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A Stationarity Principle Generating Effective Boundary Conditions for Second-Order Homogenization 生成二阶均匀化有效边界条件的平稳性原理
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-02-13 DOI: 10.1007/s10659-026-10190-8
Manon Thbaut, Basile Audoly, Claire Lestringant
{"title":"A Stationarity Principle Generating Effective Boundary Conditions for Second-Order Homogenization","authors":"Manon Thbaut,&nbsp;Basile Audoly,&nbsp;Claire Lestringant","doi":"10.1007/s10659-026-10190-8","DOIUrl":"10.1007/s10659-026-10190-8","url":null,"abstract":"<div><p>We derive an effective model for a periodic chain of linearly-elastic springs, achieving second-order accuracy in the scale separation parameter <span>(varepsilon ll 1)</span>. The chain has finite length and is made up of springs connecting both nearest- and next-nearest-neighbors: it serves as a one-dimensional prototype for higher-order periodic homogenization problems with boundaries. This type of problem has been approached by inserting two-scale expansions into the equations of equilibrium in the bulk and by matching them with boundary-layer solutions. We explore an alternative method operating at the energy level, bypassing the cumbersome matching procedure. We start from an ansatz of the microscopic displacement accounting for both boundary layers and for small-scale fluctuations in the bulk, and insert it into the discrete energy. This yields a continuous energy functional depending on the macroscopic displacement <span>(u)</span>, in the form of a series expansion in powers of <span>(varepsilon )</span>. We call it a <i>pseudo-energy</i> <span>(Phi _{varepsilon } [u])</span> as it is not positive when truncated at order <span>(varepsilon ^{2})</span>. The boundary terms in the pseudo-energy account for boundary layers in an effective way. By making the pseudo-energy stationary order by order in <span>(varepsilon )</span>, we derive the homogenized equations of equilibrium along with effective boundary conditions. We provide quantitative validation showing that the effective model is correct to second order. We point out the special form of the effective higher-order tractions, which has been overlooked in the strain-gradient theories proposed so far.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147338796","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
A Note on Hyperbolicity for Fully Implicit Theories in One Space Dimension 关于一维全隐理论双曲性的一个注记
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-02-12 DOI: 10.1007/s10659-026-10191-7
D. Sfyris, R. Bustamante
{"title":"A Note on Hyperbolicity for Fully Implicit Theories in One Space Dimension","authors":"D. Sfyris,&nbsp;R. Bustamante","doi":"10.1007/s10659-026-10191-7","DOIUrl":"10.1007/s10659-026-10191-7","url":null,"abstract":"<div><p>For theories working with fully implicit constitutive relations we find conditions for strict hyperbolicity for the system of elastodynamic equations in one space dimension. To bypass the fact that stress is only implicitly related to strain and not explicitly, we utilize three partial differential equations to form our system: the momentum equation, the compatibility equation as well as the time differentiated fully implicit constitutive law. This way we built up a system of three quasilinear partial differential equations of first order for the velocity (<span>(u_{t})</span>), the displacement gradient (<span>(u_{x})</span>) (or the strain (<span>(e)</span>)) as well as the stress (<span>(sigma )</span>). For this system we find conditions for strict hyperbolicity using the characteristic polynomial. We do that for cases where the implicit constitutive laws are either of the form <span>(f(sigma , u_{x})=0)</span>, or <span>(f(sigma , e)=0)</span>. The same method applies also to constitutive laws of the form <span>(e=g(sigma ))</span>.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147338797","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
Shock Wave Structure and Sub-Shock Formation in a Hyperbolic Nonlinear Viscoelastic Model 双曲型非线性粘弹性模型中的激波结构和次激波形成
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-02-04 DOI: 10.1007/s10659-026-10189-1
Takashi Arima, Tommaso Ruggeri, Shigeru Taniguchi
{"title":"Shock Wave Structure and Sub-Shock Formation in a Hyperbolic Nonlinear Viscoelastic Model","authors":"Takashi Arima,&nbsp;Tommaso Ruggeri,&nbsp;Shigeru Taniguchi","doi":"10.1007/s10659-026-10189-1","DOIUrl":"10.1007/s10659-026-10189-1","url":null,"abstract":"<div><p>Shock wave propagation in nonlinear viscoelastic solids is fundamental to understanding their response to rapid dynamic loading. We analyze shock wave structures within a recent symmetric hyperbolic nonlinear viscoelastic model, proposed by Ruggeri (in Int. J. Non-Linear Mech. 160, 2024) and derived within the framework of Rational Extended Thermodynamics. An explicit expression is obtained for the critical Mach number at which a smooth traveling wave loses its <span>(C^{1})</span> regularity and a sub-shock forms. For shocks propagating into an undeformed equilibrium state, the critical Mach number depends solely on the linearized material response, in particular on the ratio of the elastic moduli in the Zener constitutive model. The model admits both compressive and tensile (expansive) shock waves. Numerical simulations based on a Mooney–Rivlin elastic potential coupled with a quadratic viscous energy confirm the theoretical predictions for shock propagation in vulcanized rubber.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-026-10189-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"147336836","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
Shear Models in Finite Elasticity 有限弹性中的剪切模型
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-01-28 DOI: 10.1007/s10659-026-10187-3
Federico Oyedeji Falope, Luca Lanzoni, Angelo Marcello Tarantino
{"title":"Shear Models in Finite Elasticity","authors":"Federico Oyedeji Falope,&nbsp;Luca Lanzoni,&nbsp;Angelo Marcello Tarantino","doi":"10.1007/s10659-026-10187-3","DOIUrl":"10.1007/s10659-026-10187-3","url":null,"abstract":"<div><p>Having been studied for a long time, simple and pure shear deformation models are well known in elasticity. For small deformations, these two shear models differ only by a rigid rotation. On the contrary, for large deformations, the two models do not differ only by a rigid rotation. Therefore, in the latter situation, one cannot expect both models to fit the same constitutive properties for a prescribed form of the stored energy function. The kinematic and static differences between the two shear models, as well as their inadequacies for simulating experimental evidences, are discussed in this paper. To overcome these critical issues, a study was developed, which led quite naturally to the definition of a new shear deformation model, here called <i>purely angular shear</i>, based on the direct extension of the linearized pure shear model. The new model, characterized by a simple and immediate physical meaning, is particularly suitable for matching experimental tests.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-026-10187-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146082716","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
Some New Methods of Solving Boundary Value Problems in Linearized Elasticity and for a Class of Nonlinear Elastic Body 求解线性化弹性和一类非线性弹性体边值问题的几种新方法
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-01-27 DOI: 10.1007/s10659-026-10188-2
R. Bustamante, O. Orellana
{"title":"Some New Methods of Solving Boundary Value Problems in Linearized Elasticity and for a Class of Nonlinear Elastic Body","authors":"R. Bustamante,&nbsp;O. Orellana","doi":"10.1007/s10659-026-10188-2","DOIUrl":"10.1007/s10659-026-10188-2","url":null,"abstract":"<div><p>A new formulation is proposed for linearized elastic solids, which can be used for the analysis of boundary value problems. This formulation is based on considering both the displacement field and the stress tensor as main variables for the problem, solving in parallel the equation of motion and the constitutive equation (expressing the linearized strain as a function of the stresses) to find such unknown variables. Some boundary value problems are solved using separation of variables for the fully dynamic case for isotropic bodies. The application of the above method is briefly considered for anisotropic bodies, and also for a relatively new class of constitutive equation, wherein the linearized strain is a nonlinear function of the stress.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146082412","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
Internal Constraints and Gauge Relations in the Theory of Uniaxial Nematic Elastomers 单轴向列弹性体理论中的内部约束和规范关系
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-01-26 DOI: 10.1007/s10659-025-10185-x
Andrea Pastore, Alfio Grillo, Eliot Fried
{"title":"Internal Constraints and Gauge Relations in the Theory of Uniaxial Nematic Elastomers","authors":"Andrea Pastore,&nbsp;Alfio Grillo,&nbsp;Eliot Fried","doi":"10.1007/s10659-025-10185-x","DOIUrl":"10.1007/s10659-025-10185-x","url":null,"abstract":"<div><p>We apply the formalism of analytical mechanics for constrained systems to reformulate the equilibrium theory of uniaxial nematic elastomers, allowing for constitutive dependence on the gradient <span>(boldsymbol{G})</span> of the director <span>(boldsymbol{n})</span>. In this setting, inextensibility is enforced by requiring that <span>(|boldsymbol{n}|^{2}=1)</span> and that <span>(boldsymbol{G}^{top }boldsymbol{n}=boldsymbol{0})</span>. Starting from these constraints, and using the principle of virtual work within a thermomechanically consistent framework, we derive boundary-value problems for determining equilibrium configurations. We show that the original formulation yields an underdetermined system for the Lagrange multiplier fields unless ancillary gauge conditions are imposed. To resolve this indeterminacy, we introduce two effective Lagrange multiplier fields: one defined in the interior of the referential region and the other on that portion of the boundary where the director traction is prescribed.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146082484","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
Editorial for Special Collection on Liquid Crystal Elastomers and Their Theory 液晶弹性体及其理论特刊编辑
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2026-01-21 DOI: 10.1007/s10659-026-10186-4
L. Angela Mihai, Carl D. Modes
{"title":"Editorial for Special Collection on Liquid Crystal Elastomers and Their Theory","authors":"L. Angela Mihai,&nbsp;Carl D. Modes","doi":"10.1007/s10659-026-10186-4","DOIUrl":"10.1007/s10659-026-10186-4","url":null,"abstract":"","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2026-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146027123","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
Logarithmic Convexity, Continuous Dependence and Uniqueness in Elastodynamics with Higher Gradients 高梯度弹性动力学的对数凸性、连续相关性和唯一性
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2025-12-17 DOI: 10.1007/s10659-025-10184-y
Brian Straughan
{"title":"Logarithmic Convexity, Continuous Dependence and Uniqueness in Elastodynamics with Higher Gradients","authors":"Brian Straughan","doi":"10.1007/s10659-025-10184-y","DOIUrl":"10.1007/s10659-025-10184-y","url":null,"abstract":"<div><p>We investigate Hölder continuous dependence in theories of linear elastodynamics, assuming the elastic coefficients are not sign-definite. This is important with modern products such as auxetic materials where Poisson’s ratio may be negative. This study focusses on a class of linear elastic bodies where there are gradients of the strain and second gradients of the strain, and we also analyse a theory of elastodynamics with strain gradients where voids are also present in the body.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-025-10184-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145778579","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
Hyperelastic Stability Landscape: A Check for Hill Stability of isotropic, incompressible Hyperelasticity depending on Material Parameters 超弹性稳定性景观：基于材料参数的各向同性不可压缩超弹性山体稳定性检验
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2025-12-17 DOI: 10.1007/s10659-025-10183-z
Herbert Baaser
{"title":"Hyperelastic Stability Landscape: A Check for Hill Stability of isotropic, incompressible Hyperelasticity depending on Material Parameters","authors":"Herbert Baaser","doi":"10.1007/s10659-025-10183-z","DOIUrl":"10.1007/s10659-025-10183-z","url":null,"abstract":"<div><p>In this paper, we describe a uniform and standardized approach for analytically verifying the stability of isotropic, incompressible hyperelastic material models. Here, we address <i>stability</i> as fulfillment of the <span>Hill</span> condition – i.e. the positive definiteness of the material modulus in the <span>Kirchhoff</span> stress – log–strain relation. For incompressible material behavior, all mathematically and mechanically possible deformations lie within a range bounded, on the one hand, by uniaxial states and, on the other hand, by biaxial states; shear deformation states lie in between. This becomes particularly clear when the possible states are represented in the invariant plane. This very representation is now also used to visualize the regions of unstable material behavior depending on the selected strain energy function and the respective data set of material parameters. This demonstrates how, for some constellations of energy functions, with appropriate selection or calibration of parameters, stable and unstable regions can be observed. If such cases occur, it is no longer legitimate to use them to initiate, for example, finite element simulations. This is particularly striking when, for example, a fit appears stable in uniaxial tension, but the same parameter set for shear states results in unstable behavior without this being specifically investigated. The presented approach can reveal simple indicators for this. Nevertheless, the <i>simple shear</i> deformation, where the principal axes lag behind the deformation <span>(gamma =tan alpha )</span> of the shear angle <span>(alpha )</span>, i.e. the rotation tensor <span>(textbf{R} neq textbf{I})</span>, still represents a special case that requires extra investigations. This is especially true given that all shear components of the logarithmic strains themselves exhibit a non–monotonic behavior with respect to the deformation angle.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2025-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-025-10183-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145778580","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
Non-linear Oscillations of a Hyperelastic Cylindrical Tube Through Lie Point Symmetry 超弹性圆柱管通过李点对称的非线性振荡
IF 1.4 3区 工程技术
Journal of Elasticity Pub Date : 2025-12-09 DOI: 10.1007/s10659-025-10169-x
Ritika Bahukhandi, Kriti Arya
{"title":"Non-linear Oscillations of a Hyperelastic Cylindrical Tube Through Lie Point Symmetry","authors":"Ritika Bahukhandi,&nbsp;Kriti Arya","doi":"10.1007/s10659-025-10169-x","DOIUrl":"10.1007/s10659-025-10169-x","url":null,"abstract":"<div><p>The second-order differential equation for the non-linear radial oscillations of a transversely-isotropic hyperelastic tube has been postulated and derived based on certain requirements for the strain-energy function and the applied pressure. Lie point symmetry analysis has been used for the non-linear radial oscillatory system composed of neo-Hookean material to address the challenges in solving this equation. A comparison is conducted between the differential equations before and after the Lie transformation. Using Lie point symmetries, we demonstrate that the non-linear differential equations of the transversely-isotropic hyperelastic tube enhance non-periodic oscillations, which can contribute to the prediction of material reliability. This article aims to provide a comprehensive introduction and an application overview in the field of dynamical systems.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"158 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2025-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145730048","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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