Keywords: finite deformations, unilateral materials, finite element, total lagrangian, updated lagrangian.

In the paper it is described a constitutive model characteristic for materials that have low tensile resistance like masonry, unreiforced concrete, mortar, sand, soft rock. Commonly, for these materials, their tension resistance is neglected and they are approximated with the model of No Tension Materials (NTM) for which tension stresses are not admissible. Therefore the model takes in account the unilateral behaviour of the material and so it can be extended to light tissues where compression stress states are not admissible.

The model shown here is formulated in the field of finite deformations in order to account ether for large rigid motions, that occur in the displacement field, and for large deformations that could be present in those zones where strain concentration occur, for example prior to the formation of shear bands (ex. sands). The model extends to the finite deformation field the equivalent model formulated in small deformations [1] and has the same characteristic of reversibility because the absence of dissipation in the anelastic process is supposed. From this assumption it follows that the rate of anelastic deformations is orthogonal to the stress tensor. Ideally the deformation process is split in its elastic and its anelastic part introducing an intermediate (eventually fictitious) stress free configuration, so that the deformation gradient is multiplicatively split in an elastic and anelastic part, while the velocity of deformation tensor is additively decomposed [2]. The velocity of deformation measures used in the model satisfy the request of objectivity and are defined for to be dual in power to the stress tensors defined in each configuration. To this end, for example, the velocity of deformation tensor defined in the intermediate configuration is the symmetric part of the covariant form of the gradient of velocity tensor that is obtained pre-multiplying the mixed gradient of velocity by the elastic convective metric of the intermediate configuration [3,4].

In the paper it is reported a full set of constitutive relations that are based on the hypothesis of absence of dissipation in the anelastic part of the process, on the inadmissibility of tension stresses and on a hyper elastic relation between the stresses and the elastic deformations. The latter depends also on the third invariant of the elastic right Cauchy-Green deformation tensor, for accounting for not isochoric deformations. The model has been integrated (in the Gauss points) performing a step linearisation of the rate of deformation (material derivative of the right Cauchy-Green deformation tensor) and of the Lie derivative of the Kirchhoff stress tensor. A Newton type integration scheme has been used and the expression of the tangent stiffness matrix has been explicitly found.

A FE implementation of the model is discussed in the paper, using both a Total Lagrangian (TL) and an Updated Lagrangian (UL) formulation [5,6]. The TL formulation is based on stress and velocity measures defined in the reference fixed configuration, while the UL formulation is based on the same objects defined in the current configuration. For both the formulations are reported explicit expressions for equivalent nodal forces and for geometric and material stiffness matrices. In [6] it isshown that these formulations coincide.

Some applications have shown that the model indeed extends to large deformations the results obtained using the NTM model in small deformations. The numerical formulation appears to be robust, although modifications are needed for improving the convergence properties, that decay when a mechanism arises in the structure.

1

M. Cuomo, G. Ventura, "A complementary energy formulation of no-tension masonry-like solids", Comp. Meth. In Applied Mech. Engnr., 189 (1), 313- 339, 2000. doi:10.1016/S0045-7825(99)00298-4

2

Simo, J.C. and Hughes, T.J.R., Computational Inelasticity, Springer 1998.

3

C. Miehe, E. Stein, A canonical model of multiplicative elasto-plasticity: formulation and aspects of the numerical implementation, European J. of Mechanics A/Solids, 11, 25-44, (1982).

4

Marsden, J.E. and Hughes, T.J.R., Mathematical Foundations of Elasticity, Dover, 1983.

5

Belytschko T., Liu W.K., Moran B., Nonlinear Finite Elements for Continua and Structures; John Wiley & Sons Ltd, Chichester, England., 2000.

6

Cuomo M., Fagone M., A Finite Element Formulation for Unilateral Materials in Finite Deformations; AIMETA, XV Congresso Nazionale, Taormina, 2001.

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