M. Cuomo

A new thermodynamically consistent continuum model for hardening plasticity coupled with damage

A phenomenological model for hardening-softening elasto-plasticity coupled with damage is presented. Specific kinematic internal variables are used to describe the mechanical state of the system. These, in the hypothesis of infinitesimal changes of configuration, are partitioned in the sum of a reversible and an irreversible part. The constitutive equations, developed in the framework of the Generalised Standard Material Model, are derived for reversible processes from an internal energy functional, postulated as the sum of the deformation energy and of the hardening energy both coupled with damage, while for irreversible phenomena from a dissipation functional. Performing duality transformations, the conjugated potentials of the complementary elastic energy and of the complementary dissipation are obtained. From the latter a generalised elastic domain in the extended space of stresses and thermodynamic forces is derived. The model, which is completely formulated in the space of actual stresses, is compared with other formulations based on the concept of effective stresses in the case of isotropic damage. It is observed that such models are consistent only for particular choices of the damage coupling. Finally, the predictions of the proposed model for some simple processes are analysed.

An enriched finite element for crack opening and rebar slip in reinforced concrete members

Object of the paper is the simulation of reinforced concrete bars behaviour, accounting for crack opening and concrete-rebar slippage. A macro beam element with a single uniform reinforcement is studied in details in the uniaxial case. Distinct constitutive hypotheses are formulated for the materials. The CEB-FIP Model Code 90 rules the behaviour of the materials interface that is assumed to be fully dissipative. Steel is supposed to behave elastoplastically with hardening. Crack opening in the concrete matrix is introduced by means of a strong discontinuity approach (SDA). All the relevant equations of the problem are variationally derived from a mixed energy functional. Two enhancements of the enriched kinematics, based on polynomial or exponential shape functions, respectively, are compared with the usual SDA enhancement. As an alternative approach, high-order interpolation of the displacement field based on B-splines, both for steel and concrete, is proposed. These functions appear to be adequate in reproducing rapidly varying fields, like the stress gradients occurring in the shear lag problem near the boundaries or where slips and/or cracks occur. Their use allow to use few macro-element instead of the very dense meshing required in those areas by the traditional FE interpolations.

Stress rate formulation for elastoplastic models with internal variables based on augmented Lagrangian regularisation

The constitutive laws of elasto-plasticity with internal variables are described through the definition of suitable dual potentials, which include various hardening models. A family of variational principles for inelastic problems is obtained using convex analysis tools. The structural problem is analysed using the complementary energy (Prager-Hodge) functional. The functional is regularised introducing an Augmented Lagrangian Regularisation for the indicator function of the elastic domain so that a smooth optimisation problem is obtained. In the numerical solution the discretised problem is reformulated in a finite step form using a fully implicit integration scheme and the functional is redefined in the space of the self-equilibrated nodal stresses, after enforcing satisfaction of the equilibrium equations in a weak form. Numerical tests have shown good performance on the part of the algorithm, which approaches the converged solution for a considerably smaller number of elements as compared with other algorithms. The method is equally available for perfect or hardening plasticity.

A complementary energy formulation of no tension masonry–like solids

In the present work a complementary energy formulation for the solution of the static equilibrium problem of no-tension solids is presented. The main feature of the work is systematic use of the augmented Lagrangian approach for the regularisation of the nonsmooth and nondifferentiable anelastic deformation potential. The proposed formulation is applied to derive a finite element formulation using the subspace of the self-equilibrated stresses as primal variables. The code implements the augmented Lagrangian algorithm for the determination of the inelastic deformations. Some numerical examples are included to illustrate the performance of the proposed approach.

Complementary energy approach to contact problems based on consistent augmented Lagrangian formulation

A stress formulation for frictionless contact problems between deformable bodies is proposed. Linear compatibility equations are assumed, while the constitutive relations are supposed nonlinear, yet Reversible, i.e., ruled by a convex strain potential. The relevant contact rules are formulated in terms of concave conjugated potentials, whose superdifferentials yield the constitutive laws for the unilateral contact interface. Generalization of the mixed Hellinger-Reissner functional, and of the functionals of total potential energy and of complementary energy are formulated. The last one is used for numerical developments. The functional is regularized by means of an augmented Lagrangian function. Solution to the saddle point problem arising from the regularization is obtained in the subspace of self-equilibrated stresses only, using equilibrium equations for condensing out the complementary stresses. In the paper, some examples of more complex unilateral contact relations are also presented.

Meso-scale simulation of concrete multiaxial behaviour

Concrete is a heterogeneous structural material whose constitutive behaviour is strictly depending on the mechanical properties of the aggregates and the mortar. Its behaviour is often macroscopically characterised by assigning homogenised mechanical properties. A number of methods are devoted to the prediction of the mechanical properties of the composite material by means of meso-scale analysis. The paper concerns a new meso-scale model of cementitious materials. The numerical description of the meso-scale structure is attained using a random method that allocates at each Gauss point of the finite element discretisation of the Representative Volume Element a specific phase of the mixture: aggregate, mortar and void. Each phase is characterised by a specific constitutive model. The method is tested with numerical simulations of cyclic uniaxial compression tests and of multiaxial compression tests, the latter leading to the generation of a biaxial strength domain, for different values of the confinement pressure, that is found to be in good agreement with concrete experimental domains.

An explicit formulation of the Green's operator for general one-dimensional structures

A general procedure for the determination of the explicit expression of the Greens function of an ordinary differential operator is described in the paper, that requires only the knowledge of the null spaces of the operator and its adjoint. Although the main results concerning the structure of the Greens function are known in the literature, the proposed approach is suitable for symbolic mathematics systems and has been addressed in particular for frame structures including various beam models. The general expression of all the influence functions is given and some analytical results are presented.

Comparison of two forms of strain decomposition in an elastic-plastic damaging model for concrete

In the paper a new modified form of the constitutive equations for concrete in the general framework for elastic-plastic damaging models proposed by the authors in (Contrafatto and Cuomo 2006 J. Plast. 22 2273–300) is presented. The modification concerns the definition of the internal energy potential. In the original paper the expression of the elastic energy potential depends on the sign of the trace of the elastic strain tensor. In the new formulation a decomposition of the strain tensor in its positive and negative component by means of a basis-free representation in terms of eigenprojections is used. As a consequence a different evolution of damage, affecting in a different way the tensile and compressive component of the strain tensor, is obtained. The two models belong to the class of continuum scalar damage models and are developed within the context of simple materials. The new model, first formulated in an arbitrary cartesian coordinate system, is presented in a principal axes representation, in order to reduce the algebraic complexity of the expressions and to make easier the analysis of simple load processes, while the treatment of the constitutive equations for the general case will be an object of future developments. A comparison between the predictions of the two models is performed by means of the analysis of some loading processes. The new formulation is able to overcome some drawbacks of the original model, especially in the tensile regime. In contrast, in the compressive regime, for which already the original formulation yielded satisfactory results, no change was detected.

Continuum model of microstructure induced softening for strain gradient materials

Gradient plasticity has been introduced to predict localization bands induced by softening or damage, and has been extended to micromorphic models used for materials with microstructure and scale effects. Motivated by the consideration that in complex materials there may exist different mechanisms for the evolution of anelastic deformation at the small and at the large scale, and that they may interact, in the paper a model of second gradient material with two different evolution laws for the macroscopic and for the microscopic plastic strain is proposed, the latter related to the presence of second gradient deformation. The model is intended to be used for metamaterials, and its relevant properties can be obtained from an homogenization analysis at the microlevel.

An Augmented Lagrangian Formulation for the Analysis of No-Tension Structures with Unilateral Supports

Object of the paper is a class of structural problems that will be abstractly defined by the following three sets of equations: the compatibility equation C(u) = e, where the operator C:u —>D is assumed linear (and so is its dual C’ that relates the internal stress a to the external actions f ∈ u’); the constitutive equation f (e) = σ, assumed to be a monotone lower semi-continuos map from D to D’. Therefore the map is obtained from a generalized potential, that turns out to be lower semi-continuos and convex. This means that the material behaviour is reversible; the external force field, supposed to derive from a functional that is assumed to be lower semi-continuos and convex. The functional defines also the (non-linear) static and kinematic boundary conditions (eventually unilateral).