Abstract In this paper, a closed form solution is proposed on bending of uniformly loaded rectangular plates with adjacent edges clamped and the two others simply supported (CCSS). Deflection surface can be expressed by a double sine series referred to as Navier solution. Numerical results of deflection, bending moments agree with existing solutions including solutions of finite element model (FEM).
{"title":"A closed form solution for uniformly loaded rectangular plates with adjacent edges clamped and the two others simply supported (CCSS)","authors":"Ming Jin","doi":"10.1002/zamm.202200583","DOIUrl":"https://doi.org/10.1002/zamm.202200583","url":null,"abstract":"Abstract In this paper, a closed form solution is proposed on bending of uniformly loaded rectangular plates with adjacent edges clamped and the two others simply supported (CCSS). Deflection surface can be expressed by a double sine series referred to as Navier solution. Numerical results of deflection, bending moments agree with existing solutions including solutions of finite element model (FEM).","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"4 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135136732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the current study, wave analysis in porous thermoelastic plates with microtemperature subject to insulated or isothermal boundaries is the main focus. The governing equations, after expressing for the two‐dimensional case, are converted into nondimensional quantities. For further simplification, potential functions have been used. The normal mode analysis technique is applied to solve the problem. This technique is an influential methodology that furnishes suitable solutions with no presumed limitations. For stress‐free insulated, or isothermal boundaries, the frequency equations for symmetric and skew‐symmetric modes of wave propagation are derived. The attributes of waves (phase velocity, attenuation coefficient, specific loss, and penetration depth) are computed numerically and presented graphically to show the impacts of porosity and microtemperature. Particular cases of porous thermoelastic plates and thermoelastic plates with microtemperatures are presented. The present problem provides a theoretical foundation for the design and development of new composite materials and surface acoustic devices.
{"title":"Wave analysis in porous thermoelastic plate with microtemperature","authors":"Pooja Rani, Rajneesh Kumar, Geeta Partap","doi":"10.1002/zamm.202300558","DOIUrl":"https://doi.org/10.1002/zamm.202300558","url":null,"abstract":"Abstract In the current study, wave analysis in porous thermoelastic plates with microtemperature subject to insulated or isothermal boundaries is the main focus. The governing equations, after expressing for the two‐dimensional case, are converted into nondimensional quantities. For further simplification, potential functions have been used. The normal mode analysis technique is applied to solve the problem. This technique is an influential methodology that furnishes suitable solutions with no presumed limitations. For stress‐free insulated, or isothermal boundaries, the frequency equations for symmetric and skew‐symmetric modes of wave propagation are derived. The attributes of waves (phase velocity, attenuation coefficient, specific loss, and penetration depth) are computed numerically and presented graphically to show the impacts of porosity and microtemperature. Particular cases of porous thermoelastic plates and thermoelastic plates with microtemperatures are presented. The present problem provides a theoretical foundation for the design and development of new composite materials and surface acoustic devices.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135725545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Vitaly Nikolaevich Paimushin, Vyacheslav Anatolievich Firsov, Viktor Mikhailovich Shishkin, Ruslan Kamilevich Gazizullin
Abstract A method of experimental study of forced bending vibrations of unfixed spans of continuous thin‐walled structural element fixed on rigid support elements of finite dimensions has been developed using instrumental means for registration of vibration acceleration amplitudes. The technique is realized on beams with different types of their fixing under the loading of one of the beam section with an external force that varies in time according to a harmonic law. Experimental dependences of amplitude values of vibration accelerations in specific points of certain spans of the beam on the frequency of external load have been obtained. These results indicate the passage of vibrations through the fixed sections due to the transformation of bending vibration modes of the loaded section into longitudinal transverse‐shift vibrations of the fixed sections during the transition across the boundary from the unfixed section to the fixed one (from the fixed to the unfixed section). For the theoretical study of the described phenomenon a transformational model of the flat beam deformation is constructed on the basis of the refined Timoshenko model with additional account of the deformability of the section with fixation to a rigid support element of finite length on one of its face surfaces. The corresponding equations of motion of the unfixed and fixed sections are derived. The corresponding kinematic and force conditions of their conjugation are formulated. On their basis, analytical solutions of two specific problems are found in the linear approximation. One‐dimensional finite elements for modeling the dynamic response of continuous beam of the considered class are also constructed. Numerical experiments have been carried out for a flat beam made of aluminum alloys D16AT, AMg‐6 and unidirectional fiber reinforced plastic. The presence of a significant transformation of the stress‐strain state parameters of the considered beams at the transition across the boundary from unfixed to fixed sections and a significant increase in the level of transverse tangential stresses on the fixed section of the beam in the vicinity of the interface of the unfixed section with the fixed one have been revealed.
{"title":"Transformational deformation models of continuous thin‐walled structural elements with support elements of finite sizes: Theoretical foundations, computational, and physical experiments","authors":"Vitaly Nikolaevich Paimushin, Vyacheslav Anatolievich Firsov, Viktor Mikhailovich Shishkin, Ruslan Kamilevich Gazizullin","doi":"10.1002/zamm.202300214","DOIUrl":"https://doi.org/10.1002/zamm.202300214","url":null,"abstract":"Abstract A method of experimental study of forced bending vibrations of unfixed spans of continuous thin‐walled structural element fixed on rigid support elements of finite dimensions has been developed using instrumental means for registration of vibration acceleration amplitudes. The technique is realized on beams with different types of their fixing under the loading of one of the beam section with an external force that varies in time according to a harmonic law. Experimental dependences of amplitude values of vibration accelerations in specific points of certain spans of the beam on the frequency of external load have been obtained. These results indicate the passage of vibrations through the fixed sections due to the transformation of bending vibration modes of the loaded section into longitudinal transverse‐shift vibrations of the fixed sections during the transition across the boundary from the unfixed section to the fixed one (from the fixed to the unfixed section). For the theoretical study of the described phenomenon a transformational model of the flat beam deformation is constructed on the basis of the refined Timoshenko model with additional account of the deformability of the section with fixation to a rigid support element of finite length on one of its face surfaces. The corresponding equations of motion of the unfixed and fixed sections are derived. The corresponding kinematic and force conditions of their conjugation are formulated. On their basis, analytical solutions of two specific problems are found in the linear approximation. One‐dimensional finite elements for modeling the dynamic response of continuous beam of the considered class are also constructed. Numerical experiments have been carried out for a flat beam made of aluminum alloys D16AT, AMg‐6 and unidirectional fiber reinforced plastic. The presence of a significant transformation of the stress‐strain state parameters of the considered beams at the transition across the boundary from unfixed to fixed sections and a significant increase in the level of transverse tangential stresses on the fixed section of the beam in the vicinity of the interface of the unfixed section with the fixed one have been revealed.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818726","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract New exact solutions to large constrained torsion of highly elastic soft tubes are obtained from a new hyper‐elastic constitutive model. This new model is established with a new form of the multiaxial elastic potential toward simultaneously matching multiple data sets for three benchmark modes, including the uniaxial extension of a bar, the equi‐biaxial extension of a thin plate and the plane‐strain extension of a plate strip. With the new model, exact closed‐form solutions to the stress responses for these benchmark modes are first derived and shown to be of decoupled nature, and, then, an accurate analysis for the converse Poynting effect of twisted highly elastic tubes with fixed ends is presented with a new technique in treating nonlinear coupling complexities. The main novelties are incorporated as follows. First, multiple data sets for three benchmark modes can be accurately and automatically matched with no need to identify any unknown parameters; next, the nonlinear coupled system of the governing equations for four variables in a twisted tube, including the averaged radius and the wall‐thickness as well as the axial constrained stress and the shear stress, can be reduced to treating a single nonlinear equation; and, accordingly, the nonlinear coupling complexities can be worked out and exact results are obtainable.
{"title":"An accurate and parameter‐free analysis for the converse Poynting effect in large constrained torsion of highly elastic soft tubes","authors":"Long‐Xu Tan, Jia Kang, Quan‐Pu Liu, Si‐Yu Wang, Otto Bruhns, Heng Xiao","doi":"10.1002/zamm.202300532","DOIUrl":"https://doi.org/10.1002/zamm.202300532","url":null,"abstract":"Abstract New exact solutions to large constrained torsion of highly elastic soft tubes are obtained from a new hyper‐elastic constitutive model. This new model is established with a new form of the multiaxial elastic potential toward simultaneously matching multiple data sets for three benchmark modes, including the uniaxial extension of a bar, the equi‐biaxial extension of a thin plate and the plane‐strain extension of a plate strip. With the new model, exact closed‐form solutions to the stress responses for these benchmark modes are first derived and shown to be of decoupled nature, and, then, an accurate analysis for the converse Poynting effect of twisted highly elastic tubes with fixed ends is presented with a new technique in treating nonlinear coupling complexities. The main novelties are incorporated as follows. First, multiple data sets for three benchmark modes can be accurately and automatically matched with no need to identify any unknown parameters; next, the nonlinear coupled system of the governing equations for four variables in a twisted tube, including the averaged radius and the wall‐thickness as well as the axial constrained stress and the shear stress, can be reduced to treating a single nonlinear equation; and, accordingly, the nonlinear coupling complexities can be worked out and exact results are obtainable.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"26 18","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We study a mathematical model describing 3D viscoelastic fluids with memory, fractional viscosity, and regularized by means of a horizontal anisotropic filter. This regularization is obtained through the action of the inverse of the horizontal Helmholtz operator, and the system is considered in a fully periodic space‐domain Ω. After introducing a controlled version of such a model, we take into account for it a suitable Galerkin approximation scheme. Exploiting the Hilbert uniqueness method, we establish the exact controllability of the finite‐dimensional Galerkin system.
{"title":"On the exact controllability of a Galerkin scheme for 3D viscoelastic fluids with fractional Laplacian viscosity and anisotropic filtering","authors":"Luca Bisconti, Davide Catania","doi":"10.1002/zamm.202300056","DOIUrl":"https://doi.org/10.1002/zamm.202300056","url":null,"abstract":"Abstract We study a mathematical model describing 3D viscoelastic fluids with memory, fractional viscosity, and regularized by means of a horizontal anisotropic filter. This regularization is obtained through the action of the inverse of the horizontal Helmholtz operator, and the system is considered in a fully periodic space‐domain Ω. After introducing a controlled version of such a model, we take into account for it a suitable Galerkin approximation scheme. Exploiting the Hilbert uniqueness method, we establish the exact controllability of the finite‐dimensional Galerkin system.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"33 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The surface effect significantly affects the mechanical response of nanoscale materials. In this work, we introduce Gurtin‐Murdoch surface elasticity theory into Mindlin plate theory to study the axisymmetric vibration characteristic of graded porous circular nanoplate in a temperature field. The main structure is a porous graded material with uniform or non‐uniform porosity distribution in its thickness. The numerical shooting technique was applied to solve the governing differential equations derived from Hamilton's principle. Responses for the vibration characteristic of the graded porous nanoplate for both clamped and simply supported boundary conditions were obtained. The numerical results show that when the surface effect is removed, the classical results of Mindlin circular plate will be obtained. Natural frequencies and mode shapes varying with thermal and porosity coefficients were present graphically. And then the effects of surface material properties on the axisymmetric vibration characteristic of the nanoplates were discussed in detail. The parametric study examined the influence of porosity coefficient, porosity distribution mode, surface elastic parameters, and residual surface stress on the vibration characteristics of porous circular nanoplates.
{"title":"The influence of surface effects on axisymmetric vibration of graded porous Mindlin circular nanoplate in a temperature field","authors":"Qinglu Li, Haikun Zhang, Siyao Wang, Jinghua Zhang","doi":"10.1002/zamm.202300590","DOIUrl":"https://doi.org/10.1002/zamm.202300590","url":null,"abstract":"Abstract The surface effect significantly affects the mechanical response of nanoscale materials. In this work, we introduce Gurtin‐Murdoch surface elasticity theory into Mindlin plate theory to study the axisymmetric vibration characteristic of graded porous circular nanoplate in a temperature field. The main structure is a porous graded material with uniform or non‐uniform porosity distribution in its thickness. The numerical shooting technique was applied to solve the governing differential equations derived from Hamilton's principle. Responses for the vibration characteristic of the graded porous nanoplate for both clamped and simply supported boundary conditions were obtained. The numerical results show that when the surface effect is removed, the classical results of Mindlin circular plate will be obtained. Natural frequencies and mode shapes varying with thermal and porosity coefficients were present graphically. And then the effects of surface material properties on the axisymmetric vibration characteristic of the nanoplates were discussed in detail. The parametric study examined the influence of porosity coefficient, porosity distribution mode, surface elastic parameters, and residual surface stress on the vibration characteristics of porous circular nanoplates.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135870855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We analyse a numerical scheme for a system arising from a novel description of the standard elastic–perfectly plastic response. The elastic–perfectly plastic response is described via rate‐type equations that do not make use of the standard elastic‐plastic decomposition, and the model does not require the use of variational inequalities. Furthermore, the model naturally includes the evolution equation for temperature. We present a low order discretisation based on the finite element method. Under certain restrictions on the mesh we subsequently prove the existence of discrete solutions, and we discuss the stability properties of the numerical scheme. The analysis is supplemented with computational examples.
{"title":"Numerical approximation of a thermodynamically complete rate‐type model for the elastic–perfectly plastic response","authors":"Pablo Alexei Gazca‐Orozco, Vít Průša, Karel Tůma","doi":"10.1002/zamm.202300030","DOIUrl":"https://doi.org/10.1002/zamm.202300030","url":null,"abstract":"Abstract We analyse a numerical scheme for a system arising from a novel description of the standard elastic–perfectly plastic response. The elastic–perfectly plastic response is described via rate‐type equations that do not make use of the standard elastic‐plastic decomposition, and the model does not require the use of variational inequalities. Furthermore, the model naturally includes the evolution equation for temperature. We present a low order discretisation based on the finite element method. Under certain restrictions on the mesh we subsequently prove the existence of discrete solutions, and we discuss the stability properties of the numerical scheme. The analysis is supplemented with computational examples.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":" 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136158671","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present study examines the lateral thermal vibrations of a nanobeam subjected to an axial motion while being influenced by a sinusoidal thermal load. The partial differential equation describing the system was obtained using the extended Hamiltonian principle. Also considered were the Euler‐Bernoulli (EB) beam, the nonlocal couple stress theory, and Eringen's nonlocal elasticity model. This type of axially moving beam has multiple applications in design, including industrial, civil, structural, chemical, and computer engineering. The Laplace transform technique is utilized to translate partial differential equations into a thermoelastic differential equation of the sixth order. This study investigates the impact of nanobeam size and velocity on thermo‐mechanical characteristics. To explore the impacts of axial velocity, pulse width, nonlocal index, material length scale coefficient, and phase lag coefficients on the examined studied fields, such as lateral vibration and temperature change for the moving nanobeam are calculated. The specified factors were discovered to impact the flexibility and dynamic response of the nanobeam substantially.
{"title":"Effect of non‐local modified couple stress theory on the responses of axially moving thermoelastic nano‐beams","authors":"Ahmed E. Abouelregal","doi":"10.1002/zamm.202200233","DOIUrl":"https://doi.org/10.1002/zamm.202200233","url":null,"abstract":"Abstract The present study examines the lateral thermal vibrations of a nanobeam subjected to an axial motion while being influenced by a sinusoidal thermal load. The partial differential equation describing the system was obtained using the extended Hamiltonian principle. Also considered were the Euler‐Bernoulli (EB) beam, the nonlocal couple stress theory, and Eringen's nonlocal elasticity model. This type of axially moving beam has multiple applications in design, including industrial, civil, structural, chemical, and computer engineering. The Laplace transform technique is utilized to translate partial differential equations into a thermoelastic differential equation of the sixth order. This study investigates the impact of nanobeam size and velocity on thermo‐mechanical characteristics. To explore the impacts of axial velocity, pulse width, nonlocal index, material length scale coefficient, and phase lag coefficients on the examined studied fields, such as lateral vibration and temperature change for the moving nanobeam are calculated. The specified factors were discovered to impact the flexibility and dynamic response of the nanobeam substantially.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135216799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Two weakly nonlinear integral equations of motion commonly used in the literature to study the nonlinear dynamics of straight and initially not rectilinear Euler‐Bernoulli beams, respectively, are further investigated. Attention is focused on the process known as “static condensation”, which consists of neglecting the axial inertia in the exact, fully nonlinear system of equations of motion to determine the axial displacement as a function of the transversal one. The novelty of the paper relies on showing that, contrarily to expectation and somehow surprisingly, the integral equation for beams with a not rectilinear initial configuration cannot be obtained by the static condensation process starting from the exact, fully nonlinear, equations of motion, apart from a very particular and specific case. On the contrary, it is confirmed the well‐known result that for rectilinear beams the integral equation can be obtained by the static condensation. This highlights a major difference between the two integral equations in terms of reliability and allows us a better understanding of the integral equation of rectilinear beams, underlying its stronger mathematical background than the classical counterpart for not straight beams (i.e., its being obtainable from the exact, fully nonlinear, equations of motion via static condensation), which provides it with a “special” behavior and makes it more trustworthy.
{"title":"On the static condensation of initially not rectilinear beams","authors":"Stefano Lenci, Sergey Sorokin","doi":"10.1002/zamm.202300668","DOIUrl":"https://doi.org/10.1002/zamm.202300668","url":null,"abstract":"Abstract Two weakly nonlinear integral equations of motion commonly used in the literature to study the nonlinear dynamics of straight and initially not rectilinear Euler‐Bernoulli beams, respectively, are further investigated. Attention is focused on the process known as “static condensation”, which consists of neglecting the axial inertia in the exact, fully nonlinear system of equations of motion to determine the axial displacement as a function of the transversal one. The novelty of the paper relies on showing that, contrarily to expectation and somehow surprisingly, the integral equation for beams with a not rectilinear initial configuration cannot be obtained by the static condensation process starting from the exact, fully nonlinear, equations of motion, apart from a very particular and specific case. On the contrary, it is confirmed the well‐known result that for rectilinear beams the integral equation can be obtained by the static condensation. This highlights a major difference between the two integral equations in terms of reliability and allows us a better understanding of the integral equation of rectilinear beams, underlying its stronger mathematical background than the classical counterpart for not straight beams (i.e., its being obtainable from the exact, fully nonlinear, equations of motion via static condensation), which provides it with a “special” behavior and makes it more trustworthy.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"100 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135168766","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The present research investigates lateral stability of a functionally graded nanobeam using Eringen's differential nonlocal elasticity model under rigid (clamped, pinned, free) and deformable (lateral, rotational restraints) boundary conditions. Sigmoid and power law have been employed as grading laws to study the influence of the material distribution on the snap‐buckling analysis of a nanobeam with arbitrary boundary conditions. Moreover, Fourier sine series with Stokes’ transformation are employed to investigate the effects of boundary conditions on the stability response of nanobeams embedded in a Pasternak foundation. A parametric study has been performed to investigate the effect of deformable boundaries, Pasternak foundation and small‐scale parameters on the stability response of the nanobeam and the results have been presented in a series of tables and figures. It has been observed that consideration of the small‐scale parameter, Pasternak foundation, deformable boundaries and functionally grading index (of sigmoid and power‐law) are essential while analyzing the static stability response. The obtained analytical results may be used as benchmarks in future researches of functionally graded nanobeams embedded in an elastic medium.
{"title":"Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions","authors":"Büşra Uzun, Mustafa Özgür Yaylı","doi":"10.1002/zamm.202300569","DOIUrl":"https://doi.org/10.1002/zamm.202300569","url":null,"abstract":"Abstract The present research investigates lateral stability of a functionally graded nanobeam using Eringen's differential nonlocal elasticity model under rigid (clamped, pinned, free) and deformable (lateral, rotational restraints) boundary conditions. Sigmoid and power law have been employed as grading laws to study the influence of the material distribution on the snap‐buckling analysis of a nanobeam with arbitrary boundary conditions. Moreover, Fourier sine series with Stokes’ transformation are employed to investigate the effects of boundary conditions on the stability response of nanobeams embedded in a Pasternak foundation. A parametric study has been performed to investigate the effect of deformable boundaries, Pasternak foundation and small‐scale parameters on the stability response of the nanobeam and the results have been presented in a series of tables and figures. It has been observed that consideration of the small‐scale parameter, Pasternak foundation, deformable boundaries and functionally grading index (of sigmoid and power‐law) are essential while analyzing the static stability response. The obtained analytical results may be used as benchmarks in future researches of functionally graded nanobeams embedded in an elastic medium.","PeriodicalId":23924,"journal":{"name":"Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik","volume":"8 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135567647","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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