Journal of Elasticity最新文献

The celebrated experiments of J. H. Poynting in 1909 have given rise to a vast literature regarding an interesting feature of the nonlinear response of soft solids. Poynting conducted a series of experiments on metal wires and found that loaded wires lengthen when twisted. Thus to maintain a constant length in such experiments, a compressive axial force would need to be applied at the ends of the specimen. This is the classical (positive) Poynting effect. Another example of such an effect arises when a soft material specimen is being laterally sheared or rotated between two platens. The necessity to apply a compressive lateral normal force in order to maintain the relative distance between the platens is also often referred to as a Poynting-type effect. Both effects are inherently nonlinear phenomena. In recent years, a large body of experimental and theoretical work on the Poynting effect has been carried out. In particular, a reverse Poynting effect has been investigated where the cylinder contracts under torsion unless a tensile axial force is applied or in the case of the lateral shear problem, the platens tend to draw together laterally unless a tensile lateral normal force is applied. The purpose of the present article is to review recent research findings on both of these effects for soft materials.

The material correspondence formulation plays an essential role in connecting the Classical Continuum Mechanics and Peridynamics. In this paper, we analyze the material correspondence formulation in both the generalized two-parameter bond-based and state-based Peridynamics with a particular emphasis on the material symmetry principles. We discover a “trade-off balance” law between material symmetry and Poisson’s ratio in Peridynamics. Specifically, in the generalized two-parameter bond-based Peridynamics, the Poisson’s ratio limitation is eliminated, but the symmetry of the homogenized fourth-order material tensor in this model differs from that in Classical Continuum Mechanics. This asymmetry in the material tensor leads to energy incompatibility between the bond-based Peridynamics and Classical Continuum Mechanics. Furthermore, it can be proved that this incompatible energy has an upper bound and approaches zero as the characteristic length of the non-local interaction domain vanishes. In the case of the state-based Peridynamics, the symmetry of material tensors aligns with Classical Continuum Mechanics. However, the material correspondence formulation imposes a lower bound constraint on the Poisson’s ratio for the state-based Peridynamics. Inspired by this ‘trade-off balance’ law in Peridynamics, we propose a novel continuum model that maintains symmetry consistency. The proposed model integrates local and non-local energy into a single energy functional. By employing the Hamilton’s variational principle, we derive the governing equations with exact force boundary conditions. Unlike Peridynamics, the proposed model exerts the force boundary on the outer surface of the solids. We demonstrate that the proposed model is asymptotically compatible with Classical Continuum Mechanics. Wave dispersion analysis shows that the proposed model does not exhibit zero-energy mode oscillations.

Herein we take a first step towards merging nonlinear elasticity with the two non-relativistic Galilean-covariant limits of electromagnetism, namely the electric limit and the magnetic limit, the results of which we call Galilean electroelasticity and Galilean magnetoelasticity, respectively. Using the first law of thermodynamics for dynamical adiabatic processes, we derive, for systems (with zero free-charge and free-current densities) which undergo such processes, the internal energy density function and its associated constitutive equations in Galilean electroelasticity and magnetoelasticity, respectively. Each of the two internal energy density functions (per unit reference volume) thus obtained agrees with one of the two total energy density functions introduced by Dorfmann and Ogden in their work on electro-elastostatics and magneto-elastostatics, respectively. For linear polarizable and magnetizable dielectrics, Galilean-invariant expressions of the Maxwell stress are obtained for the electric limit and for the magnetic limit, respectively.

Linear elasticity comprises the fundamental branch of continuum mechanics that is extensively used in modern structural analysis and engineering design. In view of this concept, the displacement field provides a measure of how solid materials deform and become internally stressed due to prescribed loading conditions, a fact which is associated with linear relationships between the components of strain and stress, respectively. The mathematical characteristics of these dyadic fields are combined within the Hooke’s law via the stiffness tetratic tensor, which embodies either the isotropic or the anisotropic behavior, exhibited by materials with linear properties. In fact, Hooke’s law is incorporated into the general law of Newton that actually defines the principal spatial and temporal second-order non-homogeneous partial differential equation for the displacement. In this study, we construct handy closed-form solutions for Newton’s law in the Cartesian regime, implying time-independence and considering the case of absence of body forces. Towards this direction, our aim is twofold, in the sense that an efficient analytical technique is introduced that generates homogeneous polynomial solutions of the displacement field for both the typical isotropic and the cubic-type anisotropic structure in the invariant Cartesian geometry. The reliability of the presented methodology is verified by reducing the results for each polynomial degree from the anisotropic to the isotropic eigenspace, in terms of a simple transformation, while we demonstrate our theory with an important application, wherein the effect of a prescribed force on an isotropic half-space to the neighboring half-space of cubic anisotropy is examined.

The comprehensive incompressible strain energy function devised in a preceding contribution (J. Elast. 153:219–244, 2023) is extended in this work for application to the finite deformation of isotropic compressible rubber-like materials. Based on the two established approaches in the literature for constructing a compressible strain energy function (W) from the incompressible counterpart, two models are developed and presented: one model is developed on using a (Jleft (= lambda _{1} thinspace lambda _{2} thinspace lambda _{3} right )) term added to the general functional form of the incompressible model; and the second model on using the isochoric, or modified, principal stretches (left (bar{lambda }_{a}right )), (a = 1,2,3), in the functional form of the incompressible model, to account for the deviatoric contribution (W_{dev}). The volumetric input (W_{vol}) is considred as an additive part. Each model is then simultaneously fitted to extant multi-axial experimental datasets, and the favourable correlation between the models’ predictions and the experimental data is demonstrated. Exemplar challenging individual datasets including a shear-softening behaviour exhibited by an elastomeric foam are also considered, whereby the excellent predictions of the said behaviours by the models will be illustrated. The compatibility of both models with the kinematics of slight compressibility will also be discussed and presented.

Helical vessels are widely observed in many pathological tissues, such as solid tumors, healing wounds, and varicose veins. However, it remains yet unclear how originally healthy and straight vessels transit to helical shapes. Here, we combine theoretical analysis and numerical simulations to investigate the helical buckling of growing vessels embedded in matrix. Based on linear stability analysis, we predict the critical growth strain that induces three-dimensional (3D) helical and two-dimensional (2D) sinusoidal buckling of vessels. This critical growth strain is regulated by the geometry of the vessel and the modulus ratio between the vessel and the matrix. A phase diagram is established to reveal the dependence of helical and sinusoidal modes on the vessel thickness and modulus ratio. Finite element simulations are performed to validate the theoretical prediction of the critical growth strain and further track the postbuckling evolution of growing vessels. The pitch of helix and the long and short axis of projected cross-section of vessels are characterized with increasing growth strain. Our findings elucidate the mechanism underlying abnormal formation of helical vessels, in consistency with the observations in tumors and varicose veins. This study could also inspire mechanics-based technologies for diseases diagnosis.

Tissue homeostasis, the biological process of maintaining a steady state in tissue via control of cell proliferation and death, is essential for the development, growth, maintenance, and proper function of living organisms. Disruptions to this process can lead to serious diseases and even death. In this study, we use the vertex model for the cell-level description of tissue mechanics to investigate the impact of the tissue environment and local mechanical properties of cells on homeostasis in confined epithelial tissues. We find a dynamic steady state, where the balance between cell divisions and removals sustains homeostasis, and characterise the homeostatic state in terms of cell count, tissue area, homeostatic pressure, and the cells’ neighbour count distribution. This work, therefore, sheds light on the mechanisms underlying tissue homeostasis and highlights the importance of mechanics in its control.

In this paper, a novel non-local thermoviscoelasticity model incorporates non-Fourier effects within a fractional calculus framework. It focuses on the thermal impact on a non-simple nanobeam subjected to ramp-type heat loading. The study investigates the thermoelastic behavior of a viscoelastic nanoscale rectangular beam based on the non-local Euler-Bernoulli beam theory (EBBT) under thermal heating conditions. This paper uses integral transformation methods to derive closed-form solutions for temperature, bending moments, deflection, and thermal stress. These solutions are initially formulated in the Laplace domain and then converted into the time domain using the Gaver-Stehfest algorithm. Numerical results for silicon nitride are analyzed and graphically visualized with Mathematica software. The study examines the effects of relaxation time, ramping time parameters, and fractional order parameters across various fields, comparing the findings with previously published literature. This research highlights the complex interplay between thermal and mechanical responses in nanobeams and provides new insights into the behavior of viscoelastic materials under non-local and fractional thermoelastic conditions.

We generalize the classical lattice dynamics model to a fractal setting by writing the basic equations of crystal dynamics according to a so-called LOSA model based on the “product-like fractal measure” concept. The fractal model is used to investigate the solutions to the dynamics of a Bravais lattice. The key role is played by so-called fractal elastic waves, where the mathematical solutions to wave equations have been obtained and analyzed, accounting for the underlying fractal geometry. By comparing the theoretical results with available experimental data graphically, we obtain a good match, say, for a diamond. Our numerical analysis estimates the fractal dimension of the order of (D approx 2.52) which is in agreement with research studies on fractal phononic crystals. Attenuation of waves in cubic solids for small fractal dimensions has been observed. Such results justify the naming of this study as an estimation of the fractal dimension of phononic crystals.

We consider a Kelvin–Voigt model for viscoelastic second-grade materials, where the elastic and the viscous stress tensor both satisfy frame-indifference. Using a rigidity estimate by Ciarlet and Mardare (J. Math. Pures Appl. 104:1119–1134, 2015), existence of weak solutions is shown by means of a frame-indifferent time-discretization scheme. Further, the result includes viscous stress tensors which can be calculated by nonquadratic polynomial densities. Afterwards, we investigate the long-time behavior of solutions in the case of small external loading and initial data. Our main tool is the abstract theory of metric gradient flows (Ambrosio et al. in Gradient Flows in Metric Spaces and in the Space of Probability Measures, Birkhäuser, Basel, 2005).