. Anal. 49 (2011) 346–367. | DOI | MR | Zbl

W. Prager, An Introduction to Plasticity. Addison-Wesley series in the engineering sciences: Mechanics and thermodynamics. Addison-Wesley Pub. Co. (1959). | MR | Zbl

T. Rockafellar, Convex Analysis. Vol. 224 of Grundlehren der Mathematischen Wissenschaften. Princeton University Press, 2nd edition (1983). | MR

M.A. Save, C.E. Massonnet and G. de Saxce, Plastic Limit Analysis of Plates, Shells, and Disks, vol. 43. North Holland, 1997. | MR | Zbl

J.J. Telega, Epi-limit on HB and homogenization of heterogeneous plastic plates. Nonlinear Anal. 25 (1995) 499–529. | DOI | MR | Zbl

R. Temam, Mathematical Problems in Plasticity. Gauthier-Villars Paris (1985). | MR

S. Zhou, Y. Liu and S. Chen, Upper bound limit analysis of plates utilizing the C1 natural element method. Comput. Mech. 50 (2012) 543–561. | DOI | MR | Zbl