Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

An experimental study on the mechanical properties of rat brain tissue using different stress–strain definitions

There are different stress–strain definitions to measure the mechanical properties of the brain tissue. However, there is no agreement as to which stress–strain definition should be employed to measure the mechanical properties of the brain tissue at both the longitudinal and circumferential directions. It is worth knowing that an optimize stress–strain definition of the brain tissue at different loading directions may have implications for neuronavigation and surgery simulation through haptic devices. This study is aimed to conduct a comparative study on different results are given by the various definitions of stress–strain and to recommend a specific definition when testing brain tissues. Prepared cylindrical samples are excised from the parietal lobes of rats’ brains and experimentally tested by applying load on both the longitudinal and circumferential directions. Three stress definitions (second Piola–Kichhoff stress, engineering stress, and true stress) and four strain definitions (Almansi–Hamel strain, Green-St. Venant strain, engineering strain, and true strain) are used to determine the elastic modulus, maximum stress and strain. The highest non-linear stress–strain relation is observed for the Almansi–Hamel strain definition and it may overestimate the elastic modulus at different stress definitions at both the longitudinal and circumferential directions. The Green-St. Venant strain definition fails to address the non-linear stress–strain relation using different definitions of stress and triggers an underestimation of the elastic modulus. The results suggest the application of the true stress–true strain definition for characterization of the brain tissues mechanics since it gives more accurate measurements of the tissue’s response using the instantaneous values.     

基于二阶理论的弹性约束变截面悬臂梁刚度与稳定性分析

从计入二阶效应的挠曲微分方程出发,对惯性矩沿轴向二次变化的变截面Bernoulli-Euler梁在弹性约束下的刚度和稳定性进行了分析,推导了在弹性约束下变截面悬臂梁在复合载荷作用下的挠度和稳定性的精确表达式,给出轴向压力引起的挠度影响系数。在极端情况下,该文公式可相应退化为根部固支的变截面梁及等截面梁之刚度与稳定表达式。将该文的计算结果与用ANSYS软件密分单元的计算结果进行分析比较,分析比较结果验证了该文推导的刚度和稳定性表达式的正确性,该文方法可广泛应用于弹性约束下变截面悬臂梁的刚度和稳定性分析。     

Elastic wave propagation in a functionally graded nanocomposite reinforced by carbon nanotubes employing meshless local integral equations (LIEs)

The transient dynamic analysis of displacement field and elastic wave propagation in finite length functionally graded nanocomposite reinforced by carbon nanotubes are carried out using local integral equations (LIEs) based on meshless local Petrov–Galerkin (MLPG) method. The distribution of the aligned carbon nanotubes (CNTs) is assumed to vary as three kinds of functionally graded distributions as well as uniform distribution (UD) through radial direction of axisymmetric reinforced cylindrical composites. The mechanical properties are simulated using a micro-mechanical model in volume fraction form. A unit step function is used as a test function in the local weak form, which leads to local integral equations (LIEs). The analyzed domain is divided into small subdomains with a circular shape. The radial basis functions are used for approximation of the spatial variation of field variables. For treatment of time variations, the Laplace-transform technique is utilized. The 2D propagation of elastic waves through 2D domain is illustrated for various kinds of carbon nanotubes distributions. The time histories of displacement fields are studied in detail for various kinds of carbon nanotube distributions in reinforced cylindrical composites.     

Experimental and numerical study of stress–strain state of pressurised cylindrical shells with external defects

Work presents the experimental technique with using of strain gauges for determining of strain distribution near longitudinal external defects of semielliptical shape in pressurised cylindrical shells and an appropriate procedure of numerical calculation based on a finite element method (FEM) for assessment of strain state near such defects. Numerical assessment of elastic stress distribution at bottom of external semielliptical notch in pressurised cylindrical shell is in a good agreement with the data received on the base of different analytical models. Here, the FEM results and data based on Glinka-Newport model are the most close for maximal stress. A comparison an experimental and calculation results showed on their acceptable coincidence and this fact gives the ground for using the numerical calculations instead labour-intensive and expensive experimental tests.     

Finite twist and extension of a cylindrical elastic membrane

Summary We consider a circular cylindrical membrane subjected to longitudinal extension and twist. The associated equilibrium deformation is considered to be axisymmetric and the analysis is based on a direct two-dimensional formulation. Wrinkling of the membrane is taken into account in an approximate way by introducing arelaxed strain energy function derived from the neo-Hookean strain energy for isotropic elastic solids. Analytical formulae for wrinkled parts of the membrane are used to corroborate the results of a numerical treatment of the full boundary value problem.Dedicated to the memory of A. C. Pipkin     

Influence of fibre taper on the work of fibre pull-out in short fibre composite fracture

A model has been formulated to determine the work of pull-out, U, of an elastic fibre as it shear-slides out of a plastic matrix in a fractured composite. The fibres considered in the analysis have the following shapes: uniform cylinder and ellipsoidal, paraboloidal or conical tapers. Energy transfer at the fibre–matrix interface is described by an energy density parameter which is defined as the ratio of U to the fibre surface area. The model predicts that the energy required to pull out a tapered fibre is small because the energy transfer at the fibre–matrix interface to overcome friction is small. In contrast, the pull-out energy of a uniform cylindrical fibre is large because the energy transfer is large. The pull-out energies of the paraboloidal and ellipsoidal fibres lay between those for the uniform cylindrical and the conical fibres. With the exception of the uniform cylindrical fibre which yields a constant energy density, tapered fibres yield expressions for the energy density which depend on the fibre axial ratio, q. In particular, the energy density increases as q increases but converges at large q. By defining the critical axial ratio, q ₀, as the limit beyond which u is independent of the fibre slenderness, our model predicts the value of q ₀ to be about 10. These results are applied to explain the mechanisms regulating fibre composite fracture.     

Elastic solutions of a cylindrical rod containing periodically distributed inclusions with axisymmetric eigenstrains

Summary. In this paper, we give the elastic solution for a special type of microstructure – a circular cylindrical rod containing periodically distributed inclusions along its axial direction. Each inclusion has the same uniform axisymmetric transformation strain (eigenstrain). Analytical elastic solutions are obtained for the displacements, stresses and elastic strain energy of the rod. The effects of microstructure and its evolution (growth of inclusions) on the elastic stress and strain fields as well as the strain energy of the rod are quantitatively demonstrated. As a result of such microstructure evolution nominal stress-strain relation with strain softening is derived for a rod under uniaxial tension.     

Response of metal matrix laminated cylindrical panels to combined uniform temperature change and edge displacements

Summary A quasistatic response of metal matrix laminated cylindrical panels subjected to a combination of uniform thermal loading and applied edge displacements is addressed. To treat the problem, an approach based on a micro-to-macro analysis is proposed. A micromechanical analysis enables one to account for the thermo-viscoplastic behavior of the metallic matrix and its interaction with the thermo-elastic fibers, and provides the overall constitutive behavior of metal matrix composite (MMC). This is further employed in a structural analysis to obtain the response of simply supported MMC laminated panels.Results are presented for unidirectional and antisymmetric angle-ply SiC/Ti panels, both geometrically perfect and imperfect. The elastic and viscoplastic material properties of Ti matrix are considered as temperature-dependent. The effect of mechanical pre-loading on the following thermal response is illustrated. The influence of the panel curvature, length-to-thickness ratio, amplitude of initial geometrical imperfections, lamination angle, and the effect of heating followed by cooling are investigated. Comparisons with the corresponding elastic solutions, obtained by neglecting the inelastic effects in the metallic matrix, are presented.     

Modeling the effective elastic properties of nanocomposites with circular straight CNT fibers reinforced in the epoxy matrix

In the present study, the consistent effective elastic properties of straight, circular carbon nanotube epoxy composites are derived using the micromechanics theory. The CNT composites are known to provide high stiffness and elastic properties when the shape of the fibers is cylindrical and straight. Accordingly, in the present work, the effective elastic moduli of composite are newly obtained for straight, circular CNTs aligned in the specified direction as well as distributed randomly in the matrix. In this direction, novel analytical expressions are proposed for four cases of fiber property. First, aligned, and straight CNTs are considered with transverse isotropy in fiber coordinates, and the composite properties are also transversely isotropic in global coordinates. The short comings in the earlier developments are effectively addressed by deriving the consistent form of the strain tensor and the stiffness tensor of the CNT nanocomposite. Subsequently, effective relations for composites reinforced with aligned, straight CNTs but fibers isotropic in local coordinates are newly developed under hydrostatic loading. The effect of the unsymmetric Eshelby tensor for cylindrical fibers on the overall properties of the nanocomposite is included by deriving the strain concentration tensors. Next, the random distribution of CNT fibers in the matrix is studied with fibers being transversely isotropic as well as isotropic when CNT nanocomposites are subjected to uniform loading. The corresponding relations for the effective elastic properties are newly derived. The modeling technique is validated with results reported, and the variations in the effective properties for different CNT volume fractions are presented.     

Constitutive equation for a class of non-simple elastic materials

Summary Initial yield is the upper limit of the purely elastic deformation behaviour of an elasticplastic solid. Thus the choice of the constitutive equation describing the purely elastic deformation behaviour determines the initial yield function. The constitutive equation of a simple elastic material is only compatible with von Mises yield criterion, a conclusion which applies also to the classical infinitesimal theory. A more general form of constitutive equation for an elastic material is formulated by way of the concept of a stress loading function, the proposed constitutive equation being quadratic in the stress. The two loading coefficients associated with the stress loading function are assumed to be deriveable from a generalised isotropic yield criterion which is now assumed to hold over the entire range of deformation, and in this context is referred to as the stress intensity function. The proposed constitutive equation has the same representation in terms of the left Cauchy-Green deformation tensor as that for a simple elastic material. Using the Cayley-Hamilton theorem, this representation is rearranged and expressed in terms of a measure of finite strain which is defined to be one quarter of the difference between the left Cauchy-Green deformation tensor and its inverse. In this way the strain properties of the proposed constitutive equation are formulated by way of the concept of a strain response function. The three response coefficients associated with the strain response function are assumed to be deriveable from a generalised, isotropic, strain intensity function. The predictions of the proposed constitutive equation are considered in the context of the combined stressing of a thin sheet of incompressible material. In this way, it is shown that the proposed constitutive equation is not limited in the same way as the constitutive equation of a simple elastic material.     

Combined loading micromechanical analysis of a unidirectional composite

A microscopic region of a unidirectional composite is modelled by a finite element micromechanical analysis using a generalized plane strain formulation, but including longitudinal shear loading. The analysis is capable of treating elastic, transversely isotropic fibre materials, as well as isotropic, elastoplastic matrix materials. Matrix material properties are considered to be temperature- and/or moisture-dependent. The longitudinal shear loading capability permits the analysis of the shear response of unidirectional composites in the fibre direction. In conjunction with a laminated plate point stress analysis, the present micromechanical analysis has been used to predict the stress/strain response into the inelastic range of a graphite/epoxy [±45]_4s laminate. Available experimental data for various environmental conditions indicate excellent agreement with the analytical predictions.     

A facet-like shell theory

A linear theory for facet-like thin elastic shells is derived where strain/displacement, curvature change/displacement and constitutive relations appear the same as for flat plates. Application of Koiter's arguments shows that the theory is a valid first approximation. The theory is of interest for limiting cases of faceted finite element analysis of smooth shells. Although the final equations of facet-like shell theory do not have quite as simple a form as more conventional equations it is possible that their derivation from equations for flat plates may appeal to engineers. A specialization of the equations is given to circular cylindrical shells where four simple illustrative examples show no essential differences with results from more conventional theory.     

Nonlinear inelastic uniform torsion of bars by BEM

In this paper the elastic–plastic uniform torsion analysis of simply or multiply connected cylindrical bars of arbitrary cross-section taking into account the effect of geometric nonlinearity is presented employing the boundary element method. The stress–strain relationship for the material is assumed to be elastic–plastic–strain hardening. The incremental torque–rotation relationship is computed based on the finite displacement (finite rotation) theory, that is the transverse displacement components are expressed so as to be valid for large rotations and the longitudinal normal strain includes the second-order geometric nonlinear term often described as the “Wagner strain”. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross-section’s torsional rigidity is evaluated exactly without using the so-called Saint-Venant’s torsional constant. The torsional rigidity of the cross-section is evaluated directly employing the primary warping function of the cross-section depending on both its shape and the progress of the plastic region. A boundary value problem with respect to the aforementioned function is formulated and solved employing a BEM approach. The influence of the second Piola–Kirchhoff normal stress component to the plastic/elastic moment ratio in uniform inelastic torsion is demonstrated. The developed procedure retains most of the advantages of a BEM solution over a pure domain discretization method, although it requires domain discretization, which is used only to evaluate integrals.     

Finite strain elastic closed form solutions for some axisymmetric problems

Summary In this paper, rate type constitutive equations using the Jaumann and 2nd Piola-Kirchhoff stress rate are used to develop axisymmetric finite strain, elastic, closed form solutions for a variety of loading conditions. We examined several cases comprizing compression-extension loading conditions, simple shear and cavity expansion conditions. Results from small strain analyses are used to indicate strain ranges for which such analyses will provide satisfactory solutions. For all the cases examined, except simple shear, the 2nd Piola-Kirchhoff stress rate does not appear to be a suitable stress rate to describe a material which follows a rate-type constitutive equation for strains greater than about 40%. The Jaumann stress rate solution shows oscillatory shear stress for axisymmetric simple shear similar to that found earlier by many other authors for rectangular (or cuboidal) condition. Negative excess pore water pressure (suction) at the cavity wall during the expansion of a cylindrical cavity was also observed in Jaumann stress rate solution.     

Experimental Determination of Inflatable,Braided Tube Constitutive Properties

The primary objective of this study was to conduct constitutive tests of relatively large diameter inflatable, braided fabric tubes at different inflation pressures and braid angles in order to quantify the longitudinal modulus, in‐plane shear modulus and effective lamina stiffness properties. The stiffness properties quantified here are of high interest because the same braided fabric tubes have been used in the construction of test articles for a major, multi‐year, ground based test campaign led by the United States National Aeronautics and Space Administration. These properties are also input directly into high‐fidelity yet computationally intensive 3D shell‐based finite‐element simulations of the large, inflatable structures. Experimental methods employed during this study included tension–torsion testing, uniaxial tension testing of individual fibre tows, and uniaxial tension testing of gas bladder coupons. Digital image correlation was used to measure all of the geometric information that is necessary to perform netting theory calculations. The test results indicate that fabric in‐plane shear modulus is highly dependent on both braid angle and inflation pressure, but that longitudinal stiffness is quite small and relatively unaffected by braid angle and pressure. In addition to advancing the state‐of‐the art in experimental constitutive property determination of inflatable, braided fabric, this study includes the development of a method to back calculate lamina properties from the experimental results that are suitable for use as input to commonly used finite‐element programmes. The digital image correlation data revealed spatial variation of shear strain that was important to consider when computing the gross shear stiffness. Digital image correlation data also captured the braid surface flattening with increasing inflation pressure, which supports the fibre de‐crimping theory.     

Micro-mechanical analysis of splitting failure in concrete reinforced with fiber reinforced plastic rods

The present investigation provides a micro-mechanical model for the splitting failure analysis of fiber reinforced plastic (FRP) reinforced concrete members subjected to longitudinal tensile stresses. The model consists of three co-axial cylinders: (a) the inner elastic FRP rod; (b) the mid cracked part of concrete; and (c) the outer elastic part of concrete. The anisotropic properties of reinforcement, the compatibility of longitudinal strain at interface and the effect of Poisson's ratio of concrete are taken into account in the analysis. The method can be used to predict the stress distributions in the hybrid structure and the relations between the growth of cracks and the applied end forces. It is found that the number of splitting cracks and the material properties of the anisotropic FRP rods are not the dominant factors in splitting failure. It is also observed that neglecting Poisson's ratio of cracked concrete may under-estimate stresses in the hybrid structure.     

Homogenized mechanical behavior of composite micro-structures including micro-cracking and contact evolution

The aim of this paper is to study the effects of micro-cracking on the homogenized constitutive properties of elastic composite materials. To this end a novel micro-mechanical approach based on homogenization techniques and fracture mechanics concepts, is proposed and an original J-integral formulation is established for composite micro-structures. Accurate non-linear macroscopic constitutive laws are developed for a uniaxial and a shear macro-strain path by taking into account changes in micro-structural configuration owing to crack growth and crack face contact. Numerical results, carried out by coupling a finite element formulation and an interface model, are applied to a porous composite with edge cracks and a debonded short fiber-reinforced composite. The composite micro-structure is controlled by the macroscopic strain and the micro-to-macro transition, settled in a variational formulation, is obtained for three types of boundary conditions, i.e. linear displacements, uniform tractions and periodic fluctuations and anti-periodic tractions. The accuracy of the determined macroscopic constitutive properties to represent the failure characteristics of locally periodic defected composites is also investigated in terms of energy release rate predictions, by comparisons between a direct analysis and homogenization approaches. Results highlight the dependence of the macroscopic constitutive law for a micro-structure with evolving defects on both the macro-strain path and the type of boundary conditions and the capability of the proposed model to provide a failure model for a composite material undergoing micro-cracking and contact.     

Modeling the elastic properties of carbon nanotube array/polymer composites

In this work, the elastic properties of single-walled carbon nanotube (SWNT) arrays and their composites are investigated. The properties of twisted SWNT nano-arrays or ropes of circular cross-section are predicted through a finite element analysis by applying proper boundary conditions to the model and using the strain energy method. The nano-array properties are then used to describe the properties of twisted SWNT nano-array/polymer composites. The effect of volume fraction and aspect ratio of the nano-array as reinforcement for dilute polymer composite systems are examined for aligned and random reinforcement distribution using conventional micromechanics. Finally, elastic properties of the twisted SWNT nano-array/polymer composites are compared to the results from constitutive model of individual nanotube-reinforced polymer composites.