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We propose paradigmatic examples to show how material damage phenomena can be efficiently described as a solid-solid phase transition. Starting from the pioneering work of J.L. Ericksen (J. Elast. 5(3):191–201, 1975) and the extensions of R.L. Fosdick and other authors to three-dimensional non linear elasticity, we describe the insurgence of damage as a hard → soft transition between two material states (damage and undamaged) characterized by two different energy wells. We consider the two separate constitutive assumptions of a simple Neo-Hookean type damageable material and a more complex microstructure inspired damageable Gent type material with variable limit threshold of the first invariant. In both cases we study two different deformation shear classes, one homogeneous and the other one inhomogeneous and obtain fully analytic description of the system damage response under cyclic loading. The considered constitutive assumptions and deformation classes are aimed at attaining fully analytic descriptions. On the other hand, we remark that the proposed, Griffith type, variational approach of damage, based on two different energy density functions for the damaged and undamaged material phases, and a resulting non (rank-one) convex energy, can be extended to systems with more complex energy functions, possibly with a larger number of wells representing an increasing degree of damage.

In this work it was generally discussed the issue of the seismic behavior of buildings with resistant walls in reinforced concrete, in addition to the issue of the safety assessment of existing buildings , more and more major current topic in recent years. Specifically, are then shown the results obtained from the evaluation of seismic safety carried out on the headquarters of the Emilia-Romagna region, which is made with a seismic-resistant cylindrical walls in circular and semi-circular section. To perform this assessment of safety has also discussed the problem of the calculation of the shear strength in the walls with circular and semi-circular section making the point about the state of the art scientific research and proposing a simplified method of calculation of said shear resistance. Finally, we applied the method of "Direct Displacement Based Design" (D.D.B.D.) in the structure analyzed by checking if the application of this design method could provide more quickly a value for estimate the safety factor of the building.