In this paper we study the flow of a viscous incompressible conducting fluid through a corrugated channel filled with a porous medium. The fluid flow in the channel is under the action of the transverse magnetic field and driven by the pressure drop between the channel?s edges. Using boundary-layer analysis, we derive a higher-order asymptotic model taking into account the inertia and roughness-induced effects on the filtration velocity.
{"title":"A note on the MHD flow in a porous channel","authors":"E. Marušić‐Paloka, Igor Pažanin","doi":"10.2298/tam220103004m","DOIUrl":"https://doi.org/10.2298/tam220103004m","url":null,"abstract":"In this paper we study the flow of a viscous incompressible conducting fluid through a corrugated channel filled with a porous medium. The fluid flow in the channel is under the action of the transverse magnetic field and driven by the pressure drop between the channel?s edges. Using boundary-layer analysis, we derive a higher-order asymptotic model taking into account the inertia and roughness-induced effects on the filtration velocity.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84058050","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Georgios K. Tairidis, I. Ntintakis, G. Drosopoulos, Panagiotis Koutsianitis, G. Stavroulakis
Materials with negative Poisson?s ratio are called auxetics and they present enhanced properties (e.g. damping, indentation resistance, fracture toughness and impact resistance) under external loadings. The auxetic properties are derived from peculiar-shaped microstructures, such as starshaped frames. In the present investigation, several applications are studied using auxetic microstructures. Finite element models are developed for dynamic analysis. First, an application related to auxetic microstructures, for the core of structural panels, is presented. Next, the use of auxetic materials in armor plates in dynamic bullet penetration problems is considered. Finally, a numerical simulation for wind turbines blades, with aluminum foam, polymeric foam and the proposed auxetic material is carried out. The numerical results demonstrate that the use of auxetic microstructures results in improved dynamic response of the system in comparison to conventional materials.
{"title":"Auxetic metamaterials subjected to dynamic loadings","authors":"Georgios K. Tairidis, I. Ntintakis, G. Drosopoulos, Panagiotis Koutsianitis, G. Stavroulakis","doi":"10.2298/tam211103002t","DOIUrl":"https://doi.org/10.2298/tam211103002t","url":null,"abstract":"Materials with negative Poisson?s ratio are called auxetics and they present enhanced properties (e.g. damping, indentation resistance, fracture toughness and impact resistance) under external loadings. The auxetic properties are derived from peculiar-shaped microstructures, such as starshaped frames. In the present investigation, several applications are studied using auxetic microstructures. Finite element models are developed for dynamic analysis. First, an application related to auxetic microstructures, for the core of structural panels, is presented. Next, the use of auxetic materials in armor plates in dynamic bullet penetration problems is considered. Finally, a numerical simulation for wind turbines blades, with aluminum foam, polymeric foam and the proposed auxetic material is carried out. The numerical results demonstrate that the use of auxetic microstructures results in improved dynamic response of the system in comparison to conventional materials.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78608500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Numerical simulations of the dynamics of soft biological tissues are highly non-trivial because tissues generally exhibit complex biological response to external and internal actions, including large deformations and remodeling. Combining the advantages of globally implicit approach (GIA) solvers with the general applicability of the semi-implicit General Plasticity Algorithm (GPA), introduced by some of us some years ago, we present a new, efficient plasticity algorithm, which we call Bio Mechanics Basis Plasticity Algorithm (BMBPA). This is fully implicit, based on a nested Newton solver, and naturally suited for massively parallel computations. The Bilby-Kr?ner-Lee (BKL) multiplicative decomposition of the deformation gradient tensor is employed to introduce the unknowns of our model. We distinguish between global and local unknowns, associated with local and global equations, which are connected by means of a resolution function. The BMBPA asks for very few conditions to be applied and thus can be easily employed to solve several types of biological and biomechanical problems. We demonstrate the efficacy of BMBPA by performing two numerical experiments of a monophasic model of fiber-reinforced tissues. In one case, we consider the shear-compression test of a cubic specimen of tissue, while, in the other case, we focus on the unconfined compression test of a cylinder. The BMBPA is capable of solving the deformation and the remodeling of anisotropic biological tissues by employing a computation time of hours, while the GPA, applied to the same problems as the BMBPA, needs a substantially longer amount of time. All computations were performed in parallel and, within all tests, the performance of the BMBPA displayed substantially higher than the one of the GPA. The results of our simulations permit to study the overall mechanical behavior of the considered tissue and enable further investigations in the field of tissue biomechanics.
{"title":"An efficient algorithm for biomechanical problems based on a fully implicit nested Newton solver","authors":"Markus M. Knodel, Stefano Di, A. Nägel, A. Grillo","doi":"10.2298/tam221115012k","DOIUrl":"https://doi.org/10.2298/tam221115012k","url":null,"abstract":"Numerical simulations of the dynamics of soft biological tissues are highly non-trivial because tissues generally exhibit complex biological response to external and internal actions, including large deformations and remodeling. Combining the advantages of globally implicit approach (GIA) solvers with the general applicability of the semi-implicit General Plasticity Algorithm (GPA), introduced by some of us some years ago, we present a new, efficient plasticity algorithm, which we call Bio Mechanics Basis Plasticity Algorithm (BMBPA). This is fully implicit, based on a nested Newton solver, and naturally suited for massively parallel computations. The Bilby-Kr?ner-Lee (BKL) multiplicative decomposition of the deformation gradient tensor is employed to introduce the unknowns of our model. We distinguish between global and local unknowns, associated with local and global equations, which are connected by means of a resolution function. The BMBPA asks for very few conditions to be applied and thus can be easily employed to solve several types of biological and biomechanical problems. We demonstrate the efficacy of BMBPA by performing two numerical experiments of a monophasic model of fiber-reinforced tissues. In one case, we consider the shear-compression test of a cubic specimen of tissue, while, in the other case, we focus on the unconfined compression test of a cylinder. The BMBPA is capable of solving the deformation and the remodeling of anisotropic biological tissues by employing a computation time of hours, while the GPA, applied to the same problems as the BMBPA, needs a substantially longer amount of time. All computations were performed in parallel and, within all tests, the performance of the BMBPA displayed substantially higher than the one of the GPA. The results of our simulations permit to study the overall mechanical behavior of the considered tissue and enable further investigations in the field of tissue biomechanics.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83026085","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper deals with a body having a random 3D-distribution of two-phase inclusions: spheroidal mutually parallel voids as well as differently oriented reinforcing parallel stiff spheroidal short fibers. By the effective field approach the effective stiffness fourth-order tensor is formulated and found numerically. Simultaneous and sequential embeddings of inclusions are compared. Damage evolution is described by modified Vakulenko?s approach to endochronic thermodynamics. A brief account of the problem of effective elastic symmetry is given. The results of the theory are applied to the damage-elasto-viscoplastic strain of reactor stainless steel AISI 316H.
{"title":"On inelasticity of damaged quasi-rate-independent orthotropic materials","authors":"M. Mićunović, L. Kudrjavceva","doi":"10.2298/tam211007001m","DOIUrl":"https://doi.org/10.2298/tam211007001m","url":null,"abstract":"The paper deals with a body having a random 3D-distribution of two-phase inclusions: spheroidal mutually parallel voids as well as differently oriented reinforcing parallel stiff spheroidal short fibers. By the effective field approach the effective stiffness fourth-order tensor is formulated and found numerically. Simultaneous and sequential embeddings of inclusions are compared. Damage evolution is described by modified Vakulenko?s approach to endochronic thermodynamics. A brief account of the problem of effective elastic symmetry is given. The results of the theory are applied to the damage-elasto-viscoplastic strain of reactor stainless steel AISI 316H.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90795122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A new way of looking at symmetries is proposed, especially regarding their role in the stability of two-body motions in the Newtonian and the Hookean potentials, the two selected by Bertrand?s theorem. The role of the number of spatial dimensions is also addressed.
{"title":"Symmetries and stability of motions in the Newtonian and the Hookean potentials","authors":"C. Carimalo","doi":"10.2298/tam220213005c","DOIUrl":"https://doi.org/10.2298/tam220213005c","url":null,"abstract":"A new way of looking at symmetries is proposed, especially regarding their role in the stability of two-body motions in the Newtonian and the Hookean potentials, the two selected by Bertrand?s theorem. The role of the number of spatial dimensions is also addressed.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72476347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the engineering praxis, Newton?s law of viscosity and Fourier?s heat conduction law are applied to describe thermomechanical processes of fluids. Despite several successful applications, there are some obscure and unexplored details, which are partly answered in this paper using the methodology of irreversible thermodynamics. Liu?s procedure is applied to derive the entropy production rate density, in which positive definiteness is ensured via linear Onsagerian equations; these equations are exactly Newton?s law of viscosity and Fourier?s heat conduction law. The calculations point out that, theoretically, the transport coefficients (thermal conductivity and viscosity) can also depend on the gradient of the state variables in addition to the wellknown dependence of the state variables. This gradient dependency of the transport coefficients can have a significant impact on the modeling of such phenomena as welding, piston effect or shock waves.
{"title":"Gradient-dependent transport coefficients in the Navier-Stokes-Fourier system","authors":"Mátyás Szücs, R. Kovács","doi":"10.2298/tam221005009s","DOIUrl":"https://doi.org/10.2298/tam221005009s","url":null,"abstract":"In the engineering praxis, Newton?s law of viscosity and Fourier?s heat conduction law are applied to describe thermomechanical processes of fluids. Despite several successful applications, there are some obscure and unexplored details, which are partly answered in this paper using the methodology of irreversible thermodynamics. Liu?s procedure is applied to derive the entropy production rate density, in which positive definiteness is ensured via linear Onsagerian equations; these equations are exactly Newton?s law of viscosity and Fourier?s heat conduction law. The calculations point out that, theoretically, the transport coefficients (thermal conductivity and viscosity) can also depend on the gradient of the state variables in addition to the wellknown dependence of the state variables. This gradient dependency of the transport coefficients can have a significant impact on the modeling of such phenomena as welding, piston effect or shock waves.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87949629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Inverse Dynamics is used to calculate the forces and moments in the joints of multibody systems investigated in fields such as Biomechanics or Robotics. In a didactic spirit, this paper begins with an overview of the derivations of the kinematical and dynamical equations of rigid bodies from the point of view of modern Continuum Mechanics. Then, it introduces a matrix formulation for the solution of Inverse Dynamics problems and, finally, reports a simple two-dimensional example of application to a problem in Biomechanics.
{"title":"Inverse dynamics in rigid body mechanics","authors":"S. Federico, Mawafag F. Alhasadi","doi":"10.2298/tam221109011f","DOIUrl":"https://doi.org/10.2298/tam221109011f","url":null,"abstract":"Inverse Dynamics is used to calculate the forces and moments in the joints of multibody systems investigated in fields such as Biomechanics or Robotics. In a didactic spirit, this paper begins with an overview of the derivations of the kinematical and dynamical equations of rigid bodies from the point of view of modern Continuum Mechanics. Then, it introduces a matrix formulation for the solution of Inverse Dynamics problems and, finally, reports a simple two-dimensional example of application to a problem in Biomechanics.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74959778","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The production industries have repeatedly combated the problem of system modelling. Successful control of a system depends mainly on the exactness of the mathematical model that predicts its dynamic. Different types of studies are very common in the complicated challenges involving the estimations and approximations in describing nonlinear machines are based on a variety of studies. This article examines the behaviour and stability of holonomic mechanical system in the arbitrary parameter sets and functional configuration of forces. Differential equations of the behaviour are obtained for the proposed system on the ground of general mechanical theorems, kinetic and potential energies of the system. Lagrange?s equations of the first and second kind are introduced, as well as the representation of the system in the generalized coordinates and in Hamilton?s equations. In addition to the numerical calculations applied the system, the theoretical structures and clarifications on which all of the methods rely on are also presented. Furthermore, static equilibriums are found via two different approaches: graphical and numerical. Above all, stability of motion of undisturbed system and, later, the system that works under the action of an external disturbance was inspected. Finally, the stability of motion is reviewed through Lagrange-Dirichlet theorem, and Routh and Hurwitz criteria. Linearized equations are obtained from the nonlinear ones, and previous conclusions for the stability were proved.
{"title":"Modelling and stability analysis of the nonlinear system","authors":"Mitra Vesović, R. Radulović","doi":"10.2298/tam211101003v","DOIUrl":"https://doi.org/10.2298/tam211101003v","url":null,"abstract":"The production industries have repeatedly combated the problem of system modelling. Successful control of a system depends mainly on the exactness of the mathematical model that predicts its dynamic. Different types of studies are very common in the complicated challenges involving the estimations and approximations in describing nonlinear machines are based on a variety of studies. This article examines the behaviour and stability of holonomic mechanical system in the arbitrary parameter sets and functional configuration of forces. Differential equations of the behaviour are obtained for the proposed system on the ground of general mechanical theorems, kinetic and potential energies of the system. Lagrange?s equations of the first and second kind are introduced, as well as the representation of the system in the generalized coordinates and in Hamilton?s equations. In addition to the numerical calculations applied the system, the theoretical structures and clarifications on which all of the methods rely on are also presented. Furthermore, static equilibriums are found via two different approaches: graphical and numerical. Above all, stability of motion of undisturbed system and, later, the system that works under the action of an external disturbance was inspected. Finally, the stability of motion is reviewed through Lagrange-Dirichlet theorem, and Routh and Hurwitz criteria. Linearized equations are obtained from the nonlinear ones, and previous conclusions for the stability were proved.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74979515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Nevena Rosic, Danilo Karličić, Milan Cajić, M. Lazarevic
Wave attenuation, filtering and guiding is an ongoing topic of scientific research, as there are many opportunities for improvement of existing solutions in modern industry. One of the recent advancements has been made with the use of non-reciprocal metamaterials. Certain properties of metamaterials have made them suitable for use in various engineering fields. In this study, we investigate non-reciprocal wave propagation behavior in coupled thin beams phononics, due to time-modulation of material properties and axial loads. We compare the results for the beams which are interconnected with Winkler?s type of elastic layers and elastic or viscoelastic Pasternak layers. An analytic approach is used to discover directional band gaps and investigate wave propagation through these systems of beams, at relevant excitation frequencies. The proposed framework can be exploited in further analysis of phononic systems based on multiple beams coupled through different mediums and structural elements modeled with higher-order beam theories.
{"title":"Parametrically excited unidirectional wave propagation in thin beam phononics","authors":"Nevena Rosic, Danilo Karličić, Milan Cajić, M. Lazarevic","doi":"10.2298/tam221030010r","DOIUrl":"https://doi.org/10.2298/tam221030010r","url":null,"abstract":"Wave attenuation, filtering and guiding is an ongoing topic of scientific research, as there are many opportunities for improvement of existing solutions in modern industry. One of the recent advancements has been made with the use of non-reciprocal metamaterials. Certain properties of metamaterials have made them suitable for use in various engineering fields. In this study, we investigate non-reciprocal wave propagation behavior in coupled thin beams phononics, due to time-modulation of material properties and axial loads. We compare the results for the beams which are interconnected with Winkler?s type of elastic layers and elastic or viscoelastic Pasternak layers. An analytic approach is used to discover directional band gaps and investigate wave propagation through these systems of beams, at relevant excitation frequencies. The proposed framework can be exploited in further analysis of phononic systems based on multiple beams coupled through different mediums and structural elements modeled with higher-order beam theories.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78010997","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper deals with stability of solution for a one-dimensional model of Rao-Nakra sandwich beam with Kelvin-Voigt damping and time delay given by ?1h1utt ? E1h1uxx ? k(?u + v + awx) ? auxxt ? ?uxxt( ? , t ? ?) = 0, ?3h3vtt ? E3h3vxx + k(?u + v + ?wx) ? bwxxt = 0, ? hwtt + EIwxxxx ? k?(?u + v + ?wx)x ? cwxxt = 0. A sandwich beam is an engineering model that consists of three layers: two stiff outer layers, bottom and top faces, and a more compliant inner layer called ?core layer?. Rao-Nakra system consists of three layers and the assumption is that there is no slip at the interface between contacts. The top and bottom layers are wave equations for the longitudinal displacements under Euler-Bernoulli beam assumptions. The core layer is one equation that describes the transverse displacement under Timoshenko beam assumptions. By using the semigroup theory, the well-posedness is given by applying the Lumer-Phillips Theorem. Exponential stability is proved by employing the Gearhart-Huang-Pr?ss? Theorem.
本文研究具有Kelvin-Voigt阻尼和时滞的Rao-Nakra夹层梁一维模型解的稳定性。E1h1uxx吗?k (?U + v + awx) ?auxxt吗?uxxt (???) = 0， ?3h3vtt ?x + x + kU + v + ?wx ?BWXXT = 0， ?hwtt + EIwxxxx ?k ? (?U + v + ?wx)x ?CWXXT = 0。夹层梁是一种由三层组成的工程模型:两个坚硬的外层，底部和顶部，以及一个更柔顺的内层，称为“核心层”。Rao-Nakra系统由三层组成，假设触点之间的界面没有滑移。顶层和底层是欧拉-伯努利梁假设下纵向位移的波动方程。核心层是在Timoshenko梁假设下描述横向位移的一个方程。利用半群理论，利用Lumer-Phillips定理给出了半群的适定性。利用Gearhart-Huang-Pr?ss?定理。
{"title":"Stability of solution for Rao-Nakra sandwich beam model with Kelvin-Voigt damping and time delay","authors":"V. Cabanillas, C. Raposo, L. Potenciano-Machado","doi":"10.2298/tam210502006c","DOIUrl":"https://doi.org/10.2298/tam210502006c","url":null,"abstract":"This paper deals with stability of solution for a one-dimensional model of Rao-Nakra sandwich beam with Kelvin-Voigt damping and time delay given by ?1h1utt ? E1h1uxx ? k(?u + v + awx) ? auxxt ? ?uxxt( ? , t ? ?) = 0, ?3h3vtt ? E3h3vxx + k(?u + v + ?wx) ? bwxxt = 0, ? hwtt + EIwxxxx ? k?(?u + v + ?wx)x ? cwxxt = 0. A sandwich beam is an engineering model that consists of three layers: two stiff outer layers, bottom and top faces, and a more compliant inner layer called ?core layer?. Rao-Nakra system consists of three layers and the assumption is that there is no slip at the interface between contacts. The top and bottom layers are wave equations for the longitudinal displacements under Euler-Bernoulli beam assumptions. The core layer is one equation that describes the transverse displacement under Timoshenko beam assumptions. By using the semigroup theory, the well-posedness is given by applying the Lumer-Phillips Theorem. Exponential stability is proved by employing the Gearhart-Huang-Pr?ss? Theorem.","PeriodicalId":44059,"journal":{"name":"Theoretical and Applied Mechanics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85609572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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