Journal of Elasticity最新文献

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.