Mircea Bîrsan

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• Prof. Dr. habil.
• Professor at University of Duisburg-Essen

Detailed derivations of the Legendre-Hadamard necessary conditions for energy-minimizing states of fiber-reinforced three-dimensional solids and two-dimensional shells are presented. The underlying conceptual framework is Cosserat elasticity theory in which the Cosserat rotation field controls the orientation of the embedded fibers. This is partial...

Convexity conditions of the Legendre–Hadamard and quasiconvexity type are derived in the context of a theory for fiber-reinforced shells based on Cosserat elasticity. These furnish two-dimensional shell-theoretic versions of their three-dimensional counterparts.

The quasiconvexity inequality associated with energy minimizers is derived in the context of a nonlinear Cosserat elasticity theory for fiber-reinforced elastic solids in which the intrinsic flexural and torsional elasticities of the fibers are taken into account explicitly. The derivation accounts for non-standard kinematic constraints, associated...

We investigate discretizations of a geometrically nonlinear elastic Cosserat shell with nonplanar reference configuration originally introduced by B\^irsan, Ghiba, Martin, and Neff in 2019. The shell model includes curvature terms up to order 5 in the shell thickness, which are crucial to reliably simulate high-curvature deformations such as near-f...

We derive the linear elastic Cosserat shell model of order O ( h ⁵ ) in the shell thickness h . To this aim, we linearise the geometrically nonlinear elastic Cosserat shell model established previously. Using Korn‐type inequalities for shells, the coercivity of the energy functional and the Lax‐Milgram theorem, we prove the existence and uniqueness...

To emphasize the main aspects of our procedure in as simple a manner as possible, we start with the theory of thin flat plates. This is based on classical linear elasticity under the assumption that the three-dimensional body is generated by the parallel translation of a flat midsurface. Accordingly the complexities associated with nonlinear elasti...

Here we develop the nonlinear theories of plates and shells, and show how Koiter’s shell theory emerges in the framework of our dimension reduction procedure for nonlinearly elastic materials.

In this chapter, we describe the deformation of continua and define the strain and stress tensors. Then, we review the main results of the differential geometry of surfaces in the Euclidean space.

The background on tensor analysis acquired in the first two chapters is used in the present chapter to cast the three-dimensional theory of nonlinear elasticity in a curvilinear-coordinate setting. This furnishes an immediate application of these ideas to a topic of mechanical significance and sets the stage for our subsequent work on elastic shell...

The classical theory of plate buckling is shown here to emerge from our dimension reduction procedure applied to incremental elasticity theory, concerned with the linearized theory or small deformations superposed upon large. Plate buckling theory emerges as the leading-order-in-thickness model when the underlying pre-stress scales appropriately wi...

In this paper we consider geometrically nonlinear 6-parameter shell models. We establish some existence proofs by the direct methods of the calculus of variations. In contrast to more classical approaches, we also investigate models up to order h5 in the shell thickness, where the form of the equations is determined by a dimensional descent from a...

In this paper we derive the linear elastic Cosserat shell model incorporating in the variational problem effects up to order O(h5)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d...

Starting from three-dimensional linear elasticity and performing the dimensional reduction by integration over the thickness, we derive a general form of the areal strain energy density for elastic shells. To obtain the new constitutive model, we do not approximate the deformation fields as polynomials in the thickness coordinate, but rather we kee...

We present a theory for thin elastic plates reinforced by a single family of continuously distributed fibers. The novelty of the model is the assignment of intrinsic flexural and torsional elasticity to the embedded fibers, regarded as material curves. Our derivation is based on a dimension reduction from the three-dimensional theory to obtain a pl...

In this chapter we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant’s problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. We present a general solution procedure to determine closed-form solutions for the extension, bending,...

In this chapter, we review the main notations and results of tensor analysis in Euclidean space using curvilinear coordinates. These will be useful in the formulation of shell and plate models.

In this chapter the dimension reduction procedure is extended to curved shells, again in the context of linear elasticity. The resulting model is used to obtain some simple solutions to problems of practical interest.

Concerning a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating higher order effects, we prove the coercivity of the proposed strain energy density for shells. This result is useful to show the existence of minimizers for the energy functional using the direct methods of the calculus of variations. Then, we linea...

In this paper we derive the linear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$ as a particular case of the recently introduced geometrically nonlinear elastic Cosserat shell model. The existence and uniqueness of the solution is proven in suitable admissible sets. To this end, inequalities of K...

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order \(O(h^{5})\) in the shell thickness \(h\). We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the \(O(h^{5})\) and \(O(h^{3})\) case and we draw some con...

Starting from a Cosserat-type model for curved rods, we derive analytical expressions for the effective stiffness coefficients of multilayered composite beams with an arbitrary number of layers. For this purpose, we employ the comparison with analytical solutions of some bending, torsion, and extension problems for three-dimensional beams and rods....

In this paper, we present a general method to derive the explicit constitutive relations for isotropic elastic 6-parameter shells made from a Cosserat material. The dimensional reduction procedure extends the methods of the classical shell theory to the case of Cosserat shells. Starting from the three-dimensional Cosserat parent model, we perform t...

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order \(O(h^{5})\) in the shell thickness \(h\). The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensiona...

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solut...

We consider a recently introduced geometrically nonlinear elastic Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. We develop the corresponding geometrically nonlinear constrained Cosserat shell model, we show the existence of minimizers for the $O(h^5)$ and $O(h^3)$ case and we draw some connections to ex...

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosserat 6-parameter shell model. The proof of the main theorem is based on the direct methods of the calculus of variations using essentially the convexity of the energy in the nonlinear strain and curvature measures. We first show the existence of the solut...

We present a new geometrically nonlinear Cosserat shell model incorporating effects up to order $O(h^5)$ in the shell thickness $h$. The method that we follow is an educated 8-parameter ansatz for the three-dimensional elastic shell deformation with attendant analytical thickness integration, which leads us to obtain completely two-dimensional sets...

Starting from the three-dimensional Cosserat elasticity, we derive a two-dimensional model for isotropic elastic shells. For the dimensional reduction, we employ a derivation method similar to that used in classical shell theory, as presented systematically by Steigmann (Koiter’s shell theory from the perspective of three-dimensional nonlinear elas...

Starting from the three-dimensional Cosserat elasticity, we derive a two-dimensional model for isotropic elastic shells. For the dimensional reduction, we employ a derivation method similar to that used in classical shell theory, as presented systematically by Steigmann in [J. Elast. \textbf{111}: 91-107, 2013]. As a result, we obtain a geometrical...

In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant's problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We...

In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant’s problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We...

Using a geometrically motivated 8‐parameter ansatz through the thickness, we reduce a three‐dimensional shell‐like geometrically nonlinear Cosserat material to a fully two‐dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytic...

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytic...

Using a geometrically motivated 8-parameter ansatz through the thickness, we reduce a three-dimensional shell-like geometrically nonlinear Cosserat material to a fully two-dimensional shell model. Curvature effects are fully taken into account. For elastic isotropic Cosserat materials, the integration through the thickness can be performed analytic...

We consider the Cosserat shell approach under finite rotations. The Cosserat shell features an additional, in principle independent orthogonal frame. In this setting we establish a novel curvature tensor which we call the shell dislocation density tensor. For this variant, we derive the equations and in a hyperelastic context we show existence of m...

We describe the 6-parameter Cosserat shell model and introduce a novel shell-curvature measure. The model is able to quantitatively describe the intricate wrinkling pattern of a thin elastic sheet under shear.

Presentation given by Mircea Birsan at the conference ”Emerging Trends in Applied Mathematics and Mechanics” in Perpignan, France

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general six-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements (GFEs), which leads to an objective discrete model which naturally allows arbitrarily large rotation...

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 bendin...

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 bendin...

We reconsider the geometrically nonlinear Cosserat model for a uniformly con- vex elastic energy and write the equilibrium problem as a minimization problem. Applying the direct methods of the calculus of variations we show the existence of minimizers. We present a clear proof based on the coercivity of the elastically stored energy density and on...

Nowadays modern composites with various internal structures are used for manufacturing of structural parts of airplanes and in other branches of engineering. They have a layered structure or can be produced as functionally graded materials (FGM). For specific applications some porosity is necessary to satisfy technological process requirements. In...

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The kinematical model is similar to the general 6-parameter resultant shell model with drilling rotations. The discretization uses geodesic finite elements, which leads to an objective discrete model which naturally allows arbitrarily large rotations. Finite...

We analyze geometrically non-linear isotropic elastic shells and prove the existence of minimizers. In general, the model takes into account the effect of drilling rotations in shells. For the special case of shells without drilling rotations we present a representation theorem for the strain energy function.

In this paper we employ a Cosserat model for rod-like bodies and study the governing equations of thin thermoelastic porous rods. We apply the counterpart of Korn’s inequality in the three-dimensional elasticity theory to prove existence and uniqueness results concerning the solutions to boundary value problems for thermoelastic porous rods, both i...

The 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 iden...

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. Concerni...

We investigate sandwich composite beams using a direct approach which models slender bodies as deformable curves endowed with a certain microstructure. We derive general formulas for the effective stiffness coefficients of composite elastic beams made of several non-homogeneous materials. A special attention is given to sandwich beams with foam cor...

Let y1, y2, y3, a1, a2, a3 > 0 be such that y1 y2 y3 = a1 a2 a3 and y1 + y2 + y3 >= a1 + a2 + a3, y1 y2 + y2 y3 + y1 y3 >= a1 a2 + a2 a3 + a1 a3. Then the following inequality holds (log y1)^2 + (log y2)^2 + (log y3)^2 >= (log a1)^2 + (log a2)^2 + (log a3)^2. This can also be stated in terms of real positive definite 3x3-matrices P1, P2: If their d...

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 (...

This paper presents a thermodynamic theory for elastic rods using the model of directed curves. In this model, the thin rod-like bodies are described as deformable curves with a triad of rigidly rotating vectors attached to each point. To account for the thermal effects in rods, we introduce two independent temperature fields: the absolute temperat...

We consider the general model of 6-parametric elastic plates, in which the rotation tensor field is an independent kinematic field. In this context we show the existence of global minimizers to the mini-mization problem of the total potential energy.

In this paper we show the existence of global minimizers for the geometrically exact, non-linear equations of elastic plates, in the framework of the general 6-parametric 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 (...

In this paper we employ the direct approach to the theory of rods and beams, which is based on the deformable curve model with a triad of rotating directors attached to each point. We show that this model (also called directed curve) is an efficient approach for analyzing the deformation of elastic beams with a complex material structure. Thus, we...

In this paper, we consider thin rods modeled by the direct approach, in which the rod-like body is regarded as a one-dimensional continuum (i.e., a deformable curve) with a triad of rigidly rotating orthonormal vectors attached to each material point. In this context, we present a model for porous thermoelastic curved rods, having natural twisting...

In this paper we present a mathematical study of the equations of motion for orthotropic thermoelastic simple shells. We use a direct approach to the mechanics of thin shells, in which the shell-like body is modeled as a deformable surface endowed with a triad of orthonormal vectors connected with each material point. The thermal effects are descri...

We consider the direct approach to the theory of rods, in which the thin body is modelled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. In this context, we employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. First, we derive the dynamica...

In this paper we study the equilibrium of cylindrical elastic shells under the action of resultant forces and moments on the end edges. We employ the linear theory of Cosserat surfaces to describe the deformation of anisotropic and inhomogeneous cylindrical shells with arbitrary (open or closed) cross-section. In this context, we prove a minimum en...

We establish a general inequality of Cauchy-Schwarz type. We present this new inequality in both the discrete and the integral forms. The integral version of this inequality appears in the study of mechanical properties of thin elastic rods.

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...

We employ the theory of elastic materials with voids to describe the mechanical behavior of porous rods. In this purpose, we consider the direct approach to the theory of rods, in which the thin body is modeled as a deformable curve with a triad of rigidly rotating orthonormal vectors attached to every material point. For orthotropic and homogeneou...

In the framework of the linear theory of simple elastic shells we study some properties of the solutions to the boundary-initial-value problem for general orthotropic and inhomogeneous materials. Using the method of logarithmic convexity, we prove the continuous dependence of solutions on the external body loads and initial data. Then, for a certai...

The theory of simple shells is a surface-related Cosserat model for thin elastic shells. In this direct approach, each material point is connected with a triad of rigidly rotating directors. This paper presents a study of the governing equations for orthotropic elastic simple shells in the framework of the linearized theory. We establish the unique...

We investigate the deformation of general anisotropic and inhomogeneous shells, under the action of a given temperature distribution. We assume that the temperature field is a polynomial in the axial coordinate, and we establish the displacements produced by the prescribed thermal field. The results are obtained in the framework of the linear theor...

We consider the problem of thermal stresses in cylindrical elastic shells, modelled as Cosserat surfaces. In the theory of Cosserat shells, the thermal effects are described generally by means of two temperature fields. The problem consists in finding the equilibrium of the shell under the action of a given temperature distribution. The temperature...

We investigate the static deformation of cylindrical elastic shells, using the theory of Cosserat surfaces. We consider anisotropic and inhomogeneous cylindrical shells with arbitrary (open or closed) cross-section. The constitutive coefficients are assumed to be independent of the axial coordinate. In the context of linearized theory, we determine...

This paper investigates the equilibrium of cylindrical elastic shells under the action of prescribed body loads, external loads on the lateral edges, and resultant forces and moments on the end edges. We consider anisotropic and inhomogeneous cylindrical shells with arbitrary (open or closed) cross-sections, and we employ the linear theory of Cosse...

We consider the relaxed Saint-Venant’s problem in the linear theory of elastic shells made from a porous material. For our purpose, we use the model of Cosserat surfaces and the Nunziato-Cowin theory of elastic materials with voids [1].We extend the method employed in [2], [3] for the case of Cosserat shells with two porosity fields: one field char...

This paper is concerned with the linear theory of anisotropic and inhomogeneous Cosserat elastic shells. We establish the inequalities of Korn’s type which hold on Cosserat surfaces. Using these inequalities, we prove the existence of the solution to the variational equations in the elastostatics of Cosserat shells. For the dynamic problems, we emp...

We consider a problem of thermal stresses in cylindrical Cosserat elastic shells made from a material with voids. The cylindrical shells have arbitrary cross-sections. The problem consists in finding the equilibrium of the shell under the action of a given temperature distribution. We assume that the temperature field is independent of the axial co...

This paper is concerned with the linear theory of porous Cosserat thermoelastic shells. We consider anisotropic and inhomogeneous shells and establish the existence and uniqueness of solution to the boundary–initial–value problem associated to the dynamic deformations. To this aim, we employ the semigroup of linear operators theory.

We study the equilibrium of cylindrical Cosserat elastic shells under the action of body loads and tractions and couples distributed along its edges. The shells have arbitrary open or closed cross-sections and are made from an isotropic and homogeneous material. On the end edges, the appropriate resultant forces and resultant moments are prescribed...

We study differential equations which govern linear deformations of thin elastic shells made from a porous material. In our approach, we employ the theory of Cosserat elastic surfaces with voids. On the basis of inequalities of Korn’s type for Cosserat surfaces, we prove existence and uniqueness results for the solution of boundary value problems,...

In this paper, we consider the linear theory of Cosserat elastic shells made from an homogeneous and anisotropic material. We study the equilibrium of cylindrical shells loaded by contact forces and couples acting on the end edges. We prove the principle of Saint-Venant in the following formulation: if a system of loads has zero resultant force and...

We employ 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 initial-boundary value problem. Then we establish some existence and uniqueness results concerning the solution in both equilibrium and dynamic theor...

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 characte...

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the...

In this paper we investigate the boundary-initial-value problem of the dynamic linear theory for thermoelastic Cosserat shells with voids. We prove a reciprocity relation and derive a uniqueness theorem. Then, we study the continuous dependence of the solution on external body loads and heat supply and on initial data. A variational characterizatio...

This paper is concerned with the dynamic theory for bending of ther-moelastic plates of Mindlin–type made from a material with voids. We establish a representation of Galerkin type for the solution of the field equations. Then, we study the time–harmonic oscillations of porous thermoelastic plates. Finally, the propagation of flexural waves in an i...

The linear theory of Cosserat surfaces is employed to study Saint-Venant's problem for cylindrical shells of arbitrary cross-section. We prove minimum energy characterizations for the solution of the relaxed Saint-Venant's problem previously obtained. Then, we determine the global measures of strain appropriate to extension, bending, torsion and fl...

In this paper we investigate the mechanical behavior of Cosserat shells made from a material with voids. We formulate Saint-Venant’s problem for cylindrical shells and determine the solution of the relaxed problem. Then, we apply the theoretical results to study the deformation of circular cylindrical shells. We also compare the solution of Saint-V...

In the context of the linear theory of thermoelastic materials with voids, we study the dynamic bending of Mindlin plates. We formulate the corresponding initial-boundary value problem and establish a reciprocal theorem. Then we present two variational theorems of Gurtin type.

This paper is concerned with the linear theory of thin elastic shells modelled as Cosserat surfaces. We formulate the corresponding Saint-Venant's problem with respect to cylindrical shells. Then, we determine the solution of the relaxed problem for both open and closed cylindrical surfaces. The edge curves perpendicular to the generator are not ne...

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 co...

The paper studies the field equations governing the bending of plates in the theory of elastic materials with voids. First, we present a representation of Galerkin type for the solution of the equilibrium equations of isotropic and homogeneous materials. Then, we use this solution to determine the fundamental solutions. Finally, an integral represe...

This paper concerns the theory of elastic materials with voids, cf. the paper of J. W. Nunziato and S. C. Cowin [Arch. Ration. Mech. Anal. 72, 175-201 (1979; Zbl 0444.73018)]. In the first part of the paper we establish a nonlinear theory of thermoelastic shells with voids. We consider the shells modeled as Cosserat surfaces. Then, we derive the eq...