Intensity of singular stress at the fiber end in a hexagonal array of fibers

To evaluate the mechanical strength of fiber-reinforced composites it is necessary to consider singular stresses at the end of fibers because they cause crack initiation, propagation, and final failure. The singular stress field is controlled by generalized stress intensity factor (GSIF) defined at the fiber end. In this study, periodic and zigzag arrays of cylindrical inclusions under longitudinal tension are considered in comparison with the results for a single fiber. The unit cell region is approximated as an axi-symmetric cell; then, the body force method is applied, which requires the stress and displacement fields due to ring forces in infinite bodies having the same elastic constants as those of the matrix and inclusions. The given problem is solved on the superposition of two auxiliary problems under different boundary conditions. To obtain the GSIF accurately, the unknown body force densities are expressed as piecewise smooth functions using fundamental densities and power series. Here, the fundamental densities are chosen to represent the symmetric stress singularity, and the skew-symmetric stress singularity. The GSIFs are systematically calculated with varying the elastic modulus ratio and spacing of fibers. The effects of volume fraction and spacing of fibers are discussed in fiber-reinforced plastics.     

横向力作用下悬臂梁固定端应力分布研究

在弹性力学Hamilton体系中,利用解析法,考虑圣维南原理所覆盖的解,对横向力作用下悬臂梁固定端应力分布问题进行研究,并对计算结果进行分析。研究结果表明,辛体系解析法采用对偶的二类变量求解,能很好地处理各种复杂边界条件,并且对此类问题的分析具有优越性,计算精度较高。该方法对其他边界问题的研究也具有指导意义。     

Elastic and elastic–plastic singular fields around a crack-tip in particulate-reinforced composites with progressive debonding damage

This paper deals with elastic and elastic–plastic singular fields around a crack-tip in particulate-reinforced composites with debonding damage of particle-matrix interface. Numerical analyses are carried out on a crack-tip field in elastic-matrix and elastic–plastic-matrix composites reinforced with elastic particles, using a finite element method developed based on an incremental damage theory of particulate-reinforced composites. A particle volume fraction and interfacial strength between particles and matrix of the composites are parametrically changed. In the elastic-matrix composites, a unique elastic singular field is created on the complete damage zone in the vicinity of a crack-tip in addition to the conventional elastic singular field on the no damage zone. The macroscopic stress level around a crack-tip is reduced by the debonding damage while the microscopic stress level of the matrix remains unchanged. In the elastic–plastic-matrix composites, the damage zone develops in addition to the plastic zone due to matrix plasticity, and both the macroscopic and microscopic stress revels around a crack-tip are reduced by the debonding damage. It is concluded from the numerical results that the toughening due to damage could be expected in the elastic–plastic-matrix composites, while it is questionable in the elastic-matrix composites.     

Local solution of the stress and strain fields in the necking section of cylindrical bars under uniaxial tension

The problem of finding the stress and strain fields over the minimum cross section of necked cylindrical bars under uniaxial tensile load has been solved locally using a new fast numerical method. The scheme delivers both the accuracy of the finite element analysis and the applicability of simple closed-form analytical solutions. The required inputs are the distributions of curvature radii for both isostatic and material lines. It is numerically observed that the mathematical formulas available in the literature fail to adequately predict these distributions. Introducing the stress normalized strain-hardening rate as the most decisive parameter affecting the curvature radii, a database and interpolation technique have been developed in order to estimate the necessary information based on the results of the previously FE analyzed samples. Finally, a practical case has been solved and compared with the FE results.     

A study on stress intensity factors and singular stress fields of three-dimensional interface crack

By using the finite-part integral concepts and limit technique, the hypersingular integrodifferential equations of three-dimensional (3D) planar interface crack were obtained; then the dominant-part analysis of 2D hypersingular integral was further used to investigate the stress fields near the crack front theoretically, and the accurate formulae were obtained for the singular stress fields and the complex stress intensity factors. After that, a numerical method is proposed to solve the hypersingular integrodifferential equations of 3D planar interface crack, and the problem of elliptical planar crack is then considered to show the application of the method. The numerical results obtained are satisfactory. Project supported by the Foundation of Solid Mechanics Open Research Laboratory of State Education Commission at Tongji University and the National Natural Science Foundation.     

Meso cell model of fiber reinforced composite: Interface stress statistics and debonding paths

The primary goal of this work is to develop an efficient analytical tool for the computer simulation of progressive damage in the fiber reinforced composite (FRC) materials and thus to provide the micro mechanics-based theoretical framework for a deeper insight into fatigue phenomena in them. An accurate solution has been obtained for the micro stress field in a meso cell model of fibrous composite. The developed method combines the superposition principle, Kolosov–Muskhelishvili’s technique of complex potentials and Fourier series expansion. By using the properly chosen periodic potentials, the primary boundary-value problem stated on the multiple-connected domain has been reduced to an ordinary, well-posed set of linear algebraic equations. The meso cell can include up to several hundred inclusions which is sufficient to account for the micro structure statistics of composite. The presented numerical examples demonstrate an accuracy and high numerical efficiency of the method which makes it to be a promising tool for studying progressive damage in FRCs. By averaging over a number of random structure realizations, the statistically meaningful results have been obtained for both the local stress and effective elastic moduli of disordered fibrous composite. A special attention has been paid to the interface stress statistics and the fiber debonding paths development, which appear to correlate well with the experimental observations.     

Theoretical analysis on the local critical stress and size effect for interfacial debonding in particle reinforced rheological materials

The mechanisms of interfacial debonding of particle reinforced rheological materials are studied. Based on an energy criterion, a simple formula of local critical stress for interfacial debonding is derived and expressed in terms of the interfacial energy. The particle size effect on interface debonding can then be analyzed easily owing to the fact that critical stress is inversely proportional to the square root of particle radius. By taking PP/CaCO₃ system as an example, the present energy criterion is compared with the mechanical debonding criterion, and it is found that under the condition that bond strength is equal to matrix strength and particle radius not over 0.2μm, the mechanical debonding criterion can be automatically satisfied if the energy criterion is satisfied. A relation between critical time and interface energy is calculated by using the energy criterion. The influences of the particle volume fraction and the parlicle size, the loading rate and the relaxation time of the matrix on the critical time of interfacial debonding are also discussed. Supported by the National Natural Science Foundation of China (19632030 and 19872007) and Natural Science Foundation of Jiangsu Province.     

Failure surface of epoxy-modified fiber-reinforced composites under transverse tension and out-of-plane shear

The mechanical behavior of uniaxially fiber-reinforced composites with a ductile rubber-toughened epoxy matrix was studied through the finite element analysis of a RVE of the composite microstructure. The fibers were represented by elastic and isotropic solids, while the rubber-modified epoxy matrix behaved as a elasto-viscoplastic solid. The matrix flow stress followed the model developed by Jeong [Jeong, H.-Y., 2002. A new yield function and a hydrostatic stress-controlled void nucleation model for porous solids with pressure-sensitive matrices. International Journal of Solids and Structures 39, 1385–1403.], which included the inherent pressure-sensitivity of the yield stress in the epoxy matrix, the damage due to the cavitation of the rubber particles and subsequent void growth, and the particular features of elastic–viscoplastic behavior in glassy polymers, particularly the intrinsic softening upon yield followed by hardening. Composites with either perfect or weak fiber/matrix interfaces (the latter introduced through cohesive elements) were studied to assess the influence of interface strength on the composite behavior. Simulations under transverse tension and out-of-plane shear were carried out to establish the effect of loading conditions on the dominant deformation and failure micromechanisms. In addition, the corresponding failure locus was obtained and compared with the predictions of current phenomenological failure criteria for composites. The range of validity of these criteria and the areas for further improvement were established by comparison with the numerical results.     

悬链线竖向集中力作用下的构形分析和张力计算

目前,悬链线在竖向集中力和均布荷载共同作用下的构形分析和受力计算的理论仍不完善。针对这一问题,通过引入悬链线的几何约束方程、力平衡方程和超越方程,建立了竖向集中力与均布荷载共同作用下的非线性方程组。采用牛顿迭代法求解方程组,得到了悬链线的构形和受力情况。为了验证理论计算的正确性,进行了算例和试验验证。结果表明,算例的计算结果与文献结论保持一致,试验测得的构形和水平张力大小与理论计算的构形和水平张力大小吻合较好。本文的理论计算可以更加简单精确地计算出悬链线在竖向集中力和均布荷载共同作用下的构形和受力情况,为实际工程提供重要的理论指导。     

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.     

Uniform tension of a semi-infinite plate with a crack at an end of a stiffened edge

Summary The semi-infinite plate which is rigidly stiffened at a part on the boundary and with a crack originating from an end of the stiffened edge is analyzed as a mixed boundary value problem in a plane elastic problem. A complex variable method and a rational mapping function are used for the analysis. A closed solution is obtained. The rational mapping function is formed as a sum of fractional expressions. Distributions of stress and displacement in the neighbourhood of the crack and the stiffened edge are investigated for the state before and after occuring of a crack. Stress intensity factors which are important in linear fracture mechanics are obtained for various crack lengths.

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Übersicht Eine Halbscheibe, die an einem Teil ihrer Berandung verstärkt ist und einen Haarriß hat, der von dem einen Ende des verstärkten Randes ausgeht, wird als ein gemischtes Randwertproblem innerhalb eines Scheibenproblems analysiert. Eine Methode komplexer Veränderlicher und eine Abbildungsfunktion werden zur Analyse verwendet. Eine geschlossene Lösung läßt sich angeben. Die Abbildungsfunktion wird als Summe von Partialbruchzerlegungen dargestellt. Spannungs- und Verschiebungs verteilung in der Umgebung des Risses werden berechnet. Der verstärkte Rand wird auf den Zustand vor und nach dem Entstehen eines Risses untersucht. Spannungsintensitätsfaktoren, die in der linear-elastischen Bruchmechanik von Bedeutung sind, werden für verschiedene Rißlängen ausgerechnet.

     

自由端受磁力吸引悬臂梁的非线性振动

研究了含分式函数的非线性磁力系统的受迫振动。磁铁吸引力横向作用在悬臂梁自由端处。根据含有非线性边界的连续体模型,给出了非平凡静平衡位形。以稳定位形作坐标变换后,计算了相应的固有频率,并求得微幅受迫振动稳态响应的近似解析解。研究了不同磁铁间距与频响曲线的关系,结果表明磁力方向和大小对它们均有较大影响。数值结果与解析结果吻合得很好。     

Transient temperature fields and residual stress fields of metallic materials under welding

Based on the situation of welding thermal conduction and thermo-elasto-plasticityresearch,Ms paper explores some problems in this field.First,the boundary elementmethod for nonlinear problems is improved by linearization of nonlinear problems and usedin welding thermal conduction analysis.Second,the thermo-elasto-plastic finite elementmethod is used for the welding stress calculation,in which the phase transformation isconsidered by the"equivalent linear expansion coefficient method".The comparison of the calculated results with experimental data shows that themethods provided in this paper are available.     

A continuum approach to the analysis of the stress field in a fiber reinforced composite with a transverse crack

The stress field in a periodically layered composite with an embedded crack oriented in the normal direction to the layering and subjected to a tensile far-field loading is obtained based on the continuum equations of elasticity. This geometry models the 2D problem of fiber reinforced materials with a transverse crack. The analysis is based on the combination of the representative cell method and the higher-order theory. The representative cell method is employed for the construction of Green’s functions for the displacements jumps along the crack line. The problem of the infinite domain is reduced, in conjunction with the discrete Fourier transform, to a finite domain (representative cell) on which the Born–von Karman type boundary conditions are applied. In the framework of the higher-order theory, the transformed elastic field is determined by a second-order expansion of the displacement vector in terms of local coordinates, in conjunction with the equilibrium equations and these boundary conditions. The accuracy of the proposed approach is verified by a comparison with the analytical solution for a crack embedded in a homogeneous plane.Results show the effects of crack lengths, fiber volume fractions, ratios of fiber to matrix Young’s moduli and matrix Poisson’s ratio on the resulting elastic field at various locations of interest. Comparisons with the predictions obtained from the shear lag theory are presented.     

Analytical representation of the non-square-root singular stress field at a finite angle sharp notch

The stress field near the tip of a finite angle sharp notch is singular. However, unlike a crack, the order of the singularity at the notch tip is less than one-half. Under tensile loading, such a singularity is characterized by a generalized stress intensity factor which is analogous to the mode I stress intensity factor used in fracture mechanics, but which has order less than one-half. By using a cohesive zone model for a notional crack emanating from the notch tip, we relate the critical value of the generalized stress intensity factor to the fracture toughness. The results show that this relation depends not only on the notch angle, but also on the maximum stress of the cohesive zone model. As expected the dependence on that maximum stress vanishes as the notch angle approaches zero. The results of this analysis compare very well with a numerical (finite element) analysis in the literature. For mixed-mode loading the limits of applicability of using a mode I failure criterion are explored.     

Determination of the stress intensity factors at the ends of two collinear transverse shear cracks under steady vibrations

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