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A Hidden Convexity of Nonlinear Elasticity 非线性弹性的隐藏凸性
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-07-16 DOI: 10.1007/s10659-024-10081-w
Siddharth Singh, Janusz Ginster, Amit Acharya
{"title":"A Hidden Convexity of Nonlinear Elasticity","authors":"Siddharth Singh,&nbsp;Janusz Ginster,&nbsp;Amit Acharya","doi":"10.1007/s10659-024-10081-w","DOIUrl":"10.1007/s10659-024-10081-w","url":null,"abstract":"<div><p>A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive ‘Hadamard instability;’ this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"975 - 1014"},"PeriodicalIF":1.8,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-024-10081-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141722266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Second-Order Multiscale Model for Finite-Strain Poromechanics Based on the Method of Multiscale Virtual Power 基于多尺度虚拟力方法的有限应变孔力学二阶多尺度模型
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-07-10 DOI: 10.1007/s10659-024-10077-6
José Luís Medeiros Thiesen, Bruno Klahr, Thiago André Carniel, Pablo Javier Blanco, Eduardo Alberto Fancello
{"title":"A Second-Order Multiscale Model for Finite-Strain Poromechanics Based on the Method of Multiscale Virtual Power","authors":"José Luís Medeiros Thiesen,&nbsp;Bruno Klahr,&nbsp;Thiago André Carniel,&nbsp;Pablo Javier Blanco,&nbsp;Eduardo Alberto Fancello","doi":"10.1007/s10659-024-10077-6","DOIUrl":"10.1007/s10659-024-10077-6","url":null,"abstract":"<div><p>A second-order multiscale theory based on the concept of a Representative Volume Element (RVE) is proposed to link a classical poromechanical model at the RVE scale to a high-order poromechanical model at the macro-scale in the context of finite-strain kinematics. The proposed theory is carefully derived from the Principle of Multiscale Virtual Power, which is a generalization of the Hill-Mandel Principle of Macrohomogeneity. The coupled governing equations of the low-scale and the homogenization rules for the flux and stress-like quantities are obtained by means of standard variational arguments. The main theoretical result is that the minimally constrained space for the pore pressure field allows for non-zero net fluid flow across the RVE boundaries, unlike first-order theories. The direct consequence of this finding is that the present theory can be consistently applied in cases where the low-scale (RVE level) exhibits substantial volume changes (swelling or shrinking) as a consequence of the evolution of the macro-scale kinematics. Details of formulation development and expression for the homogenized tangent operators are presented for those interested in the computational implementation of the model.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"917 - 954"},"PeriodicalIF":1.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Kinematics of Growth of Regular Boundaries of Bodies into Fractals 论物体规则边界向分形增长的运动学
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-07-10 DOI: 10.1007/s10659-024-10080-x
Vladimir Gol’dshtein, Reuven Segev
{"title":"On the Kinematics of Growth of Regular Boundaries of Bodies into Fractals","authors":"Vladimir Gol’dshtein,&nbsp;Reuven Segev","doi":"10.1007/s10659-024-10080-x","DOIUrl":"10.1007/s10659-024-10080-x","url":null,"abstract":"<div><p>Generalizing smooth volumetric growth to the singular case, using de Rham currents and flat chains, we demonstrate how regular boundaries of bodies may evolve to fractals.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"955 - 974"},"PeriodicalIF":1.8,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-024-10080-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141584624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Axisymmetric Contact Problem for a Homogeneous Space with a Circular Disk-Shaped Crack Under Static Friction 静摩擦力作用下带有圆盘形裂缝的均质空间的轴对称接触问题
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-07-09 DOI: 10.1007/s10659-024-10078-5
V. Hakobyan, A. Sahakyan, H. A. Amirjanyan, L. Dashtoyan
{"title":"Axisymmetric Contact Problem for a Homogeneous Space with a Circular Disk-Shaped Crack Under Static Friction","authors":"V. Hakobyan,&nbsp;A. Sahakyan,&nbsp;H. A. Amirjanyan,&nbsp;L. Dashtoyan","doi":"10.1007/s10659-024-10078-5","DOIUrl":"10.1007/s10659-024-10078-5","url":null,"abstract":"<div><p>The paper considers an axisymmetric stress state of a homogeneous elastic space with a circular disc-shaped crack, one of the edges of which is pressed into a cylindrical circular stamp with static friction. It is assumed that the contact zone is considered under the generalized law of dry friction, i.e. tangential contact stresses are proportional to normal contact pressure, while the proportionality coefficient depends on the radial coordinates of the points of the contacting surfaces and is directly proportional to them. Considering the fact that in this case the Abel images of contact stresses are also related in a similar way, the solution of the problem, with the help of rotation operators and theory of analytical functions, is reduced to an inhomogeneous Riemann problem for two functions and the closed solution in quadratures is constructed. A numerical analysis was carried out and regularities of changes in both normal and shear real contact stresses, as well as rigid displacement of the stamp depending on the physical and geometric parameters were revealed.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"899 - 916"},"PeriodicalIF":1.8,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141569990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Rayleigh-Type Waves in Nonlocal Micropolar Thermoelastic Half-Space Containing Void Pores 包含空隙的非局部微波热弹性半空间中的瑞利波
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-07-05 DOI: 10.1007/s10659-024-10079-4
Suraj Kumar, S. K. Tomar
{"title":"Rayleigh-Type Waves in Nonlocal Micropolar Thermoelastic Half-Space Containing Void Pores","authors":"Suraj Kumar,&nbsp;S. K. Tomar","doi":"10.1007/s10659-024-10079-4","DOIUrl":"10.1007/s10659-024-10079-4","url":null,"abstract":"<div><p>Propagation of Rayleigh-type waves is investigated in a half-space composed of nonlocal micropolar thermoelastic material containing void pores. Dispersion relation is derived for a mechanically stress-free and thermally insulated boundary surface of the half-space. The particle motion during the propagation of the waves is found to follow elliptical path. Numerical computations for a specific material are performed to analyze the characteristics of propagating Rayleigh-type waves in detail. Comparison between the phase speed and corresponding attenuation coefficient in some particular cases is also carried out. The effect of various parameters on the characteristics of waves in question is also studied.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"877 - 898"},"PeriodicalIF":1.8,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions 关于带有位移边界条件的立方到四方相变的规模问题
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-07-02 DOI: 10.1007/s10659-024-10075-8
Angkana Rüland, Antonio Tribuzio
{"title":"On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions","authors":"Angkana Rüland,&nbsp;Antonio Tribuzio","doi":"10.1007/s10659-024-10075-8","DOIUrl":"10.1007/s10659-024-10075-8","url":null,"abstract":"<div><p>We provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the <i>order of lamination</i> of the affine displacement data determines the complexity of the microstructure. As in (Rüland and Tribuzio in ESAIM Control Optim. Calc. Var. 29:68, 2023) we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"837 - 875"},"PeriodicalIF":1.8,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-024-10075-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141514280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Elastodynamics of Multilattices: Field Equations of the Linear Theory as a First Order System 多网格的弹性力学：作为一阶系统的线性理论场方程
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-06-27 DOI: 10.1007/s10659-024-10076-7
D. Sfyris, G. I. Sfyris
{"title":"Elastodynamics of Multilattices: Field Equations of the Linear Theory as a First Order System","authors":"D. Sfyris,&nbsp;G. I. Sfyris","doi":"10.1007/s10659-024-10076-7","DOIUrl":"10.1007/s10659-024-10076-7","url":null,"abstract":"<div><p>For a one dimensional 2-lattice we write down the momentum equation and the equation ruling the shift vector for the dynamic case as a first order system and characterize it in terms of its hyperbolicity. We use similar arguments for problems of increasing difficulty and tackle: one dimensional 3-lattices, three dimensional 2-lattices, three dimensional 3-lattices and finally the most general case of three dimensional <span>((n+1))</span>-lattices. Our approach is confined to the geometrically and materially linear elastodynamic theory and is valid for generic anisotropic materials that have the multilattice structure. The main finding is that, with the assumptions adopted, the presence of each shift vector is related with zero eigenvalues when the system is written as a first order system.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"813 - 835"},"PeriodicalIF":1.8,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Elastic Solids with Strain-Gradient Elastic Boundary Surfaces 具有应变梯度弹性边界面的弹性固体
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-06-14 DOI: 10.1007/s10659-024-10073-w
C. Rodriguez
{"title":"Elastic Solids with Strain-Gradient Elastic Boundary Surfaces","authors":"C. Rodriguez","doi":"10.1007/s10659-024-10073-w","DOIUrl":"10.1007/s10659-024-10073-w","url":null,"abstract":"<div><p>Recent works have shown that in contrast to classical linear elastic fracture mechanics, endowing crack fronts in a brittle Green-elastic solid with Steigmann-Ogden surface elasticity yields a model that predicts bounded stresses and strains at the crack tips for plane-strain problems. However, singularities persist for anti-plane shear (mode-III fracture) under far-field loading, even when Steigmann-Ogden surface elasticity is incorporated. This work is motivated by obtaining a model of brittle fracture capable of predicting bounded stresses and strains for all modes of loading. We formulate an exact general theory of a three-dimensional solid containing a boundary surface with strain-gradient surface elasticity. For planar reference surfaces parameterized by flat coordinates, the form of surface elasticity reduces to that introduced by Hilgers and Pipkin, and when the surface energy is independent of the surface covariant derivative of the stretching, the theory reduces to that of Steigmann and Ogden. We discuss material symmetry using Murdoch and Cohen’s extension of Noll’s theory. We present a model small-strain surface energy that incorporates resistance to geodesic distortion, satisfies strong ellipticity, and requires the same material constants found in the Steigmann-Ogden theory. Finally, we derive and apply the linearized theory to mode-III fracture in an infinite plate under far-field loading. We prove that there always exists a unique classical solution to the governing integro-differential equation, and in contrast to using Steigmann-Ogden surface elasticity, our model is consistent with the linearization assumption in predicting finite stresses and strains at the crack tips.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"769 - 797"},"PeriodicalIF":1.8,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503946","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
An Elliptical Incompressible Liquid Inclusion in a Compressible Hyperelastic Solid of Harmonic Type 谐波型可压缩超弹性固体中的椭圆形不可压缩液体包涵体
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-06-14 DOI: 10.1007/s10659-024-10074-9
Xu Wang, Peter Schiavone
{"title":"An Elliptical Incompressible Liquid Inclusion in a Compressible Hyperelastic Solid of Harmonic Type","authors":"Xu Wang,&nbsp;Peter Schiavone","doi":"10.1007/s10659-024-10074-9","DOIUrl":"10.1007/s10659-024-10074-9","url":null,"abstract":"<div><p>A complete closed-form solution is derived to the finite plane elasticity problem associated with an elliptical incompressible liquid inclusion embedded in an infinite compressible hyperelastic matrix of harmonic type subjected to uniform remote in-plane Piola stresses. The internal uniform hydrostatic Piola stresses and deformation gradients within the elliptical incompressible liquid inclusion are obtained. The internal uniform hydrostatic tension and hoop stress on the matrix side along the liquid-solid interface for various shapes of the elliptical liquid inclusion and different elastic constants of the matrix under various remote loadings are calculated.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"799 - 811"},"PeriodicalIF":1.8,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141341738","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Homogenization of a Soft Elastic or Perfectly Viscoplastic Material Reinforced by Fibers 纤维增强软弹性或完美粘塑性材料的均质化
IF 1.8 3区 工程技术
Journal of Elasticity Pub Date : 2024-05-20 DOI: 10.1007/s10659-024-10070-z
Michel Bellieud
{"title":"Homogenization of a Soft Elastic or Perfectly Viscoplastic Material Reinforced by Fibers","authors":"Michel Bellieud","doi":"10.1007/s10659-024-10070-z","DOIUrl":"10.1007/s10659-024-10070-z","url":null,"abstract":"<div><p>We study the effective properties of a linear elastic or perfectly viscoplastic composite consisting of a soft matrix reinforced by possibly very stiff fibers. The effective material is characterized by the emergence of a discrepancy between the displacement in the fibers and the overall averaged displacement, giving rise to a concentration strain in the matrix. The effective energy stored in the fibers is a combination of stretching, bending and torsional energies. This work completes and corrects results previously obtained by the author and coworkers, who failed to notice the torsional contribution.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 3","pages":"739 - 767"},"PeriodicalIF":1.8,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141123036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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