Pub Date : 2024-07-26DOI: 10.1177/10812865241258377
Vu Thi Ngoc Anh, Pham Chi Vinh, Tran Thanh Tuan
In this paper, we establish the transfer matrix for an orthotropic elastic layer made of nanomaterial that is modeled by the weakly nonlocal elasticity model. This model is proposed recently and different from other existing nonlocal models of elasticity it has been proved to be well-posed for any problem of plane waves. Since the established transfer matrix is totally explicit, it is a convenient tool in investigating various problems of plane waves propagating in layered nonlocal elastic media. To prove this point, we use it to derive the dispersion equation of Lamb waves propagating in a composite plate consisting of two weakly nonlocal orthotropic layers. The dispersion equation in explicit form of Lamb waves has been derived, and it is shown that the derivation of this equation is much more simple than the one using the traditional technique. The obtained explicit dispersion equation will be used to establish techniques monitoring the health of layered nanostructures.
{"title":"Transfer matrix for a weakly nonlocal elastic layer and Lamb waves in layered nonlocal composite plates","authors":"Vu Thi Ngoc Anh, Pham Chi Vinh, Tran Thanh Tuan","doi":"10.1177/10812865241258377","DOIUrl":"https://doi.org/10.1177/10812865241258377","url":null,"abstract":"In this paper, we establish the transfer matrix for an orthotropic elastic layer made of nanomaterial that is modeled by the weakly nonlocal elasticity model. This model is proposed recently and different from other existing nonlocal models of elasticity it has been proved to be well-posed for any problem of plane waves. Since the established transfer matrix is totally explicit, it is a convenient tool in investigating various problems of plane waves propagating in layered nonlocal elastic media. To prove this point, we use it to derive the dispersion equation of Lamb waves propagating in a composite plate consisting of two weakly nonlocal orthotropic layers. The dispersion equation in explicit form of Lamb waves has been derived, and it is shown that the derivation of this equation is much more simple than the one using the traditional technique. The obtained explicit dispersion equation will be used to establish techniques monitoring the health of layered nanostructures.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"19 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776125","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}
Pub Date : 2024-07-26DOI: 10.1177/10812865241258723
Arkadi Berezovski
Depending on whether the experiments are quasi-static or fast dynamics, the measured twin boundary velocity values range from zero to the material’s sound speed. The twin boundary velocity is not yet predicted theoretically in the continuum mechanics framework. The extension of continual description is provided in the paper by means of internal variables. It is shown that a diffusional slow motion of twin boundaries can be represented using a single internal variable. The dual internal variable technique is employed for the description of the fast dynamics of twin boundaries.
{"title":"An internal variable model of macroscopic twin boundary dynamics","authors":"Arkadi Berezovski","doi":"10.1177/10812865241258723","DOIUrl":"https://doi.org/10.1177/10812865241258723","url":null,"abstract":"Depending on whether the experiments are quasi-static or fast dynamics, the measured twin boundary velocity values range from zero to the material’s sound speed. The twin boundary velocity is not yet predicted theoretically in the continuum mechanics framework. The extension of continual description is provided in the paper by means of internal variables. It is shown that a diffusional slow motion of twin boundaries can be represented using a single internal variable. The dual internal variable technique is employed for the description of the fast dynamics of twin boundaries.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"69 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776118","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}
Pub Date : 2024-07-26DOI: 10.1177/10812865241252375
Alexander M Khludnev
We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other.
{"title":"Non-coercive problems for elastic plates with thin junction","authors":"Alexander M Khludnev","doi":"10.1177/10812865241252375","DOIUrl":"https://doi.org/10.1177/10812865241252375","url":null,"abstract":"We consider a non-coercive boundary value problem for two elastic Kirchhoff–Love plates connected to each other by a thin junction. The non-coercivity of the problem is due to the Neumann-type conditions imposed at the external boundaries of the plates. A solution existence is proved for suitable given external forces. Passages to limits are justified as a rigidity parameter of the junction tends to infinity and to zero. We prove that the model corresponding to the first limit case describes an equilibrium of elastic plates with a thin rigid junction; the second limit model fits to the equilibrium state of two elastic plates independent of each other.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"27 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776119","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}
Pub Date : 2024-07-26DOI: 10.1177/10812865241259134
Péter Máté, András Szekrényes
Cylindrical shells curved in only one direction show an interesting behaviour when bent, especially if they remain completely in the elastic region and do not undergo plastic forming. This can be observed in their most common application: the measuring tape. They can be coiled easily because of the loss of stability of their cross-sections, which makes transportation of long shells efficient. This property could be very useful if one could use such a one-way curved shell as a beam, which could be transported and deployed easily. The aim of this study is to observe the behaviour of such a shell, under pure bending load, with special emphasis on the stability loss of the cross-section. In this paper, analytical, semi-analytical, and finite-element methods are used for the description of the shell. The solution derived here uses a variable cross-section Euler–Bernoulli beam model combined with elements of the Kirchhoff plate theory without the shallow shell assumption. It is assumed that the cross-section remains circular and does not change its length. For a universal description, dimensionless parameters and variables are introduced. The semi-analytical investigation revealed that the snap-through ability of the shell may not exist for certain cross-sections which can be presented on a stability map. The derived model reveals the existence of a limiting point between the cross-section deformation modes for larger cross-section angles. In the article, ready-to-use analytical and semi-analytical solutions are given for the critical load and stability map of these shells, which are compared to similar shallow shell models from the literature and the finite-element solution of the problem. The finite-element method also revealed that for a dimensionless description, a length-cross-section radius parameter should be introduced to describe the three-point bending scenario.
{"title":"Stability analysis of thin cylindrical shells under pure and three-point bending","authors":"Péter Máté, András Szekrényes","doi":"10.1177/10812865241259134","DOIUrl":"https://doi.org/10.1177/10812865241259134","url":null,"abstract":"Cylindrical shells curved in only one direction show an interesting behaviour when bent, especially if they remain completely in the elastic region and do not undergo plastic forming. This can be observed in their most common application: the measuring tape. They can be coiled easily because of the loss of stability of their cross-sections, which makes transportation of long shells efficient. This property could be very useful if one could use such a one-way curved shell as a beam, which could be transported and deployed easily. The aim of this study is to observe the behaviour of such a shell, under pure bending load, with special emphasis on the stability loss of the cross-section. In this paper, analytical, semi-analytical, and finite-element methods are used for the description of the shell. The solution derived here uses a variable cross-section Euler–Bernoulli beam model combined with elements of the Kirchhoff plate theory without the shallow shell assumption. It is assumed that the cross-section remains circular and does not change its length. For a universal description, dimensionless parameters and variables are introduced. The semi-analytical investigation revealed that the snap-through ability of the shell may not exist for certain cross-sections which can be presented on a stability map. The derived model reveals the existence of a limiting point between the cross-section deformation modes for larger cross-section angles. In the article, ready-to-use analytical and semi-analytical solutions are given for the critical load and stability map of these shells, which are compared to similar shallow shell models from the literature and the finite-element solution of the problem. The finite-element method also revealed that for a dimensionless description, a length-cross-section radius parameter should be introduced to describe the three-point bending scenario.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"19 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776120","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}
Pub Date : 2024-07-26DOI: 10.1177/10812865241257850
Jacobo Ayensa-Jiménez, Javier Orera-Echeverría, Manuel Doblare
Nonlinear materials are often difficult to model with classical state model theory because they have a complex and sometimes inaccurate physical and mathematical description, or we simply do not know how to describe such materials in terms of relations between external and internal variables. In many disciplines, neural network methods have emerged as powerful tools to identify very complex and nonlinear correlations. In this work, we use the very recently developed concept of physically guided neural networks with internal variables (PGNNIVs) to discover constitutive laws using a model-free approach and training solely with measured force–displacement data. PGNNIVs make a particular use of the physics of the problem to enforce constraints on specific hidden layers and are able to make predictions without internal variable data. We demonstrate that PGNNIVs are capable of predicting both internal and external variables under unseen loading scenarios, regardless of the nature of the material considered (linear, with hardening or softening behavior and hyperelastic), unravelling the constitutive law of the material hence explaining its nature altogether, endowing the method with some explanatory character that distances it from the traditional black box approach.
{"title":"Predicting and explaining nonlinear material response using deep physically guided neural networks with internal variables","authors":"Jacobo Ayensa-Jiménez, Javier Orera-Echeverría, Manuel Doblare","doi":"10.1177/10812865241257850","DOIUrl":"https://doi.org/10.1177/10812865241257850","url":null,"abstract":"Nonlinear materials are often difficult to model with classical state model theory because they have a complex and sometimes inaccurate physical and mathematical description, or we simply do not know how to describe such materials in terms of relations between external and internal variables. In many disciplines, neural network methods have emerged as powerful tools to identify very complex and nonlinear correlations. In this work, we use the very recently developed concept of physically guided neural networks with internal variables (PGNNIVs) to discover constitutive laws using a model-free approach and training solely with measured force–displacement data. PGNNIVs make a particular use of the physics of the problem to enforce constraints on specific hidden layers and are able to make predictions without internal variable data. We demonstrate that PGNNIVs are capable of predicting both internal and external variables under unseen loading scenarios, regardless of the nature of the material considered (linear, with hardening or softening behavior and hyperelastic), unravelling the constitutive law of the material hence explaining its nature altogether, endowing the method with some explanatory character that distances it from the traditional black box approach.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"69 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776124","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}
Pub Date : 2024-07-25DOI: 10.1177/10812865241257268
Luis Dorfmann, José Merodio, Raimondo Penta, Prashant Saxena
{"title":"In recognition of the 80th birthday of Ray Ogden","authors":"Luis Dorfmann, José Merodio, Raimondo Penta, Prashant Saxena","doi":"10.1177/10812865241257268","DOIUrl":"https://doi.org/10.1177/10812865241257268","url":null,"abstract":"","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"53 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776127","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}
Pub Date : 2024-06-22DOI: 10.1177/10812865241257534
Xu Wang, Peter Schiavone
We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.
{"title":"A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion","authors":"Xu Wang, Peter Schiavone","doi":"10.1177/10812865241257534","DOIUrl":"https://doi.org/10.1177/10812865241257534","url":null,"abstract":"We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"2 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505237","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}
Pub Date : 2024-06-21DOI: 10.1177/10812865241250015
Xin-Lin Gao
Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.
{"title":"Critical velocities of a three-layer composite tube incorporating the rotary inertia and material anisotropy","authors":"Xin-Lin Gao","doi":"10.1177/10812865241250015","DOIUrl":"https://doi.org/10.1177/10812865241250015","url":null,"abstract":"Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"24 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529206","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}
Pub Date : 2024-06-02DOI: 10.1177/10812865241253208
Rashmi Prasad, Roushan Kumar
This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining.
{"title":"Some qualitative results in hyperbolic two-temperature generalized thermoelasticity","authors":"Rashmi Prasad, Roushan Kumar","doi":"10.1177/10812865241253208","DOIUrl":"https://doi.org/10.1177/10812865241253208","url":null,"abstract":"This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"35 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253819","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}
Pub Date : 2024-06-01DOI: 10.1177/10812865241253520
Xinze Guo, Kemin Zhou
In the design optimization of fiber-reinforced composites, spatial material discontinuity is considered intractable within the manufacturing reality imposed by advanced technologies. This paper presents a topological optimization framework based on truss-like material to design composite structures with continuous fiber. Specifically, the fiber morphology at the scattered design points, which controls the orientation and volume fraction, is taken as design variables. Using the Shepard interpolant scheme, the fiber morphology at any given computational point is interpolated by scattered design variables within a certain circular influence domain. The employed interpolation inherently ensures the spatial continuity and range-restricted of the physical field in an element-independent manner. Since separating the design variable field and analysis mesh on two independent sets of points, this method is well suited for using a sparse design variable field. The computational savings are compelling due to the reduced number of design variables without significantly restricting the design freedom. Numerical instability such as checkerboard and mesh dependencies vanished as no intermediate densities are suppressed in optimization. The continuous fiber-reinforced composites (CFRCs) in the form of truss-like continua are ready to be manufactured with the aid of a simple post-processing. Several numerical examples are investigated to demonstrate the feasibility and effectiveness of the proposed formulation and numerical techniques.
{"title":"Topology optimization of continuous fiber-reinforced composites using Shepard interpolation and its design variable reduction","authors":"Xinze Guo, Kemin Zhou","doi":"10.1177/10812865241253520","DOIUrl":"https://doi.org/10.1177/10812865241253520","url":null,"abstract":"In the design optimization of fiber-reinforced composites, spatial material discontinuity is considered intractable within the manufacturing reality imposed by advanced technologies. This paper presents a topological optimization framework based on truss-like material to design composite structures with continuous fiber. Specifically, the fiber morphology at the scattered design points, which controls the orientation and volume fraction, is taken as design variables. Using the Shepard interpolant scheme, the fiber morphology at any given computational point is interpolated by scattered design variables within a certain circular influence domain. The employed interpolation inherently ensures the spatial continuity and range-restricted of the physical field in an element-independent manner. Since separating the design variable field and analysis mesh on two independent sets of points, this method is well suited for using a sparse design variable field. The computational savings are compelling due to the reduced number of design variables without significantly restricting the design freedom. Numerical instability such as checkerboard and mesh dependencies vanished as no intermediate densities are suppressed in optimization. The continuous fiber-reinforced composites (CFRCs) in the form of truss-like continua are ready to be manufactured with the aid of a simple post-processing. Several numerical examples are investigated to demonstrate the feasibility and effectiveness of the proposed formulation and numerical techniques.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"42 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141193874","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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