Pub Date : 2023-10-18DOI: 10.1177/10812865231196296
Henryk Petryk
Incremental energy minimization is revisited as a method of determining an incremental solution for rate-independent dissipative solids undergoing isothermal quasi-static deformation. The incremental minimization is applied to the total internal energy of the compound thermodynamic system that consists of a deforming body with internal variables, a conservative loading device, and an ambient heat reservoir. It is shown that the difference between the virtual and actual dissipation rates plays a fundamental role in this minimization, which is related to thermodynamic extremal principles of local and global type. The analysis is carried out within the gradient plasticity framework with the energetic forces derived as the variational derivative of the Helmholtz free energy depending on the spatial gradient of arbitrary internal variables. Specifications are given for existing models of gradient plasticity.
{"title":"On thermodynamic extremal principles in gradient plasticity with energetic forces","authors":"Henryk Petryk","doi":"10.1177/10812865231196296","DOIUrl":"https://doi.org/10.1177/10812865231196296","url":null,"abstract":"Incremental energy minimization is revisited as a method of determining an incremental solution for rate-independent dissipative solids undergoing isothermal quasi-static deformation. The incremental minimization is applied to the total internal energy of the compound thermodynamic system that consists of a deforming body with internal variables, a conservative loading device, and an ambient heat reservoir. It is shown that the difference between the virtual and actual dissipation rates plays a fundamental role in this minimization, which is related to thermodynamic extremal principles of local and global type. The analysis is carried out within the gradient plasticity framework with the energetic forces derived as the variational derivative of the Helmholtz free energy depending on the spatial gradient of arbitrary internal variables. Specifications are given for existing models of gradient plasticity.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135884460","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 : 2023-10-16DOI: 10.1177/10812865231193427
Samuel Amstutz, Nicolas van Goethem
This work deals with the modeling of solid continua undergoing incompatible deformations due to the presence of microscopic defects like dislocations. Our approach relies on a geometrical description of the medium by the strain tensor and the representation of internal efforts using zeroth- and second-order strain gradients in an infinitesimal framework. At the same time, energetic arguments allow to monitor the corresponding moduli. We provide mathematical and numerical results to support these ideas in the framework of isotropic constitutive laws.
{"title":"A second-order model of small-strain incompatible elasticity","authors":"Samuel Amstutz, Nicolas van Goethem","doi":"10.1177/10812865231193427","DOIUrl":"https://doi.org/10.1177/10812865231193427","url":null,"abstract":"This work deals with the modeling of solid continua undergoing incompatible deformations due to the presence of microscopic defects like dislocations. Our approach relies on a geometrical description of the medium by the strain tensor and the representation of internal efforts using zeroth- and second-order strain gradients in an infinitesimal framework. At the same time, energetic arguments allow to monitor the corresponding moduli. We provide mathematical and numerical results to support these ideas in the framework of isotropic constitutive laws.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077518","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 : 2023-10-15DOI: 10.1177/10812865231188931
HA Erbay, KR Rajagopal, G Saccomandi, Y Şengül
It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation.
{"title":"Dispersive transverse waves for a strain-limiting continuum model","authors":"HA Erbay, KR Rajagopal, G Saccomandi, Y Şengül","doi":"10.1177/10812865231188931","DOIUrl":"https://doi.org/10.1177/10812865231188931","url":null,"abstract":"It is well known that propagation of waves in homogeneous linearized elastic materials of infinite extent is not dispersive. Motivated by the work of Rubin, Rosenau, and Gottlieb, we develop a generalized continuum model for the response of strain-limiting materials that are dispersive. Our approach is based on both a direct inclusion of Rivlin–Ericksen tensors in the constitutive relations and writing the linearized strain in terms of the stress. As a result, we derive two coupled generalized improved Boussinesq-type equations in the stress components for the propagation of pure transverse waves. We investigate the traveling wave solutions of the generalized Boussinesq-type equations and show that the resulting ordinary differential equations form a Hamiltonian system. Linearly and circularly polarized cases are also investigated. In the case of unidirectional propagation, we show that the propagation of small-but-finite amplitude long waves is governed by the complex Korteweg–de Vries (KdV) equation.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135758630","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 : 2023-10-14DOI: 10.1177/10812865231199067
K Koocheki, S Pietruszczak
This paper deals with mesoscale analysis of masonry structures, which involves fracture propagation in brick units as well as along the masonry joints. The brick–mortar interfaces are incorporated in standard finite elements by employing a constitutive law with embedded discontinuity. Macrocracks in bricks are modelled in a discrete way using the same methodology, without any a-priori assumptions regarding their orientation. The proposed approach is computationally efficient as it does not explicitly require the discretization of joints. The accuracy of the approximation is first assessed by comparing the solution with a detailed mesoscale model incorporating interface elements. Later, a numerical study is conducted involving simulation of various experimental tests on small masonry assemblages, as well as single-leaf masonry walls, with running bond pattern, subjected to in-plane loading. The results clearly demonstrate the predictive abilities of the proposed simplified approach.
{"title":"Mesoscale analysis of fracture process in brick masonry structures","authors":"K Koocheki, S Pietruszczak","doi":"10.1177/10812865231199067","DOIUrl":"https://doi.org/10.1177/10812865231199067","url":null,"abstract":"This paper deals with mesoscale analysis of masonry structures, which involves fracture propagation in brick units as well as along the masonry joints. The brick–mortar interfaces are incorporated in standard finite elements by employing a constitutive law with embedded discontinuity. Macrocracks in bricks are modelled in a discrete way using the same methodology, without any a-priori assumptions regarding their orientation. The proposed approach is computationally efficient as it does not explicitly require the discretization of joints. The accuracy of the approximation is first assessed by comparing the solution with a detailed mesoscale model incorporating interface elements. Later, a numerical study is conducted involving simulation of various experimental tests on small masonry assemblages, as well as single-leaf masonry walls, with running bond pattern, subjected to in-plane loading. The results clearly demonstrate the predictive abilities of the proposed simplified approach.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135803180","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 : 2023-10-13DOI: 10.1177/10812865231201623
Roger Bustamante
Shear stresses for a two-dimensional (2D) beam are calculated modifying the classical method developed by Jouravski, for the case the linearized strain tensor is assumed to be a nonlinear function of the Cauchy stresses. Two problems are studied, namely, the case of a cantilever beam with a point load on its free edge (considering a rectangular and a circular cross-section) and the three-point flexural test for a beam of rectangular cross-section. Numerical results are obtained for the particular case of a bimodular constitutive model for rock, and the results for the shear stresses are compared with the predictions of the classical theory of strength of materials for such problems, assuming a linearized elastic body.
{"title":"Approximate determination of shear stresses for a 2D beam made of a non-Green elastic solid","authors":"Roger Bustamante","doi":"10.1177/10812865231201623","DOIUrl":"https://doi.org/10.1177/10812865231201623","url":null,"abstract":"Shear stresses for a two-dimensional (2D) beam are calculated modifying the classical method developed by Jouravski, for the case the linearized strain tensor is assumed to be a nonlinear function of the Cauchy stresses. Two problems are studied, namely, the case of a cantilever beam with a point load on its free edge (considering a rectangular and a circular cross-section) and the three-point flexural test for a beam of rectangular cross-section. Numerical results are obtained for the particular case of a bimodular constitutive model for rock, and the results for the shear stresses are compared with the predictions of the classical theory of strength of materials for such problems, assuming a linearized elastic body.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135858325","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 : 2023-10-11DOI: 10.1177/10812865231199460
Daniel J Arrigo, Travis C Chism
The governing equations for plane deformations of isotropic incompressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. As far as we are aware, the Varga material is the only material in which results of any generality are known. In this paper, we show that for the Varga material, the full governing equations are in fact linearizable.
{"title":"Plane deformations of the Varga material","authors":"Daniel J Arrigo, Travis C Chism","doi":"10.1177/10812865231199460","DOIUrl":"https://doi.org/10.1177/10812865231199460","url":null,"abstract":"The governing equations for plane deformations of isotropic incompressible hyperelastic materials are highly nonlinear, and consequently, very few exact solutions are known. As far as we are aware, the Varga material is the only material in which results of any generality are known. In this paper, we show that for the Varga material, the full governing equations are in fact linearizable.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136210853","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 : 2023-10-11DOI: 10.1177/10812865231193735
L Angela Mihai
Motivated by the need for new materials and green energy production and conversion processes, a class of mathematical models for liquid crystal elastomers (LCEs) integrated within a theoretical charge pump electrical circuit is considered. The charge pump harnesses the chemical and mechanical properties of LCEs transitioning from the nematic to isotropic phase when illuminated or heated to generate higher voltage from a lower voltage supplied by a battery. For the material constitutive model, purely elastic and neoclassical-type strain energy densities applicable to a wide range of monodomain nematic elastomers are combined, while elastic and photothermal responses are decoupled to make the investigation analytically tractable. By varying the model parameters of the elastic and neoclassical terms, it is found that LCEs are more effective than rubber when used as dielectric material within a charge pump capacitor.
{"title":"A theoretical model for power generation via liquid crystal elastomers","authors":"L Angela Mihai","doi":"10.1177/10812865231193735","DOIUrl":"https://doi.org/10.1177/10812865231193735","url":null,"abstract":"Motivated by the need for new materials and green energy production and conversion processes, a class of mathematical models for liquid crystal elastomers (LCEs) integrated within a theoretical charge pump electrical circuit is considered. The charge pump harnesses the chemical and mechanical properties of LCEs transitioning from the nematic to isotropic phase when illuminated or heated to generate higher voltage from a lower voltage supplied by a battery. For the material constitutive model, purely elastic and neoclassical-type strain energy densities applicable to a wide range of monodomain nematic elastomers are combined, while elastic and photothermal responses are decoupled to make the investigation analytically tractable. By varying the model parameters of the elastic and neoclassical terms, it is found that LCEs are more effective than rubber when used as dielectric material within a charge pump capacitor.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"244 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136063817","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 : 2023-10-09DOI: 10.1177/10812865231190845
Soheil Moradalizadeh, Heiko Topol, Hasan Demirkoparan, Andrey Melnikov, Bernd Markert, José Merodio
We consider a tubular membrane whose mechanical behavior is defined by a strain energy density function that combines the neo-Hookean model with the Demiray model. Both, the neo-Hookean and the Demiray models are isotropic material models, which found their application in the modeling of the mechanical behavior of biological soft tissue. This tubular membrane is subjected to an inner pressure, an axial stretch, and a change in the material volume due to swelling or deswelling. The interplay between the cylinder geometry and the loading conditions and different instability modes, namely, bulging bifurcation, bending bifurcation, and prismatic bifurcation, are studied. It is shown that a change in the material volume has a strong effect on the occurrence of these bifurcation modes because a change in the material volume may stabilize the cylinder against a particular bifurcation mode and may trigger another bifurcation mode at the same time. Despite the restriction to isotropic material behavior, this article shows that the material response to pressure, axial stretch, and material volume change is quite complex.
{"title":"Remarks on bifurcation of an inflated and extended swellable isotropic tube","authors":"Soheil Moradalizadeh, Heiko Topol, Hasan Demirkoparan, Andrey Melnikov, Bernd Markert, José Merodio","doi":"10.1177/10812865231190845","DOIUrl":"https://doi.org/10.1177/10812865231190845","url":null,"abstract":"We consider a tubular membrane whose mechanical behavior is defined by a strain energy density function that combines the neo-Hookean model with the Demiray model. Both, the neo-Hookean and the Demiray models are isotropic material models, which found their application in the modeling of the mechanical behavior of biological soft tissue. This tubular membrane is subjected to an inner pressure, an axial stretch, and a change in the material volume due to swelling or deswelling. The interplay between the cylinder geometry and the loading conditions and different instability modes, namely, bulging bifurcation, bending bifurcation, and prismatic bifurcation, are studied. It is shown that a change in the material volume has a strong effect on the occurrence of these bifurcation modes because a change in the material volume may stabilize the cylinder against a particular bifurcation mode and may trigger another bifurcation mode at the same time. Despite the restriction to isotropic material behavior, this article shows that the material response to pressure, axial stretch, and material volume change is quite complex.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135094260","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 : 2023-10-08DOI: 10.1177/10812865231193697
Liyuan Wang, Hongmei Wu, Zhiying Ou
In this study, we present a novel surface model that utilizes surface energy density to predict the surface effect in nanobeam contact problems. To address the issue of a finite-length elastic nanobeam being indented by a rigid cylindrical indenter, we propose an equivalent substitution method. This method allows us to formulate analytical relations between the load and contact half-width for two different boundary cases. The explicit expressions of the contact-zone width, the pressure distribution in the contact zone, the deflection outside the contact zone, and the load–displacement relation are obtained for the nanobeam with surface effect and are compared with classical results in detail. The results show that the influence of the surface effect is very significant for nanobeam contact behavior, especially when the half-width of the contact zone increases and the contact zone becomes two independent symmetric strips. It is also found that the length-height ratio of nanobeam and the end support conditions have a fairly obvious effect on the normalized pressure distribution, which deviates significantly from the one predicted by the classical results due to the surface effect. However, for a given beam length and indenter radius, the ratio of the width of the contact zone to the beam thickness is almost constant, independent of the indenter load and beam boundary conditions. Meanwhile, the model predicts that the contact pressure distribution after the normalization of the average indentation pressure is almost independent of the indentation load and beam boundary conditions, but obviously depends on the surface effect parameters. The present method and result should be helpful not only to the measurement of mechanical properties of the indentation nanobeam but also to the design of the nanobeam-based devices.
{"title":"Contact behaviors involving a nanobeam with surface effect by a rigid indenter","authors":"Liyuan Wang, Hongmei Wu, Zhiying Ou","doi":"10.1177/10812865231193697","DOIUrl":"https://doi.org/10.1177/10812865231193697","url":null,"abstract":"In this study, we present a novel surface model that utilizes surface energy density to predict the surface effect in nanobeam contact problems. To address the issue of a finite-length elastic nanobeam being indented by a rigid cylindrical indenter, we propose an equivalent substitution method. This method allows us to formulate analytical relations between the load and contact half-width for two different boundary cases. The explicit expressions of the contact-zone width, the pressure distribution in the contact zone, the deflection outside the contact zone, and the load–displacement relation are obtained for the nanobeam with surface effect and are compared with classical results in detail. The results show that the influence of the surface effect is very significant for nanobeam contact behavior, especially when the half-width of the contact zone increases and the contact zone becomes two independent symmetric strips. It is also found that the length-height ratio of nanobeam and the end support conditions have a fairly obvious effect on the normalized pressure distribution, which deviates significantly from the one predicted by the classical results due to the surface effect. However, for a given beam length and indenter radius, the ratio of the width of the contact zone to the beam thickness is almost constant, independent of the indenter load and beam boundary conditions. Meanwhile, the model predicts that the contact pressure distribution after the normalization of the average indentation pressure is almost independent of the indentation load and beam boundary conditions, but obviously depends on the surface effect parameters. The present method and result should be helpful not only to the measurement of mechanical properties of the indentation nanobeam but also to the design of the nanobeam-based devices.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135199987","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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