Pub Date : 2024-01-04DOI: 10.1177/10812865231213321
KP Soldatos
{"title":"Author’s response to Shariff et al. [1]: Basic errors in couple-stress hyperelasticity articles","authors":"KP Soldatos","doi":"10.1177/10812865231213321","DOIUrl":"https://doi.org/10.1177/10812865231213321","url":null,"abstract":"","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"49 3","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139386596","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 : 2024-01-04DOI: 10.1177/10812865231213311
M. Shariff, R. Bustamante, J. Merodio
We highlight the basic errors found in a related set of couple-stress hyperelasticity articles and evince some statements are incorrect, which suggest that the results obtained in these set of articles are questionable.
{"title":"Basic errors in couple-stress hyperelasticity articles","authors":"M. Shariff, R. Bustamante, J. Merodio","doi":"10.1177/10812865231213311","DOIUrl":"https://doi.org/10.1177/10812865231213311","url":null,"abstract":"We highlight the basic errors found in a related set of couple-stress hyperelasticity articles and evince some statements are incorrect, which suggest that the results obtained in these set of articles are questionable.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"51 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139385079","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 : 2024-01-03DOI: 10.1177/10812865231214262
Heiko Topol, Alejandro Font, Andrey Melnikov, Jesús Lacalle, M. Stoffel, J. Merodio
We consider the bifurcation and post-bifurcation of an extended and inflated circular cylindrical membrane under limited extensibility of its constituents. First, for illustration of the limited extensibility effect, a membrane made of the (isotropic) Gent model is briefly analyzed. Second, the membrane is considered to be made of an isotropic ground substance reinforced with fibers symmetrically arranged in two helically distributed families which are mechanically equivalent. The mechanical behavior of the fibers in a similar way to the (isotropic) Gent model is taken to reflect fiber limited extensibility. In particular, the materials under consideration are NH models augmented with two functions called reinforcing models, each one accounting for unidirectional reinforcement. For a specific material, since both families are mechanically equivalent, the reinforcing models are equal. The nature of the anisotropy considered, i.e., the reinforcing model, can depend only on the stretch in the fiber direction or can depend on the fiber stretch and also can have an influence on the shear response of the material (i.e., it also captures shearing in the fiber direction). The limitation associated with the material anisotropic deformability as well as the arrangement of the material constituents is discussed with respect to the initiation of bulging and necking for the membrane. The subsequent consequences for configurations in equilibrium during post-bifurcation are also studied in detail and a variety of results are given in terms of the limited extensibility (deformability) of the fibers as well as their mechanical response and arrangement (fiber winding angle).
{"title":"On the inflation, bulging/necking bifurcation and post-bifurcation of a cylindrical membrane under limited extensibility of its constituents","authors":"Heiko Topol, Alejandro Font, Andrey Melnikov, Jesús Lacalle, M. Stoffel, J. Merodio","doi":"10.1177/10812865231214262","DOIUrl":"https://doi.org/10.1177/10812865231214262","url":null,"abstract":"We consider the bifurcation and post-bifurcation of an extended and inflated circular cylindrical membrane under limited extensibility of its constituents. First, for illustration of the limited extensibility effect, a membrane made of the (isotropic) Gent model is briefly analyzed. Second, the membrane is considered to be made of an isotropic ground substance reinforced with fibers symmetrically arranged in two helically distributed families which are mechanically equivalent. The mechanical behavior of the fibers in a similar way to the (isotropic) Gent model is taken to reflect fiber limited extensibility. In particular, the materials under consideration are NH models augmented with two functions called reinforcing models, each one accounting for unidirectional reinforcement. For a specific material, since both families are mechanically equivalent, the reinforcing models are equal. The nature of the anisotropy considered, i.e., the reinforcing model, can depend only on the stretch in the fiber direction or can depend on the fiber stretch and also can have an influence on the shear response of the material (i.e., it also captures shearing in the fiber direction). The limitation associated with the material anisotropic deformability as well as the arrangement of the material constituents is discussed with respect to the initiation of bulging and necking for the membrane. The subsequent consequences for configurations in equilibrium during post-bifurcation are also studied in detail and a variety of results are given in terms of the limited extensibility (deformability) of the fibers as well as their mechanical response and arrangement (fiber winding angle).","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"142 18","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139387469","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 : 2024-01-02DOI: 10.1177/10812865231217640
M. Shirani, Mircea Bîrsan, D. Steigmann
The quasiconvexity inequality associated with energy minimizers is derived in the context of a nonlinear Cosserat elasticity theory for fiber-reinforced elastic solids in which the intrinsic flexural and torsional elasticities of the fibers are taken into account explicitly. The derivation accounts for non-standard kinematic constraints, associated with the materiality of the embedded fibers, connecting the deformation gradient and the Cosserat rotation field.
{"title":"Quasiconvexity in a model of fiber-reinforced solids based on Cosserat elasticity theory","authors":"M. Shirani, Mircea Bîrsan, D. Steigmann","doi":"10.1177/10812865231217640","DOIUrl":"https://doi.org/10.1177/10812865231217640","url":null,"abstract":"The quasiconvexity inequality associated with energy minimizers is derived in the context of a nonlinear Cosserat elasticity theory for fiber-reinforced elastic solids in which the intrinsic flexural and torsional elasticities of the fibers are taken into account explicitly. The derivation accounts for non-standard kinematic constraints, associated with the materiality of the embedded fibers, connecting the deformation gradient and the Cosserat rotation field.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"142 24","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139452956","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 : 2023-12-18DOI: 10.1177/10812865231208016
Davide Ambrosi, L. Deorsola, Stefano Turzi, Marta Zoppello
This paper investigates the role of mechanics in the morphogenesis of the annulus of the mitral valve. We represent the annulus in its embryonic stage as an elastic ring and we perform a mechanical simulation of the development process applying a distributed torque on the rod: because of the mechanical action of the other growing cardiac chambers on the atrio-ventricular region, it departs from a planar circular shape. The numerical integration of the mathematical rod model subject to a bending load yields a shape very near to the one reported in the medical literature as anatomical reference for healthy patients. To make the comparison quantitative, we illustrate a numerical approach to match two curves in 3D defining their distance in a proper mathematical way. Such a methodology is first applied to compare the annular shape resulting from the mechanical model with an anatomical reference “master” shape and it is then applied to set to clinical data extracted from MRI for a cohort of healthy patients. The good agreement among anatomical master description, numerical mechanical model, and clinical data supports our speculation about a possible role of mechanics in determining the shape of the mitral valve.
{"title":"The shape of the mitral annulus: A hypothesis of mechanical morphogenesis","authors":"Davide Ambrosi, L. Deorsola, Stefano Turzi, Marta Zoppello","doi":"10.1177/10812865231208016","DOIUrl":"https://doi.org/10.1177/10812865231208016","url":null,"abstract":"This paper investigates the role of mechanics in the morphogenesis of the annulus of the mitral valve. We represent the annulus in its embryonic stage as an elastic ring and we perform a mechanical simulation of the development process applying a distributed torque on the rod: because of the mechanical action of the other growing cardiac chambers on the atrio-ventricular region, it departs from a planar circular shape. The numerical integration of the mathematical rod model subject to a bending load yields a shape very near to the one reported in the medical literature as anatomical reference for healthy patients. To make the comparison quantitative, we illustrate a numerical approach to match two curves in 3D defining their distance in a proper mathematical way. Such a methodology is first applied to compare the annular shape resulting from the mechanical model with an anatomical reference “master” shape and it is then applied to set to clinical data extracted from MRI for a cohort of healthy patients. The good agreement among anatomical master description, numerical mechanical model, and clinical data supports our speculation about a possible role of mechanics in determining the shape of the mitral valve.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":" 17","pages":""},"PeriodicalIF":2.6,"publicationDate":"2023-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138994827","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 : 2023-12-16DOI: 10.1177/10812865231213408
Cheng Huang, Ming Dai
The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when [Formula: see text] (where R is the radius of the inclusion, while [Formula: see text] and [Formula: see text] are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.
{"title":"Circular inclusion with a refined linearized version of Steigmann–Ogden model","authors":"Cheng Huang, Ming Dai","doi":"10.1177/10812865231213408","DOIUrl":"https://doi.org/10.1177/10812865231213408","url":null,"abstract":"The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when [Formula: see text] (where R is the radius of the inclusion, while [Formula: see text] and [Formula: see text] are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"5 4p2","pages":""},"PeriodicalIF":2.6,"publicationDate":"2023-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138968074","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 : 2023-12-14DOI: 10.1177/10812865231207401
R. Bustamante, K. Rajagopal, Oscar Orellana
The circumferential shear of a nonlinear isotropic incompressible elastic annulus is studied using the neo-Hookean, Ogden constitutive relations in addition to a new constitutive relation for the Hencky strain in terms of the Cauchy stress. The predictions of the three constitutive relations to the specific boundary value problem are delineated. In view of the predictions being quite distinct between the new constitutive relation studied and that for the Ogden constitutive relation, it would be worthwhile to carry out an experiment to determine the efficacy of the models.
{"title":"Circumferential shear for an incompressible non-Green elastic cylindrical annulus","authors":"R. Bustamante, K. Rajagopal, Oscar Orellana","doi":"10.1177/10812865231207401","DOIUrl":"https://doi.org/10.1177/10812865231207401","url":null,"abstract":"The circumferential shear of a nonlinear isotropic incompressible elastic annulus is studied using the neo-Hookean, Ogden constitutive relations in addition to a new constitutive relation for the Hencky strain in terms of the Cauchy stress. The predictions of the three constitutive relations to the specific boundary value problem are delineated. In view of the predictions being quite distinct between the new constitutive relation studied and that for the Ogden constitutive relation, it would be worthwhile to carry out an experiment to determine the efficacy of the models.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"2017 9","pages":""},"PeriodicalIF":2.6,"publicationDate":"2023-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139001703","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 : 2023-12-08DOI: 10.1177/10812865231209975
MA Güler, Y. Alinia, E. Radi
This study investigates the contact mechanics of a thin film laying on an elastic substrate within the context of couple-stress elasticity. It aims to introduce the effects of material internal length scale, which has proven an effective way of modeling structures at micro- to nano-scales, allowing to capture their size-dependent behavior. Specifically, stress analysis for a thin film bonded to a couple-stress elastic half-space is considered under plane strain loading conditions by assuming that both shear stress and couple tractions are exchanged between the thin film and the substrate. The problem is converted to a singular integral equation, which is solved by expanding the shear stress tractions as a Chebyshev series. The results show that the introduction of couple tractions decreases the shear stress tractions and the axial load in the thin film. When the characteristic length is sufficiently small, but still finite, the results for classical elastic behavior are approached.
{"title":"Couple-stress effects in a thin film bonded to a half-space","authors":"MA Güler, Y. Alinia, E. Radi","doi":"10.1177/10812865231209975","DOIUrl":"https://doi.org/10.1177/10812865231209975","url":null,"abstract":"This study investigates the contact mechanics of a thin film laying on an elastic substrate within the context of couple-stress elasticity. It aims to introduce the effects of material internal length scale, which has proven an effective way of modeling structures at micro- to nano-scales, allowing to capture their size-dependent behavior. Specifically, stress analysis for a thin film bonded to a couple-stress elastic half-space is considered under plane strain loading conditions by assuming that both shear stress and couple tractions are exchanged between the thin film and the substrate. The problem is converted to a singular integral equation, which is solved by expanding the shear stress tractions as a Chebyshev series. The results show that the introduction of couple tractions decreases the shear stress tractions and the axial load in the thin film. When the characteristic length is sufficiently small, but still finite, the results for classical elastic behavior are approached.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"45 36","pages":""},"PeriodicalIF":2.6,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138588574","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 : 2023-12-08DOI: 10.1177/10812865231210188
L. A. Mihai
{"title":"A mathematical mechanics mixed tape: Editor’s foreword to special issue in honor of Alain Goriely","authors":"L. A. Mihai","doi":"10.1177/10812865231210188","DOIUrl":"https://doi.org/10.1177/10812865231210188","url":null,"abstract":"","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"45 43","pages":""},"PeriodicalIF":2.6,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138588691","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 : 2023-12-07DOI: 10.1177/10812865231204972
L. Greco, Domenico Castello, Massimo Cuomo
We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation ( slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.
{"title":"An objective and accurate G1-conforming mixed Bézier FE-formulation for Kirchhoff–Love rods","authors":"L. Greco, Domenico Castello, Massimo Cuomo","doi":"10.1177/10812865231204972","DOIUrl":"https://doi.org/10.1177/10812865231204972","url":null,"abstract":"We present a new invariant numerical formulation suitable for the analysis of Kirchhoff–Love rod model based on the smallest rotation map for which the two kinematic descriptors are the placement of the centroid curve and the rotation angle that describes the orientation of the cross-section. In order to guarantee objectivity with respect to a rigid body motion, the spherical linear interpolation ( slerp) for the rotations is introduced. A new hierarchical interpolation for the rotation angle of the cross-section is proposed, composed by the drilling rotation (with respect to the unit tangent of the centroid curve) contained in the geodetic interpolation of the ends’ rotations plus an additional polynomial correction term. This correction term is needed in order to enhance the accuracy of the proposed finite element (FE) formulation. The FE model implicitly guarantees the G1 continuity conditions, thanks to a change in the parametrization of the centroid curve that leads to a generalization of the one-dimensional (1D) Hermitian interpolation. A mixed formulation is used in order to avoid locking and improve the computational efficiency of the method. A symmetric tangent stiffness operator is obtained performing the second covariant derivative of the mixed functional for which the Levi-Civita connection of the configurations manifold of the rod is needed. The objectivity, the robustness, and accuracy of the G1-conforming formulation are confirmed by means of several numerical examples.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"79 5","pages":""},"PeriodicalIF":2.6,"publicationDate":"2023-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138590683","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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