Journal of Elasticity最新文献

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).

Recently, an Eulerian formulation of a nonlinear thermomechanical Cosserat theory of a 3D continuum enriched with a deformable triad of director vectors was developed for anisotropic elastic-inelastic response. To study the influence of the directors on size-dependent response the small deformation purely mechanical equations for this Cosserat continuum are used to formulate and solve the problem of plane-strain pure bending of a circular tube of an elastically isotropic incompressible Cosserat material. Examples present the influences of the stiffness to deformations of the directors and the intrinsic length in the formulation.

This paper presents analytical formulations for systematically deriving the solutions of Biot’s poroelasticity in saturated multilayered media of either full-space or halfspace extents. The number of the saturated multilayer media is either (n+2) for full-space extent or (n+1) for halfspace extent, where (n) is a positive or zero integer. The applied loadings include the internal forces and liquid source for full-space and both internal and external loadings for halfspace region with eight cases of four boundary conditions. The mathematical tools for the formulations are classical and include the two-dimensional Fourier transform, the Hankel transform, Laplace transform as well as linear algebra. The solutions are expressed in matrix forms and each matrix is explicitly expressed with clear physical meaning and well-defined elements. The matrix solutions in the Fourier and Laplace transform domains are axially symmetric about the vertical axis. The internal and boundary conditions can be four-dimensional and the matrix solutions in the physical domain are also four-dimensional.

Following Hill and Leblond, the aim of our work is to show, for isotropic nonlinear elasticity, a relation between the corotational Zaremba–Jaumann objective derivative of the Cauchy stress (sigma ), i.e.

and a constitutive requirement involving the logarithmic strain tensor. Given the deformation tensor (F = mathrm {D}varphi ), the left Cauchy-Green tensor (B = F , F^{T}), and the strain-rate tensor (D = operatorname{sym}(dot{F} , F^{-1})), we show that

where (operatorname{Sym}^{++}_{4}(6)) denotes the set of positive definite, (minor and major) symmetric fourth order tensors. We call the first inequality of (1) “corotational stability postulate” (CSP), a novel concept, which implies the True-Stress True-Strain strict Hilbert-Monotonicity (TSTS-M⁺) for (B mapsto sigma (B) = widehat{sigma}(log B)), i.e.

A similar result, but for the Kirchhoff stress (tau = J , sigma ) has been shown by Hill as early as 1968. Leblond translated this idea to the Cauchy stress (sigma ) but only for the hyperelastic case. In this paper we expand on the ideas of Hill and Leblond, extending Leblond calculus to the Cauchy elastic case.

X-ray diffraction methods are an established technique to analyze residual stresses in polycrystalline materials. Using diffraction, lattice plane distances are measured, from which residual stresses can be calculated by using diffraction elastic constants which can be inferred from experimental measurements or calculated based on micromechanical model assumptions. We consider two different generalizations of existing micromechanical models for the case of texture-free, i.e. statistically isotropic, single-phase polycrystals. The first is based on the singular approximation method of classical micromechanics, from which existing Voigt, Reuss, Hashin-Shtrikman and self-consistent methods are recovered. The second approach, which is newly proposed in this work, is based on the micromechanical Maximum Entropy Method. Both approaches are applied to the problem of calculating diffraction elastic constants of texture-free cubic polycrystals and are found to be consistent with each other in that case. Full-field FFT simulations are used to validate the analytical models by simulating X-ray diffraction measurements of copper. In the simulative setting, many sources of experimental measurement error are not present, which results in a particularly accurate validation of theoretical bounds and approximations. The first core result of the paper is a formulation of diffraction elastic constants for texture-free polycrystals in terms of the macroscopically measurable effective shear modulus. These diffraction elastic constants can be adapted to the properties of a given material sample. The second core result is the validation of the Maximum Entropy Method for X-ray diffraction stress analysis of texture-free single-phase materials as a preliminary step before extending the method to textured and multi-phase materials.