Mircea Bîrsan

Existence of minimizers in the geometrically non-linear 6-parameter resultant shell theory with drilling rotations:

This paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells with drilling degrees of freedom. This theory establishes a general model for shells, which is characterized by two independent kinematic fields: the translation vector and the rotation tensor. Thus, the kinematical structure of 6-parameter shells is identical to that of Cosserat shells. We show the existence of global minimizers for the geometrically non-linear 2D equations of elastic shells. The proof of the existence theorem is based on the direct methods of the calculus of variations essentially using the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze the particular cases of isotropic shells, orthotropic shells and composite shells separately.

A BENDING THEORY OF POROUS THERMOELASTIC PLATES

This article is concerned with a plate theory for thermoelastic materials with voids. We establish the field equations for the bending of thermoelastic thin plates made from an isotropic and homogeneous material in the context of the dynamic linear theory. Then, we present a uniqueness theorem with no definiteness assumptions on the constitutive coefficients. Finally, by means of the logarithmic convexity method, we obtain continuous dependence results and study the instability of the solutions.

Existence Theorems in the Geometrically Non-linear 6-Parameter Theory of Elastic Plates

In this paper we show the existence of global minimizers for the geometrically non-linear equations of elastic plates, in the framework of the general 6-parameter shell theory. A characteristic feature of this model for shells is the appearance of two independent kinematic fields: the translation vector field and the rotation tensor field (representing in total 6 independent scalar kinematic variables). For isotropic plates, we prove the existence theorem by applying the direct methods of the calculus of variations. Then, we generalize our existence result to the case of anisotropic plates.

On a Thermodynamic Theory of Porous Cosserat Elastic Shells

This paper presents a theory for porous thermoelastic shells using the model of Cosserat surfaces and the Nunziato–Cowin theory for materials with voids. To describe the porosity of the thin body, we introduce two scalar fields: one field accounts for the changes in volume fraction along the middle surface of the shell, and the other field characterizes the porosity variations along the shells thickness. First, we postulate the principles of thermodynamics for these two-dimensional continua and we obtain the equations of the nonlinear theory. Then, we consider the linearized theory and prove the uniqueness of solution to the boundary initial value problem with no definiteness assumption on the constitutive coefficients. Finally, we consider the deformation of isotropic and homogeneous shells and determine the constitutive coefficients for Cosserat surfaces, by comparison with the results obtained from the three-dimensional approach to shell theory.

Sum of squared logarithms - an inequality relating positive definite matrices and their matrix logarithm

AbstractLet y1,y2,y3,a1,a2,a3∈(0,∞) be such that y1y2y3=a1a2a3 and y1+y2+y3≥a1+a2+a3,y1y2+y2y3+y1y3≥a1a2+a2a3+a1a3. Then (logy1)2+(logy2)2+(logy3)2≥(loga1)2+(loga2)2+(loga3)2. This can also be stated in terms of real positive definite 3×3-matrices P1, P2: If their determinants are equal, detP1=detP2, then trP1≥trP2andtrCofP1≥trCofP2⟹∥logP1∥F2≥∥logP2∥F2, where log is the principal matrix logarithm and ∥P∥F2=∑i,j=13Pij2 denotes the Frobenius matrix norm. Applications in matrix analysis and nonlinear elasticity are indicated.MSC:26D05, 26D07.

Cosserat-Type Rods

In this chapter we discuss a Cosserat-type theory of rods. Cosserat-type rod theories are based on the consideration of a rod base curve as a deformable directed curve, that is a curve with attached deformable or non-deformable (rigid) vectors (directors), or based on the derivation of one-dimensional (1D) rod equations from the three-dimensional (3D) micropolar (Cosserat) continuum equations. In the literature are known theories of rods kinematics of which described by introduction of the translation vector and additionally p deformable directors or one deformable director or three unit orthogonal each other directors. The additional vector fields of directors describe the rotational (in some special cases additional) degrees of freedom of the rod. The aim of the chapter is to present a Cosserat-type theory of rods and to show various applications.

Thermoelastic Deformations of Cylindrical Multi-Layered Shells Using a Direct Approach

In this article we study the deformation of thermo-elastic multi-layered shells, using a Cosserat model. By this direct approach, the shell-like bodies are modeled as deformable surfaces with a triad of rigidly rotating directors assigned to every point. The thermal effects are described with the help of two independent temperature fields. Concerning cylindrical orthotropic layered shells, we establish a general solution procedure for a class of thermal stresses problems. These analytical solutions are compared in some special cases with the corresponding three-dimensional solutions and thus, the thermo-elastic coupling coefficients for shells are identified in terms of the material/geometrical parameters of the layers. Finally, we present a comparison between our theoretical results and the numerical solutions obtained by a finite element analysis of a 3-layered cylindrical shell.

On the dislocation density tensor in the Cosserat theory of elastic shells

We consider the Cosserat continuum in its finite strain setting and discuss the dislocation density tensor as a possible alternative curvature strain measure in three-dimensional Cosserat models and in Cosserat shell models. We establish a close relationship (one-to-one correspondence) between the new shell dislocation density tensor and the bending-curvature tensor of 6-parameter shells.

Analysis of the Deformation of Multi-layered Orthotropic Cylindrical Elastic Shells Using the Direct Approach

In this paper we analyze the deformation of cylindrical multi-layered elastic shells using the direct approach to shell theory. In this approach, the thin shell-like bodies are modeled as deformable surfaces with a triad of vectors (directors) attached to each point. This triad of directors rotates during deformation and describes the rotations of the thickness filament of the shell. We consider a general set of constitutive equations which can model orthotropic multi-layered shells. For this type of shells we investigate the equilibrium of thin-walled tubes (not necessarily circular) subjected to external body loads and to resultant forces and moments applied to the end edges. We present a general procedure to derive the analytical solution of this problem. We consider that the external body loads are given polynomials in the axial coordinate, which coefficients can be arbitrary functions of the circumferential coordinate. We illustrate our method in the case of circular cylindrical three-layered shells and obtain the solution in closed form. For isotropic shells, the solution is in agreement with classical known results.

On the Bending Equations for Elastic Plates with Voids

In this paper we employ the Nunziato–Cowin theory for elastic materials with voids in order to investigate the bending of plates made from a porous material. We first present the fundamental equations and formulate the boundary initial value problem. Then, we establish some existence and uniqueness results concerning the solution in both equilibrium and dynamic theory. Finally, we apply the theory presented to solve a bending problem for a circular plate with voids.