Mircopolar Model of Fracture for Composite Material

Failure of a composite is a complex process accompanied by irreversible changes in the microstructure of the material. Microscopic mechanisms are known of the accumulation of damage and failure of the type of localized and multiple ruptures of the fibers delamination along interphase boundaries, and also mechanisms associated with fracture of fibers. In this work, we propose a mathematical model of the local mechanism of failure of a composite material randomly reinforced with a system of short fibers. We implement the Cosserat moment model of crack tip for filament material, reinforced with whiskers or in fiber- reinforced polycrystalline materials. It is assumed that the angular distribution of the fibers is isotropic and the elastic characteristics of the fibers are considerably higher than the elastic constants of the matrix. We implement the homogenization procedure for the effective Cosserat constants similarly to the effective elastic constants. The singular solution in the vicinity of the crack tip in the Cosserat moment model is found. Using this solution, we examine the bending stresses in the filaments due to effective moment stresses in the material. The constructed model describes the phenomenon of fracture of the fibers occurring during crack propagation in those composites. The following assumptions are used as the main hypotheses for the micromechanical model. The matrix contains a nucleation crack. When the load is increased the crack grows and its boundary comes into contact with the reinforcing fibers. A further increase of the stress causes bending of the fiber. When~the fiber curvature reaches a specific critical value, the fiber ruptures. If the stress at infinity is given, the fibers no longer delay the development of failure during crack propagation The degree of bending distortion of the fiber in the vicinity of the boundary of the crack is determined by the moment model of the material. The necessity to take into account the moment stresses in the failure theory of the reinforced material was stressed in [Muki and Sternberg (1965) Zeitschrift f angew Math und Phys 16:611–615; Garajeu and Soos (2003) Math Mech Solids 8(2):189–218; Ostoja-Starzewski et al (1999) Mech Res Commun 26:387–396]. The moment Cosserat stresses were accounted also for inhomogeneous biomechanical materials by Buechner and Lakes (2003) Bio Mech Model Mechanobiol 1: 295–301. We should also mention the important methodological studies [Sternberg and Muki (1967) J Solids Struct 1:69–95; Atkinson and Leppington (1977) Int J Solids Struct 13: 1103–1122] concerned with the moment stresses in homogeneous fracture mechanics.     

弹性力学空间轴对称问题通解的研究

本文证明了横观各向同性弹性力学空间轴对称问题的通解是完备的.这里φ满足同时得到了轴对称问题一个新的完备通解这里φ满足     

APPLICATION OF VECTOR TO SMALL DEFORMATION

ＡＰＰＬＩＣＡＴＩＯＮ　ＯＦ　ＶＥＣＴＯＲ　ＴＯ　ＳＭＡＬＬ　ＤＥＦＯＲＭＡＴＩＯＮＺｈａｎｇＳｈｅｎｘｕｅ（张慎学）（ＲｅｃｅｉｖｅｄＤｅｃ．１６，１９９４，Ｍａｙ２８，１９９５，ＣｏｍｍｕｎｉｃａｔｅｄｂｙＣｈｉｅｎＷｅｉｚａｎｇ）Ａｂｓｔｒａｃ...     

Sufficient conditions for strong ellipticity for a class of anisotropic materials

We provide sufficient conditions for strong ellipticity for a general class of anisotropic hyperelastic materials. This general class includes as a subclass transversely isotropic materials. Our sufficient conditions require that the first partial derivatives of the reduced-stored energy function satisfy some simple inequalities and that the second partial derivatives satisfy a convexity condition. We also characterize a restricted type of strong ellipticity for a subclass of transversely isotropic materials undergoing pure homogeneous deformations. We apply our results to a model of soft tissue from the biomechanics literature.     

The Stress Response of Functionally Graded Isotropic Linearly Elastic Rotating Disks

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic solid circular disks or cylinders, rotating at constant angular velocity about a central axis. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic problem for a homogeneous isotropic rotating solid disk or cylinder is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. For the case when the Young"s modulus has a power-law dependence on the radial coordinate, explicit exact solutions are obtained. It is shown that the stress response of the inhomogeneous disk (or cylinder) is significantly different from that of the homogeneous body. For example, the maximum radial and hoop stresses do not, in general, occur at the center as in the case for the homogeneous material. Furthermore, for the case where the Young"s modulus increases with radial distance from the center, it is shown that radially symmetric solutions exist provided the rate of growth of the Young"s modulus is, at most, cubic in the radial variable. It is also shown for the general inhomogeneous isotropic case how the material inhomogeneity may be tailored so that the radial and hoop stress are identical throughout the disk. This revised version was published online in August 2006 with corrections to the Cover Date.     

Simple Torsion of Isotropic, Hyperelastic, Incompressible Materials with Limiting Chain Extensibility

The purpose of this research is to investigate the simple torsion problem for a solid circular cylinder composed of isotropic hyperelastic incompressible materials with limiting chain extensibility. Three popular models that account for hardening at large deformations are examined. These models involve a strain-energy density which depends only on the first invariant of the Cauchy–Green tensor. In the limit as a polymeric chain extensibility tends to infinity, all of these models reduce to the classical neo-Hookean form. The main mechanical quantities of interest in the torsion problem are obtained in closed form. In this way, it is shown that the torsional response of all three materials is similar. While the predictions of the models agree qualitatively with experimental data, the quantitative agreement is poor as is the case for the neo-Hookean material. In fact, by using a global universal relation, it is shown that the experimental data cannot be predicted quantitatively by any strain-energy density which depends solely on the first invariant. It is shown that a modification of the strain energies to include a term linear in the second invariant can be used to remedy this defect. Whether the modified strain-energies, which reflect material hardening, are a feasible alternative to the classic Mooney–Rivlin model remains an open question which can be resolved only by large strain experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stability of a rectangular plate under biaxial tension

The stability problem of a rectangular plate undergoing uniform biaxial in-plane tensile strain is solved using the three-dimensional equations of nonlinear elasticity. The surfaces of the plate are stress-free, and special boundary conditions that allow one to separate variables in the linearized equilibrium equations are specified on the lateral surfaces. For three particular models of incompressible materials, the critical curves are constructed and the instability region is determined in the plane of the loading parameters (the multiplicities of elongations of the plate material in the unperturbed equilibrium state). The numerical results show that for thin plates loaded by tensile stresses, the size and shape of the instability region depend only slightly on the relation among the length, width, and thickness of the plate. Based on the results obtained, a simple approximate stability criterion is proposed for an elastic plate under tensile loads. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 94–103, July–August, 2007.     

The Cauchy Relations in Linear Elasticity Theory

In linear elasticity, we decompose the elasticity tensor into two irreducible pieces with 15 and 6 independent components, respectively. The vanishing of the piece with 6 independent components corresponds to the Cauchy relations. Thus, for the first time, we recognize the group-theoretical underpinning of the Cauchy relations.     

Second order effects in an elastic half-space acted upon by a non-uniform normal load

A closed form solution to the second order elasticity problem, when an isotropic compressible elastic half-space undergoes a deformation owing to a non-uniformly distributed normal load, is presented. The method of integral transform is employed and the case when loading is distributed, in accordance with Hertz's law, is discussed. The limiting solution for incompressible isotropic elastic material is also derived. Numerical calculations for the second order elastic material for the displacement and the normal stress in thez-direction are carried out. It is found that, in comparision to the linear elastic case, the displacement increases and the normal stress decreases in the second order elastic material

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Sommario Si presenta una soluzione del secondo ordine in forma chiusa del problema di un semi-spazio elastico isotropo compressibile soggetto a un carico normale distribuito non uniformemente, ottenuta mediante il metodo delle trasformate integrali. Si discute il caso del carico distribuito secondo la legge di Hertz; si deriva anche la soluzione limite per materiale elastico isotropo incompressibile. Si ottengono numericamente alcuni risultati relativi a spostamenti e tensioni normali nella direzione z: si trova che, a paragone del caso elastico lineare, gli spostamenti aumentano e le tensioni diminuiscono nel materiale elastico del secondo ordine.

     

高速液—固碰撞时固体弹性和液体可压缩性对撞击压力的影响

本文联立求解可压缩平面液体撞击弹性固体表面时液体激波面及液固相界面上的物理过程控制方程组,导出了激波速度、撞击压力和固体表面变形速度的计算公式.导出的公式中包括了表征液体可压缩性的撞击马赫数M_0和表征固体弹性的液固声阻抗比Γ.通过计算液体撞击PMMA,钙钠玻璃、ZnSe和1Cr13时的撞击压力,本文具体分析了M_0和Γ对撞击压力产生的综合影响。