Equilibrium shapes of inflated inextensible membranes

This paper proposes an effective method for directly determining the final equilibrium shapes of closed inextensible membranes subjected to internal pressures. With reference to new high-performance textile materials, we assume that the mechanical response of a fabric membrane can be accurately represented by regarding it as a two-state material. In the active state, the membrane is subject to tensile stresses and is virtually inextensible; vice versa, in the passive state it is unable to sustain any compressive stress, so it contracts freely. Equilibrium of the membrane in the final configuration is enforced by recourse to the minimum total potential energy principle. The Lagrange multipliers method is used to solve the minimum problem by accounting for the aforesaid nonlinear constitutive law. The set of governing equations is solved for the unknown coordinates of the equilibrium surface points. Closed form solutions are obtained for fully wrinkled cylindrical and axisymmetric membranes under homogeneous boundary conditions, while a simple iterative procedure is used to numerically solve cases of axisymmetric membranes under various inhomogeneous boundary conditions. The soundness of the proposed method is verified by comparing the results with solutions available in the literature.     

Analysis of pulled axisymmetric membranes with wrinkling

The nonlinear behavior of an axisymmetric hyperelastic membrane subjected to pulling forces is analyzed. The membrane is considered to be ideal in the sense that it cannot carry compressive stress resultants. If the membrane has a positive initial Gaussian curvature, the pulling gives rise to wrinkles which form over parts of the surface. The full nonlinear equations governing the membrane behavior in the doubly tense and in the wrinkled regions are formulated, and then solved using a numerical integration procedure. Solutions for various examples are presented, with Hookean and neo-Hookean constitutive behavior. These include a few examples of wrinkled membranes with positive initial Gaussian curvatures, and one example of a membrane with a negative initial Gaussian curvature, where no wrinkles are formed.     

Exact solution of functionally graded thick cylinder with finite length under longitudinally non-uniform pressure

Elastic analysis of a functionally graded thick-walled cylindrical pressure vessel is analytically studied in the present research. Gradation is considered for all mechanical properties along the thickness direction based on a power function. The constitutive relations are developed in the general cylindrical coordinate system for an axisymmetric pressurized cylinder. For simulation of these two deformation components, first order shear deformation theory is considered. The FG cylinder is subjected to longitudinally non-uniform pressure along the length of the cylinder. The present problem is applicable for simulation of non-uniform pressurized cylinder by fluids or gases.     

Elastic instabilities in rubber

Materials that undergo large elastic deformations can exhibit novel instabilities. Several examples are considered here: development of an aneurysm on inflating a cylindrical rubber tube; non-uniform stretching on inflating a spherical balloon; expansion of small cavities in rubber blocks when they are subjected to a critical amount of triaxial tension or when they are supersaturated with a dissolved gas; wrinkling of the surface of a block at a critical amount of compression; and the sudden formation of “knots” on twisting stretched cylindrical rods. These various deformations are analyzed in terms of simple strain energy functions using Rivlin's theory of large elastic deformations. The theoretical results are then compared with experimental measurements of the onset of unstable states. Such comparisons provide new tests of Rivlin's theory and, at least in principle, critical tests of proposed strain energy functions for rubber. Moreover, the onset of highly non-uniform deformations has serious implications for the fatigue life and fracture resistance of rubber components.     

压力敏感材料中的空洞长大研究及其在页岩及高分子材料中的应用

主要研究压力敏感材料中含内压的空洞长大,如页岩或者高分子材料。采用数值方法研究含内压空洞的对称和非对称球形和柱形胞元的宏观力学行为。结果表明,压力敏感性及其空洞内压将极大影响空洞的形核与长大。在球形胞元情形中未出现柱形胞元的单轴拉伸现象。将胞元有限变形的数值计算结果与基于近期提出的考虑压力敏感材料中空洞长大的塑形力学模型的分析结果进行了对比。     

Homogenization estimates for fiber-reinforced elastomers with periodic microstructures

This work presents a homogenization-based constitutive model for the mechanical behavior of elastomers reinforced with aligned cylindrical fibers subjected to finite deformations. The proposed model is derived by making use of the second-order homogenization method [Lopez-Pamies, O., Ponte Castañeda, P., 2006a. On the overall behavior, microstructure evolution, and macroscopic stability in reinforced rubbers at large deformations: I—theory. J. Mech. Phys. Solids 54, 807–830], which is based on suitably designed variational principles utilizing the idea of a “linear comparison composite.” Specific results are generated for the case when the matrix and fiber materials are characterized by generalized Neo-Hookean solids, and the distribution of fibers is periodic. In particular, model predictions are provided and analyzed for fiber-reinforced elastomers with Gent phases and square and hexagonal fiber distributions, subjected to a wide variety of three-dimensional loading conditions. It is found that for compressive loadings in the fiber direction, the derived constitutive model may lose strong ellipticity, indicating the possible development of macroscopic instabilities that may lead to kink band formation. The onset of shear band-type instabilities is also detected for certain in-plane modes of deformation. Furthermore, the subtle influence of the distribution, volume fraction, and stiffness of the fibers on the effective behavior and onset of macroscopic instabilities in these materials is investigated thoroughly.     

Wrinkling of Orthotropic Membranes: An Analysis by the Polar Method

In the present paper, a simple membrane model based on the wrinkle strain approach is revisited with the aim of examining how the material elastic constants affect the static response of anisotropic membranes when wrinkling is taken into account. Employing the polar method, we analyze the role played by the polar moduli, which enable expressing the elasticity matrix components of an anisotropic material in terms of its invariant quantities. With reference to orthotropic materials, we first address the issue of membrane susceptibility to wrinkling by investigating the influence of the three polar parameters characterizing the anisotropic part of the constitutive law. The stress and strain states at any given point in a wrinkled membrane are analyzed by searching for explicit expressions for the principal values of stress and wrinkle strain. Finally, a comparison between our results and those obtained by a numerical solution available in the literature is made in the basic case of a membrane subjected to a pure shear strain state.     

Ratcheting and wrinkling of tubes due to axial cycling under internal pressure: Part II analysis

Part I presented a set of experiments in which pressurized tubes were cycled axially under stress control about a compressive mean stress. This loading history causes biaxial ratcheting involving compressive axial strain and expansion of the diameter of the tube. The compressive strain in turn induces the initiation and growth of axisymmetric wrinkles. Persistent cycling resulted in localization of the wrinkles and collapse. In Part II the problem is first modeled as a shell with initial axisymmetric imperfections while the biaxial ratcheting of the material is modeled using the Dafalias–Popov two-surface nonlinear kinematic hardening model. It is demonstrated that when suitably calibrated this modeling framework reproduces the prevalent ratcheting deformations and the evolution of wrinkling including the conditions at collapse accurately for all experiments. The calibrated model is then used to evaluate the ratcheting behavior of pipes under thermal-pressure cyclic loading histories experienced by axially restrained pipelines.     

Axisymmetric indentation of curved elastic membranes by a convex rigid indenter

Motivated by applications to seed germination, we consider the transverse deflection that results from the axisymmetric indentation of an elastic membrane by a rigid body. The elastic membrane is fixed around its boundary, with or without an initial pre-stretch, and may be initially curved prior to indentation. General indenter shapes are considered, and the load-indentation curves that result for a range of spheroidal tips are obtained for both flat and curved membranes. Wrinkling may occur when the membrane is initially curved, and a relaxed strain-energy function is used to calculate the deformed profile in this case. Applications to experiments designed to measure the mechanical properties of seed endosperms are discussed.     

On the Wrinkling of Anisotropic Elastic Membranes

The existence and uniqueness of a relaxed energy function for anisotropic elastic membranes is established. It is shown that the extra strains during wrinkling abide by a normality rule akin to that of metal plasticity, and that throughout such wrinkling processes both the strain energy and the Piola-Kirchhoff stresses remain constant. This revised version was published online in August 2006 with corrections to the Cover Date.     

Capillary instabilities in a thin nematic liquid crystalline fiber embedded in a viscous matrix

$m$ to take into account non-axisymmetric modes. Capillary instabilities in nematic fibers reflect the anisotropic nature of liquid crystals, such as the orientation contribution to the surface elasticity and surface bending stresses. Surface gradients of bending stresses provide additional anisotropic contributions to the capillary pressure that may renormalize the classical displacement and curvature forces that exist in any fluid fiber. The exact nature (stabilizing and destabilizing) and magnitude of the renormalization of the displacement and curvature forces depend on the nematic orientation and the anisotropic contribution to the surface energy, and accordingly capillary instabilities may be axisymmetric or non-axisymmetric, with finite or unbounded wavelengths. Thus, the classical fiber-to-droplet transformation is one of several possible instability pathways while others include surface fibrillation. The contribution of the viscosity ratio to the capillary instabilities of a thin nematic fiber in a viscous matrix is analyzed by two parameters, the fiber and matrix Ohnesorge numbers, which represent the ratio between viscous and surface forces in each phase. The capillary instabilities of a thin nematic fiber in a viscous matrix are suppressed by increasing either fiber or matrix Ohnesorge number, but estimated droplet sizes after fiber breakup in axisymmetric instabilities decrease with increasing matrix Ohnesorge number. Received November 26, 2001 / Published online May 21, 2002 Communicated by Epifanio Virga, Pavia     

Dynamic multi-axial behavior of shape memory alloy nanowires with coupled thermo-mechanical phase-field models

The objective of this paper is to provide new insight into the dynamic thermo-mechanical properties of shape memory alloy (SMA) nanowires subjected to multi-axial loadings. The phase-field model with Ginzburg–Landau energy, having appropriate strain based order parameter and strain gradient energy contributions, is used to study the martensitic transformations in the representative 2D square-to-rectangular phase transformations for FePd SMA nanowires. The microstructure and mechanical behavior of martensitic transformations in SMA nanostructures have been studied extensively in the literature for uniaxial loading, usually under isothermal assumptions. The developed model describes the martensitic transformations in SMAs based on the equations for momentum and energy with bi-directional coupling via strain, strain rate and temperature. These governing equations of the thermo-mechanical model are numerically solved simultaneously for different external loadings starting with the evolved twinned and austenitic phases. We observed a strong influence of multi-axial loading on dynamic thermo-mechanical properties of SMA nanowires. Notably, the multi-axial loadings are quite distinct as compared to the uniaxial loading case, and the particular axial stress level is reached at a lower strain. The SMA behaviors predicted by the model are in qualitative agreements with experimental and numerical results published in the literature. The new results reported here on the nanowire response to multi-axial loadings provide new physical insight into underlying phenomena and are important, for example, in developing better SMA-based MEMS and NEMS devices     

Hybrid method for determining the fraction of plastic work converted to heat

The fraction of plastic work converted to heat is typically measured either by nearly isothermal experiments, in which the thermal energy is measured during a deformation experiment with a calorimeter, or by adiabatic experiments, in which the thermal energy is determined from the temperature rise, measured either during the test or immediately after the test by dropping the sample into a calorimeter. In the present work, the temperature is measured with a single fine-wire thermocouple. The restriction to adiabatic loadings is relaxed by using a hybrid method that combines the measurements with finite difference simulations to calculate the heat losses that occur during the test. These heat losses are then accounted for in the final energy balance to determine the fraction of plastic work converted to heat. The method is applied to annealed 302 stainless steel. The results show that the fraction of plastic work converted to heat is a decreasing function ranging from 0.7 to 0.4 over a tensile strain range of 0 to 0.15. An analysis of the restrictions to this method and of the potential errors is given.     

Contact force and mechanical loss of multistage cable under tension and bending

A theoretical model for calculating the stress and strain states of cabling structures with different loadings has been developed in this paper. We solve the problem for the first-and second-stage cable with tensile or bending strain. The contact and friction forces between the strands are presented by two-dimensional contact model. Several theo-retical models have been proposed to verify the results when the triplet subjected to the tensile strain, including contact force, contact stresses, and mechanical loss. It is found that loadings will affect the friction force and the mechanical loss of the triplet. The results show that the contact force and mechanical loss are dependent on the twist pitch. A shorter twist pitch can lead to higher contact force, while the trend of mechanical loss with twist pitch is compli-cated. The mechanical loss may be reduced by adjusting the twist pitch reasonably. The present model provides a simple analysis method to investigate the mechanical behaviors in multistage-structures under different loads.     

冲击荷载与火灾联合作用下SFRC梁的力学行为

为了探究冲击荷载与火灾联合作用下钢纤维混凝土(steel fiber reinforced concrete, SFRC)梁的力学性能,联合应用高性能落锤试验系统、四点弯曲实验装置与装配式电炉开展了4根SFRC梁的冲击实验与高温恒载实验,观察了其破坏模式并记录了跨中位移和钢筋应变的时程曲线,探讨了冲击损伤SFRC梁的抗火性能。此外,在实验研究的基础上,考虑材料的应变率强化效应及温度软化效应,建立数值模型,首先对梁进行冲击加载模拟,并以冲击模拟结果为初始状态,采用热-力“顺序”耦合方法,对冲击加载与高温恒载联合作用下SFRC梁的力学行为进行了三维宏观有限元数值模拟。同时,考虑混凝土内部结构非均质性的影响,采用类似步骤,开展了细观模拟。宏/细观模拟结果与实验结果的良好吻合验证了本文数值方法的合理性与有效性,并体现了细观方法的优越性。研究发现,冲击能量较小时,SFRC梁在冲击荷载作用下,尽管局部混凝土开裂,梁整体残余变形较小,抗火性能有一定程度的下降;随着钢纤维掺量增大,混凝土基体抗剪强度增大,SFRC梁在冲击荷载作用下的开裂形态由弯剪裂缝并存向以弯曲裂缝为主转变;冲击损伤SFRC梁在高温恒载作用下裂缝分布较为集中,且发生脆性破坏。     

A consideration on pre- and post-instability of an axisymmetric elastic beam subjected to axial leakage flow

The dynamical behavior of an axisymmetric elastic beam subjected to axial leakage flow is investigated numerically and experimentally. The coupled equations of motion for a fluid and a beam structure are derived using the Navier–Stokes equation for an axial leakage flow-path and the Euler–Bernoulli beam theory. Performing complex eigenvalue analysis, the variation of the dynamic behavior during pre- and post-instability is investigated with respect to increasing axial leakage flow velocity. Also, an experiment was performed to determine the critical velocity of the unstable dynamic behavior of an axisymmetric elastic beam confined in a concentric cylinder subjected to axial leakage flow through a small annulus, and to measure the variation of the dynamic behavior on pre- and post-instability when the unstable phenomenon with the lower predominant frequency is shifted to the higher one. The relationships between the axial flow velocities and the unstable phenomena are clarified for the transition from the lower mode to the higher mode by comparing the theoretical calculations with experimental observations. Especially, the generation of traveling waves and the energy balance for the distortion of vibration response in the axial direction are discussed and considered at the transition region of the complex coupled vibration response of an axisymmetric elastic beam subjected to an axial leakage flow. Numerical and experimental results are found to be in quite good agreement.     

Mechanics of axisymmetric wavy surface adhesion: JKR–DMT transition solution

Recent work on the mechanics of detachment of a rigid sphere from an elastic axisymmetric wavy surface in the presence of JKR adhesion has shown that the presence of small-amplitude waviness introduces instabilities into the detachment process which dissipate mechanical energy. These instabilities result in interface toughening and strengthening; both the external work and peak force required for separation of a wavy interface are higher than those for a flat interface. In this paper, we summarize the key dimensionless parameters governing axisymmetric wavy surface adhesion in the JKR regime. We then proceed to derive a solution for the JKR–DMT adhesion transition for the axisymmetric wavy surface contact problem using a Maugis–Dugdale cohesive zone formulation. The phenomenon of interface toughening and strengthening due to the presence of surface waviness is seen to be restricted primarily to the JKR adhesion regime.     

Onset of macroscopic instabilities in fiber-reinforced elastomers at finite strain

In this paper, we investigate theoretically the possible development of instabilities in fiber-reinforced elastomers (and other soft materials) when they are subjected to finite-strain loading conditions. We focus on the physically relevant class of “macroscopic” instabilities, i.e., instabilities with wavelengths that are much larger than the characteristic size of the underlying microstructure. To this end, we make use of recently developed homogenization estimates, together with a fundamental result of Geymonat, Müller and Triantafyllidis linking the development of these instabilities to the loss of strong ellipticity of the homogenized constitutive relations. For the important class of material systems with very stiff fibers and random microstructures, we derive a closed-form formula for the critical macroscopic deformation at which instabilities may develop under general loading conditions, and we show that this critical deformation is quite sensitive to the loading orientation relative to the fiber direction. The result is also confronted with classical estimates (including those of Rosen) for laminates, which have commonly been used as two-dimensional (2-D) approximations for actual fiber-reinforced composites. We find that while predictions based on laminate models are qualitatively correct for certain loadings, they can be significantly off for other more general 3-D loadings. Finally, we provide a parametric analysis of the effects of the matrix and fiber properties and of the fiber volume fraction on the onset of instabilities for various loading conditions.     

Crack growth instability studied by the strain energy density theory

The strain energy density theory has successfully been used to address the problem of material damage and structural failure in problems of engineering interest. The theory makes use of the strain energy density function, dW/dV, and focuses attention in its stationary values. The directions of crack growth and yielding are determined from the minimum and maximum values of dW/dV, respectively, along the circumference of a circle centered at the point of failure initiation. Failure by crack growth or yielding takes place when these values of dW/dV become equal to their critical values which are material constants. In the present work the basic principles of the strain energy density theory were reviewed. Furthermore, this theory was used to study three problems of structural failure, namely the problem of slow stable growth of an inclined crack in a plate subjected to uniaxial tension, the problem of fracture instability of a plate with a central crack and two notches, and the problem of unstable crack growth in a circular disc subjected to two equal and opposite forces. The results of stress analysis were combined with the strain energy density theory to obtain the whole history of crack growth from initiation to instability. A length parameter was introduced to define the fracture instability of a mechanical system. Fracture trajectories were obtained for fast unstable crack propagation.