The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibility, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as Subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space of all the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibility, a family of non-smooth func̀tionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem. © 1993.

Variational formulations of non-linear and non-smooth structural problems / Romano, Giovanni; Rosati, Luciano; MAROTTI DE SCIARRA, Francesco. - In: INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS. - ISSN 0020-7462. - STAMPA. - 28:2(1993), pp. 195-208.

Variational formulations of non-linear and non-smooth structural problems

ROMANO, GIOVANNI;ROSATI, LUCIANO;MAROTTI DE SCIARRA, FRANCESCO
1993

Abstract

The inverse problem of variational calculus is addressed with reference to structural models governed by non-linear field equations and monotone multi-valued constitutive operators. For such a class of models a non-smooth analysis must be necessarily carried out. The concept of consistency of non-linear strain operators is first recalled in view of a Lagrangian formulation of equilibrium. The structural problem is then recast in terms of a single structural operator which encompasses the field and constitutive equations by means of two sub-operators. The first one, which accounts for equilibrium and compatibility, is proved to be conservative and its potential explicitly derived. The second one is assumed to be conservative since it embodies multi-valued constitutive relations which are expressed as Subdifferentials of convex functionals. The original problem is then amenable to a weak formulation and, recalling recent results on the potential theory of monotone multi-valued operators, a constructive method for the variational formulation of problems expressed in terms of conservative multi-valued operators is presented. The structural operator is accordingly integrated in the product space of all the state variables to get the expression of the associated potential. Further, by enforcing constraint relations and kinematic compatibility, a family of non-smooth func̀tionals is derived and the related stationarity conditions are suitably defined starting from the concept of local subdifferential. Finally, it is shown that the stationarity of each of these functionals yields back an operator form of the structural problem. © 1993.
1993
Variational formulations of non-linear and non-smooth structural problems / Romano, Giovanni; Rosati, Luciano; MAROTTI DE SCIARRA, Francesco. - In: INTERNATIONAL JOURNAL OF NON-LINEAR MECHANICS. - ISSN 0020-7462. - STAMPA. - 28:2(1993), pp. 195-208.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11588/484461
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