Two topics will be discussed here, which usually are not put in direct relationship: nonlocal approaches and the existence of strain-softening. It will be shown how some of the expedients used for treating strain-softening by avoiding the numerical instability associated with a material tangent stiffness matrix that ceases to be positive definite are equivalent to introducing a scale separation between the load-displacement and stress-strain laws. That is, they are equivalent to employing a relationship between effective stress and effective strain that is, not necessarily, softening. The identification procedure of the effective law is based just on the assumption that we must separate the material (stress-strain) from the structure (load-displacement) scale, since strain-softening is not a real material property but the result of inhomogeneous deformation caused by the experimental technique. This procedure provides the first experimental evidence against the existence of strain-softening in concrete and identifies a monotone strictly nondecreasing effective law. However, we could argue that this law is not interesting from the numerical point of view, since, being a local law, it is not suitable for modelling problems in which the size-effect is involved. As far as nonlocality is concerned, it will be argued that there is no evident physical justification for incorporating a length scale into the constitutive relationships. This need is due to the limit process on the basis of any differential formulation and may be avoided by using a discrete formulation. The Cell Method (CM) is the only formulation that is truly discrete, at present. This method allows us to express the governing equations in discrete form directly. Several numerical results will be provided here, which allows us to conclude that it is possible to perform a nonlocal analysis by means of the CM and the effective law, without any need to incorporate nonlocal effects in the constitutive equations.
E. Ferretti (2009). Ability of the Cell Method of modelling nonlocality.
Ability of the Cell Method of modelling nonlocality
FERRETTI, ELENA
2009
Abstract
Two topics will be discussed here, which usually are not put in direct relationship: nonlocal approaches and the existence of strain-softening. It will be shown how some of the expedients used for treating strain-softening by avoiding the numerical instability associated with a material tangent stiffness matrix that ceases to be positive definite are equivalent to introducing a scale separation between the load-displacement and stress-strain laws. That is, they are equivalent to employing a relationship between effective stress and effective strain that is, not necessarily, softening. The identification procedure of the effective law is based just on the assumption that we must separate the material (stress-strain) from the structure (load-displacement) scale, since strain-softening is not a real material property but the result of inhomogeneous deformation caused by the experimental technique. This procedure provides the first experimental evidence against the existence of strain-softening in concrete and identifies a monotone strictly nondecreasing effective law. However, we could argue that this law is not interesting from the numerical point of view, since, being a local law, it is not suitable for modelling problems in which the size-effect is involved. As far as nonlocality is concerned, it will be argued that there is no evident physical justification for incorporating a length scale into the constitutive relationships. This need is due to the limit process on the basis of any differential formulation and may be avoided by using a discrete formulation. The Cell Method (CM) is the only formulation that is truly discrete, at present. This method allows us to express the governing equations in discrete form directly. Several numerical results will be provided here, which allows us to conclude that it is possible to perform a nonlocal analysis by means of the CM and the effective law, without any need to incorporate nonlocal effects in the constitutive equations.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
