Analytical Solutions for Bending Curved Beams with Different Moduli in Tension and Compression

When materials that exhibit different mechanical behaviors in tension and compression must be analyzed, Ambartsumyan's bimodular model for isotropic materials can be adopted. It deals with the principal stress state in a point, which is particularly important in the analysis and design of structures. In this article, an equivalent section method is used to transform the bimodular curved beam into a classical one with singular modulus; consequently, the simplified solution for bending stresses may be easily determined only by changing a few parameters relating to section characteristics. For the determination of the unknown neutral layer, a perturbation method is used to obtain the explicit expression. Based on the known neutral layer, a stress function method is used to obtain the elasticity solution for stresses and displacements via boundary conditions and continuity conditions. Based on the elasticity solution, an initial stresses problem in a bimodular multiply-connected body is considered. The comparison between two solutions shows that the simplified solution agrees very well with the elasticity one. Moreover, the inclusion of shear stress and the application of the equivalent section method in reinforced-concrete curved beams are also discussed. The results indicate that the bimodularity of materials has definite influences on the bending behavior of a bimodular curved beam.     

A perturbation solution of von-Kármán circular plates with different moduli in tension and compression under concentrated force

By modifying classical von-Kármán equations, we established bimodular von-Kármán equations of thin plates with different moduli in tension and compression. Adopting central deflection as a perturbation parameter, we used a perturbation method to solve the equations under various boundary conditions, including rigidly clamped, loosely clamped, simply hinged, and simply supported. The relation of load versus central deflection and stress formulas were derived via the perturbation solution obtained. The numerical simulation also shows that the perturbation solution based on central deflection is overall valid. The results indicate that when the compressive modulus of materials is greater than the tensile one, the bearing capacity of the plate will be further strengthened, which should be considered in the analysis and design of plate-like structures with obvious bimodular effect. Moreover, by comparing with the case under uniformly distributed load, the plate-membrane transition under centrally concentrated force presents discontinuity to some extent.     

双模量复合材料层板的大挠度分析

本文用动力松弛法(DRM)分析了双模量复合材料层板的大挠度弯曲问题。文中介绍了求解该问题的主要公式和步骤;对轻度双模量材料及高度双模量材料的两层正交铺层矩形板在正弦载荷和均匀载荷作用下的挠度和内力进行了数值计算,并把其结果与小挠度结果及单模量结果做了比较,讨论了大挠度分析的必要性及模量性质对大挠度分析的影响。      

用Timoshenko梁修正理论研究功能梯度材料梁的动力响应

采用Timoshenko梁修正理论研究了功能梯度材料梁的动力响应问题,利用静力方程确定了功能梯度材料梁的中性轴位置,在此基础上应用Timoshenko梁修正理论建立了功能梯度材料梁的振动方程,求得其自振频率表达式及其在简谐荷载作用下强迫振动的解析解。讨论分析了中性面位置、梯度指数等因素对功能梯度材料梁的动力响应的影响,并用有限元法验证了Timoshenko梁修正理论。通过实例计算,得到了中性轴位置对功能梯度材料梁动力响应有较大影响的结论。     

Analytical solution of a cantilever functionally graded beam

The problem of a cantilever functionally graded beam subjected to different loads is studied. In terms of Airy stress function a general two-dimensional solution is presented for a cantilever functionally graded beam, assuming that all the elastic moduli of the material have the same variations along the beam-thickness direction. Explicit expressions of analytical solutions to some specific examples under different boundary conditions are obtained to demonstrate the usefulness of the proposed general solution technique. This solution will be useful in analyzing functionally graded beam with arbitrary variations of material properties and it can serve as a basis for establishing simplified functionally graded beam theories.     

Analytical solution for a functionally graded beam with arbitrary graded material properties

The plane stress problem of an orthotropic functionally graded beam with arbitrary graded material properties along the thickness direction is investigated by the displacement function approach for the first time. A general two-dimensional solution is obtained for a functionally graded beam subjected to normal and shear tractions of arbitrary form on the top and bottom surfaces and under various end boundary conditions. For isotropic case explicit solutions are given to some specific through-the-thickness variations of Young’s modulus such as exponential model, linear model and reciprocal model. The influence of different grade models on the stress and displacement fields are illustrated in numerical examples. These analytical solutions can serve as a basis for establishing simplified theories and evaluating numerical solutions of functionally graded beams.     

Longitudinal vibration analysis of strain gradient bars made of functionally graded materials (FGM)

Longitudinal free vibration analysis of axially functionally graded microbars is investigated on the basis of strain gradient elasticity theory. Functionally graded materials can be defined as nonhomogeneous composites which are obtained by combining of two different materials in order to obtain a new desired material. In this study, material properties of microbars are assumed to be smoothly varied along the axial direction. Rayleigh–Ritz solution technique is utilized to obtain an approximate solution to the free longitudinal vibration problem of strain gradient microbars for clamped–clamped and clamped-free boundary conditions. A parametric study is carried out to show the influences of additional material length scale parameters, material ratio, slenderness ratio and ratio of Young’s modulus on natural frequencies of axially functionally graded microbars.     

考虑材料的拉压模量不同及应变软化特性的柱形孔扩张问题

建立了拉压模量不同及应变软化特性材料的柱形孔扩张理论。对于岩土类材料,提出用a及b分别作为拉压模量不同和软化特征的控制参数,运用不同模量弹性理论及应力跌落软化模型推导了Tresca和Mohr-Coulomb材料柱形圆孔扩张问题的应力及位移解。分析了不同模量及软化特性材料对柱形孔扩张的影响,结果表明:圆孔极限扩张压力,塑性区的发展规律,应力场,位移场等均随着模量参数a和软化系数b的变化而变化,因此若采用经典的弹性理论及传统的不考虑应变软化来对岩土类的工程材料进行设计计算,必会带来较大的误差。     

覆盖层为功能梯度材料弹性半平面中的Love波

对均匀各向同性弹性半平面上覆盖一层功能梯度材料中存在的Love波的频散问题进行了研究,给出了Love波频散方程的一般形式。利用WKBJ近似理论,给出了功能梯度材料层的位移、应力近似解析解,导出了Love波WKBJ近似频散方程的一般形式。该文以功能梯度材料层的剪切弹性模量和质量密度沿厚度方向均为指数函数变化为例,进行了实例计算和分析,给出了频散曲线,讨论了Love波在功能梯度材料覆盖层弹性半平面中传播的一般性质。这些结论对无损检测和反问题分析方法的改进提供理论依据。     

Contact problem of a functionally graded layer resting on a Winkler foundation

The contact problem for a functionally graded layer supported by a Winkler foundation is considered using linear elasticity theory in this study. The layer is loaded by means of a rigid cylindrical punch that applies a concentrated force in the normal direction. Poisson’s ratio is taken as constant, and the elasticity modulus is assumed to vary exponentially through the thickness of the layer. The problem is reduced to a Cauchy-type singular integral equation with the use of Fourier integral transform technique and the boundary conditions of the problem. The numerical solution of the integral equation is performed by using Gauss–Chebyshev integration formulas. The effect of the material inhomogeneity, stiffness of the Winkler foundation and punch radius on the contact stress, the contact area and the normal stresses are given.     

Evolution of Phase Strains During Tensile Loading of Bovine Cortical Bone

Synchrotron X‐ray scattering is used to measure average strains in the two main nanoscale phases of cortical bone – hydroxyapatite (HAP) platelets and collagen fibrils – under tensile loading at body temperature (37 °C) and under completely hydrated conditions. Dog‐bone shaped specimens from bovine femoral cortical bone were prepared from three anatomical quadrants: anterio‐medial, anterio‐lateral, and posterio‐lateral. The apparent HAP and fibrillar elastic moduli – ratios of tensile stress as applied externally and phase strains as measured by diffraction – exhibit significant correlations with the (i) femur quadrant from which the samples are obtained, (ii) properties obtained at the micro‐scale using micro‐computed tomography, i.e., microstructure, porosity and attenuation coefficient, and (iii) properties at the macro‐scale using thermo‐gravimetry and tensile testing, i.e., volume fraction and Young's modulus. Comparison of these tensile apparent moduli with compressive apparent moduli (previously published for samples from the same animal and tested under the same temperature and irradiation conditions) indicates that collagen deforms plastically to a greater extent in tension. Greater strains in the collagen fibril and concomitant greater load transfer to the HAP result in apparent moduli that are significantly lower in tension than in compression for both phases. However, tensile and compressive Young's moduli measured macroscopically are not significantly different during uniaxial testing.     

功能梯度材料Timoshenko型剪切梁的自由振动分析

基于Timoshenko梁理论,研究各向异性功能梯度材料梁的自由振动。假设材料参数沿梁厚度方向按同一函数规律变化,建立了功能梯度材料梁的振动方程,求得简支条件下其自振频率表达式。通过算例,给出指数函数梯度变化Timoshenko梁的自振频率和模态图,结果表明不同梯度变化对材料结构动力响应有较大影响。该方法为发展功能梯度材料梁的设计与数值计算提供了理论依据。     

Deflection of thick beams of multimodular materials

A transfer-matrix analysis is presented for determining the static behaviour of thick beams of ‘multimodular materials’ (i.e. materials which have different elastic behaviour in tension and compression, with nonlinear stress–strain curves approximated as piecewise linear, with four or more segments). To validate the transfer-matrix method results, a closed-form solution is also presented for cases in which the neutral-surface location is constant along the beam axis. Numerical results for axial displacement, transverse deflection, bending slope, bending moment, transverse shear, axial force and location of neutral surface are presented for multimodular and bimodular models of unidirectional aramid cordrubber. The transfer-matrix method results agree very well with the closed-form solutions.     

Compressional behaviour of carbon fibres

Spectroscopic-mechanical studies have been conducted on a range of carbon fibres by bonding single filaments on the top surface of a cantilever beam. Such a loading configuration allows the acquisition of the Raman spectrum of carbon fibres and the derivation of the Raman frequency strain dependence in tension and compression. Strain hardening phenomena in tension and strain softening phenomena in compression were closely observed. The differences in the slopes of the Raman frequency versus applied strain curves in tension and compression respectively, have been used to obtain good estimates of the compression moduli. A method of converting the fibre Raman frequency versus strain data into stress-strain curves in both tension and compression, is demonstrated. Values of fibre stress and fibre modulus at failure in compression compare exceptionally well with corresponding estimates deduced from full composite data. The mode of failure in compression has been found to depend upon the carbon fibre structure. It is demonstrated that certain modifications in the manufacturing technology of PAN-based fibres can lead to fibres which show resistance to catastrophic compressive failure without significant losses in the fibre compressive modulus.     

正常使用状态下RUHTCC梁的弯曲变形预测

该文以有效截面惯性矩为参数,提出了正常使用状态下钢筋增强超高韧性水泥基复合材料(RUHTCC)梁的挠度简化计算方法。与钢筋混凝土梁不同,RUHTCC梁需要考虑受拉区UHTCC在弯曲开裂后的承载能力,理论分析表明基于拉压区面积矩平衡与基于内力平衡且假定压应力线性分布所得的中和轴高度计算公式不同,而后者符合中和轴的物理意义。采用UHTCC的非线性拉、压本构模型以及受压区边缘压应变对应的割线模量,得到基于内力平衡的RUHTCC梁完全开裂截面的中和轴高度以及惯性矩公式。其次,有效惯性矩公式中的常系数m修正为随配筋率变化的函数,得到适用于RUHTCC梁的有效惯性矩公式。采用该文提出的挠度简化计算方法,对比预测与实测的屈服前荷载-挠度曲线,两者基本一致。     

Impulsive responses of functionally graded material bars due to collision

We investigated the ability of functionally graded materials (FGMs) to absorb impact energy by mathematically analyzing the impulsive responses of functionally graded (FG) bars colliding with a homogeneous bar on the basis of Laplace transformation and calculated by using numerical transformation and its inversion. Young’s modulus in the FG bar was assumed to be proportional to the square of its density, which was similar to foam materials. Results showed that maximum impact loads were not strongly dependent on the distribution of material property in the FG bar. In the FG bar with increasing modulus from the impact end to the fixed end, much larger compressive stress and even large tensile stress occurred near the fixed end compared with the ones in the homogeneous bar. In the FG bar with decreasing modulus from the impact end, the compressive stress was approximately the same as the one in the homogeneous bar, and the history of the stress varied regularly.     

Nonlinear large deflection buckling analysis of compression rod with different moduli

Many novel materials exhibit a property of different elastic moduli in tension and compression. One such material is graphene, a wonder material, which has the highest strength yet measured. Investigations on buckling problems for structures with different moduli are scarce. To address this new problem, first, the nondimensional expression of the relation between offset of neutral axis and deflection curve is derived based on the phased integration method, and then using the energy method, load–deflection relation of the rod is determined; second, based on the improved constitutive model for different moduli, large deformation finite element formulations are developed, and combined with the arc-length method, finite element iterative program for rods with different moduli is established to obtain buckling critical loads; third, material mechanical properties testing of graphite, which is the raw material of graphene, is performed to measure the tensile and compressive elastic moduli; moreover, buckling tests are also conducted to investigate the buckling behavior of this kind of graphite rod. By comparing the calculation results of the energy method and finite element method with those of laboratory tests, the analytical model and finite element numerical model are demonstrated to be accurate and reliable. The results show that it may lead to unsafe results if the classic theory was still adopted to determine the buckling loads of those rods composed of a material having different moduli. The proposed models could provide a novel approach for further investigation of nonlinear mechanical behavior for other structures with different moduli.     

拉压不同模量材料的参变量变分原理和有限元方法

对具有拉伸和压缩不同模量的材料,建立了平面静力问题的参变量变分原理.基于参变量变分原理,并结合有限元方法,将拉压不同模量平面问题转化为互补问题求解.经典的Lemke算法被用于求解此互补问题.该方法避免了应力状态的假设和刚度矩阵的更新,算法稳定,且收敛速度快.     

Fatigue fracture of a rectangular block under fluctuating stress cycles

In this paper an attempt is made to extend the problem of fatigue fracture of a rectangular block under alternating tension cycles solved by Thomas [1], so as to embrace a wider range of fluctuating stress cycles created by numerically unequal tensile and compressive loads which act alternately on the block. Endurance limits have been obtained for various types of fluctuating loads and those for simple tension (compression) cycles and alternating tension cycles are deduced therefrom. Also. the slip moduli k and K and the fatigue moduli α and ω of the material are determined.     

Frequency-dependent tensile and compressive effective moduli of elastic solids with distributed penny-shaped microcracks

This paper develops micromechanics models to estimate the tensile and compressive elastic moduli of elastic solids containing randomly distributed penny-shaped microcracks. The crack faces are open under tension and closed under compression. When the crack faces are closed, they may slide against one another following Coulomb’s law of dry friction. The micromechanics models provide analytical expressions of the tensile and compressive moduli for both static and dynamic cases. It is found that the tensile and compressive moduli are different. Further, under dynamic loading, both compressive and tensile moduli are frequency dependent. As a by-product, the micromechanics models also predict wave attenuation in the dynamic case. Numerical simulations using the finite element method are conducted to validate the micromechanics models.