Due to their high non-linearity, boundary problems involving materials with no resistance to tension, known as no-tension materials, are quite exacting to solve.

Moreover, the no-tension hypothesis of the material's response requires that the solution be characterized by a stress tensor that is negative semidefinite in every point of the continuum. Such a solution is guaranteed by satisfying precise conditions expressed via the two stress invariants T1 and T2: the trace has to be minor or equal to zero, the determinant major or equal to zero.

The solution defines three different regions within R, designated as R1, R2 and R3. In R1, where the trace is negative and the determinant is positive, the material's behavior is bilateral linear elastic; in R3, where the trace and the determinant are nil, the material is completely cracked and therefore offers no resistance at all; the situation being as if it did not exist at all; in R3, where the trace is negative and the determinant is nil, the stress tensor degenerates and the region is characterized throughout by a negative and a nil principal stresses [1].

In general, the solution to a no-tension problem can only be determined through the use of numerical calculation techniques, given that explicit solutions can be obtained only for very simple cases.

A good deal of literature has been devoted to solution methods based on the use of finite elements to deal with the displacements [2,3]. Although built on extremely refined mathematical models, such methods suffer from an problem inherent in the chosen approach, which leads to instability in terms of convergence. In fact, for particular load conditions, even if the static solution is unique, the kinematic one is indeterminate.

The numerical approach proposed in the following, on the other hand, is based on a simple matrix formulation of the elastic problem regarding a continuum under plane stress discretized into triangular constant-stress finite elements without bending stiffness; it consists in an iterative procedure applied only to the static parameters, in conformity with the equilibrium conditions.

The matrix formulation of the problem facilitates adding the necessary conditions regarding the admissibility of the stress state, in conformity with equilibrium of the structure.

Although a number of solution methods have been proposed in the literature, the approach presented herein has the advantage of circumventing the computational difficulties due to such non-linearity by resolving a sequence of linear problems, the solution to each of which is guaranteed by efficient use of the projection operator. In other words, what is proposed is an iterative procedure that, while satisfying the equilibrium conditions, makes appropriate corrections to the linear-elastic solution in order to make it converge on a final result congruent with the defined conditions of the material's lack of tensile resistance.

The two-dimensional problems are studied under the hypothesis of a plane stress state, and the method applied to continua discretized via three-node finite elements under constant stress. The effectiveness of the method is then demonstrated by resolving some simple example problems.

1

Di Pasquale S., "New trends in the analysis of masonry structures", Meccanica, 27, 173-184, 1992. doi:10.1007/BF00430043

2

Lucchesi, M., Padovani, C., Pagni, A., "A numerical methods for solving equilibrium problems of masonry-like solids", Meccanica, 29, 175-193, 1994. doi:10.1007/BF01007500

3

Romano, G., Sacco, E., "No tension materials: constitutive equations and structural analysis", Atti dell'Istituto di Scienza delle Costruzioni, Internal Report, 350, 1984, (in italian).

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £125 +P&P)