Fracture dynamics problems of mode I semi-infinite crack for anisotropic orthotropic body

By means of the theory of complex functions, fracture dynamics problems of mode I semi- infinite crack for anisotropic orthotropic body were researched. Analytical solutions of stress, displacement, and dynamic stress intensity factor under the action of moving increasing loads Px ³/t ³, Pt ⁴/x ³, respectively, are very easily obtained utilizing the approaches of self-similar functions. In the light of relevant material’s coefficients, the alterable rule of dynamic stress intensity factor was depicted very well. The correlative closed solutions are attained based on the Riemann–Hilbert problems. After those analytical solutions were applied by the superposition principle, the solutions of discretional complex problems could be attained.     

On corner flows of Oldroyd-B fluids

Exact series solutions for planar creeping flows of Oldroyd-B fluids in the neighbourhood of sharp corners are presented and discussed. Both reentrant and non-reentrant sectors are considered. For reentrant sectors it is shown that more than one type of series solution can exist formally, one type exhibiting Newtonian-like asymptotic behaviour at the corner, away from walls, and another type exhibiting the same kind of asymptotics as an Upper Convected Maxwell (UCM) fluid. The solutions which are Newtonian-like away from walls are shown to develop non-integrable stress singularities at the walls when the no-slip velocity boundary condition is imposed. These mathematical solutions are therefore inadmissible from the physical viewpoint under no-slip conditions. An inadmissible solution, with stress singularities which are not everywhere integrable, is identified among the solutions of UCM-type. For a 270° reentrant sector the radial behaviour of the normal stress is everywhere r^−0.613. In the viscometric region near a wall, the radial normal stress σ^rr behaves like (rε)^−0.613, where ε is the angle made with the wall. In addition σ^rθ is infinite (not integrable) at the wall even when r is non-zero. Another UCM-type solution has a normal stress behaviour away from walls which is r^−0.985 for 270° sector. Again, this solution has a non-integrable stress singularity and is therefore inadmissible. Finally, for non-reentrant sectors it is shown that the flow is always Newtonian-like away from walls.     

Analytical solutions for tapering quadratic and cubic rat-holes in highly frictional granular solids

New exact analytical solutions are presented for both stress and velocity fields for a Coulomb–Mohr granular solid assuming non-dilatant double-shearing theory. The solutions determined apply to highly frictional materials for which the angle of internal friction φ is assumed equal to 90°. This major assumption is made primarily to facilitate exact analytical solutions, and it is discussed at length in the Introduction, both in the context of real materials which exhibit large angles of internal friction, and in the context of using the solutions derived here as the leading term in a regular perturbation solution involving powers of 1−sinφ. The analytical velocity fields so obtained are illustrated graphically by showing the direction of the principal stress as compared to the streamlines. The stress solutions are also exploited to determine the static stress distribution for a granular material contained within vertical boundaries and a horizontal base, which is assumed to have an infinitesimal central outlet through which material flows until a rat-hole of parabolic or cubic profile is obtained, and no further flow takes place. A rat-hole is a stable structure that may form in storage hoppers and stock-piles, preventing any further flow of material. Here we consider the important problems of two-dimensional parabolic rat-holes of profile y=ax², and three-dimensional cubic rat-holes of profile z=ar³, which are both physically realistic in practice. Analytical solutions are presented for both two and three-dimensional rat-holes for the case of a highly frictional granular solid, which is stored at rest between vertical walls and a horizontal rigid plane, and which has an infinitesimal central outlet. These solutions are bona fide exact solutions of the governing equations for a Coulomb–Mohr granular solid, and satisfy exactly the free surface condition along the rat-hole surface, but approximate frictional conditions along the containing boundaries. The analytical solutions presented here constitute the only known solutions for any realistic rat-hole geometry, other than the classical solution which applies to a perfectly vertical cylindrical cavity.     

Fracture dynamics problem on mode I semi-infinite crack

By means of the theory of complex functions, fracture dynamics problems concerning mode I semi-infinite crack were studied. Analytical solutions of stress, displacement and dynamic stress intensity factor under the action of moving increasing loads Pt ³/x ³, Px ³/t ², respectively, are very easily obtained using the ways of self-similar functions. The correlative closed solutions are attained based on the Riemann–Hilbert problems.     

NONLINEAR BENDINGS OF RECTANGULAR SYMMETRICALLY LAMINATED CROSS-PLY PLATES UNDER VARIOUS SUPPORTS

This paper studies the nonlinear bendings of rectangular symmetrically laminated cross-ply plates subjected to uniform pressure under various supports on the basis of[3] by the singular perturbation method offered in[1]. The uniformly valid N-order asymptotic solutions of the deflection and stress function are derived. Analyses and numerical solutions are given for simply supported rectangular laminates and edge displacement zero.     

Stresses in a deep beam with a central concentrated load

The stress distribution in deep beams has received the attention of mathematicians an research workers since the early days of Wilson, Stokes and Filon.^{1, 2} A general theoretical solution is not possible and particular solutions of the biharmonic equatiion for the Airy stress function have been worked out to satisfy certain loadings. In the case of a deep beam under a symmetrical pressure on the top edge, solutions by (1) Durant and Garwood,³ (2) Chow, Conway and Winter⁴ have been put forward. Systematic photoelastic experimental work appears to be scarce and the object of this article is to present for comparison with existing theories the results of three photoelastic tests on simply supported deep beams carrying a central point load. The experimental stresses and those obtained from the solutions referred to above agreed reasonably well. The normal stresses σ _x calculated from the Wilson-Stokes theory were in error for the case under consideration (the theory is not valid for (H/L)>0.4), and the simple theory of bending is adequate for the maximum tensile stress provided the span exceeds 1.5 times the depth. Results are summarized in figures showing the magnitude and direction of the principal stresses; magnitudes are in a dimensionless from, hence it is possible to adapt them fior any beam of similar depth to span ratio under the same type of loading.     

RETRACTED ARTICLE: Flow of fractional Maxwell fluid between coaxial cylinders

This paper deals with the study of unsteady flow of a Maxwell fluid with fractional derivative model, between two infinite coaxial circular cylinders, using Laplace and finite Hankel transforms. The motion of the fluid is produced by the inner cylinder that, at time t = 0⁺, is subject to a time-dependent longitudinal shear stress. Velocity field and the adequate shear stress are presented under series form in terms of the generalized G and R functions. The solutions that have been obtained satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Maxwell and Newtonian fluids are obtained as limiting cases of general solutions. Finally, the influence of the pertinent parameters on the fluid motion as well as a comparison between the three models is underlined by graphical illustrations.     

HAMILTONIAN SYSTEM AND SIMPLETIC GEOMETRY IN MECHANICS OF COMPOSITE MATERIALS (Ⅱ)——PLANE STRESS PROBLEM

Fundamental theory presented in Part (I)^[8] is used to analyze anisotropic plane stress problems. First we construct the generalized variational principle to enter Hamiltonian system and get Hamiltonian differential operator matrix; then we solve eigen problem; finally, we present the process of obtaining analytical solutions and semi-analytical solutions for anisotropic plane stress problems on rectangular area.     

Exact solutions for the unsteady rotational flow of an Oldroyd-B fluid with fractional derivatives induced by a circular cylinder

In this research article, the unsteady rotational flow of an Oldroyd-B fluid with fractional derivative model through an infinite circular cylinder is studied by means of the finite Hankel and Laplace transforms. The motion is produced by the cylinder, that after time t=0⁺, begins to rotate about its axis with an angular velocity Ωt ^p . The solutions that have been obtained, presented under series form in terms of the generalized G-functions, satisfy all imposed initial and boundary conditions. The corresponding solutions that have been obtained can be easily particularized to give the similar solutions for Maxwell and Second grade fluids with fractional derivatives and for ordinary fluids (Oldroyd-B, Maxwell, Second grade and Newtonian fluids) performing the same motion, are obtained as limiting cases of general solutions. The most important things regarding this paper to mention are that (1) we extracted the expressions for the velocity field and the shear stress corresponding to the motion of Second grade fluid with fractional derivatives as a limiting case of our general solutions corresponding to the Oldroyd-B fluid with fractional derivatives, this is not previously done in the literature to the best of our knowledge, and (2) the expressions for the velocity field and the shear stress are in the most simplified form, and the point worth mentioning is that these expressions are free from convolution product and the integral of the product of the generalized G-functions. Finally, the influence of the pertinent parameters on the fluid motion, as well as a comparison between models, is shown by graphical illustrations.     

Exact analytical solutions for axial flow of a fractional second grade fluid between two coaxial cylinders

The velocity field and the adequate shear stress corresponding to the longitudinal flow of a fractional second grade fluid, between two infinite coaxial circular cylinders, are determined by applying the Laplace and finite Hankel transforms. Initially the fluid is at rest, and at time t = 0⁺, the inner cylinder suddenly begins to translate along the common axis with constant acceleration. The solutions that have been obtained are presented in terms of generalized G functions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for ordinary second grade and Newtonian fluids are obtained as limiting cases of the general solutions. Finally, some characteristics of the motion, as well as the influences of the material and fractional parameters on the fluid motion and a comparison between models, are underlined by graphical illustrations.     

Fracture dynamics problems of orthotropic solids under anti-plane shear loading

By application of the approaches of the theory of complex functions, fracture dynamics problems of orthotropic solids under anti-plane shear loading were researched. Universal representation of analytical solutions was obtained by means of self-similar functions. The problems dealt with can be facilely transformed into Riemann–Hilbert problems by this technique, and analytical solutions of the stress, the displacement and dynamic stress intensity factor under the actions of moving increasing loads Px ²/t ² and Pt ³/x ² for the edges of asymmetrical mode III crack, respectively, were acquired. In the light of corresponding material properties, the variable rule of dynamic stress intensity factor was illustrated very well.     

An analysis on three-dimensional effects near the tip of a crack in an elastic plate

An infinite plate containing a finite through crack under tensile loading is analysed by Fourier transform based on the Kane-Mindlin kinematic assumptions for the quasi-three-dimensional deformation of plates in extension. The asymptotic expressions of stress and displacement fields near the crack tip, the variation of the stress intensity factor with the plate-thickness and the three-dimensional deformation zone near the crack tip are investigated. The results of the analysis show that, (a) the crack-tip stress and displacement fields accounting for the plate-thickness effects are different from the plane stress solutions and this is true even for extremely small parameter (=1–vh/6 a). In a very small region near the crack tip, plane strain solutions prevail; (b) the ratio of the stress intensity factor K_I to the corresponding plane stress one K_I, K_I/K _I ^o , approaches 1/(1–v²) as tends to zero; (c) plane stress solutions can give satisfactory results for points a distance from the crack tip greater than about three-fourths of the plate-thickness; (d) the linear elastic result for the zone of three-dimensional effects is approximately valid for an elasto-plastic material with linear strain-hardening when the plastic tangential mudulus E_t is not very small.The Project Supported by National Natural Science Foundation of China.     

The stress field near the front of an arbitrarily shaped crack in a three-dimensional elastic body

The problem stated in the title is investigated with special emphasis on the first three terms of the stress expansion, proportional to r ^-1/2, r ⁰=1 and r ^1/2 respectively, where r denotes the distance to the crack front. The particular case of a plane crack with a straight front and of stresses independent of the distance along the latter is studied first. It is shown that the classical plane strain and antiplane solutions must be supplemented by a few additional particular solutions to obtain the full stress expansion. The general case is then considered. The stress expansion is studied by writing the field equations (equilibrium, strain compatibility and boundary conditions) in a system of suitable curvilinear coordinates. It is shown that the number of independent constants in the stress expansion is the same as in the particular case considered previously but that the curvatures of the crack and its front and the non-uniformity of the stresses along the latter induce the appearance of corrective terms in this expansion.     

Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders

The velocity field and the associated shear stress corresponding to the torsional oscillatory flow of a second grade fluid, between two infinite coaxial circular cylinders, are determined by means of the Laplace and Hankel transforms. At time t = 0, the fluid and both the cylinders are at rest and at t = 0 + , cylinders suddenly begin to oscillate around their common axis in a simple harmonic way having angular frequencies ω 1 and ω 2 . The obtained solutions satisfy the governing differential equation and all imposed initial and boundary conditions. The solutions for the motion between the cylinders, when one of them is at rest, can be obtained from our general solutions. Furthermore, the corresponding solutions for Newtonian fluid are also obtained as limiting cases of our general solutions.     

A hybrid technique for the solution of unsteady Maxwell fluid with fractional derivatives due to tangential shear stress

The article describes the unsteady motion of viscoelastic fluid for a Maxwell model with fractional derivatives. The flow is produced by cylinder, considering time dependent quadratic shear stress ft² on Maxwell fluid with fractional derivatives. The fractional calculus approach is used in the constitutive relationship of Maxwell model. By applying Laplace transform with respect to time t and modified Bessel functions, semianalytical solutions for velocity function and tangential shear stress are obtained. The obtained semianalytical results are presented in transform domain, satisfy both initial and boundary conditions. Our solutions particularized to Newtonian and Maxwell fluids having typical derivatives. The inverse Laplace transform has been calculated numerically. The numerical results for velocity function are shown in Table by using MATLAB program and compared them with two other algorithms in order to provide validation of obtained results. The influence of fractional parameters and material constants on the velocity field and tangential stress is analyzed by graphs.     

The average stress tensor in systems with interacting particles

The derivation of an expression of the macroscopic stress tensor in terms of microscopic variables in systems of finite interacting particles is discussed from different points of view. It is shown that in volume averaging the introduction of a fictitious “interaction stress field”T ^I with special boundary conditions on the boundary of the averaging volume is needed. In ensemble averaging similar results are obtained by using a multipole expansion of the local stress and force fields. In the appropriate limiting cases, the obtained results are shown to be consistent with the results of kinetic theories of polymer solutions. Paper, presented at the First Conference of European Rheologists at Graz, April 14 – 16, 1982.     

无网格伽辽金法求解平面偶应力问题

提出采用无网格伽辽金法(EFGM)求解偶应力问题,以避免有限元求解中因C¹连续要求可能引起的不便。推导了基于二次基和移动最小二乘技术的EFGM计算公式,通过计算受轴向均匀拉伸的带中心小孔无限大板和细长杆的偶应力问题,对所提方法进行了数值验证,与解析解相比结果令人满意,此外还讨论了节点密度和权函数对计算结果的影响。数值结果表明,所提算法可有效地求解平面偶应力问题。     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

The analysis of thermal residual stresses near the apex in bonded dissimilar materials

By employing the complex variable method and constructing the particular solution sequences in the form of complex functions, all the cases of the thermal residual stress field near the apex in dissimilar materials bonded with two arbitrary angles are researched theoretically, and the corresponding classical solutions are obtained. Moreover, the primary paradox, the secondary paradox and even the triple paradox are discovered in the classical solutions and also resolved here, thereby it is confirmed that thermal residual stresses near the apex in bonded dissimilar materials probably possess the singularities of lnr (when the primary paradox occurs) , ln²r (when the secondary paradox occurs) and even ln³r (when the triple paradox occurs) . In addition, the systematic method to solve multiple paradox problems is put forward. © 1999 Elsevier Science Ltd. All rights reserved.     

Stress analysis in two dimensional electrostrictive material with an elliptic rigid conductor

This paper presents the governing equations of electrostrictive materials. The stress and electric field solutions for an infinite plate with a rigid elliptic conductor under applied load at infinity are given. The asymptotic expansions of the solution for a narrow elliptic conductor show that the stresses and the electric fields near the end of a narrow elliptic conductor possess r⁻¹ and r^−1/2 forms respectively in a local coordinate system with the origin at its focus.