P. Fuschi

Closed form solution for a nonlocal elastic bar in tension

A simple mechanical one-dimensional problem in the context of nonlocal (integral) elasticity is solved analytically. The nonlocal elastic material behaviour is described by the ‘‘Eringen model’’ whose nonlocality features all reside in the constitutive relation. This relation, of integral type, contains an attenuation function (usually assumed symmetric) aimed to capture the diffusion process of the nonlocality effects; it also exhibits a convolution format. The governing equation is a Fredholm integral equation of second kind whose analytical treatment, even for the usual choice of a symmetric kernel, is not easy to develop. In the present paper, assuming a specific shape for the attenuation function, a closed form solution in terms of strains is alternatively obtained by solving a Volterra integral equation of second kind. The latter can be easily solved with standard techniques, at least for the adopted kernel, taking also advantage from the symmetry of the solution. Such a closed form solution is an essential result to validate the effectiveness of numerical procedures aimed to solve more complex mechanical problems in the context of nonlocal elasticity. 2002 Elsevier Science Ltd. All rights reserved.

A Thermodynamic Approach to Nonlocal Plasticity and Related Variational Principles

Elastic-plastic rate-independent materials with isotropic hardening/softening of nonlocal nature are considered in the context of small displacements and strains. A suitable thermodynamic framework is envisaged as a basis of a nonlocal associative plasticity theory in which the plastic yielding laws comply with a (nonlocal) maximum intrinsic dissipation theorem. Additionally, the rate response problem for a (continuous) set of (macroscopic) material particles, subjected to a given total strain rate field, is discussed and shown to be characterized by a minimum principle in terms of plastic coefficient. This coefficient and the relevant continuum tangent stiffness matrix are shown to admit, in the region of active plastic yielding, some specific series representations. Finally, the structural rate response problem for assigned load rates is studied in relation to the solution uniqueness, and two variational principles are provided for this boundary value problem.

Theorems of restricted dynamic shakedown

Abstract Dynamic shakedown for a rate-independent material with internal variables is addressed in the hypothesis that the load values are restricted to those of a specified load history of finite or even infinite duration, thus ruling out the possibility—typical of classical shakedown theory—of indefinite load repetitions. Instead of the usual approach to dynamic shakedown, based on the bounded plastic work criterion, another approach is adopted here, based on the adaptation time criterion. Static, kinematic and mixed-form theorems are presented, which characterize the minimum adaptation time (MAT), a feature of the structure-load system, but which are also able to assess whether plastic work is finite or not in the case of infinite duration load histories, where they then prove to be equivalent to known shakedown theorems.

Studies on generalized midpoint integration in rate-independent plasticity with reference to plane stress J2-flow theory

Abstract Issues related to use of the generalized midpoint integration in rate-independent plasticity are investigated restricting attention to plane stress J 2 -flow theory. Accuracy analysis is performed by developing the iso-error maps for a representative range of loading conditions and α-values utilized in the midpoint integration rule. Consequences of the discrete enforcement of the consistency condition on the numerical resolution of the boundary value problem are discussed. A numerical example is presented to test the influence of the choice of α-value on a global accuracy and effectiveness of the solution scheme.

Structural shakedown for elastic-plastic materials with hardening saturation surface

Abstract An elastic–plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is assumed as a basis to formulate a statical Melan-type shakedown theorem. Grounding on the optimality conditions relative to the shakedown load multiplier problem for a structure subjected to cyclic loads, the impending inadaptation collapse mechanism at the shakedown limit state is analyzed and discussed. It is shown that the adopted model is able to catch ratchetting collapse mode at a structural level. Numerical results for a simple structure are finally reported.

On numerical integration of the five-parameter model for concrete

Issues related to numerical integration of the Willam-Warnke version of the five-parameter model for concrete [K.J. Willam and E.P. Warnke, Constitutive model for triaxial behaviour of concrete. IABSE Seminar on Concrete Structures Subjected to Triaxial Stresses (1974)] are discussed. The constitutive equations are numerically integrated by an algorithm based on the operator split methodology. The cutting plane method and the full Newton-Raphson method are applied in the solution of the nonlinear set of equations in the plastic corrector stage and comparative analysis is performed. Attention is restricted to the perfectly plastic behaviour of compressed concrete under plane stress conditions. Accuracy analysis is performed via iso-error maps for a representative range of loading conditions. Numerical examples are presented to test the effectiveness of the solution scheme.

Internal-variable constitutive model for rate-independent plasticity with hardening saturation surface

SummaryAn elastic-plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is presented. The existence of such a saturation surface in the internal variables space — a consequence of the boundedness of the energy that can be stored in the materials internal micro-structure — encompasses, in case of general kinematic/isotropic hardening, a one-parameter family of envelope surfaces in the stress space, which in turn is enveloped by a limit surface. In contrast to a multi-surface model, noad hoc rules are required to avoid the intersection between the yield and bounding/envelope surface. The flow laws of the proposed model are studied in case of associative plasticity with the aid of the maximum intrinsic dissipation theorem. It is shown that the material behaves like a standard one as long as its hardening state either is not saturated, or undergoes a desaturation from a saturated hardening state, whereas, for saturated hardening states not followed by desaturation, it conforms to a combined yielding law in which the static internal variable rates obey a nonlinear hardening rule similar to that of analogous models of the literature. Additionally, the material is shown to behave as a perfectly plastic material for a class of (critical) saturated hardening states for which the stress state is on the limit surface. For nonassociative material models, it is shown that, under a special choice of the plastic and saturation potentials and through a suitable parameter identification, the well-known Chaboche model is reproduced. A few numerical examples are presented to illustrate the associative material response under monotonic and cyclic loadings.

Symmetric Structures Made of a Nonlocal Elastic Material

The paper focuses on the analysis of symmetric structures in the context of nonlocal integral elasticity of Eringen-type. In particular, it highlights how the standard (local-type) concept of structural symmetry cannot be applied in a straightforward manner, but it has to be redefined involving an enlarged symmetric model of the structure. Such enlarged model is indeed able to take into account the nonlocal effects exerted on the (standard) symmetric portion of the structure chosen for the analysis by the portion neglected. The appropriate boundary conditions that have to be applied to the enlarged symmetric model for guaranteeing the exact matching between the mirrored symmetric solution and the complete one, are also discussed. Two numerical examples are solved by means of a nonlocal version of the finite element method and the results obtained are critically discussed.

LIMIT STATE EVALUATION OF STEEL-REINFORCED CONCRETE ELEMENTS BY VON MISES AND MENÉTREY–WILLAM-TYPE YIELD CRITERIA

An advanced version of a recently developed numerical limit analysis procedure for the prediction of peak loads and failure modes of steel-reinforced concrete elements is proposed. The modified procedure allows to take into account possible yielding of reinforcement thus capturing the actual behavior at the collapse of both steel and concrete. This implies a finite element (FE) modeling of the reinforced concrete (RC) elements in which concrete is governed by a Menetrey–Willam-type yield criterion, with cap in compression, while steel bars are governed by a von Mises yield criterion. The peak load of a wide range of RC structures whose behavior at ultimate state is dominated either by the concrete crushing or by the steel bars yielding is then predicted with a very good accuracy.

Mechanical testing and numerical modelling of pull-wound carbon-epoxy spinnaker poles

The paper deals with experimental testing and numerical simulation of the mechanical behaviour of multi-layer cylindrical coupons, of two different diameters, made in carbon-epoxy composite. The aim of the study is to provide a simple and effective numerical model that can be used as a design tool for structural elements having analogous geometrical and manufacturing characteristics. The numerical analysis, performed in the elastic regime with a standard finite element (FE) code, was strongly correlated with the laboratory determination of fibre-volume fractions and of some elastic parameters of the material system. Other parameters, like the shear modulus values G, were in fact appropriately chosen to calibrate the numerical FE model which was forced to reproduce the results of the initial specific ring stiffness tests carried out on pole coupons with external diameter equal to 80 mm. The model was, then, validated by comparison between the numerical results and the experimental ones obtained for coupons of 60 mm diameter.