{"title":"A Generalized Model for Large Deformations of an Elastically Isotropic Material with Elastic-Inelastic Response","authors":"M. B. Rubin","doi":"10.1007/s10659-023-10036-7","DOIUrl":"10.1007/s10659-023-10036-7","url":null,"abstract":"<div><p>The objective of this classroom note is to propose a generalized model for a compressible elastically isotropic material with elastic-inelastic response. The model is generalized for an exponential Fung-type strain energy with a sum of new higher order elastic distortional deformation invariants that can model elastic-inelastic distortional deformation. In contrast with an Ogden-type model, the coefficients of the proposed higher order invariants do not affect the small deformation response. Moreover, there is no need to determine eigenvalues and eigenvectors of the elastic distortional deformation tensor. Examples show that the equations can model preconditioning of biological tissues as well as elastic instability.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 1","pages":"325 - 332"},"PeriodicalIF":1.8,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135735410","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}
{"title":"Cauchy Relations in Linear Elasticity: Algebraic and Physics Aspects","authors":"Yakov Itin","doi":"10.1007/s10659-023-10035-8","DOIUrl":"10.1007/s10659-023-10035-8","url":null,"abstract":"<div><p>The Cauchy relations distinguish between rari- and multi-constant linear elasticity theories. These relations are treated in this paper in a form that is invariant under two groups of transformations: indices permutation and general linear transformations of the basis. The irreducible decomposition induced by the permutation group is outlined. The Cauchy relations are then formulated as a requirement of nullification of an invariant subspace. A successive decomposition under rotation group allows to define the partial Cauchy relations and two types of elastic materials. We explore several applications of the full and partial Cauchy relations in physics of materials. The structure’s deviation from the basic physical assumptions of Cauchy’s model is defined in an invariant form. The Cauchy and non-Cauchy contributions to Hooke’s law and elasticity energy are explained. We identify wave velocities and polarization vectors that are independent of the non-Cauchy part for acoustic wave propagation. Several bounds are derived for the elasticity invariant parameters.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 1","pages":"39 - 77"},"PeriodicalIF":1.8,"publicationDate":"2023-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47277993","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}
{"title":"Construction of Invariant Relations of (n) Symmetric Second-Order Tensors","authors":"Adair Roberto Aguiar, Gabriel Lopes da Rocha","doi":"10.1007/s10659-023-10031-y","DOIUrl":"10.1007/s10659-023-10031-y","url":null,"abstract":"<div><p>A methodology is presented to find either implicit or explicit relations, called syzygies, between invariants in a minimal integrity basis for <span>(n)</span> symmetric second-order tensors defined on a three-dimensional euclidean space. The methodology i) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis and ii) allows the construction of the implicit relations. The results of this investigation are important in modeling biological structures, which, in general, are non-homogeneous and made of anisotropic viscoelastic materials that are subjected to large deformations and are modeled through constitutive relations that depend on symmetric tensors.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"154 1-4","pages":"45 - 60"},"PeriodicalIF":2.0,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42042772","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}
{"title":"Edge Crack Subject to Anti-Plane Shear Wave in an Orthotropic Strip","authors":"Somashri Karan, Sourav Kumar Panja, Sanjoy Basu, Subhas Chandra Mandal","doi":"10.1007/s10659-023-10032-x","DOIUrl":"10.1007/s10659-023-10032-x","url":null,"abstract":"<div><p>In this article, the proposed model analyzed shear wave propagation through an orthotropic strip with an edge crack. Dual integral equations have been developed for solution of the governing mixed boundary value problem with the aid of Hankel transform technique. Then, the dual integral equations have been transformed into a second kind Fredholm integral equation employing Abel’s transformation. The numerical calculations of stress intensity factor and crack opening displacement are performed utilizing the Fox & Goodwin method and displayed graphically. Elastic constants of two orthotropic materials have been used to illustrate the influence of material orthotropy and normalized strip width on SIF and COD.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 1","pages":"23 - 37"},"PeriodicalIF":1.8,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45488335","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}
Ryan S. Elliott, Adair R. Aguiar, Yi-Chao Chen, Gianni Royer-Carfangi
{"title":"Foreword: In Recognition of the 85th Birthday of Roger L. Fosdick","authors":"Ryan S. Elliott, Adair R. Aguiar, Yi-Chao Chen, Gianni Royer-Carfangi","doi":"10.1007/s10659-023-10033-w","DOIUrl":"10.1007/s10659-023-10033-w","url":null,"abstract":"","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"154 1-4","pages":"1 - 2"},"PeriodicalIF":2.0,"publicationDate":"2023-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47013595","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}
{"title":"A Class of Nonlinear Elasticity Problems with No Local but Many Global Minimizers","authors":"Yury Grabovsky, Lev Truskinovsky","doi":"10.1007/s10659-023-10026-9","DOIUrl":"10.1007/s10659-023-10026-9","url":null,"abstract":"<div><p>We present a class of models of elastic phase transitions with incompatible energy wells in an arbitrary space dimension, where in a hard device an abundance of Lipschitz global minimizers coexists with a complete lack of strong local minimizers. The analysis is based on the proof that every strong local minimizer in a hard device is also a global minimizer which is applicable much beyond the chosen class of models. Along the way we show that a new demonstration of sufficiency for a subclass of affine boundary conditions can be built around a novel nonlinear generalization of the classical Clapeyron theorem.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"154 1-4","pages":"147 - 171"},"PeriodicalIF":2.0,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44163270","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}
{"title":"Correction to: Harmonic Decomposition, Irreducible Basis Tensors, and Minimal Representations of Material Tensors and Pseudotensors","authors":"Chi-Sing Man, Wenwen Du","doi":"10.1007/s10659-023-10030-z","DOIUrl":"10.1007/s10659-023-10030-z","url":null,"abstract":"","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"154 1-4","pages":"43 - 43"},"PeriodicalIF":2.0,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43405914","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}
{"title":"Decomposition of Rod Displacements via Bernoulli–Navier Displacements. Application: A Loading of the Rod with Shearing","authors":"Georges Griso","doi":"10.1007/s10659-023-10029-6","DOIUrl":"10.1007/s10659-023-10029-6","url":null,"abstract":"<div><p>Within the framework of linear elasticity, we show that any displacement of a straight rod is the sum of a Bernoulli–Navier displacement and two terms, one for shearing and the other for warping. Then, we load a straight rod so that bending and shear contribute the same to the rotations of the cross-section.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"156 1","pages":"1 - 22"},"PeriodicalIF":1.8,"publicationDate":"2023-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44725975","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}
Bhaskar Vajipeyajula, Pavitra Murru, K. R. Rajagopal
{"title":"Stress Concentration Due to the Presence of a Rigid Elliptical Inclusion in Porous Elastic Solids Described by a New Class of Constitutive Relations","authors":"Bhaskar Vajipeyajula, Pavitra Murru, K. R. Rajagopal","doi":"10.1007/s10659-023-10027-8","DOIUrl":"10.1007/s10659-023-10027-8","url":null,"abstract":"<div><p>In a large class of porous elastic solids such as cement concrete, rocks, ceramics, porous metals, biological materials such as bone, etc., the material moduli depend on density. When such materials undergo sufficiently small deformations, the usual approach of appealing to a linearized elastic constitutive relation to describe their response will not allow us to capture the dependence of the material moduli on the density, as this would imply a nonlinear relationship between the stress and the linearized strain in virtue of the balance of mass as dependence on density implies dependence on the trace of the linearized strain. It is possible to capture the dependence of the material moduli on the density, when the body undergoes small deformations, within the context of implicit constitutive relations. We study the stress concentration due to a rigid elliptic inclusion within a new class of implicit constitutive relations in which the stress and the linearized strain appear linearly, that allows us to capture the dependence of the material moduli on the density. We find that the stress concentration that one obtains employing the constitutive relation wherein the material moduli depend on the density can be significantly different from that obtained by adopting the classical linearized elastic constitutive relation to which it reduces to when the density dependence of the material moduli are ignored.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"154 1-4","pages":"255 - 273"},"PeriodicalIF":2.0,"publicationDate":"2023-07-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45993934","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}
{"title":"Anisotropy and Asymmetry of the Elastic Tensor of Lattice Materials","authors":"Huiming Yin, Chao Liu","doi":"10.1007/s10659-023-10028-7","DOIUrl":"10.1007/s10659-023-10028-7","url":null,"abstract":"<div><p>Lattice materials formed by hinged springs or linear elastic bonds may exhibit diverse anisotropy and asymmetry features of the overall elastic behavior depending on their unit cell configuration. The recently developed singum model transfers the force-displacement relationship of the springs in the lattice to the stress-strain relationship in the continuum particle, and provides the analytical form of tangential elasticity. When a pre-stress exists in the lattice, the stiffness tensor significantly changes due to the effect of the configurational stress; existing methods like the lattice spring method, relying on a scalar energy equivalence, are insufficient in such situations. Instead, a tensorial homogenization method with the new definition of singum stress and strain, should be preferred. Different lattice structures lead to different symmetries of the stiffness tensors, which are demonstrated by five lattices. When all bonds exhibit the same length, regular hexagonal, honeycomb, and auxetic lattices demonstrate that the stiffness changes from an isotropic to anisotropic, from symmetric to asymmetric tensor. When the central symmetry of the unit cell is not satisfied, the primitive cell will contain more than one singums and the Cauchy–Born rule fails by the loss of equilibrium of the single singum. A secondary stress is induced to balance the singums. Displacement gradient <span>(d_{ij}=u_{j,i})</span> is proposed to replace strain in the constitutive law for the general case because <span>(d_{12})</span> and <span>(d_{21})</span> can produce different stress states. Although the hexagonal and honeycomb lattices may exhibit isotropic behavior, for general auxetic lattices, an anisotropic and asymmetric elastic tensor is obtained with the loss of both minor and major symmetry, which is also demonstrated in a square lattice with unbalanced central symmetry and a chiral lattice. The modeling procedure and results can be generalized to three dimensions and other lattices with the anisotropic and asymmetric stiffness.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"154 5","pages":"659 - 691"},"PeriodicalIF":1.8,"publicationDate":"2023-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49524891","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}