In the hybrid-displacement model presented here, both the displacements in the domain of each element and the field of applied stresses along the kinematic boundary are directly and independently approximated. Meshing is not as complicated as it is in conventional displacement formulations since accurate solutions can be obtained by using macro-element meshes combined with effective p-refinement procedures.

Although this formulation leads to a large number of degrees of freedom, a high level of sparsity is achieved for the global stiffness matrix in the elastic regime using a complete series of orthonormal Legendre polynomials as approximation functions. Furthermore, analytical expressions are known for integrals involving these functions, which makes computations even more efficient while the stiffness matrix is computed based on physical linearity.

To model the physically non-linear concrete behaviour, two scalar isotropic continuum damage models, following the same kind of regularization technique, are chosen due to the simplicity of their implementation: the Comi and Perego's and the Mazars damage models [1,2]. Some assumptions had to be made during the course of the work, in order to simplify the problem, focusing on what is important and without compromising the proposed objectives. First of all, the hypothesis of geometrical linearity is supposed to remain valid. Temperature is not an intervenient factor; hence energy dissipation has an origin only in the mechanical mechanisms. The load is supposed to be monotonic and applied at constant speed in such a way that the analysis remains static and avoiding hysteretic phenomena characteristic of cyclic loads. The material is considered to be homogeneous. The damage models are isotropic, which means that all the entries of the elementary stiffness matrix are multiplied by the same factor as damage evolves. The constitutive model may be considered elastic in the sense that permanent strains are not considered, but just a degradation of the elastic properties, which allows the use of a secant law for the stiffness relation always regarding the origin of a stress-strain coordinate system.

A set of benchmark tests is presented to validate and illustrate the use of the hybrid-displacement finite element model. The accuracy and efficiency of the numerical model being discussed is established by comparing the numerical results obtained with those provided by other classical finite element formulations.

1

C.M. Silva, L.M.S.S. Castro, "Nonlocal damage theory in hybrid-displacement formulations", International Journal of Solids and Structures, 46, 3516-3534, 2009. doi:10.1016/j.ijsolstr.2009.05.004

2

C.M. Silva, L.M.S.S. Castro, "Continuum damage models with non-conventional finite element formulations", International Journal of Non-Linear Mechanics, 45(2), 83-99, 2010. doi:10.1016/j.ijnonlinmec.2009.09.005

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