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An Analytical Solution for the Periodically Spaced Two Collinear and Symmetric Cracks Under Remote Tension 远距离张力作用下周期性间隔的两平行对称裂缝的解析解
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-11-06 DOI: 10.1007/s10659-024-10093-6
Jiayao Hu, Fan Jin, Fan Xia, Jicheng Li

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.

本文提供了远距离拉伸条件下周期性双对偶对称裂缝阵列(P-TCSC)的解析解。为此,本文将多对偶裂纹问题表示为具有离散韧带区域的接触问题,并将控制方程表示为具有 Cauchy 型核的积分方程。推导出了裂缝开口轮廓、法向应力分布和模式 I 应力强度因子 (SIF) 的闭式表达式,在特殊条件下,这些表达式可简化为两个对偶对称裂缝 (TCSC) 或一排周期性等长等间距对偶裂缝 (PCEE) 的经典解。此外,还进行了有限元分析,以验证所获得的分析解。与 TCSC 的情况不同,结果表明 P-TCSC 的裂缝起始似乎更复杂,取决于两个非尺寸参数的组合,并进一步构建了 P-TCSC 的 SIFs 图,以给出更精确的评估。所提出的方法只需求解具有 Cauchy 型内核的积分方程,并结合相应的边界条件,而无需事先了解传统平面弹性复变法中的复势函数,可应用于塑性区评估和共线裂缝的断裂判据。
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引用次数: 0
Universal Displacements in Anisotropic Linear Cauchy Elasticity 各向异性线性考氏弹性中的普遍位移
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-11-06 DOI: 10.1007/s10659-024-10094-5
Arash Yavari, Dimitris Sfyris

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.

通用位移是指在没有体力的情况下,仅通过施加边界牵引力就能保持特定类别线性弹性材料中任何成员的位移。对于线性超弹性(绿色弹性)固体,众所周知,通用位移空间明确取决于材料的对称群,此外,对称群越大,通用位移集就越大。线性考氏弹性固体(包括作为特例的线性超弹性固体)不一定具有基本的能量函数。因此,它们的弹性常数不具备主要对称性。在本文中,我们描述了各向异性线性考西弹性的普遍位移。我们证明了这样一个结果:对于每个对称类,线性考氏弹性的通用位移集与线性超弹性的通用位移集相同。
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引用次数: 0
Axisymmetric Indentation of Circular Rigid Plate on Layered Elastic Halfspace with Transverse Isotropy 具有横向各向同性的层状弹性半空间上的圆形刚性板轴对称压痕
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-10-09 DOI: 10.1007/s10659-024-10090-9
Sha Xiao, Zhongqi Quentin Yue

This paper investigates the contact problem of a layered elastic halfspace with transverse isotropy under the axisymmetric indentation of a circular rigid plate. Fourier integral transforms and a backward transfer matrix method are used to obtain the analytical solution of the contact problem. The interaction between the rigid plate and the layered halfspace can be expressed with the standard Fredholm integral equations of the second kind. The induced elastic field in the layered halfspace can be expressed as the semi-infinite integrals of four known kernel functions. The convergence and singularity of the semi-infinite integrals near or at the surface of the layered halfspace are resolved using an isolating technique. The efficient numerical algorithms are used and developed for accurately calculating the Fredholm integral equations and the semi-infinite integrals. Numerical results show the correctness of the proposed method and the effect of layering non-homogeneity on the elastic fields in layered transversely isotropic halfspace induced by the axisymmetric indentation of a circular rigid plate.

本文研究了具有横向各向同性的层状弹性半空间在圆形刚性板的轴对称压痕作用下的接触问题。本文采用傅里叶积分变换和后向传递矩阵法求得接触问题的解析解。刚性板与层状半空间之间的相互作用可以用标准的第二类弗雷德霍姆积分方程来表示。层状半空间中的诱导弹性场可表示为四个已知核函数的半无限积分。利用隔离技术解决了半无限积分在层状半空间附近或表面的收敛性和奇异性问题。使用并开发了高效的数值算法,用于精确计算弗雷德霍姆积分方程和半无限积分。数值结果表明了所提方法的正确性,以及分层非均质性对圆形刚性板轴对称压痕诱导的分层横向各向同性半空间弹性场的影响。
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引用次数: 0
Self-Oscillations of Submerged Liquid Crystal Elastomer Beams Driven by Light and Self-Shadowing 由光和自阴影驱动的浸没液晶弹性体光束的自振荡
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-10-09 DOI: 10.1007/s10659-024-10091-8
Reza Norouzikudiani, Luciano Teresi, Antonio DeSimone

Liquid Crystal Elastomers (LCEs) are responsive materials that undergo significant, reversible deformations when exposed to external stimuli such as light, heat, and humidity. Light actuation, in particular, offers versatile control over LCE properties, enabling complex deformations. A notable phenomenon in LCEs is self-oscillation under constant illumination. Understanding the physics underlying this dynamic response, and especially the role of interactions with a surrounding fluid medium, is still crucial for optimizing the performance of LCEs. In this study, we have developed a multi-physics fluid-structure interaction model to explore the self-oscillation phenomenon of immersed LCE beams exposed to light. We consider a beam clamped at one end, originally vertical, and exposed to horizontal light rays of constant intensity focused near the fixed edge. Illumination causes the beam to bend towards the light due to a temperature gradient. As the free end of the beam surpasses the horizontal line through the clamp, self-shadowing induces cooling, initiating the self-oscillation phenomenon. The negative feedback resulting from self-shadowing injects energy into the system, with sustained self-oscillations in spite of the energy dissipation in the surrounding fluid. Our investigation involves parametric studies exploring the impact of beam length and light intensity on the amplitude, frequency, and mode of oscillation. Our findings indicate that the self-oscillation initiates above a certain critical light intensity, which is length-dependent. Also, shorter lengths induce oscillations in the beam with the first mode of vibration, while increasing the length changes the elasticity property of the beam and triggers the second mode. Additionally, applying higher light intensity may trigger composite complex modes, while the frequency of oscillation increases with the intensity of the light if the mode of oscillation remains constant.

液晶弹性体(LCE)是一种反应性材料,在受到光、热和湿度等外部刺激时会发生显著的可逆变形。光驱动尤其能对液晶弹性体的特性进行多功能控制,实现复杂的变形。LCE 的一个显著现象是在恒定光照下的自振荡。了解这种动态响应的基本物理原理,尤其是与周围流体介质相互作用的作用,对于优化 LCE 的性能仍然至关重要。在这项研究中,我们建立了一个多物理流体-结构相互作用模型,以探索浸没在光下的 LCE 光束的自振现象。我们考虑了一个一端夹紧、原本垂直的横梁,并将其暴露在聚焦于固定边缘附近的恒定强度水平光线下。由于温度梯度的作用,光照会导致光束向光线弯曲。当光束的自由端超过穿过夹具的水平线时,自阴影会导致冷却,从而引发自振现象。自阴影产生的负反馈为系统注入了能量,尽管能量在周围流体中耗散,但自振现象仍在持续。我们的研究包括参数研究,探索光束长度和光强对振幅、频率和振荡模式的影响。我们的研究结果表明,自振荡是在超过一定临界光强时开始的,而临界光强与长度有关。此外,较短的长度会在光束中引起第一振动模式的振荡,而增加长度则会改变光束的弹性特性并引发第二振动模式。此外,如果振荡模式保持不变,则振荡频率随光照强度的增加而增加。
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引用次数: 0
Residual Stress Concentration Due to Nano-Scaled Particulate Contamination at Direct Bonding Interface with Localized Material Inhomogeneity 局部材料不均匀性直接粘合界面上的纳米级微粒污染导致的残余应力集中
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-09-30 DOI: 10.1007/s10659-024-10089-2
X. W. Chen, Wendal Victor Yue

Direct bonding is an attractive technique to join material components without the use of intermediate adhesive medium. Usually, the bonding interface can experience high level of residual stress concentration due to entrapped nano-scale particulate contamination. Existing theoretical models are not capable of analyzing such residual stress concentration, since they fail to consider the localized material inhomogeneity formed between the bonding pairs as result of thermal and diffusion processes. This paper proposes a new theoretical model to analyze the residual stress concentration in the bonding interface with the consideration of localized material inhomogeneity. Following the idea of Selvadurai and Singh (Int. J. Fract. 25:69–77, 1984), the nano-scaled particulate contamination induced interfacial defect is simulated as a penny-shaped crack indented by a smooth rigid disc inclusion. This mode I crack-inclusion model is interpreted as a three-part mixed boundary value problem in the theory of elasticity, which is solved by a series expansion technique. Mathematical difficulties associated with modelling arbitrary localized material inhomogeneity are overcome by the use of the General Kelvin Solution (GKS) based method. Exact analytical solutions for the stress intensity factors (SIFs) and resultant force on the inclusion are obtained. Our results show that the inclusion-crack radius ratio and the localized material inhomogeneity can have significance effect on the residual stress concentration at the bonding interface.

直接粘接是一种极具吸引力的技术,可在不使用中间粘合介质的情况下连接材料组件。通常,由于夹带纳米级微粒污染,粘接界面会出现较高的残余应力集中。现有的理论模型无法分析这种残余应力集中,因为它们没有考虑粘合对之间由于热和扩散过程而形成的局部材料不均匀性。本文提出了一种新的理论模型,在考虑局部材料不均匀性的情况下分析键合界面的残余应力集中。根据 Selvadurai 和 Singh(Int. J. Fract. 25:69-77, 1984)的观点,纳米级微粒污染引起的界面缺陷被模拟为由光滑刚性圆盘夹杂物缩进的一分钱形裂纹。这种 I 型裂纹-夹杂模型被解释为弹性理论中的三部分混合边界值问题,并通过序列展开技术加以解决。使用基于一般开尔文解法(GKS)的方法克服了与任意局部材料不均匀性建模相关的数学困难。我们获得了应力强度因子(SIF)和包体上结果力的精确解析解。我们的结果表明,包体-裂缝半径比和局部材料不均匀性会对粘接界面的残余应力集中产生重要影响。
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引用次数: 0
De-Hydration and Remodeling of Biological Materials: Swelling Theory for Multi-Domain Bodies 生物材料的脱水和重塑:多域体的膨胀理论
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-09-25 DOI: 10.1007/s10659-024-10088-3
Michele Curatolo, R. G. M. van der Sman, Luciano Teresi

Biological materials always exhibit heterogeneous physical properties, both mechanical and chemical, which give them a rich phenomenology that poses significant challenges in the developing of effective models. The Flory–Rehner theory revolutionized our understanding of the dynamics of the liquid-polymers coupling in soft swollen gels, recognizing polymers as elastic networks stretched by the presence of liquid. Despite its foundational role, applying this theory to bodies with non uniform physical properties requires further improvements. This article proposes a unified approach to address mechano-diffusion challenges in multi-domain bodies, that is in material bodies made of regions having different chemo-mechanical properties, and focuses on the dehydration and remodeling of biological-like materials. Drawing inspiration from natural systems, we integrate principles from nonlinear mechanics and swelling theories; in particular, what is specifically new is the idea of applying the notion of the multiplicative decomposition of the strain–developed for plasticity–to model the swelling properties of a body made of two or more materials. The article gives a systematic presentation of the subject, and guides readers through key concepts and practical insights, aiming to provide a robust framework for modeling chemo-mechanical interactions. Moreover, it paves the way for the modeling of heterogenous bodies having spatially-varying properties.

生物材料总是表现出异质的物理特性,包括机械和化学特性,这赋予了它们丰富的现象学,给建立有效模型带来了巨大挑战。弗洛里-雷纳理论彻底改变了我们对软膨胀凝胶中液体-聚合物耦合动力学的理解,将聚合物视为因液体存在而被拉伸的弹性网络。尽管该理论具有奠基性作用,但将其应用于物理性质不均匀的物体仍需进一步改进。本文提出了一种统一的方法来解决多域体(即由具有不同化学机械特性的区域组成的物质体)中的机械扩散难题,并将重点放在类生物材料的脱水和重塑上。我们从自然系统中汲取灵感,整合了非线性力学和膨胀理论的原理;特别是,我们的新思路是应用为塑性而开发的应变乘法分解概念来模拟由两种或两种以上材料构成的体的膨胀特性。文章系统地介绍了这一主题,并引导读者了解关键概念和实用见解,旨在为化学-机械相互作用建模提供一个稳健的框架。此外,它还为具有空间变化特性的异质体建模铺平了道路。
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引用次数: 0
Initial Stresses in a Twisted Porous Fluid-Saturated Cylinder 扭曲多孔流体饱和圆柱体中的初始应力
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-09-12 DOI: 10.1007/s10659-024-10086-5
Alexander Suvorov

In this paper a porous fluid-saturated cylinder subjected to a finite twist deformation is analyzed. The material of the skeleton of the porous cylinder is hyperelastic of Ogden-type and assumed nearly incompressible. The twist is applied to the cylinder in a fast rate so that the fluid pressure develops in the pores of the cylinder. The main objective of this paper is to study the stresses and the fluid pressure in the cylinder over a short period of time after the twist has been applied, or to study the initial response. The analytical expressions for the stress components and the fluid pressure are derived for Ogden material with arbitrary material parameters. The quantitative picture for the stress state is given and the signs of the normal stresses are explained. The stress arising in some imaginary fibers that were initially parallel to the axis of the cylinder is obtained. The present problem is similar to the torsion problem of a totally incompressible and nonporous cylinder in a sense that the total stresses are identical in both problems. But decomposition of the total stresses into the fluid pressure and the effective stresses, which is specific for the fluid-saturated body, can be found only using the present analysis.

本文分析了受有限扭转变形影响的多孔饱和流体圆柱体。多孔圆柱体的骨架材料为奥格登型超弹性材料,假定几乎不可压缩。扭转以较快的速度施加到圆柱体上,从而在圆柱体的孔隙中产生流体压力。本文的主要目的是研究施加扭转后短时间内圆柱体中的应力和流体压力,或研究初始响应。本文推导了具有任意材料参数的奥格登材料的应力分量和流体压力的分析表达式。给出了应力状态的定量图像,并解释了法向应力的符号。得出了最初平行于圆柱体轴线的一些假想纤维中产生的应力。本问题类似于完全不可压缩的无孔圆柱体的扭转问题,从某种意义上说,两个问题中的总应力是相同的。但是,只有采用本分析方法,才能将总应力分解为流体压力和有效应力,这是流体饱和体所特有的。
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引用次数: 0
New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar 扭转刚性和 z 杆多项式逼近的新视角
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-09-11 DOI: 10.1007/s10659-024-10087-4
Adam Kraus, Brian Simanek

We consider polynomial approximations of (bar{z}) to better understand the torsional rigidity of polygons. Our main focus is on low degree approximations and associated extremal problems that are analogous to Pólya’s Conjecture for torsional rigidity of polygons. We also present some numerics in support of Pólya’s Conjecture on the torsional rigidity of pentagons.

我们考虑了 (bar{z})的多项式近似,以更好地理解多边形的扭转刚性。我们主要关注低度近似和相关的极值问题,这些问题类似于 Pólya 对多边形扭转刚性的猜想。我们还提出了一些数字来支持 Pólya 关于五边形扭转刚性的猜想。
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引用次数: 0
A Morphoelastic Shell Theory of Biological Invagination in Embryos 胚胎生物迷走的形态弹性壳理论
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-09-03 DOI: 10.1007/s10659-024-10084-7
Xiaoyi Chen, Xiang Yu, Pasquale Ciarletta

The embryo of Volvox globator, a monolayer spheroidal cell sheet, undergoes an inversion to release its flagella during the late stage of its development. This inversion, known as the type-B inversion, initiates from the equator. Other types of inversions also exist, such as the inversion from the anterior pole of Volvox carteri and the bowl-shaped inversion of Pleodorina. These inversions can be regarded as axisymmetric processes, during which complex fold patterns are generated. The invagination of the cell sheet plays a crucial role in embryonic development, and our aim is to understand this process from an interdisciplinary point of view, with a particular focus on the mechanical aspects. In this work, we first develop a morphoelastic shell theory for general deformations of biological shells, incorporating both active and passive biomechanical effects, as well as membrane and bending effects. By means of asymptotic analysis, we establish an analytical framework to study axisymmetric deformations of morphoelastic shells focusing mainly on the membrane effects. For illustrative purposes, we apply this framework to investigate the invagination of Volvox globator embryo. The underlying active stretches driving this process are derived analytically by inverse analysis of experimental data through the morphoelastic shell model. We highlight a two-order remodeling strategy that generates the observed invagination pattern: the Gaussian morphing of the cell sheet creates the first fundamental form of the stress-free folded patterns, while a secondary remodeling generates the membrane tension necessary to balance the external pressure and the second fundamental form of the invaginated pattern. This remodeling strategy unveils the complex interplay between geometry, mechanics, and biological processes during Volvox globator embryogenesis.

Volvox globator 的胚胎是一种单层球形细胞片,在其发育后期会发生倒转以释放鞭毛。这种倒转被称为 B 型倒转,从赤道开始。还存在其他类型的倒转,如 Volvox carteri 的前极倒转和 Pleodorina 的碗状倒转。这些倒转可被视为轴对称过程,在此过程中会产生复杂的褶皱图案。细胞膜的内陷在胚胎发育过程中起着至关重要的作用,我们的目标是从跨学科的角度来理解这一过程,尤其侧重于机械方面。在这项工作中,我们首先开发了一种形态弹性壳理论,用于研究生物壳的一般变形,其中包含主动和被动生物力学效应以及膜和弯曲效应。通过渐近分析,我们建立了一个分析框架来研究形态弹性壳的轴对称变形,主要侧重于膜效应。为了说明问题,我们将此框架用于研究 Volvox globator 胚胎的内陷。通过形态弹性壳模型对实验数据进行反分析,分析得出了驱动这一过程的基本活动拉伸。我们强调了产生所观察到的内陷模式的二阶重塑策略:细胞膜的高斯变形产生了无应力折叠模式的第一种基本形式,而二次重塑产生了平衡外部压力和内陷模式第二种基本形式所需的膜张力。这种重塑策略揭示了 Volvox globator 胚胎发生过程中几何、力学和生物过程之间复杂的相互作用。
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引用次数: 0
A Direct Approach to the Polar Representation of Plane Tensors 平面张量极性表示的直接方法
IF 1.8 3区 工程技术 Q2 ENGINEERING, MULTIDISCIPLINARY Pub Date : 2024-08-30 DOI: 10.1007/s10659-024-10085-6
Marco Picchi Scardaoni

We show we can derive the so called polar representation of 2D symmetric second and fourth order real tensors essentially just relying on the spectral theorem for unitary tensors in the complex field. The use of a coordinate-free approach allows us to clearly and methodically detect rotation-invariant quantities and to readily (and directly) deduce the representation theorems.

我们的研究表明,只需依靠复数场中单元张量的谱定理,我们就能推导出二维对称二阶和四阶实张量的所谓极性表示。通过使用无坐标方法,我们可以清晰、有条不紊地检测旋转不变量,并轻松(直接)推导出表示定理。
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引用次数: 0
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Journal of Elasticity
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