A Note on Limit Analysis

Quarterly of Applied Mathematics, 2010

This note presents a limit analysis for normal materials based on energy minimization. The class of normal materials includes some of those used to model masonry structures, namely, no-tension materials and materials with bounded compressive strength; it also includes the Hencky plastic materials. Considering loads L(λ) that depend affinely on the loading multiplier λ ∈ R, we examine the infimum I 0 (λ) of the potential energy I(u, λ) over the set of all admissible displacements u. Since I 0 (λ) is a concave function of λ, the set Λ of all λ with I 0 (λ) > −∞ is an interval. Each finite endpoint λ c ∈ R of Λ is called a collapse multiplier, and we interpret the loads corresponding to λ c as the loads at which the collapse of the structure occurs. We show that the standard definition of collapse based on the collapse mechanism does not capture all situations: the collapse mechanism is sufficient but not necessary for the collapse. We then examine the validity of the static and kinematic theorems of limit analysis under the present definition. We show that the static theorem holds unconditionally while the kinematic theorem holds for Hencky plastic materials and materials with bounded compressive strength. For no-tension materials it generally does not hold; a weaker version is given for this class of materials.

Mechanics Research Communications, 2010

The limit analysis approach as a direct method limit loads assessment of plastic deformable bodies is followed up to deal with plastic collapse of non-linear kinematic strain hardening materials. For this goal, the kinematic and static approaches of modern limit analysis are adopted. The non-linear kinematic hardening law is a non-associative plastic flow rule, but, it can be described by the bipotential concept. Based on this, an extension of the limit analysis approach is proposed. Limit loads, back stresses and other variables are assessed at each step by the sequential limit analysis method. Large plastic deformation could be taken into consideration by updating geometry after each sequence.

Journal of Mechanics of Materials and Structures, 2012

The kinematic and static problems of limit analysis of no-tension bodies are formulated. The kinematic problem involves the infimum of kinematically admissible multipliers, and the static problem the supremum of statically admissible multipliers. The central question of the paper is under which conditions these two numbers coincide. This involves choices of function spaces for the competitor displacements and competitor stresses. A whole ordered scale of these spaces is presented. These problems are formulated as convex variational problems considered by Ekeland and Témam. The static problem is unconditionally shown to be the dual problem (in the sense of the mentioned reference) of the kinematic problem. A necessary and sufficient condition, the normality, guarantees that the kinematic and static problems give the same result. The normality is not always satisfied, as examples show (one of which is presented here). The qualification hypothesis of Ekeland and Témam, sufficient for the equality of the static and kinematic problems, is never satisfied in the spaces of admissible displacements of bounded deformation or of functions integrable together with the gradient in the power p, 1 ≤ p < ∞. In the cases of lipschitzian displacements and of smooth displacements, the qualification hypothesis is equivalent to simple conditions that can be satisfied in the case of the pure traction problem. However, it is shown that then the space of admissible stresses must be enlarged to contain stress fields represented by finitely or countably additive tensor-valued measures.

Journal of Applied Mathematics and Mechanics, 1985

Archives of Mechanics

The paper discusses a methodology for the evaluation of shakedown and ratchet limits for an elastic perfectly plastic solid subjected to mechanical and thermal cy-cles of loading. The steady cyclic state is characterised by a minimum theorem that contains the classical shakedown theorems as a special case. For a prescribed class of kinematically admissible strain rate histories, the minimum of the functional is found by a programming method, the Linear Matching Method, which converges to the least upper bound. Three examples are given for a finite element implementa-tion, rolling contact on a half-space, the behaviour of a complex heat exchanger and the behaviour of a regular particulate metal matrix composite subjected to variable temperature.

Mathematical Models and Methods in Applied Sciences, 2005

Slender beams with small cracks described by Γ limits: a description of an elastic-perfectly plastic beam or rod is obtained as a variational limit of 2D or 3D bodies with damage at small scale satisfying the Kirchhoff kinematic restriction on the deformations.

International Journal of Plasticity, 2001

The paper considers perfectly plastic materials with a yield condition of the form È ð Þ ¼ F ij ij þ F ijk' ij k' 4 1 corresponding to a second order truncation of the tensor polynomial expression proposed by Tsai and Wu for failure criteria. Such an expression is often employed for materials exhibiting particular forms of anisotropic failure properties, including orthotropic ones, and accounts for non-symmetric strengths. The limit analysis problem is considered next. The formulation based on the kinematic theorem, reducing to the search of the constrained minimum of a convex functional, was successfully employed in the isotropic case for numerical solutions and can be extended to the present context without modifications, provided that the expression for the dissipation power as an explicit function of strain rates is available. For the material considered, this expression is established in this paper. The result is specialized to plane stress orthotropy and an example is worked out. Although extremely simple, it permits the assessment of the influence of the ratio between tensile and compressive strength and of the inclination of the orthotropy axes with respect to loading directions.

Archive for Rational Mechanics and Analysis, 1986

Mechanics Research Communications, 1995