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Papers by Wojciech Pietraszkiewicz

International Journal of Solids and Structures, 2014

ABSTRACT It is well known that distribution of displacements through the shell thickness is non-l... more ABSTRACT It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.

Shell Structures: Theory and Application, 2013

Shell Structures: Theory and Application, 2013

International Journal of Solids and Structures, 2007

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching a... more We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements. boundary value problem. Such a general, dynamically and kinematically exact, six-scalar-field theory of regular shells, formulated with regard to a non-material weighted surface of mass taken as the shell base surface, was developed by Simmonds (1983, 1998) and , and with regard to a material surface arbitrary located within the shell-like body by Stumpf (1990), Chró ścielewski et al. (1992), and . For this general shell model efficient finite element algorithms were developed and many numerical examples of equilibrium, stability, and dynamics of regular and complex shell structures were presented by Chró ścielewski et al. works it was assumed that the region of shell irregularity (e.g., branching, self-intersection, stiffening, technological junction, etc.) is small as compared with other shell dimensions and its size can be ignored in deriving the resultant 2D equilibrium conditions. However, such an assumption brings an undefinable error into the resultant dynamic continuity conditions formulated along the singular surface curves modelling the irregularity regions. Therefore, such conditions cannot be regarded as exact implications of 3D equilibrium conditions of continuum mechanics.

Thin-Walled Structures, 2015

International Journal of Solids and Structures, 2011

ABSTRACT We construct the unique two-dimensional (2D) kinematics which is work-conjugate to the e... more ABSTRACT We construct the unique two-dimensional (2D) kinematics which is work-conjugate to the exact, resultant local equilibrium conditions of the non-linear theory of branching shells. It is shown that the compatible shell displacements consist of the translation vector and rotation tensor fields defined on the regular parts of the shell base surface as well as independently on the singular surface curve modelling the shell branching. Discussing relations between limits of the translation vector and rotation tensor fields when approaching the singular curve, and analogous fields given only along the singular curve itself, several types of the junctions are described. Among them are the stiff, entirely simply connected and partly simply supported junction as well as the elastically and dissipatively deformable junction, and the non-local elastic junction. For each type of junction the explicit form of the principle of virtual work is derived.

The six-field non-linear theory of elastic shells with the phase transformation of the material i... more The six-field non-linear theory of elastic shells with the phase transformation of the material is developed. Equilibrium conditions are found from the variational principle of stationary total potential energy. New dynamic continuity conditions are derived at the movable singular surface curve modelling the phase interface. Particular forms of the continuity conditions at coherent and incoherent interface curves are given. The results are illustrated by an example of the phase transition in an infinite plate with a circular hole.

International Journal of Solids and Structures, 2007

We show how to determine the midsurface of a deformed thin shell from known geometry of the undef... more We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: a) the stretch field is found by pure algebra, b) the rotation field is obtained by solving a system of first-order PDEs, and c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.

Advanced Structured Materials, 2013

Advances in Mechanics and Mathematics, 2010

Definitions of the Lagrangian stretch and wryness tensors in the nonlinear Cosserat continuum are... more Definitions of the Lagrangian stretch and wryness tensors in the nonlinear Cosserat continuum are discussed applying three different methods. The resulting unique strain measures have several distinguishing features and are called the natural ones. They are expressed through the translation vector and either the rotation tensor or various finite rotation vector fields. The relation of the natural strain measures to those proposed in the representative literature is reviewed.

Mathematics and Mechanics of Solids, 2015

Encyclopedia of Thermal Stresses, 2014

Shell Structures. Theory and Applications, Volume contains 104 contributions from over 22 countri... more Shell Structures. Theory and Applications, Volume contains 104 contributions from over 22 countries, reflecting a wide spectrum of scientific and engineering problems of shell structures. The papers are divided into six broad groups: 1. General lectures; 2. Theoretical modeling; 3. Stability; 4. Dynamics; 5. Finite element analysis; 6. Engineering design, and will be of interest to academics and researchers, designers and engineers dealing with theoretical modelling, computerized analyses and engineering design of thin-walled structures and shell structural elements.

International Journal of Non-Linear Mechanics, 2004

International Journal of Non-Linear Mechanics, 2004

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

International Journal of Solids and Structures, 2014

ABSTRACT It is well known that distribution of displacements through the shell thickness is non-l... more ABSTRACT It is well known that distribution of displacements through the shell thickness is non-linear, in general. We introduce a modified polar decomposition of shell deformation gradient and a vector of deviation from the linear displacement distribution. When strains are assumed to be small, this allows one to propose an explicit definition of the drilling couples which is proportional to tangential components of the deviation vector. The consistent second approximation to the complementary energy density of the geometrically non-linear theory of isotropic elastic shells is constructed. From differentiation of the density we obtain the consistently refined constitutive equations for 2D surface stretch and bending measures. These equations are then inverted for 2D stress resultants and stress couples. The second-order terms in these constitutive equations take consistent account of influence of undeformed midsurface curvatures. The drilling couples are explicitly expressed by the stress couples, undeformed midsurface curvatures, and amplitudes of quadratic part of displacement distribution through the thickness. The drilling couples are shown to be much smaller than the stress couples, and their influence on the stress and strain state of the shell is negligible. However, such very small drilling couples have to be admitted in non-linear analyses of irregular multi-shell structures, e.g. shells with branches, intersections, or technological junctions. In such shell problems six 2D couple resultants are required to preserve the structure of the resultant shell theory at the junctions during entire deformation process.

Shell Structures: Theory and Application, 2013

Shell Structures: Theory and Application, 2013

International Journal of Solids and Structures, 2007

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching a... more We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements. boundary value problem. Such a general, dynamically and kinematically exact, six-scalar-field theory of regular shells, formulated with regard to a non-material weighted surface of mass taken as the shell base surface, was developed by Simmonds (1983, 1998) and , and with regard to a material surface arbitrary located within the shell-like body by Stumpf (1990), Chró ścielewski et al. (1992), and . For this general shell model efficient finite element algorithms were developed and many numerical examples of equilibrium, stability, and dynamics of regular and complex shell structures were presented by Chró ścielewski et al. works it was assumed that the region of shell irregularity (e.g., branching, self-intersection, stiffening, technological junction, etc.) is small as compared with other shell dimensions and its size can be ignored in deriving the resultant 2D equilibrium conditions. However, such an assumption brings an undefinable error into the resultant dynamic continuity conditions formulated along the singular surface curves modelling the irregularity regions. Therefore, such conditions cannot be regarded as exact implications of 3D equilibrium conditions of continuum mechanics.

Thin-Walled Structures, 2015

International Journal of Solids and Structures, 2011

ABSTRACT We construct the unique two-dimensional (2D) kinematics which is work-conjugate to the e... more ABSTRACT We construct the unique two-dimensional (2D) kinematics which is work-conjugate to the exact, resultant local equilibrium conditions of the non-linear theory of branching shells. It is shown that the compatible shell displacements consist of the translation vector and rotation tensor fields defined on the regular parts of the shell base surface as well as independently on the singular surface curve modelling the shell branching. Discussing relations between limits of the translation vector and rotation tensor fields when approaching the singular curve, and analogous fields given only along the singular curve itself, several types of the junctions are described. Among them are the stiff, entirely simply connected and partly simply supported junction as well as the elastically and dissipatively deformable junction, and the non-local elastic junction. For each type of junction the explicit form of the principle of virtual work is derived.

The six-field non-linear theory of elastic shells with the phase transformation of the material i... more The six-field non-linear theory of elastic shells with the phase transformation of the material is developed. Equilibrium conditions are found from the variational principle of stationary total potential energy. New dynamic continuity conditions are derived at the movable singular surface curve modelling the phase interface. Particular forms of the continuity conditions at coherent and incoherent interface curves are given. The results are illustrated by an example of the phase transition in an infinite plate with a circular hole.

International Journal of Solids and Structures, 2007

We show how to determine the midsurface of a deformed thin shell from known geometry of the undef... more We show how to determine the midsurface of a deformed thin shell from known geometry of the undeformed midsurface as well as the surface strains and bendings. The latter two fields are assumed to have been found independently and beforehand by solving the so-called intrinsic field equations of the non-linear theory of thin shells. By the polar decomposition theorem the midsurface deformation gradient is represented as composition of the surface stretch and 3D finite rotation fields. Right and left polar decomposition theorems are discussed. For each decomposition the problem is solved in three steps: a) the stretch field is found by pure algebra, b) the rotation field is obtained by solving a system of first-order PDEs, and c) position of the deformed midsurface follows then by quadratures. The integrability conditions for the rotation field are proved to be equivalent to the compatibility conditions of the non-linear theory of thin shells. Along any path on the undeformed shell midsurface the system of PDEs for the rotation field reduces to the system of linear tensor ODEs identical to the one that describes spherical motion of a rigid body about a fixed point. This allows one to use analytical and numerical methods developed in analytical mechanics that in special cases may lead to closed-form solutions.

Advanced Structured Materials, 2013

Advances in Mechanics and Mathematics, 2010

Definitions of the Lagrangian stretch and wryness tensors in the nonlinear Cosserat continuum are... more Definitions of the Lagrangian stretch and wryness tensors in the nonlinear Cosserat continuum are discussed applying three different methods. The resulting unique strain measures have several distinguishing features and are called the natural ones. They are expressed through the translation vector and either the rotation tensor or various finite rotation vector fields. The relation of the natural strain measures to those proposed in the representative literature is reviewed.

Mathematics and Mechanics of Solids, 2015

Encyclopedia of Thermal Stresses, 2014

Shell Structures. Theory and Applications, Volume contains 104 contributions from over 22 countri... more Shell Structures. Theory and Applications, Volume contains 104 contributions from over 22 countries, reflecting a wide spectrum of scientific and engineering problems of shell structures. The papers are divided into six broad groups: 1. General lectures; 2. Theoretical modeling; 3. Stability; 4. Dynamics; 5. Finite element analysis; 6. Engineering design, and will be of interest to academics and researchers, designers and engineers dealing with theoretical modelling, computerized analyses and engineering design of thin-walled structures and shell structural elements.

International Journal of Non-Linear Mechanics, 2004

International Journal of Non-Linear Mechanics, 2004

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009, 2009