Three types of multi-field dual-mixed variational principles using a priori non-symmetric stress field will be derived for dynamic problems of linearly elastic solids. Starting from the functional of complementary Hamilton’s principle it is shown how this variational formulation is modified with applying the Lagrange multiplier technique, to obtain a new four-field dual-mixed functional. The independent fields of this functional will be the displacement vector, the non-symmetric stress tensor, the skew-symmetric rotation tensor and the momentum vector. Then we present how to treat appropriately the six different types of initial conditions in a weak and/or strong sense by adding a new integral expression to the four-field functional. Modifying this functional through a Legendre transformation results in a new complementary energy-based five-field dual-mixed functional which allows the momentum- and velocity vectors to be independently approximated. A complementary energy-based three-field dual-mixed variational formulation is obtained by eliminating the momentum- and velocity fields through the strong enforcement of the kinematic equation and the impulse-velocity relation from the five-field principle.

Multi-field Dual-Mixed Variational Principles Using Non-symmetric Stress Field in Linear Elastodynamics

In this work, adaptive explicit and implicit time marching techniques are proposed to analyze nonlinear dynamic models. In these time marching adaptive procedures, the adopted time integration parameters modify themselves along the solution process, in accordance with the varying properties and results of the model. Thus, the time integrators are locally evaluated, assuming different values along the spatial and temporal discretizations, enabling more accurate and efficient solution techniques. Numerical results are presented along the manuscript, illustrating the good performance of the proposed space/time adaptive explicit and implicit time marching strategies for nonlinear analyses.

Nonlinear dynamic analysis considering explicit and implicit time marching techniques with adaptive time integration parameters

A direct theory of affine rods is developed from first principles. To concentrate on the central aspects of the model, we use an axiomatic format and tools from Lie group theory. To facilitate comparisons with other theories, we propose an identification procedure to derive the constitutive relations of the affine rod from those of a rod modeled as a three-dimensional body.

A Direct Theory of Affine Rods

The vibrating behaviour of thin structures is affected by prestress states. Hence, the effects of thermal prestress are important research subjects in view of ambient vibration monitoring of civil structures. The interaction between prestress, geometrically non-linear behaviour, as well as damping and its coupling with the aforementioned phenomena has to be taken into account for a comprehensive understanding of the structural behaviour. Since the literature on this subject lacks a clear procedure to derive models of thin prestressed and damped structures from 3D continuum mechanics, this paper presents a new derivation of models for thin structures accounting for generic prestress, moderate rotations and viscous damping. Although inspired by classical approaches, the proposed procedure is quite different, because of (i) the definition of a modified Hu-Washizu (H-W) functional, accounting for stress constraints associated with Lagrange multipliers, in order to derive lower-dimensional models in a convenient way; (ii) an original definition of a (mechanical and thermal) strain measure and a rotation measure enabling one to identify the main terms in the strain energy and to derive a cascade of lower-dimensional models (iii) a new definition of "strain-rotation domains" providing a clear interpretation of the classical assumptions of "small perturbations" and "small strains and moderate rotations"; (iv) the introduction of a pseudo-potential with stress constraints to account for viscous damping. The proposed procedure is applied to thin beams.

https://hal.archives-ouvertes.fr/hal-00345318/document

On the derivation of structural models with general thermomechanical prestress

The port-Hamiltonian system approach is intended to be an innovative and unifying way of modeling multiphysics systems, by expressing all of them as systems of conservation laws. Indeed, the increasing developments in recent years allow finding better control and coupling strategies. This work aimed to apply such an approach to Naghdi–Reissner’s five-kinematic-field shell model in linear elasticity, while including often-neglected higher-order intrinsic geometric coupling effects, therefore preparing the theoretical background required for the coupling (or interconnection) with an acoustic fluid model and the different types of interactions that can arise among them. The model derived thusly can be used for controller design in a wide variety of applications such as inflatable space structures, launcher tank vibration damping, payload vibration protection using smart materials, and many other related applications.

/pdf/port-hamiltonian-formulations-of-some-elastodynamics-3ujoj4an.pdf

Port-Hamiltonian Formulations of Some Elastodynamics Theories of Isotropic and Linearly Elastic Shells: Naghdi–Reissner’s Moderately Thick Shells

Nonlinear elastodynamics problems are approached from the point of view of a mixed variational principle. It is shown that four different functionals lead to different formulations of the problem with different independent fields. One of the functionals is specialized to the case of spatial beams undergoing large deformations, and small strains. The extension of these functionals to the case in which kinematic constraints are presented leads to the useful concept of dual constraints, which can be advantageously employed in treating problems involving contact, impact, and changing topology.

Mixed variational principles in space and time for elastodynamics analysis

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Mixed variational principles in space and time for elastodynamics analysis