Marina Diaco

On continuum dynamics

The theory of continuous dynamical systems is developed with an intrinsic geometric approach based on the action principle formulated in the velocity-time manifold. By endowing the finite dimensional Riemannian ambient manifold with a connection, an induced connection is naturally defined in the infinite dimensional configuration manifold of maps. The motion is shown to be governed, in the Banach configuration manifold, by a generalized Lagrange law and, in the ambient manifold, by a generalized Euler law which is independent of the Banach topology of the configuration manifold. Extended versions of Euler–Poincare law, Euler classical laws and d’Alembert law are also derived as special cases. Stress fields in the body are introduced as Lagrange’s multipliers of the rigidity constraint on virtual velocities, dual to the Lie derivative of the metric. No special assumptions are made so that any constitutive behaviors can be modeled.

Tangent stiffness of elastic continua on manifolds

Non-linear models of beams, shells and polar continua are addressed from a general point of view with the aim of providing a clear motivation of the fact that the tangent stiffness of these structural models may be nonsymmetric. Classical and polar models of continua are investigated and a critical analysis of the commonly adopted strain measures is performed. It is emphasized that the kinematic space of a polar continuum is a non-linear differentiable manifold. Accordingly, by choosing a connection on the manifold, the Hessian operator of the elastic potential is defined as the second covariant derivative of the elastic potential. The Hessian operator can be expressed as the difference between the second directional derivative along the trial and test fields and the first directional derivative in the direction of the covariant derivative of the test field along the trial field. It follows that the evaluation of the Hessian operator requires the extension of the local virtual displacement to a vector field over the non-linear kinematic manifold. In any case the tensoriality of the Hessian operator ensures that the result is independent of the choice of the extension, and its symmetry depends on whether or not the assumed connection is torsionless. Conservative and nonconservative loads are considered and it is shown that, at equilibrium points, the tangent stiffness is independent of the chosen connection on the fiber manifold and symmetry holds for conservative loads.

Well-Posedness of Mixed Formulations in Elasticity

Mixed formulations in elasticity are analysed and existence and uniqueness of the solution are discussed in the context of Hilbert space theory. New results, referred to in the analysis of elasticity problems, are proved. They are concerned with the closedness of the product of two linear operators and a projection property equivalent to the closedness of the sum of two closed, subspaces. A set of two necessary and sufficient conditions for the well-posedness of an elastic problem with a singular elastic compliance procides the most general result of this kind in linear elasticity. Sufficient criteria for the well-posedness of elastic problems in structural mechanics including the presence of supporting elastic beds are contributed and applications are exemplified.

Tangent stiffness of a Timoshenko beam undergoing large displacements

The polar model of an elastic Timoshenko beam undergoing large displacements is investigated in detail. Special emphasis is given to the problems involved in the evaluation of the tangent stiffness to provide a complete answer to the question of whether or not tangent stiffness is tensorial and symmetric.

On Torsion of Functionally Graded Elastic Beams

The evaluation of tangential stress fields in linearly elastic orthotropic Saint-Venant beams under torsion is based on the solution of Neumann and Dirichlet boundary value problems for the cross-sectional warping and for Prandtl stress function, respectively. A skillful solution method has been recently proposed by Ecsedi for a class of inhomogeneous beams with shear moduli defined in terms of Prandtl stress function of corresponding homogeneous beams. An alternative reasoning is followed in the present paper for orthotropic functionally graded beams with shear moduli tensors defined in terms of the stress function and of the elasticity of reference inhomogeneous beams. An innovative result of invariance on twist centre is also contributed. Examples of functionally graded elliptic cross sections of orthotropic beams are developed, detecting thus new benchmarks for computational mechanics.