Journal of Elasticity最新文献

{"title":"Characterization of an Overlooked Kinematical Descriptor in the Second-Gradient Hyperelastic Theory for Thin Shells","authors":"Sankalp Tiwari,&nbsp;Eliot Fried","doi":"10.1007/s10659-024-10103-7","DOIUrl":"10.1007/s10659-024-10103-7","url":null,"abstract":"<div><p>In 1978, Murdoch presented a direct second-gradient hyperelastic theory for thin shells in which the strain-energy density associated with a deformation <span>(boldsymbol{eta })</span> of a surface <span>(mathcal{S})</span> is allowed to depend constitutively on the three kinematical descriptors <span>(boldsymbol{C})</span>, <span>(boldsymbol{H})</span>, and <span>(boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G})</span>, where <span>(boldsymbol{F}=text{Grad} _{scriptscriptstyle mathcal{S}} boldsymbol{eta })</span>, <span>(boldsymbol{C}=boldsymbol{F}^{scriptscriptstyle top }boldsymbol{F})</span>, <span>(boldsymbol{H}=boldsymbol{F}^{scriptscriptstyle top }boldsymbol{L}_{ scriptscriptstyle mathcal{S}'}boldsymbol{F})</span> is the covariant pullback of the curvature tensor <span>(boldsymbol{L}_{scriptscriptstyle mathcal{S}'})</span> of the deformed surface <span>(mathcal{S}')</span>, and <span>(boldsymbol{G}=text{Grad} _{scriptscriptstyle mathcal{S}} boldsymbol{F})</span>. On the other hand, in Koiter’s direct thin-shell theory, the strain-energy density depends constitutively on only <span>(boldsymbol{C})</span> and <span>(boldsymbol{H})</span>. Due to the popularity of Koiter’s theory, the second-order tensors <span>(boldsymbol{C})</span> and <span>(boldsymbol{H})</span> are well understood and have been extensively characterized. However, the third-order tensor <span>(boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G})</span> in Murdoch’s theory is largely overlooked in the literature. We address this gap, providing a detailed characterization of <span>(boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G})</span>. We show that for <span>(boldsymbol{eta })</span> twice continuously differentiable, <span>(boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G})</span> depends solely on <span>(boldsymbol{C})</span> and its surface gradient <span>(text{Grad} _{scriptscriptstyle mathcal{S}}boldsymbol{C})</span> and does not depend on <span>(boldsymbol{L}_{scriptscriptstyle mathcal{S}'})</span>. For the special case of a conformal deformation, we find that a suitably defined strain measure corresponding to <span>(boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G})</span> depends only the conformal stretch and its surface gradient. For the further specialized case of an isometric deformation, this strain measure vanishes. An orthogonal decomposition of <span>(boldsymbol{F}^{scriptscriptstyle top }boldsymbol{G})</span> reveals that it belongs to a ten-dimensional subspace of the space of third-order tensors and embodies two independent types of non-local phenomena: one related to the spatial variations in the stretching of <span>(mathcal{S}')</span> and the other to the curvature of <span>(mathcal{S})</span>.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-024-10103-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826119","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}

{"title":"On the Modeling of Active Deformation in Biological Transversely Isotropic Materials","authors":"Giulia Giantesio,&nbsp;Alessandro Musesti","doi":"10.1007/s10659-024-10101-9","DOIUrl":"10.1007/s10659-024-10101-9","url":null,"abstract":"<div><p>Many biological materials exhibit the ability to actively deform, essentially due to a complex chemical interaction involving two proteins, actin and myosin, in the myocytes (the muscle cells). While the mathematical description of passive materials is well-established, even for large deformations, this is not the case for active materials, since capturing its complexities poses significant challenges. This paper focuses on the mathematical modeling of active deformation of biological materials, guided by the important example of skeletal muscle tissue. We will consider an incompressible and transversely isotropic material within a hyperelastic framework. Our goal is to design constitutive relations that agree with uniaxial experimental data whenever possible. Finally, we propose a novel model based on a coercive and polyconvex elastic energy density for a fiber-reinforced material; in this model, active deformation occurs solely through a change in the reference configuration of the fibers, following the mixture active strain approach. This model assumes a constant active parameter, preserving the good mathematical features of the original model while still capturing the essential deformations observed in experiments.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142826431","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 “Energies for elastic plates and shells from quadratic-stretch elasticity”","authors":"E. Vitral,&nbsp;J. A. Hanna","doi":"10.1007/s10659-024-10096-3","DOIUrl":"10.1007/s10659-024-10096-3","url":null,"abstract":"","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142821491","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":"Deformation of a Planar Ferromagnetic Elastic Ribbon","authors":"G. R. Krishna Chand Avatar,&nbsp;Vivekanand Dabade","doi":"10.1007/s10659-024-10100-w","DOIUrl":"10.1007/s10659-024-10100-w","url":null,"abstract":"<div><p>In this paper we explore the influence of magnetisation on the deformation of planar ferromagnetic elastic ribbons. We begin the investigation by deriving the leading-order magnetic energy associated with a curved planar ferromagnetic elastic ribbon. The sum of the magnetic and the elastic energy is the total energy of the ribbon. We derive the equilibrium equations by taking the first variation of the total energy. We then systematically determine and analyse solutions to these equilibrium equations under various canonical boundary conditions. We also determine the stability of the equilibrium solutions. Comparing our findings with the well-studied Euler’s elastica provides insights into the magnetic effects on the deformation behaviour of elastic ribbons. Our analysis contributes to a deeper understanding of the interplay between magnetisation and the mechanical response of planar ferromagnetic structures, and offers valuable insights for both theoretical and practical applications.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142798518","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":"Vertical Dynamic Analysis of Rigid Strip Foundation on Layered Unsaturated Media","authors":"Zhi Yong Ai,&nbsp;Li Wei Shi,&nbsp;Lei Sheng","doi":"10.1007/s10659-024-10099-0","DOIUrl":"10.1007/s10659-024-10099-0","url":null,"abstract":"<div><p>This paper analytically investigates the vertical dynamic response of a rigid strip foundation on layered unsaturated media. Using the triphasic Biot-type model and extended precise integration method, we derive the flexibility coefficient for layered unsaturated media. On this basis, by introducing the Bessel function series of the first kind, the dual integral equations of the mixed boundary value problem in this study are transformed into a set of linear equations. Finally, we obtain explicit expressions for the contact stress and vertical compliance, which are used to evaluate the soil-structure interaction. After the proposed solution is verified, several parameters are presented to study the impacts of the stratification, soil thickness, saturation degree, air-entry value and dimensionless frequency.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142753994","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":"Energy Balance and Damage for Dynamic Fast Crack Growth from a Nonlocal Formulation","authors":"Robert P. Lipton,&nbsp;Debdeep Bhattacharya","doi":"10.1007/s10659-024-10098-1","DOIUrl":"10.1007/s10659-024-10098-1","url":null,"abstract":"<div><p>A nonlocal model for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. Evolution from the elastic to the inelastic phase depends on material strength. Existence and uniqueness of the displacement-failure set pair follow from an initial value problem describing the evolution. The displacement-failure pair satisfies energy balance. The length of nonlocality <span>(epsilon )</span> is taken to be small relative to the domain in <span>(mathbb{R}^{d})</span>, <span>(d=2,3)</span>. The strain is formulated as a difference quotient of the displacement in the nonlocal model. The two point force is expressed in terms of a weighted difference quotient and delivers an evolution on a subset of <span>(mathbb{R}^{d}times mathbb{R}^{d})</span>. This evolution provides an energy balance between external energy, elastic energy, and damage energy including fracture energy. For any prescribed loading the deformation energy resulting in material failure over a region <span>(R)</span> is uniformly bounded as <span>(epsilon rightarrow 0)</span>. For fixed <span>(epsilon )</span>, the failure energy is discovered to be is nonzero for <span>(d-1)</span> dimensional regions <span>(R)</span> associated with flat crack surfaces. Calculation shows, this failure energy is the Griffith fracture energy given by the energy release rate multiplied by area for <span>(d=3)</span> (or length for <span>(d=2)</span>). The nonlocal field theory is shown to recover a solution of Naiver’s equation outside a propagating flat traction free crack in the limit of vanishing spatial nonlocality. The theory and simulations presented here corroborate the recent experimental findings of (Rozen-Levy et al. in Phys. Rev. Lett. 125(17):175501, 2020) that cracks follow the location of maximum energy dissipation inside the intact material. Simulations show fracture evolution through the generation of a traction free internal boundary seen as a wake left behind a moving strain concentration.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714648","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":"Torsion and Extension of Functionally Graded Mooney–Rivlin Cylinders","authors":"Kesna A. Fairclough,&nbsp;Romesh C. Batra","doi":"10.1007/s10659-024-10095-4","DOIUrl":"10.1007/s10659-024-10095-4","url":null,"abstract":"<div><p>We analytically study finite torsional and extensional deformations of rubberlike material circular cylinders with the two material moduli in the Mooney–Rivlin relation assumed to be continuous functions of the undeformed radius. It is shown that under null resultant axial load on the end faces the cylinder length increases upon twisting. Furthermore, when the two moduli are affine functions of the radius the inhomogeneity parameters can be found to have the maximum shear stress occur at a pre-determined interior point. Whereas the radial stress is finite at the center of a cross-section of a homogeneous material cylinder, it may have large values for an inhomogeneous material cylinder. The closed-form solutions provided herein for the two moduli having affine, power-law and exponential functions of the radius should benefit numerical analysts verify their algorithms and engineers design soft material robots for improving their performance under torsional loads.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-024-10095-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679538","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}

{"title":"An Approach to Growth Mechanics Based on the Analytical Mechanics of Nonholonomic Systems","authors":"Alfio Grillo,&nbsp;Andrea Pastore,&nbsp;Salvatore Di Stefano","doi":"10.1007/s10659-024-10092-7","DOIUrl":"10.1007/s10659-024-10092-7","url":null,"abstract":"<div><p>Motivated by the convenience, in some biomechanical problems, of interpreting the mass balance law of a growing medium as a nonholonomic constraint on the time rate of a structural descriptor known as growth tensor, we employ some results of analytical mechanics to show that such constraint can be studied variationally. Our purpose is to move a step forward in the formulation of a field theory of the mechanics of volumetric growth by defining a Lagrangian function that incorporates the nonholonomic constraint of the mass balance. The knowledge of such Lagrangian function permits, on the one hand, to deduce the dynamic equations of a growing medium as the result of a variational procedure known as Hamilton–Suslov Principle (clearly, up to non-potential generalized forces that are accounted for by extending this procedure), and, on the other hand, to study the symmetries and conservation laws that pertain to a given growth problem. While this second issue is not investigated in this work, we focus on the first one, and we show that the Euler–Lagrange equations of the considered growing medium, which describe both its motion and the evolution of the growth tensor, can be obtained by reformulating a variational method developed by other authors. We discuss the main features of this method in the context of growth mechanics, and we show how our procedure is able to improve them.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10659-024-10092-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672724","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}

{"title":"An Analytical Solution for the Periodically Spaced Two Collinear and Symmetric Cracks Under Remote Tension","authors":"Jiayao Hu,&nbsp;Fan Jin,&nbsp;Fan Xia,&nbsp;Jicheng Li","doi":"10.1007/s10659-024-10093-6","DOIUrl":"10.1007/s10659-024-10093-6","url":null,"abstract":"<div><p>The present paper provides an analytical solution for a periodic array of two collinear and symmetric cracks (P-TCSC) under remote tension. This is achieved by representing the multiple collinear cracks problem as the contact problem with discrete ligament regions, and the governing equations are obtained as integral equations with Cauchy-type kernel. Closed-form expressions are derived for the crack opening profile, normal stress distribution and mode I stress intensity factors (SIFs), which can reduce to the classical solutions of two collinear and symmetric cracks (TCSC) or a periodic row of collinear cracks with equal length and equal spacing (PCEE) under special conditions. Finite element analysis is also performed to validate the analytical solutions obtained. Different from the TCSC case, results show that crack initiation for P-TCSC seems more complicated depending on a combination of two nondimensional parameters, and a SIFs map for P-TCSC is further constructed to give a more precise evaluation. The proposed method relies solely on solving the integral equations with Cauchy-type kernel combined with the corresponding boundary conditions without a prior knowledge of the complex potential function in traditional complex variable method of plane elasticity, and it may find application in plastic zone evaluation and fracture criteria of collinear cracks.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595324","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":"Universal Displacements in Anisotropic Linear Cauchy Elasticity","authors":"Arash Yavari,&nbsp;Dimitris Sfyris","doi":"10.1007/s10659-024-10094-5","DOIUrl":"10.1007/s10659-024-10094-5","url":null,"abstract":"<div><p>Universal displacements are those displacements that can be maintained for any member of a specific class of linear elastic materials in the absence of body forces, solely by applying boundary tractions. For linear hyperelastic (Green elastic) solids, it is known that the space of universal displacements explicitly depends on the symmetry group of the material, and moreover, the larger the symmetry group the larger the set of universal displacements. Linear Cauchy elastic solids, which include linear hyperelastic solids as a special case, do not necessarily have an underlying energy function. Consequently, their elastic constants do not possess the major symmetries. In this paper, we characterize the universal displacements of anisotropic linear Cauchy elasticity. We prove the result that for each symmetry class, the set of universal displacements of linear Cauchy elasticity is identical to that of linear hyperelasticity.</p></div>","PeriodicalId":624,"journal":{"name":"Journal of Elasticity","volume":"157 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142595323","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}