Mircea Birsan | University of Duisburg-Essen (original) (raw)

Papers by Mircea Birsan

Research paper thumbnail of Lecture Notes on the Theory of Plates and Shells

Solid mechanics and its applications, 2023

Research paper thumbnail of On the characterization of drilling

We analyze geometrically non-linear isotropic elastic shells and prove the existence of minimizer... more 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.

Research paper thumbnail of Tensor Analysis in Euclidean Space Using Curvilinear Coordinates

Lecture Notes on the Theory of Plates and Shells

Research paper thumbnail of The 6-parameter Cosserat-shell model

Research paper thumbnail of The isotropic Cosserat shell model including terms up to O(h^5). Part II: Existence of minimizers

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosser... more 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 solution for the theory including O(h^5) terms. The energy allows us to show the coercivity for terms up to order O(h^5) and the convexity of the energy. Secondly, we consider only that part of the energy including O(h^3) terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including O(h^5) the conditions...

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

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells wi... more 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 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 using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

Research paper thumbnail of 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 densi... more 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.

Research paper thumbnail of On the characterization of drilling rotation in the 6-parameter resultant shell theory

We analyze geometrically non-linear isotropic elastic shells and prove the existence of minimizer... more 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.

Research paper thumbnail of Existence theorems in the geometrically non-linear 6-parametric theory of elastic plates

In this paper we show the existence of global minimizers for the geometrically exact, non-linear ... more 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 (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. We also present a detailed comparison with a previously established Cosserat plate model.

Research paper thumbnail of Numerical Treatment of a Geometrically Nonlinear Planar Cosserat Shell Model

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The k... more 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 elements of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkles of thin elastic sheets in shear.

Research paper thumbnail of Existence of minimizers in the geometrically

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells wi... more 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 6parameter 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 using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

Research paper thumbnail of On the Equations of Geometrically Nonlinear Elastic Plates with Rotational Degrees of Freedom *

We consider the general model of 6-parametric elastic plates, in which the rotation tensor field ... more 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 minimization problem of the total potential energy.

Research paper thumbnail of Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

In this paper, we present a general method to derive the explicit constitutive relations for isot... more 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 the integration over the thickness and obtain a consistent shell model of order O(h5)O(h^5)O(h5) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

Research paper thumbnail of Multilayered and FGM structural elements under mechanical and thermal loads. Part I: Comparison of finite elements and analytical models

Archives of Civil and Mechanical Engineering, 2015

ABSTRACT Nowadays modern composites with various internal structures are used for manufacturing o... more ABSTRACT 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 this paper several numerical models of structural parts of airplanes made of different composites were elaborated. The finite element (FE) analysis and the mechanical response of the structural elements were compared with previously formulated simplified analytical models 0005 and 0010. The calculations concern estimation of deformation states in: –a layered exhaust pipe covered by thermal barrier coating (TBC) subjected to an internal pressure and a temperature gradient,–a two-layered plate under thermal loading,–three-layered beams subjected to bending with non-symmetrical cross-section,–a FGM beam element subjected to bending load,–a FGM two-layered rod subjected to uniaxial tension. For each considered case a good correlation of the results using both methods – analytical and numerical ones – was obtained, which proves correctness of the theoretical derivations, 0005 and 0010. The paper provides also own general procedure for modelling of FGM in the ABAQUS.

Research paper thumbnail of Mechanical behavior of sandwich composite beams made of foams and functionally graded materials

International Journal of Solids and Structures, 2013

We investigate sandwich composite beams using a direct approach which models slender bodies as de... more 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 core, which are made of functionally graded or piecewise homogeneous materials. In the case of small deformations, the theoretical predictions are compared with experimental measurements for the three-point bending of sandwich beams, showing a very good agreement. For functionally graded sandwich columns we obtain the analytical solutions of bending, torsion and extension problems and compare them with numerical results computed by the finite element method.

Research paper thumbnail of Cylindrical Orthotropic Thermoelastic Shells Modeled by Direct Approach

Encyclopedia of Thermal Stresses, 2014

Emanating from the Boltzmann transport equation, a newly developed C-and F-processes heat conduct... more Emanating from the Boltzmann transport equation, a newly developed C-and F-processes heat conduction constitutive model and the associated dynamic thermoelasticity are described. The model acknowledges the notion of the simultaneous coexistence of both the slow Cattaneo-type C-processes and fast Fourier-type F-processes in the mechanisms of heat conduction. The formulation leads to a generalization of the macroscale in space one temperature theory for heat conduction in solids of the Jeffreys-type model, Cattaneo model, and the Fourier model for heat conduction in solids. This is unlike the Jeffreys-type phenomenological model which cannot reduce to the classical Fourier model (but only to a Fourier-like representation with relaxation), and the Jeffreys-type model cannot explain the underlying physics associated the C-and F-processes model. A generalized thermoelastic theory is described to study the dynamic thermoelastic behavior of solids with special features which can explain the classical and nonclassical dynamic thermoelastic theories. R.B. Hetnarski (ed.

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

Shell-like Structures, 2011

In this paper we analyze the deformation of cylindrical multi-layered elastic shells using the di... more 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

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

Journal of Thermal Stresses, 2013

ABSTRACT In this article we study the deformation of thermo-elastic multi-layered shells, using a... more ABSTRACT 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.

Research paper thumbnail of Deformation analysis of functionally graded beams by the direct approach

Composites Part B: Engineering, 2012

Research paper thumbnail of On the Equations of Geometrically Nonlinear Elastic Plates with Rotational Degrees of Freedom∗

We consider the general model of 6-parametric elastic plates, in which the rotation tensor field ... more 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 minimization problem of the total potential energy. MSC: 74K20, 74K25, 74G65, 74G25. keywords: elastic plates, geometrically non-linear plates, shells, existence of minimizers, Cosserat plate.

Research paper thumbnail of Lecture Notes on the Theory of Plates and Shells

Solid mechanics and its applications, 2023

Research paper thumbnail of On the characterization of drilling

We analyze geometrically non-linear isotropic elastic shells and prove the existence of minimizer... more 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.

Research paper thumbnail of Tensor Analysis in Euclidean Space Using Curvilinear Coordinates

Lecture Notes on the Theory of Plates and Shells

Research paper thumbnail of The 6-parameter Cosserat-shell model

Research paper thumbnail of The isotropic Cosserat shell model including terms up to O(h^5). Part II: Existence of minimizers

We show the existence of global minimizers for a geometrically nonlinear isotropic elastic Cosser... more 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 solution for the theory including O(h^5) terms. The energy allows us to show the coercivity for terms up to order O(h^5) and the convexity of the energy. Secondly, we consider only that part of the energy including O(h^3) terms. In this case the obtained minimization problem is not the same as those previously considered in the literature, since the influence of the curved initial shell configuration appears explicitly in the expression of the coefficients of the energies for the reduced two-dimensional variational problem and additional mixed bending-curvature and curvature terms are present. While in the theory including O(h^5) the conditions...

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

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells wi... more 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 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 using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

Research paper thumbnail of 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 densi... more 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.

Research paper thumbnail of On the characterization of drilling rotation in the 6-parameter resultant shell theory

We analyze geometrically non-linear isotropic elastic shells and prove the existence of minimizer... more 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.

Research paper thumbnail of Existence theorems in the geometrically non-linear 6-parametric theory of elastic plates

In this paper we show the existence of global minimizers for the geometrically exact, non-linear ... more 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 (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. We also present a detailed comparison with a previously established Cosserat plate model.

Research paper thumbnail of Numerical Treatment of a Geometrically Nonlinear Planar Cosserat Shell Model

We present a new way to discretize a geometrically nonlinear elastic planar Cosserat shell. The k... more 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 elements of any approximation order can be constructed. The resulting algebraic problem is a minimization problem posed on a nonlinear finite-dimensional Riemannian manifold. We solve this problem using a Riemannian trust-region method, which is a generalization of Newton's method that converges globally without intermediate loading steps. We present the continuous model and the discretization, discuss the properties of the discrete model, and show several numerical examples, including wrinkles of thin elastic sheets in shear.

Research paper thumbnail of Existence of minimizers in the geometrically

The paper is concerned with the geometrically non-linear theory of 6-parametric elastic shells wi... more 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 6parameter 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 using essentially the convexity of the energy in the strain and curvature measures. Since our result is valid for general anisotropic shells, we analyze separately the particular cases of isotropic shells, orthotropic shells, and composite shells.

Research paper thumbnail of On the Equations of Geometrically Nonlinear Elastic Plates with Rotational Degrees of Freedom *

We consider the general model of 6-parametric elastic plates, in which the rotation tensor field ... more 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 minimization problem of the total potential energy.

Research paper thumbnail of Alternative derivation of the higher-order constitutive model for six-parameter elastic shells

In this paper, we present a general method to derive the explicit constitutive relations for isot... more 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 the integration over the thickness and obtain a consistent shell model of order O(h5)O(h^5)O(h5) with respect to the shell thickness h. We derive the explicit form of the strain energy density for 6-parameter (Cosserat) shells, in which the constitutive coefficients are expressed in terms of the three-dimensional elasticity constants and depend on the initial curvature of the shell. The obtained form of the shell strain energy density is compared with other previous variants from the literature, and the advantages of our constitutive model are discussed.

Research paper thumbnail of Multilayered and FGM structural elements under mechanical and thermal loads. Part I: Comparison of finite elements and analytical models

Archives of Civil and Mechanical Engineering, 2015

ABSTRACT Nowadays modern composites with various internal structures are used for manufacturing o... more ABSTRACT 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 this paper several numerical models of structural parts of airplanes made of different composites were elaborated. The finite element (FE) analysis and the mechanical response of the structural elements were compared with previously formulated simplified analytical models 0005 and 0010. The calculations concern estimation of deformation states in: –a layered exhaust pipe covered by thermal barrier coating (TBC) subjected to an internal pressure and a temperature gradient,–a two-layered plate under thermal loading,–three-layered beams subjected to bending with non-symmetrical cross-section,–a FGM beam element subjected to bending load,–a FGM two-layered rod subjected to uniaxial tension. For each considered case a good correlation of the results using both methods – analytical and numerical ones – was obtained, which proves correctness of the theoretical derivations, 0005 and 0010. The paper provides also own general procedure for modelling of FGM in the ABAQUS.

Research paper thumbnail of Mechanical behavior of sandwich composite beams made of foams and functionally graded materials

International Journal of Solids and Structures, 2013

We investigate sandwich composite beams using a direct approach which models slender bodies as de... more 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 core, which are made of functionally graded or piecewise homogeneous materials. In the case of small deformations, the theoretical predictions are compared with experimental measurements for the three-point bending of sandwich beams, showing a very good agreement. For functionally graded sandwich columns we obtain the analytical solutions of bending, torsion and extension problems and compare them with numerical results computed by the finite element method.

Research paper thumbnail of Cylindrical Orthotropic Thermoelastic Shells Modeled by Direct Approach

Encyclopedia of Thermal Stresses, 2014

Emanating from the Boltzmann transport equation, a newly developed C-and F-processes heat conduct... more Emanating from the Boltzmann transport equation, a newly developed C-and F-processes heat conduction constitutive model and the associated dynamic thermoelasticity are described. The model acknowledges the notion of the simultaneous coexistence of both the slow Cattaneo-type C-processes and fast Fourier-type F-processes in the mechanisms of heat conduction. The formulation leads to a generalization of the macroscale in space one temperature theory for heat conduction in solids of the Jeffreys-type model, Cattaneo model, and the Fourier model for heat conduction in solids. This is unlike the Jeffreys-type phenomenological model which cannot reduce to the classical Fourier model (but only to a Fourier-like representation with relaxation), and the Jeffreys-type model cannot explain the underlying physics associated the C-and F-processes model. A generalized thermoelastic theory is described to study the dynamic thermoelastic behavior of solids with special features which can explain the classical and nonclassical dynamic thermoelastic theories. R.B. Hetnarski (ed.

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

Shell-like Structures, 2011

In this paper we analyze the deformation of cylindrical multi-layered elastic shells using the di... more 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

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

Journal of Thermal Stresses, 2013

ABSTRACT In this article we study the deformation of thermo-elastic multi-layered shells, using a... more ABSTRACT 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.

Research paper thumbnail of Deformation analysis of functionally graded beams by the direct approach

Composites Part B: Engineering, 2012

Research paper thumbnail of On the Equations of Geometrically Nonlinear Elastic Plates with Rotational Degrees of Freedom∗

We consider the general model of 6-parametric elastic plates, in which the rotation tensor field ... more 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 minimization problem of the total potential energy. MSC: 74K20, 74K25, 74G65, 74G25. keywords: elastic plates, geometrically non-linear plates, shells, existence of minimizers, Cosserat plate.