Luciano Rosati

MITC finite elements for laminated composite plates

Within the framework of the first-order shear deformation theory, 4- and 9-node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking-free, they do not have zero-energy modes and provide accurate in-plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out-of-plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright

A variational theory for finite-step elasto-plastic problems

Abstract An extended version ofgeneralized standard elasto-plastic material is considered in the framework of an internal variable theory of associated plasticity. According to a backward difference scheme for time integration of the flow rule, a finite-step structural problem is formulated in a geometrically linear range. Convex analysis and a brand new potential theory for monotone multivalued operators are shown to provide the natural mathematical setting for the derivation of the related variational formulation. A general stationarity principle is obtained and then specialized to obtain a minimum principle in terms of displacements, plastic strains and internal variables. A critical comparison with an analogous minimum principle recently proposed in literature is performed, showing the inadequacy of classical procedures in deriving non-smooth variational formulations.

An internal variable theory of inelastic behaviour derived from the uniaxial rigid-perfectly plastic law

In the general framework provided by the internal variable theories of associated inelastic behaviour the formulation of constitutive relations is addressed in this paper. Attention is focused on the basic properties of the evolution relation involving rates of internal variables and dual thermodynamic forces. It is shown that a suitable generalization of the uniaxial rigid-perfectly plastic law can be performed by introducing the definition of step-shaped constitutive maps. This definition allows us to derive a general theory of associated inelastic behaviour with its characteristic properties: convexity of the elastic locus, normality rule, existence of a sublinear dissipation functional and of a canonical yield functional. Finally the formulation of the constitutive relation in terms of yield functionals and related inelastic multipliers is discussed. The analysis is performed on the basis of a chain rule of subdifferential calculus, recently contributed by the authors, which provides an effective tool to develop the theory of Kuhn-Tucker vectors in optimization problems.

Solution procedures for J3 plasticity and viscoplasticity

Abstract A solution strategy for plasticity and viscoplasticity models with isotropic yield surfaces depending upon all the principal invariants of the stress tensor is presented. Basically, it requires the inversion of a fourth-order positive definite tensor G both for the solution of the constitutive problem and for the evaluation of the consistent tangent operator. It is proved that the assumption of isotropic elastic behaviour and the isotropy of the yield criterion entail an explicit representation formula for G −1 as linear combination of dyadic and square tensor products. Further, an analogous representation formula for the consistent tangent operator is provided. By exploiting basic composition rules between dyadic and square tensor products along with Rivlins identities for tensor polynomials, all tensor operations required to compute the coefficients of the adopted representation formula for G −1 are carried out in intrinsic form. It is thus shown that the relevant computational burden essentially amounts to solving a linear system of order three. The performances of the proposed approach are illustrated by means of some numerical examples referred to the Argyris failure criterion.

A general approach to the evaluation of consistent tangent operators for rate-independent elastoplasticity

Abstract We present a general approach to the derivation of the explicit expression of tangent operators, consistent with the finite-step integration schemes of the flow rule, for rate-independent elastoplastic models with mixed hardening. By exploiting the model of generalized standard material, we show that the algorithmic tangent operator can be obtained by inverting a suitable positive definite matrix. Use of the Sherman-Morrison-Woodbury formula allows us to reduce to a half the burden associated with such inversion and to derive an expression of the consistent tangent operator prone to computer implementation. The application of the proposed approach to the von Mises yield criterion with linear kinematic and isotropic hardening shows that the closed-form expression of the operator can be obtained by a straightforward procedure.

Variational formulations of non-linear and non-smooth structural problems

Abstract The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibility, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as Subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space of all the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibility, a family of non-smooth functionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem.

Derivatives and Rates of the Stretch and Rotation Tensors

General expressions for the derivatives and rates of the stretch and rotation tensors with respect to the deformation gradient are derived. They are both specialized to some of the formulas already available in the literature and used to derive some new ones, in three and two dimensions. Essential ingredients of the treatment are basic elements of differential calculus for tensor valued functions of tensors and recently derived results on the solution of the tensor equation A X + XA= H in the unknown X.

A tangent–secant approach to rate-independent elastoplasticity: formulations and computational issues

A general and robust solution procedure for nonlinear finite element equations in small strain elastoplastic structural problems is presented. Its peculiar feature lies in the choice of the most suitable constitutive operator to be adopted at each iteration of a generic load step in order to ensure the utmost stability and convergence rate. Namely, the consistent tangent operator is replaced by a secant one, or vice versa, whether the adopted norm of the residual does not, or does, conveniently decrease at the current iteration. The secant operator is defined as to recover the finite-step increment of the plastically admissible stress from the total, not iterative, strain increment. The original formulation of the solution procedure, consisting of alternate tangent and secant iterations, is then extended to achieve an effective coupling with line searches. The excellent performances of the two procedures are illustrated by numerical examples carried out for typical benchmark problems in plane strain and three-dimensional cases.

A displacement-like finite element model for J2 elastoplasticity: Variational formulation and finite-step solution

The displacement-like finite element formulation for finite-step J2 elastoplasticity is revisited in this paper. The classical computational strategy, according to which, plastic loading is tested at the Gauss points of each element and an independent return mapping algorithm is performed for given incremental displacements, is consistently derived from a suitably discretized version of a min-max variational principle. The sequence of solution phases to be performed within each load step adopting a full Newtons method is illustrated in detail and the importance of a correct update of the plastic strains is emphasized. It is further shown that, in order to increase the rate of convergence and the stability properties of the Newtons method, the consistent elastoplastic tangent operator must be exploited even at the first iteration of each load step subsequent to the first yielding of the structural model. This is in contrast with the traditional implementation according to which the elastic operator is used at the first iteration of each load step. The effectiveness of the present approach is shown by a set of numerical examples referred to plane strain problems.

Automatic analysis of multicell thin-walled sections

An automatic procedure is outlined for the determination of the shear centre and the evaluation of the overall state of stress in multicell thin-walled sections subject to axial force, bending moment, shearing force and torque. Graph theory is shown to be the rationale to establish a topological model of the section which is preliminary to a computer implementation of the shear stress analysis. Specifically, we exploit the main features of the Depth-First-Search graph algorithm in order to automatically determine a number of independent circuits equal to the degree of connection m of the section. The algorithm also localize the m slits which make the section open, a preliminary step for the analysis of multicell sections subject to a shearing force. Further, the evaluation of the first elastic area moment at any point of the open section is addressed by means of the Open-Section-Cut algorithm elaborated in this paper. The outlined procedure entails a considerable simplification of the analysis, since the geometrical data which need to be assigned are only the strictly necessary ones, namely the coordinates of the vertices, the branches connecting them and their thickness. A numerical example, carried out for a ships hull by means of a computer program written in Mathematica, shows the effectiveness of the proposed approach.