Closed form solutions of T-stress in plane elasticity crack problems

The T-stress is considered as an important parameter in linear elastic fracture mechanics. In this paper, several closed form solutions of T-stress in plane elasticity crack problems in an infinite plate are investigated using the complex potential theory. In the line crack case, if the applied loading is the remote stress or the concentrated forces, the T-stress can be derived from the basic field. Here, the basic field is defined as the field caused by the applied loading in the infinite plate without the crack. For the circular are crack, the T-stress can be abstracted from a known solution. For the cusp crack problems, the T-stress can be separated from the obtained stress solution for which the conformal mapping technique is used.

Analysis of two-dimensional finite solids with microcracks

A general method is presented for solving the plane elasticity problem of finite plates with multiple microcracks. The method directly accounts for the interactions between different microcracks and the effect of outer boundary of a finite plate. Analysis is based on a superposition scheme and series expansions of the complex potentials. By using the traction-free conditions on each crack surface and resultant forces relations along outer boundary, a set of governing equations is formulated. The governing equations are solved numerically on the basis of a boundary collocation procedure. The effective Young's moduli for randomly oriented cracks and parallel cracks are evaluated for rectangular plates with microcracks. The numerical results are compared with those from various micromechanics models and experimental data. These results show that the present method provides a direct and efficient approach to deal with finite solids containing multiple microcracks.

A contact crack problem in an infinite plate

The problem of an infinite plate with crack of length 2a loaded by the remote tensile stress P and a pair of concentrated forces Q is discussed. The value of the force Q for the initial contact of crack face is investigated and the contact length elevated, while the Q force increases. The problem is solved assuming that the stress intensity factor vanishes at the end point of the contact portion. By the Fredholm integral equation for the multiple cracks, the reduction of stress intensity factor due to Q is found. (C) 1999 Elsevier Science Ltd. All rights reserved.

Computation of stress intensity factors by the sub-region mixed finite element method of lines

Based on the sub-region generalized variational principle, a sub-region mixed version of the newly-developed semi-analytical 'finite element method of lines' (FEMOL) is proposed in this paper for accurate and efficient computation of stress intensity factors (SIFs) of two-dimensional notches/cracks. The circular regions surrounding notch/crack tips are taken as the complementary energy region in which a number of leading terms of singular solutions for stresses are used, with the sought SIFs being among the unknown coefficients. The rest of the arbitrary domain is taken as the potential energy region in which FEMOL is applied to obtain approximate displacements. A mixed system of ordinary differential equations (ODEs) and algebraic equations is derived via the sub-region generalized variational principle. A singularity removal technique that eliminates the stress parameters from the mixed equation system eventually yields a standard FEMOL ODE system, the solution of which is no longer singular and is simply and efficiently obtained using a standard general-purpose ODE solver. A number of numerical examples, including bi-material notches/cracks in anti-plane and plane elasticity, are given to show the generally excellent performance of the proposed method.

Orthotropy rescaling and implications for fracture in composites

For most practically important plane elasticity problems of orthotropic materials, stresses depend on elastic constants through two nondimensional combinations. A spatial rescaling has been found to reduce the orthotropic problems to equivalent problems in materials with cubic symmetry. The latter, under favorable conditions, may be approximated by isotropic materials. Consequently, solutions for orthotropic materials can be constructed approximately from isotropic material solutions or rigorously from cubic ones. The concept is developed to gain insight into the interplay between anisotropy and finite geometry. The inherent simplicity of the solutions allows a variety of technical problems to be addressed efficiently. Included are stress concentration related cracking, effective contraction of orthotropic material specimens, crack deflection onto easy fracture planes, and surface flaw induced delamination.

Study on cracked spherical shells

The local-global anatysis method is systematically extended to the fracture analysis of spherical shells. On the basis of the shallow shell theory, which takes into account transverse shear deformations, governing equations for cracked spherical shells expressed in displacement and stress functions f, F and φ are proposed, and then a general solution including Modes, Ⅰ, Ⅱ, Ⅲ for stress-strain fields at crack tip in a spherical shell is obtained, which plays the same role as Williams's expansion in plane elasticity. The numerical results for finite-size spherical shells under different boundary conditions have been obtained. Furthermore, the bulging factors are analyzed with regard to shearing stiffness and an approximate formula is given.

Elasticity, stability, and ideal strength of beta-SiC in plane-wave-based ab initio calculations

On the basis of the pseudopotential plane-wave method and the local-density-functional theory, this paper studies energetics, stress-strain relation, stability, and ideal strength of beta-SiC under various loading modes, where uniform uniaxial extension and tension and biaxial proportional extension are considered along directions [001] and [111]. The lattice constant, elastic constants, and moduli of equilibrium state are calculated and the results agree well with the experimental data. As the four SI-C bonds along directions [111], [(1) over bar 11], [11(1) over bar] and [111] are not the same under the loading along [111], internal relaxation and the corresponding internal displacements must be considered. We find that, at the beginning of loading, the effect of internal displacement through the shuffle and glide plane diminishes the difference among the four Si-C bonds lengths, but will increase the difference at the subsequent loading, which will result in a crack nucleated on the {111} shuffle plane and a subsequently cleavage fracture. Thus the corresponding theoretical strength is 50.8 GPa, which agrees well with the recent experiment value, 53.4 GPa. However, with the loading along [001], internal relaxation is not important for tetragonal symmetry. Elastic constants during the uniaxial tension along [001] are calculated. Based on the stability analysis with stiffness coefficients, we find that the spinodal and Born instabilities are triggered almost at the same strain, which agrees with the previous molecular-dynamics simulation. During biaxial proportional extension, stress and strength vary proportionally with the biaxial loading ratio at the same longitudinal strain.

A new type of boundary integral-equation for plane problems of elasticity including rotational forces

Based on the idea proposed by Hu [Scientia Sinica Series A XXX, 385-390 (1987)], a new type of boundary integral equation for plane problems of elasticity including rotational forces is derived and its boundary element formulation is presented. Numerical results for a rotating hollow disk are given to demonstrate the accuracy of the new type of boundary integral equation.

Ab initio investigation of the elasticity and stability of aluminium

On the basis of the pseudopotential plane-wave (PP-PW) method in combination with the local density functional theory (LDFT), complete stress-strain curves for the uniaxial loading and uniaxial deformation along the [001] and [111] directions, and the biaxial proportional extension along [010] and [001] for aluminium are obtained. During the uniaxial loading, certain general behaviours of the energy versus the stretch and the load versus the stretch are confirmed; in each case, there exist three special unstressed structures: f.c.c., b.c.c., and f.c.t. for [001]; f.c.c., s.c., and b.c.c. for [111]. Using stability criteria, we find that all of these states are unstable, and always occur together with shear instability, except the natural f.c.c. structure. A Pain transformation from the stable f.c.c. structure to the stable b.c.c. configuration cannot be obtained by uniaxial compression along any equivalent [001] and [111] direction. The tensile strengths are similar for the two directions. For the higher energy barrier of the [111] direction, the compressive strength is greater than that for the [001] direction. With increase in the ratio of the biaxial proportional extension, the stress and tensile strength increase; however, the critical strain does not change significantly. Our results add to the existing ab initio database for use in fitting and testing interatomic potentials.

Elasticity, band-gap bowing, and polarization of AlxGa1-xN alloys

Elastic constants, the bulk modulus, Young's modulus, band-gap bowing coefficients, spontaneous and piezoelectric polarizations, and piezoelectric coefficients of hexagonal AlxGa1-xN ternary alloys are calculated using first-principles methods. The fully relaxed structures and the structures subjected to homogeneous biaxial and uniaxial tension are investigated. We show that the biaxial tension in the plane perpendicular to the c axis and the uniaxial tension along the c axis all reduce the bulk modulus, whereas they reduce and enhance Young's modulus, respectively. We find that the biaxial and uniaxial tension can enhance the bowing coefficients. We also find that the biaxial tension can enhance the total polarization, while the uniaxial tension will suppress the total polarization. (C) 2008 American Institute of Physics.

Complex potentials and singular integral equation for curve crack problem in antiplane elasticity

Four types of the fundamental complex potential in antiplane elasticity are introduced: (a) a point dislocation, (b) a concentrated force, (c) a dislocation doublet and (d) a concentrated force doublet. It is proven that if the axis of the concentrated force doublet is perpendicular to the direction of the dislocation doublet, the relevant complex potentials are equivalent. Using the obtained complex potentials, a singular integral equation for the curve crack problem is introduced. Some particular features of the obtained singular integral equation are discussed, and numerical solutions and examples are given.

Experimental Study on Vortex-Induced Vibrations of a Pipeline Near a Plane Boundary in Steady Flow

对单向水流作用下近壁管道横向涡激振动进行了实验模拟,重点探讨了管道与壁面间隙比(e/D)对管道涡激振动幅值和涡激振动频率响应特性的影响规律.实验结果表明,管道与壁面间隙宽度对管道涡激振动特性有较明显影响.在较大间隙比(e/D＞0.66)下,管道振幅随着Vr数的增大先快速增长到最大值,然后平缓下降;在振动初期(即Vr数较小时),管道振动频率变化基本符合Strouhal规律;在振动中后期(即Vr数较大时),管道振动频率变化不符合Strouhal规律,而在管道固有频率附近缓慢增长.在较小间隙比(e/D＜0.30)下,管道振幅随Vr数的增大先平缓上升到最大值,随后较快速下降;在振动初期,管道振动频率变化不遵循Strouhal规律;在整个振动范围内,与较大间隙比情况相比,随着Vr数增加,管道振动频率增长幅度明显较大.

Micro-shock waves generated inside a fluid jet impinging on plane wall

An Nd:glass laser pulse (18 ns, 1.38 J) is focused in a tiny area of about 100-mum diam under ambient conditions to produce micro-shock waves. The laser is focused above a planar surface with a typical standoff distance of about 4 mm, The laser energy is focused inside a supersonic circular jet of carbon dioxide gas produced by a nozzle with internal diameter of 2.9 mm and external diameter of 8 mm, Nominal value of the Mach number of the jet is around 2 with the corresponding pressure ratio of 7.5 (stagnation pressure/static pressure at the exit of the nozzle), The interaction process of the micro-shock wave generated inside the supersonic jet with the plane wall is investigated using double-pulse holographic interferometry. A strong surface vortex field with subsequent generation of a side jet propagating outward along the plane wail is observed. The interaction of the micro-shock wave with the cellular structure of the supersonic jet does not seem to influence the near surface features of the flowfield. The development of the coherent structures near the nozzle exit due to the upstream propagation of pressure waves seems to be affected by the outward propagating micro-shock wave. Mach reflection is observed when the micro-shock wave interacts with the plane wall at a standoff distance of 4 mm, The Mach stem is slightly deflected, indicating strong boundary-layer and viscous effects near the wall. The interaction process is also simulated numerically using an axisymmetric transient laminar Navier-Stokes solver. Qualitative agreement between experimental and numerical results is good.

An Analytical Solution for Three-Dimensional Diffraction of Plane P-Waves by a Hemispherical Alluvial Valley with Saturated Soil Deposits

An analytical solution for the three-dimensional scattering and diffraction of plane P-waves by a hemispherical alluvial valley with saturated soil deposits is developed by employing Fourier-Bessel series expansion technique. Unlike previous studies, in which the saturated soil deposits were simulated with the single-phase elastic theory, in this paper, they are simulated with Biot's dynamic theory for saturated porous media, and the half space is assumed as a single-phase elastic medium. The effects of the dimensionless frequency, the incidence angle of P-wave and the porosity of soil deposits on the surface displacement magnifications of the hemispherical alluvial valley are investigated. Numerical results show that the existence of a saturated hemispherical alluvial valley has much influence on the surface displacement magnifications. It is more reasonable to simulate soil deposits with Biot's dynamic theory when evaluating the displacement responses of a hemispherical alluvial valley with an incidence of P-waves.

The relationship of SIF between plate and plane fracture problems and the effect of the plate thickness on SIF

In the present paper, it is shown that the zero series eigenfunctions of Reissner plate cracks/notches fracture problems are analogous to the eigenfunctions of anti-plane and in-plane. The singularity in the double series expression of plate problems only arises in zero series parts. In view of the relationship with eigen-values of anti-plane and in-plane problem, the solution of eigen-values for Reissner plates consists of two parts: anti-plane problem and in-plane problem. As a result the corresponding eigen-values or the corresponding eigen-value solving programs with respect to the anti-plane and in-plane problems can be employed and many aggressive SIF computed methods of plane problems can be employed in the plate. Based on those, the approximate relationship of SIFs between the plate and the plane fracture problems is figured out, and the effect relationship of the plate thickness on SIF is given.