K.T. Chau

Finite solid circular cylinders subjected to arbitrary surface load. Part I — Analytic solution

This paper presents a general framework for obtaining analytic solutions for finite elastic isotropic solid cylinders subjected to arbitrary surface load. The method of solution uses the displacement function approach to uncouple the equations of equilibrium. The most general solution forms for the two displacement functions for solid cylinders are proposed in terms of infinite series, with z- and θ-dependencies in terms of trigonometric and hyperbolic functions, and with r-dependency in terms of Bessel and modified Bessel functions of the first kind of fractional order. All possible combinations of odd and even dependencies of θ and z are included; and the curved boundary loads are expanded into double Fourier Series expansion, while the end boundary loads are expanded into Fourier–Bessel expansion. It is showed analytically that only one set of the end boundary conditions needs to be satisfied. A system of simultaneous equations for the unknown constants is given independent of the type of the boundary loads. This new approach provides the most general theory for the stress analysis of elastic isotropic solid circular cylinders of finite length. Application of the present solution to the stress analysis for the double-punch test is presented in Part II of this study.

A new boundary integral formulation for plane elastic bodies containing cracks and holes

This paper presents a new boundary integral formulation for a plane elastic body containing an arbitrary number of cracks and holes. The body is assumed to be linear elastic and isotropic, but can be of either finite or infinite extend. The cracks inside the body can be either internal or edge crack, and either straight or curvilinear; and the holes can be of arbitrary number and shape. Starting from Somigliana formula, we obtain a system of boundary integral equations by applying integration by parts. In complex variables notation, the stress and displacement components can be expressed in terms of Muskhelishvilis analytic functions, which are in turn written as functions of boundary traction and displacement data in the form of Cauchy integral. The complex boundary integral equations for traction involve only singularity of order 1/1, where r is the distance measured from the singular boundary points, and no hypersingular terms appear. This new boundary integral formulation provides an effective basis in solving problems both analytically and numerically. To illustrate the validity of our new integral formulation, a number of classical problems are re-examined analytically using the present formulation: (i) an infinite body containing a circular hole subject to far field biaxial stress, internal pressure, and a point force on the holes boundary respectively; and (ii) an infinite body containing a circular-are crack under remote uniaxial tension. To illustrate the applicability of the present formulation for boundary element method analysis, two numerical examples for the interactions between two collinear cracks are considered and the results agree well with the existing solutions by Chandra et al. (1995) for the case of finite rectangular plates and with Isida (cited in p. 195 of Murakami, 1987) for the case of infinite plates

Spherically isotropic, elastic spheres subject to diametral point load strength test

Abstract This paper presents an analytic solution for the stress concentrations within a spherically isotropic, elastic sphere of radius R subject to diametral point load strength test. The method of solution uses the displacement potential approach together with the Fourier–Legendre expansion for the boundary loads. For the case of isotropic sphere, our solution reduces to the solution by Hiramatsu and Oka, 1966 and agrees well with the published experimental observations by Frocht and Guernsey (1953) . A zone of higher tensile stress concentration is developed near the point loads, and the difference between this maximum tensile stress and the uniform tensile stress in the central part of the sphere increases with E / E ′ (where E and E ′ are the Youngs moduli governing axial deformations along directions parallel and normal to the planes of isotropy, respectively) , G ′/ G (where G and G ′ are the moduli governing shear deformations in the planes of isotropy and the planes parallel to the radial direction) , and ν / ν ′ (where ν and ν ′ are the Poissons ratios characterizing transverse reduction in the planes of isotropy under tension in the same plane and under radial tension, respectively) . This stress difference, in general, decreases with the size of loading area and the Poissons ratio.

Landslides modeled as bifurcations of creeping slopes with nonlinear friction law

Abstract This paper investigates landslide as a consequence of the unstable slide of an initially stationary or creeping slope triggered by a small perturbation, such as the effect due to rainfall. Motivated by Skemptons (1985) observation on clay and siltstone containing low clay fraction, the one state variable friction law proposed by Ruina (1983), in which shear resistance along the slip surface ( τ ) depends on both sliding history (or the state) and the creeping velocity of the slope ( V ), is extended to model both fluid-saturated rock and soil joints. Presuming that the slope is shallow and infinitely long, a system of three coupled nonlinear first-order differential equations, relating τ , V and u (displacement of the slip surface), is formulated. Linear stability analysis suggests that an equilibrium state (or a critical point) of a slope can be classified as either an asymptotically stable spiral point, an asymptotically stable improper point, or an asymptotically unstable saddle point in the dimensionless shear stress-velocity ( s - v ) phase plane, depending on the state parameters on the slip surface. For the special case that steady state τ is insensitive to V and the gravitational pull equals the threshold shear stress ( τ 0 ), the critical point becomes a neutrally stable equilibrium line in the s - v phase plane; trajectories in the s - v phase plane are obtained analytically for this case. Periodic solution (or Hopf bifurcation) for the system is ruled out based on physical grounds. A fully nonlinear numerical analysis is done for all possible scenarios when a finite perturbation is imposed to the equilibrium state. As the slip continues and erosion due to rainfall occurs, nonlinear parameters of the slip surface may evolve such that a previously stable slope may become unstable (i.e. bifurcation occurs) when a small perturbation is imposed. Thus, the present analysis offers a plausible explanation to why slope failure occurs at a particular rainfall, which is not the largest in the history of the slope.

An infinite plane loaded by a rivet of a different material

Abstract This paper derives a closed-form solution for the stress distributions in an infinite plane loaded by a rivet of a different material under either plane stress or plane strain condition. A distinctive feature of the present analysis is that the rivet load is modelled by distributed body forces over the section of the rivet, in contrast to the commonly-used assumption of a concentrated load acting at the centre of the rivet. Two body force potentials are introduced to model the cases of conservative, uniform distributed force (Loading Case I) and non-conservative, non-uniform distributed force (Loading Case II), which is similar to those caused by shear force on a circular section. Our results show that the normal contact stress decreases with both the stiffness ratio ζ = μ 2 / μ 1 ( μ 1 and μ 2 are the shear moduli for the plane and rivet, respectively) and the frictional coefficient μ between the plane and rivet; conversely, the shear contact stress increases with both μ and ζ . The normal contact stress for Loading Case I is larger than that for Loading Case II, while the opposite conclusion applies to the shear contact stress; their differences are more apparent for larger ζ . Larger values of ζ and μ result in higher maximum hoop stress and the corresponding location of maximum hoop stress deviates farther from the edge of contact zone; and the maximum hoop stress resulted from Loading Case II is larger than that induced by Loading Case I. The hoop stress at the rivet hole agrees well with experimental results by Coker and Filon [Coker, E. G. and Filon, L. N. G. (1931). A Treatise on Photoelasticity, Cambridge University Press, Cambridge], Frocht [Frocht, M. M. (1949). Photoelasticity , Vol. 1, Wiley,NY], Nisida and Saito [Nisida, M. and Saito, H. (1966). Stress distributions in a semi-infinite plate due to a pin determined by interferometric method. Experimental Mechanics 6 , 273–279] and Hyer and Liu [Hyer, M. W. and Liu, D. (1984). Stresses in pin-loaded orthotropic plates using photoelasticity. NASA contractor report , CR-172498, NASA, USA]. In general, a compression zone ( π > { θ } > θ 1 ) and a tension zone ( θ 1 > { θ } > 0) in hoop stress are observed, where θ is measured from the direction of the resultant rivet force and the typical value of θ 1 is about 160°. For the case of a rigid rivet with high friction, a second compressive zone near θ = 0 is observed; this differs from all previous theoretical studies, but agrees with the experimental observation by Frocht [Frocht, M. M. (1949). Photoelasticity , vol. 1, Wiley, NY].

Finite solid circular cylinders subjected to arbitrary surface load. Part II — Application to double-punch test

Abstract This paper derives the stress distributions within a finite isotropic solid circular cylinder of diameter 2b and length 2h under the double-punch test, which was introduced by Chen (1970) for the determination of the indirect tensile strength of concrete. The stresses induced by the two rigid circular punches of diameter 2a at the top and bottom of the solid cylinder are modeled by considering contact problem. The general stress analysis discussed in a companion paper (Part I) is used to obtain the stress field within the solid. In general, tensile stress concentrations are developed beneath the punches compared to the roughly uniform tensile stress at the central portion of the axis of the cylinder. The maximum tensile stress in the tensile zone decreases with the increase of Poisson’s ratio and a/b, but is roughly independent of h/b. For small Poisson’s ratio (say about 0.1) and a/b (say smaller than 0.1), the assertion made by Chen, 1970 , Marti, 1989 that a uniform tensile stress field, similar to that of the Brazilian test, is developed along the axis of symmetry is incorrect. The tensile strength interpreted from the present analysis is found comparable to the formula proposed by Bortolotti (1988) for a/b>0.2 and agrees well with the experimental data, and thus provides an improvement over Chen’s (1970) original formula.

A new analytic solution for the diametral point load strength test on finite solid circular cylinders

Abstract This paper presents an exact analytic solution for a finite isotropic circular cylinder of diameter D and length 2 L subjected to the diametral point load strength test (PLST). Two displacement functions are introduced to uncouple the equations of equilibrium, and two new series expressions for these functions are proposed in terms of the Bessel and modified Bessel functions of the first kind, the trigonometric functions and the hyperbolic functions. The contact stresses between the curved surface of the cylinder and the spherical heads of the indentors are expanded into double Fourier series expansion in order to match the limiting values of the stress field on the boundary. Our numerical results show that tensile stress concentrations are developed near the point loads, compared to the roughly uniform tensile stress at the central portion of the line between the two point loads. The pattern of tensile stress distribution along this line resembles that obtained for a sphere under the diametral PLST (Chau, K.T., Wei, X.X., 1999. Int. J. Solids and Struct. 36 (29), 4473–4496) and a cylinder under the axial PLST (Wei, X.X., et al., 1999. J. Eng. Mech. ASCE 125 (12), 1349–1357). The maximum tensile stress decreases with the increase of Poisson’s ratio, the contact area, the radius of the spherical heads of the indentors, but increases with the diameter of the cylinder. It also decreases drastically with the increase of L / D for short cylinders (say L / D L / D is long enough (say L / D >0.7). Both the predicted size and shape effects of specimens on the diametral PLST agree with our experimental observations.

Arbitrarily oriented crack near interface in piezoelectric bimaterials

Arbitrarily oriented crack near interface in piezoelectric bimaterials is considered. After deriving the fundamental solution for an edge dislocation near the interface, the present problem can be expressed as a system of singular integral equations by modeling the crack as continuously distributed edge dislocations. In the paper, the dislocations are described by a density function defined on the crack line. By solving the singular integral equations numerically, the dislocation density function is determined. Then, the stress intensity factors (SIFs) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Subsequently, the influences of the interface on crack tip SIFs, EDIF, and the mechanical strain energy release rate (MSERR) are investigated. The J-integral analysis in piezoelectric bimaterals is also performed. It is found that the path-independent of J1-integral and the path-dependent of J2-integral found in no-piezoelectric bimaterials are still valid in piezoelectric bimaterials.

Youngs modulus interpreted from plane compressions of geomaterials between rough end blocks

Abstract This note derives an approximate expression of the true Youngs modulus of a rectangular solid under plane compression between two rough end blocks, provided that the Poissons ratio ν of the solid is known. The friction between the loading platens and the ends of the specimen is assumed to be large enough to restrain slippage at the contact. By using the function space concept of Prager and Synge (1947) , a correction factor λ with calculable error is obtained which can be multiplied to the apparent Youngs modulus (i.e., the one obtained by assuming uniform stress field) to yield the true Youngs modulus; it is evaluated numerically for 0 ⩽ ν ⩽ 0.49 and 0 ⩽ η ⩽ 3 (where η = b⧸h with b and h being the half width and half length of the specimen) . In general, λ increases with ν and η for both plane strain and plane stress compressions. Within this range of ν and η , λ may vary from 0.37–1.0 for the plane strain case and from 0.84–1.0 for the plane stress case. Thus, the assumption of uniform stress field may lead to erroneous interpretation of the Youngs modulus. When the special case of ν = 1⧸3 and η = 1 is considered, we obtain λ = 0.9356, which compares well with 0.9359 obtained by Greenberg and Truell, 1948 .