ABSTRACT
In plane stress analysis the non zero components of stress tensor, and the respective components of strain, are related by the ortothropic stiffness matrix D. Its rank is equal to 3 and D is defined by the Young modulus in the vertical and horizontal direction (E^ and E2), the Poisson ratio v\i and the shear modulus G\I. The evaluation of the stress follows a Predictor-Corrector scheme:
Prediction:
Correction:
in which er° is the initial stress, cr* represents the predicted stress, o is the corrected stress, AE is the total strain increment and A£J the variation of its inelastic part (unknown). Inelastic strains occur only when eq. (1) is violated by the predicted stress. The eq. (1) can be rewritten for the fc-th plane as:
where n* is the normal to the plane and r* the distance between the plane and the origin at the beginning of the analysis step. The predicted stress er* can stay outside of one or more planes. If more than one plane has been violated by the predicted stress <r* (saying M the number of those planes), the inelastic strain variation needs to be evaluated by a sum over the contribution coming from each plane. Introducing an associated plasticity criterion with Ae; normal to the yield surface,
3.1 Damage Evolution Law
An isotropic damage law is here adopted in order to model the material decreasing stiffness. In [8] a brittle material damage is modeled considering a part of the total inelastic deformation as irreversible. Here one starts from a plastic material and a ratio a (with 0 < a < 1) is introduced to obtain the inelastic strain tensor EJ (considered as reversible) and the plastic strain tensor sp (non-reversible), through coaxial laws:
At the end of the step, after the evaluation of the corrected stress <r, one has the following expression for the total strain:
in which CQ = Djj"1 is the compliance matrix at the beginning of the solution step. The increment of material compliance during the step is then related to the reversible part of the inelastic strain variation as follows:
Then from eq. (12) one obtains:
where C is the final compliance matrix. Introducing two unit vectors n and m, coaxial respectively to stress and damage strain variations, as:
and a scalar quantity p:
the expression of the variation AC of the compliance matrix is:
4. NUMERICAL ANALYSES
In order to check the reliability of the model proposed, some simple analyses have been performed. Imposing a cyclic strain history along direction 2 on a single finite element, the plot of fig. 3 is obtained. It is possible to note that the model describe correctly the different compression and tensile strength of the material. In fig. 4 the response due to increasing level of aii strain in two direction is showed. In fig. 4b the plane 9 (shear stress limit) has been made inactive: can be seen how the maximum shear stress increases, towards high incompatible values. A first conclusion is that the
Figure 6: Cyclic shear test, (a) plane 1 failure (tension), (b) plane 6 failure (compression), (c) plane 9 failure (shear)
5. Papa, E. and Nappi, A. 'Modellazione numerica di strutture murarie soggette a carichi ciclici', in La meccanica delle murature tra teoria e progetto, L. Gambarotta ed., Universita di Messina, Italy, 1996, pp. 441-450 (in Italian).
