Paweł Szeptyński | Cracow University of Technology (original) (raw)

Papers by Paweł Szeptyński

MATERIALS, 2025

In this study, we investigated the vibration of adhesively bonded composite cantilevers consisti... more In this study, we investigated the vibration of adhesively bonded composite cantilevers consisting of two beech wood lamella and a bondline of flexible polyurethane. The beams had a constant total height, while the thickness of the adhesive layer varied. We analyzed both the driven and free vibration of a single cantilever beam and a cantilever with an additional mass attached to its end. The eigenfrequencies were determined using Fourier analysis of a sweep load response, the response to an impact load excited using
an impact hammer, and the response observed via the manual displacement of the beam’s tip. The system’s damping was estimated according to the recorded logarithmic decrement. Theoretical estimates of the fundamental natural frequency were obtained using the γ-method and employing a linear elastic theory of composite beams. A numerical modal analysis was carried out using the finite element method. Upon comparing the results of our experiments with the numerical estimates and theoretical predictions, a fair agreement was found.

MATERIALS, 2024

This study investigated the optimal shape of glue-laminated timber beams using an analytical mod... more This study investigated the optimal shape of glue-laminated timber beams using an analytical model of a slender beam, taking into account the anisotropy of its strength properties as well as boundary conditions at the oblique bottom face of the beam. A control theory problem was formulated in order to optimize the shape of the modeled beam. Two optimization tasks were considered: minimizing material usage ( V min ) for a fixed load-carrying capacity (LCC) of the beam and maximizing load-bearing capacity ( Q max ) for a given volume of the beam. The optimal solution was found using Pontryagin’s maximum principle (PMP). Optimal shapes were determined using Dircol v. 2.1 software and then adjusted according to a 3D finite element analysis (FEA) performed in Abaqus. The final shapes obtained through this procedure were used in the CNC-based production of three types of nine beams: three reference rectangular beams, three V min beams, and three Q max beams. All specimens were subjected to a four-point bending test. The experimental results were contrasted with theoretical assumptions. Optimization reduced material usage by ca. 12.9% while preserving approximately the same LCC. The maximization of LCC was found to be rather unsuccessful due to the significant dependence of the beams’ response on the highly variable mechanical properties of GLT.

ARCHIVES OF MECHANICS, 2024

The paper presents a linear elastic one-dimensional Discrete Layer–Wise (DLW) analytical model of... more The paper presents a linear elastic one-dimensional Discrete Layer–Wise (DLW) analytical model of a composite girder consisting of two beams bonded together with a layer of a flexible adhesive. The model takes into account both longitudinal and transverse deformation of component beams, the First Order Shear Deformation Theory (FSDT) for these adherends as well as extensibility of the adhesive layer. A system of governing equations is derived and a general solution is found with the use of the method of generalized eigenvectors. Two examples are analyzed both with the use of the considered 1D analytical model and a 3D Finite Element Analysis (FEA) in order to validate predictions of the introduced theory. Satisfactory agreement is found between theoretical and numerical results.

Inżynieria i Budownictwo, 2024

JOURNAL OF THEORETICAL AND APPLIED MECHANICS, 2024

The paper presents a beam theory for composite girders consisting of two beams joined together wi... more The paper presents a beam theory for composite girders consisting of two beams joined together with an adhesive layer. The height of the bottom beam is considered variable. The governing equations are suitable for formulation of a shape optimization problem in terms of control theory. The use of Pontryagin’s maximum principle enables finding an optimal solution satisfying necessary optimality conditions. The presented optimization approach allows for including issues which cannot be accounted for by commercial topology optimization software. The introduced theory provides an estimated solution, which is then validated by an analysis of a 3D finite element model.

Materials, 2023

Creep behavior of Cross-Laminated-Timber (CLT) beams with a finite-thickness layer of flexible ad... more Creep behavior of Cross-Laminated-Timber (CLT) beams with a finite-thickness layer of flexible adhesives is investigated. Creep tests were carried out for all component materials as well as for the composite structure itself. Three-point bending creep tests were performed for spruce planks and for CLT beams, and uniaxial compression creep tests were performed for two flexible polyurethane adhesives: Sika ® PS and Sika ® PMM. All materials are characterized with the use of the three-element Generalized Maxwell Model. The results of creep tests for component materials were used in elaboration of the Finite Element (FE) model. The problem of linear theory of viscoelasticity was solved numerically with the use of the Abaqus software. Obtained results of Finite Element Analysis (FEA) are compared with experimental results.

ARCHIVES OF CIVIL ENGINEERING, 2023

The problem of optimal design of a steel plated girder according to the Eurocode 3 is considered... more The problem of optimal design of a steel plated girder according to the Eurocode 3 is considered. Code regulations admit the Finite Element analysis (FEA) in designing plated structures with variable cross-sections. A technique of determining an approximate solution to the optimization problem is presented. It is determined a solution of a control theory optimization task, in which Eurocode requirements regarding the Ultimate Limit State (bearing capacity, local and global stability) as well as Serviceability Limit State (flexural rigidity) are used as appropriate inequality constraints. Static analysis is performed within the framework of linear elasticity and Bernoulli-Euler beam theory making an account for second-order effects due to prescribed imperfections. Obtained solutions, after regularization, may be used for direct verification with the use of FEA or as the first guess for iterative topology optimization algorithms. Code requirements governing the determination of optimal shape are visualized in the constraint activity diagram, which is a proposed tool for analysis of optimization process.

Composite Structures, 2022

A linear elastic model of a bent multilayer composite beam is considered. Closed-form analytical ... more A linear elastic model of a bent multilayer composite beam is considered. Closed-form analytical solution is derived with the use of method of generalized eigenvectors. Deformation predicted by the analytical model is compared with the results of a three-dimensional linear Finite Element Analysis as well as with the results of experiments performed on three types of multilayer composite beams.

Archives of Metallurgy and Materials, 2000

An Extension of Burzyński Hypothesis of Material Effort Accounting for the Third Invariant of Str... more An Extension of Burzyński Hypothesis of Material Effort Accounting for the Third Invariant of Stress Tensor The aim of the paper is to propose an extension of the Burzyński hypothesis of material effort to account for the influence of the third invariant of stress tensor deviator. In the proposed formulation the contribution of the density of elastic energy of distortion in material effort is controlled by Lode angle. The resulted yield condition is analyzed and possible applications and comparison with the results known in the literature are discussed.

Polymers, 2021

Practical aspects of modelling of flexible adhesives with the energy conjugate measures ofstress ... more Practical aspects of modelling of flexible adhesives with the energy conjugate measures ofstress and strain of the Darijani–Naghdabadi (D-N) family are discussed. A possibility of descriptionof materials exhibiting non-linear physical characteristics with the use of non-linear geometricrelationships and linear elastic constitutive law is considered. Nominal stress vs. stretch relations arespecified in cases of simple tension and simple shear with the use of the Kirchhoff–de Saint-Venantelastic potential and D-N energy conjugate stress and strain measures. Obtained theoretical estimateswere compared with experimental results of simple tension and simple shear tests performed onSika PM polyurethane (Cracow, Sika Poland). The deformation rate was fixed in order to minimizethe influence of viscosity. Values of parametersα,βin the definition of the D-N strain tensor wereoptimized in order to provide good agreement between model predictions and experimental results.Observed discrepancies indicate that the proposed approach is not appropriate for constitutivemodelling of the PM polymer. The presented approach is proposed to be used as a simple designmodel providing practical formulas describing the behavior of materials of non-linear characteristicsin chosen mechanical states. Admissible values of exponentsα,βare discussed regarding its bijectivityin a limited range of variation of principal stretches.

Composite Structures, 2021

This paper discusses the issue of a lap joint in the form of a thinfilm attached to a rigid base ... more This paper discusses the issue of a lap joint in the form of a thinfilm attached to a rigid base with the use of alayer offlexible adhesive exhibiting a non‐linear constitutive relation between shear stress and distortionalstrain, which is approximated by the second‐order power law. Similarity theory is used to perform a qualitativeanalysis of the influence of magnitude of non‐linearity on the performance of the system. The introduced sim-ilarity numbers are proposed to be used in formulation of simple design tools. The analysis is performed bysolving the derived governing equations for characteristic cases in a numerical way combining the fourth‐order Runge‐Kutta algorithm and iterative shooting ruled by a steepest descent method. The results are com-pared with the results of the plane stressfinite element analysis. It can be observed that within considered mod-els the influence of the material non‐linearity is of minor importance and in some cases it may be almostentirely surpassed by local effects resulting from boundary conditions. For relatively small load intensity itbecomes negligibly small.

Composite Structures, 2020

Three analytical one-dimensional linear elastic models of composite laminated beams are considere... more Three analytical one-dimensional linear elastic models of composite laminated beams are considered – composite Bernoulli-Euler beam (BE), composite Timoshenko beam (T) and multilayer sandwich beam model (MS). They are compared with results obtained via finite element method for a two-dimensional model in plane stress state. Overall system stiffness is verified with experimental data obtained for two statical configurations – three-point bending and four-point bending. The first configuration concerned 8 types of three-panel cross-laminated timber (CLT) beams accounting for various materials and thickness of timber panels as well as various materials and thickness of adhesive layer, while the second one concerned 5 types of two-panel aluminium laminated beams accounting for different thickness of adhesive layer. Simplified multilayer sandwich model is found to be in good accordance with FEA results and with experimental data, while simple BE and T models are shown to provide erroneous estimates.

An approximate thickness optimization of a rectangular Kirchhoff-Love plate with variable stiffne... more An approximate thickness optimization of a rectangular Kirchhoff-Love plate with variable stiffness under uniform load is performed in this paper. The authors propose an original method for formulating problems of optimal design for plate structures of variable thickness. Partial discretization, which is described in this paper, reduces the number of independent variables in the problem formulation to only one, making the problem possible to solve via application of the Pontryagin’s minimum principle. The optimization problem relates to the search for the optimal plate thickness distributions, which provides the minimum structural volume of the material used while simultaneously meeting all constraint conditions. The optimal design task is formulated as a control theory problem, maintaining the formal structure of the minimum principle, and then is transformed into a two-point boundary value problem. Such an approximate solution, meeting all necessary optimality conditions, is found by using Dircol software for a chosen illustrative example.

Influence of non-uniformity of cracking on calculation of deflection of reinforced-concrete eleme... more Influence of non-uniformity of cracking on calculation of deflection of reinforced-concrete elements, according to eurocode 2 Wpływ nierównomierności zarysowania na obliczanie ugięć elementów żelbetowych według eurokodu 2 Abstract Formulae for the deflection of two examples of isostatic systems (simply supported beam, cantilever) were derived, accounting for influence of distribution of bending moments on cracking and beam stiffness distribution, according to EC2. Numerical analysis of the problem for the same two examples as well as for two further hyperstatic systems was performed using an iterative FEM algorithm. The influence of non-uniformity of cracking on deflection and distribution of bending moments was shown to be negligible in typical practical design problems, therefore also an estimation of deflection on the basis of the distribution of bending moments and the value of the factor α k obtained from linear solution (before redistribution) is shown to be justified.

A general proposition of an energy-based limit condition for anisotropic materials exhibiting str... more A general proposition of an energy-based limit condition for anisotropic materials exhibiting strength-differential effect (SDE) based on spectral decomposition of elasticity tensors and the use of scaling pressure-dependent functions is specified for the case of orthotropic materials. A detailed algorithm (based on classical solutions of cubic equations) for the determination of elastic ei-genstates and eigenvalues of the orthotropic stiffness tensor is presented. A yield condition is formulated for both two-dimensional and three-dimensional cases. Explicit formulas based on simple strength tests are derived for parameters of criterion in the plane case. The application of both criteria for the description of yielding and plastic deformation of metal sheets is discussed in detail. The plane case criterion is verified with experimental results from the literature.

An investigation for directions of extreme – maximum or minimum – values of the longitudinal and ... more An investigation for directions of extreme – maximum or minimum – values of the longitudinal and transverse stiffness moduli as well as of the limit uniaxial and limit shear stresses in anisotropic linear elastic solids is performed in the paper.
The cases of cubic symmetry (regular crystal system) and of volumetrically isotropic cylindrical symmetry (hexagonal
crystal system with additional constraints) are considered. The systems of non-linear equations for the components
of the versors of investigated directions are derived with use of the spectral decomposition of the elasticity (stiffness and
compliance) tensors

In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It... more In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It is based on the concept of influence functions introduced by Burzyński [3] and on the concept of decomposition of elastic energy density introduced by Rychlewski [18]. A new proposition enables description of a wide class of linearly elastic materials of arbitrary symmetry exhibiting strength differential effect.

General form of yield condition for isotropic and homogeneous bodies is considered in the paper. ... more General form of yield condition for isotropic and homogeneous bodies is considered in the paper. In the space of principal stresses, the limit condition is graphically represented by a proper regular surface which is assumed here to be at least of C2 class. Due to Drucker’s Postulate, the yield surface should be convex. General form of convexity condition of the considered surface is derived using methods of differential geometry. Parametrization of the yield surface is given, the first and the second derivatives of the position vector with respect to the chosen parameters are calculated, what enables determination of the tangent and unit normal vectors at given point, and also determination of the first and the second fundamental form of the considered surface. Finally the Gaussian and mean curvatures, which are given by the coefficients of the first and the second fundamental form as the invariants of the shape operator, are found. Convexity condition of the considered surface expressed in general in terms of the mean and Gaussian curvatures, is formulated for any form of functions determining the character of the surface.

A proposition of an energy-based hypothesis of material effort for isotropic materials exhibiting... more A proposition of an energy-based hypothesis of material effort for isotropic materials exhibiting strength-differential (SD) effect, pressure-sensitivity and Lode angle dependence is discussed. It is a special case of a general hypothesis proposed by the authors in [11] for anisotropic bodies, based on Burzyński’s concept of influence functions [2] and Rychlewski’s concept of elastic energy decomposition [16]. General condition of the convexity of the yield surface
is introduced, and its derivation is given in the second part of the paper. Limit condition is specified for Inconel 718 alloy, referring to the experimental results published by Iyer and Lissenden [7].

Some misstatements appearing in the final form of the failure criterion formulation, derived from... more Some misstatements appearing in the final form of the failure criterion formulation, derived from Burzyński’s hypothesis of material effort for anisotropic bodies, which haven’t been noticed in the literature as yet, are pointed out and discussed. Alternative interpretations of the results obtained by Burzyński are presented. Propositions of different formulation of the failure criterion, basing on original ideas of Burzyński, are given.

MATERIALS, 2025

In this study, we investigated the vibration of adhesively bonded composite cantilevers consisti... more In this study, we investigated the vibration of adhesively bonded composite cantilevers consisting of two beech wood lamella and a bondline of flexible polyurethane. The beams had a constant total height, while the thickness of the adhesive layer varied. We analyzed both the driven and free vibration of a single cantilever beam and a cantilever with an additional mass attached to its end. The eigenfrequencies were determined using Fourier analysis of a sweep load response, the response to an impact load excited using
an impact hammer, and the response observed via the manual displacement of the beam’s tip. The system’s damping was estimated according to the recorded logarithmic decrement. Theoretical estimates of the fundamental natural frequency were obtained using the γ-method and employing a linear elastic theory of composite beams. A numerical modal analysis was carried out using the finite element method. Upon comparing the results of our experiments with the numerical estimates and theoretical predictions, a fair agreement was found.

MATERIALS, 2024

This study investigated the optimal shape of glue-laminated timber beams using an analytical mod... more This study investigated the optimal shape of glue-laminated timber beams using an analytical model of a slender beam, taking into account the anisotropy of its strength properties as well as boundary conditions at the oblique bottom face of the beam. A control theory problem was formulated in order to optimize the shape of the modeled beam. Two optimization tasks were considered: minimizing material usage ( V min ) for a fixed load-carrying capacity (LCC) of the beam and maximizing load-bearing capacity ( Q max ) for a given volume of the beam. The optimal solution was found using Pontryagin’s maximum principle (PMP). Optimal shapes were determined using Dircol v. 2.1 software and then adjusted according to a 3D finite element analysis (FEA) performed in Abaqus. The final shapes obtained through this procedure were used in the CNC-based production of three types of nine beams: three reference rectangular beams, three V min beams, and three Q max beams. All specimens were subjected to a four-point bending test. The experimental results were contrasted with theoretical assumptions. Optimization reduced material usage by ca. 12.9% while preserving approximately the same LCC. The maximization of LCC was found to be rather unsuccessful due to the significant dependence of the beams’ response on the highly variable mechanical properties of GLT.

ARCHIVES OF MECHANICS, 2024

The paper presents a linear elastic one-dimensional Discrete Layer–Wise (DLW) analytical model of... more The paper presents a linear elastic one-dimensional Discrete Layer–Wise (DLW) analytical model of a composite girder consisting of two beams bonded together with a layer of a flexible adhesive. The model takes into account both longitudinal and transverse deformation of component beams, the First Order Shear Deformation Theory (FSDT) for these adherends as well as extensibility of the adhesive layer. A system of governing equations is derived and a general solution is found with the use of the method of generalized eigenvectors. Two examples are analyzed both with the use of the considered 1D analytical model and a 3D Finite Element Analysis (FEA) in order to validate predictions of the introduced theory. Satisfactory agreement is found between theoretical and numerical results.

Inżynieria i Budownictwo, 2024

JOURNAL OF THEORETICAL AND APPLIED MECHANICS, 2024

The paper presents a beam theory for composite girders consisting of two beams joined together wi... more The paper presents a beam theory for composite girders consisting of two beams joined together with an adhesive layer. The height of the bottom beam is considered variable. The governing equations are suitable for formulation of a shape optimization problem in terms of control theory. The use of Pontryagin’s maximum principle enables finding an optimal solution satisfying necessary optimality conditions. The presented optimization approach allows for including issues which cannot be accounted for by commercial topology optimization software. The introduced theory provides an estimated solution, which is then validated by an analysis of a 3D finite element model.

Materials, 2023

Creep behavior of Cross-Laminated-Timber (CLT) beams with a finite-thickness layer of flexible ad... more Creep behavior of Cross-Laminated-Timber (CLT) beams with a finite-thickness layer of flexible adhesives is investigated. Creep tests were carried out for all component materials as well as for the composite structure itself. Three-point bending creep tests were performed for spruce planks and for CLT beams, and uniaxial compression creep tests were performed for two flexible polyurethane adhesives: Sika ® PS and Sika ® PMM. All materials are characterized with the use of the three-element Generalized Maxwell Model. The results of creep tests for component materials were used in elaboration of the Finite Element (FE) model. The problem of linear theory of viscoelasticity was solved numerically with the use of the Abaqus software. Obtained results of Finite Element Analysis (FEA) are compared with experimental results.

ARCHIVES OF CIVIL ENGINEERING, 2023

The problem of optimal design of a steel plated girder according to the Eurocode 3 is considered... more The problem of optimal design of a steel plated girder according to the Eurocode 3 is considered. Code regulations admit the Finite Element analysis (FEA) in designing plated structures with variable cross-sections. A technique of determining an approximate solution to the optimization problem is presented. It is determined a solution of a control theory optimization task, in which Eurocode requirements regarding the Ultimate Limit State (bearing capacity, local and global stability) as well as Serviceability Limit State (flexural rigidity) are used as appropriate inequality constraints. Static analysis is performed within the framework of linear elasticity and Bernoulli-Euler beam theory making an account for second-order effects due to prescribed imperfections. Obtained solutions, after regularization, may be used for direct verification with the use of FEA or as the first guess for iterative topology optimization algorithms. Code requirements governing the determination of optimal shape are visualized in the constraint activity diagram, which is a proposed tool for analysis of optimization process.

Composite Structures, 2022

A linear elastic model of a bent multilayer composite beam is considered. Closed-form analytical ... more A linear elastic model of a bent multilayer composite beam is considered. Closed-form analytical solution is derived with the use of method of generalized eigenvectors. Deformation predicted by the analytical model is compared with the results of a three-dimensional linear Finite Element Analysis as well as with the results of experiments performed on three types of multilayer composite beams.

Archives of Metallurgy and Materials, 2000

An Extension of Burzyński Hypothesis of Material Effort Accounting for the Third Invariant of Str... more An Extension of Burzyński Hypothesis of Material Effort Accounting for the Third Invariant of Stress Tensor The aim of the paper is to propose an extension of the Burzyński hypothesis of material effort to account for the influence of the third invariant of stress tensor deviator. In the proposed formulation the contribution of the density of elastic energy of distortion in material effort is controlled by Lode angle. The resulted yield condition is analyzed and possible applications and comparison with the results known in the literature are discussed.

Polymers, 2021

Practical aspects of modelling of flexible adhesives with the energy conjugate measures ofstress ... more Practical aspects of modelling of flexible adhesives with the energy conjugate measures ofstress and strain of the Darijani–Naghdabadi (D-N) family are discussed. A possibility of descriptionof materials exhibiting non-linear physical characteristics with the use of non-linear geometricrelationships and linear elastic constitutive law is considered. Nominal stress vs. stretch relations arespecified in cases of simple tension and simple shear with the use of the Kirchhoff–de Saint-Venantelastic potential and D-N energy conjugate stress and strain measures. Obtained theoretical estimateswere compared with experimental results of simple tension and simple shear tests performed onSika PM polyurethane (Cracow, Sika Poland). The deformation rate was fixed in order to minimizethe influence of viscosity. Values of parametersα,βin the definition of the D-N strain tensor wereoptimized in order to provide good agreement between model predictions and experimental results.Observed discrepancies indicate that the proposed approach is not appropriate for constitutivemodelling of the PM polymer. The presented approach is proposed to be used as a simple designmodel providing practical formulas describing the behavior of materials of non-linear characteristicsin chosen mechanical states. Admissible values of exponentsα,βare discussed regarding its bijectivityin a limited range of variation of principal stretches.

Composite Structures, 2021

This paper discusses the issue of a lap joint in the form of a thinfilm attached to a rigid base ... more This paper discusses the issue of a lap joint in the form of a thinfilm attached to a rigid base with the use of alayer offlexible adhesive exhibiting a non‐linear constitutive relation between shear stress and distortionalstrain, which is approximated by the second‐order power law. Similarity theory is used to perform a qualitativeanalysis of the influence of magnitude of non‐linearity on the performance of the system. The introduced sim-ilarity numbers are proposed to be used in formulation of simple design tools. The analysis is performed bysolving the derived governing equations for characteristic cases in a numerical way combining the fourth‐order Runge‐Kutta algorithm and iterative shooting ruled by a steepest descent method. The results are com-pared with the results of the plane stressfinite element analysis. It can be observed that within considered mod-els the influence of the material non‐linearity is of minor importance and in some cases it may be almostentirely surpassed by local effects resulting from boundary conditions. For relatively small load intensity itbecomes negligibly small.

Composite Structures, 2020

Three analytical one-dimensional linear elastic models of composite laminated beams are considere... more Three analytical one-dimensional linear elastic models of composite laminated beams are considered – composite Bernoulli-Euler beam (BE), composite Timoshenko beam (T) and multilayer sandwich beam model (MS). They are compared with results obtained via finite element method for a two-dimensional model in plane stress state. Overall system stiffness is verified with experimental data obtained for two statical configurations – three-point bending and four-point bending. The first configuration concerned 8 types of three-panel cross-laminated timber (CLT) beams accounting for various materials and thickness of timber panels as well as various materials and thickness of adhesive layer, while the second one concerned 5 types of two-panel aluminium laminated beams accounting for different thickness of adhesive layer. Simplified multilayer sandwich model is found to be in good accordance with FEA results and with experimental data, while simple BE and T models are shown to provide erroneous estimates.

An approximate thickness optimization of a rectangular Kirchhoff-Love plate with variable stiffne... more An approximate thickness optimization of a rectangular Kirchhoff-Love plate with variable stiffness under uniform load is performed in this paper. The authors propose an original method for formulating problems of optimal design for plate structures of variable thickness. Partial discretization, which is described in this paper, reduces the number of independent variables in the problem formulation to only one, making the problem possible to solve via application of the Pontryagin’s minimum principle. The optimization problem relates to the search for the optimal plate thickness distributions, which provides the minimum structural volume of the material used while simultaneously meeting all constraint conditions. The optimal design task is formulated as a control theory problem, maintaining the formal structure of the minimum principle, and then is transformed into a two-point boundary value problem. Such an approximate solution, meeting all necessary optimality conditions, is found by using Dircol software for a chosen illustrative example.

Influence of non-uniformity of cracking on calculation of deflection of reinforced-concrete eleme... more Influence of non-uniformity of cracking on calculation of deflection of reinforced-concrete elements, according to eurocode 2 Wpływ nierównomierności zarysowania na obliczanie ugięć elementów żelbetowych według eurokodu 2 Abstract Formulae for the deflection of two examples of isostatic systems (simply supported beam, cantilever) were derived, accounting for influence of distribution of bending moments on cracking and beam stiffness distribution, according to EC2. Numerical analysis of the problem for the same two examples as well as for two further hyperstatic systems was performed using an iterative FEM algorithm. The influence of non-uniformity of cracking on deflection and distribution of bending moments was shown to be negligible in typical practical design problems, therefore also an estimation of deflection on the basis of the distribution of bending moments and the value of the factor α k obtained from linear solution (before redistribution) is shown to be justified.

A general proposition of an energy-based limit condition for anisotropic materials exhibiting str... more A general proposition of an energy-based limit condition for anisotropic materials exhibiting strength-differential effect (SDE) based on spectral decomposition of elasticity tensors and the use of scaling pressure-dependent functions is specified for the case of orthotropic materials. A detailed algorithm (based on classical solutions of cubic equations) for the determination of elastic ei-genstates and eigenvalues of the orthotropic stiffness tensor is presented. A yield condition is formulated for both two-dimensional and three-dimensional cases. Explicit formulas based on simple strength tests are derived for parameters of criterion in the plane case. The application of both criteria for the description of yielding and plastic deformation of metal sheets is discussed in detail. The plane case criterion is verified with experimental results from the literature.

An investigation for directions of extreme – maximum or minimum – values of the longitudinal and ... more An investigation for directions of extreme – maximum or minimum – values of the longitudinal and transverse stiffness moduli as well as of the limit uniaxial and limit shear stresses in anisotropic linear elastic solids is performed in the paper.
The cases of cubic symmetry (regular crystal system) and of volumetrically isotropic cylindrical symmetry (hexagonal
crystal system with additional constraints) are considered. The systems of non-linear equations for the components
of the versors of investigated directions are derived with use of the spectral decomposition of the elasticity (stiffness and
compliance) tensors

In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It... more In the paper a new proposition of an energy-based hypothesis of material effort is introduced. It is based on the concept of influence functions introduced by Burzyński [3] and on the concept of decomposition of elastic energy density introduced by Rychlewski [18]. A new proposition enables description of a wide class of linearly elastic materials of arbitrary symmetry exhibiting strength differential effect.

General form of yield condition for isotropic and homogeneous bodies is considered in the paper. ... more General form of yield condition for isotropic and homogeneous bodies is considered in the paper. In the space of principal stresses, the limit condition is graphically represented by a proper regular surface which is assumed here to be at least of C2 class. Due to Drucker’s Postulate, the yield surface should be convex. General form of convexity condition of the considered surface is derived using methods of differential geometry. Parametrization of the yield surface is given, the first and the second derivatives of the position vector with respect to the chosen parameters are calculated, what enables determination of the tangent and unit normal vectors at given point, and also determination of the first and the second fundamental form of the considered surface. Finally the Gaussian and mean curvatures, which are given by the coefficients of the first and the second fundamental form as the invariants of the shape operator, are found. Convexity condition of the considered surface expressed in general in terms of the mean and Gaussian curvatures, is formulated for any form of functions determining the character of the surface.

A proposition of an energy-based hypothesis of material effort for isotropic materials exhibiting... more A proposition of an energy-based hypothesis of material effort for isotropic materials exhibiting strength-differential (SD) effect, pressure-sensitivity and Lode angle dependence is discussed. It is a special case of a general hypothesis proposed by the authors in [11] for anisotropic bodies, based on Burzyński’s concept of influence functions [2] and Rychlewski’s concept of elastic energy decomposition [16]. General condition of the convexity of the yield surface
is introduced, and its derivation is given in the second part of the paper. Limit condition is specified for Inconel 718 alloy, referring to the experimental results published by Iyer and Lissenden [7].

Some misstatements appearing in the final form of the failure criterion formulation, derived from... more Some misstatements appearing in the final form of the failure criterion formulation, derived from Burzyński’s hypothesis of material effort for anisotropic bodies, which haven’t been noticed in the literature as yet, are pointed out and discussed. Alternative interpretations of the results obtained by Burzyński are presented. Propositions of different formulation of the failure criterion, basing on original ideas of Burzyński, are given.

A Short Introduction to the Theory of Plasticity, 2020

Wprowadzenie do teorii plastyczności, 2020

Szczegółowe omówienie podstawowych zagadnień teorii sprężystości., 2020

W szczegółowy sposób omówione zostały założenia ogólnej nieliniowej teorii sprężystości i wyprowa... more W szczegółowy sposób omówione zostały założenia ogólnej nieliniowej teorii sprężystości i wyprowadzenia podstawowych elementów mechaniki ośrodka ciągłego, a także przytoczone zostały dowody fundamentalnych twierdzeń rozważanej teorii w ramach kinematyki, statyki i dynamiki ciał odkształcalnych. Omówiono kwestie formułowania związków konstytutywnych dla ciał izotropowych i anizotropowych zarówno o liniowej, jak i nieliniowej charakterystyce, ze szczególnym uwzględnieniem zagadnienia energetycznego sprzężenia miar naprężenia i odkształcenia oraz zastosowania rozkładu widmowego tensorów sprężystości. Osobno omówiono liniowy wariant teorii z różnymi jego sformułowaniami oraz metodami analitycznymi, m.in. zastosowaniem funkcji naprężeń, potencjałów, metod półodwrotnych oraz analizy zespolonej. Rozważania te poszerzono o opis teorii ogólniejszych – mechanikę ośrodka mikropolarnego Cosseratów oraz mechanikę ośrodka mikromorficznego Mindlina. W ramach teorii liniowej przedstawiono matematyczny opis zjawisk elastooptyki, propagacji płaskich i sferycznych fal sprężystych, a także propagacji sprężystych fal powierzchniowych. Przytoczono i udowodniono twierdzenia wariacyjne teorii liniowej oraz omówiono ich zastosowanie w ramach numerycznych metod rozwiązywania zagadnień tej teorii. Przeanalizowano rozwiązania trzynastu klasycznych zagadnień liniowej teorii sprężystości oraz zagadnień pokrewnych, ze szczególnym uwzględnieniem zagadnień dynamicznych. Całość uzupełniona została rozbudowanym dodatkiem poświęconym rachunkowi tensorowemu i absolutnemu rachunkowi różniczkowemu.

Praca doktorska poświęcona opracowaniu kryterium stanu granicznego dla anizotropowych materiałów ... more Praca doktorska poświęcona opracowaniu kryterium stanu granicznego dla anizotropowych materiałów liniowo-sprężystych wykazujących różnicę wielkości naprężenia granicznego w zależności od jego znaku. Zaproponowane kryterium bazuje na koncepcji Rychlewskiego rozkładu głównego energii sprężystej oraz koncepcji Burzyńskiego wykorzystania zależnych od aktualnego stanu naprężenia funkcji skalujących kombinację elementów tego rozkładu. Zebrano i omówiono klasyczne kryteria dla materiałów anizotropowych oraz przeanalizowano i porównano tzw. "kryteria energetyczne". Przeprowadzono dowód twierdzenia Rychlewskiego dla wielowymiarowej przestrzeni stanów bezpiecznych. Szczegółowo scharakteryzowano poszczególne przypadki symetrii sprężystych oraz odpowiadające im postacie kryterium stanu granicznego.