Effect of notch depth on strain-concentration factor of rectangular bars with a single-edge notch under pure bending

The FEM is employed to study the effect of notch depth on a new strain-concentration factor (SNCF) for rectangular bars with a single-edge notch under pure bending. The new SNCF $K_{\epsilon }^{\mathrm{new}}$ is defined under the triaxial stress state at the net section. The elastic SNCF increases as the net-to-gross thickness ratio h₀/H₀ increases and reaches a maximum at h₀/H₀ = 0.8. Beyond this value of h₀/H₀ it rapidly decreases to the unity with h₀/H₀. Three notch depths were selected to discuss the effect of notch depth on the elastic–plastic SNCF; they are the extremely deep notch (h₀/H₀ = 0.20), the deep notch (h₀/H₀ = 0.60) and the shallow notch (h₀/H₀ = 0.95). The new SNCF increases from its elastic value to the maximum as plastic deformation develops from the notch root. The maximum $K_{\epsilon }^{\mathrm{new}}$ of the shallow notch is considerably greater than that of the deep notch. The elastic $K_{\epsilon }^{\mathrm{new}}$ of the shallow notch is however less than that of the deep notch. Plastic deformation therefore has a strong effect on the increase in $K_{\epsilon }^{\mathrm{new}}$ of the shallow notch. The variation in $K_{\epsilon }^{\mathrm{new}}$ with M/M_Y, the ratio of bending moment to that at yielding at the notch root, is slightly dependent up to the maximum $K_{\epsilon }^{\mathrm{new}}$ for the shallow notch. This dependence is remarkable beyond the maximum $K_{\epsilon }^{\mathrm{new}}$. On the other hand, the variation in $K_{\epsilon }^{\mathrm{new}}$ with M/M_Y is independent of the stress–strain curve for the deep and extremely deep notches.     

Scale-by-scale energy budget in a turbulent boundary layer over a rough wall

Hot-wire velocity measurements are carried out in a turbulent boundary layer over a rough wall consisting of transverse circular rods, with a ratio of 8 between the spacing (w) of two consecutive rods and the rod height (k). The pressure distribution around the roughness element is used to accurately measure the mean friction velocity ($U_{\tau }$) and the error in the origin. It is found that $U_{\tau }$ remained practically constant in the streamwise direction suggesting that the boundary layer over this surface is evolving in a self-similar manner. This is further corroborated by the similarity observed at all scales of motion, in the region $0.2⩽y/\delta ⩽0.6$, as reflected in the constancy of Reynolds number ($R_{\lambda }$) based on Taylor’s microscale and the collapse of Kolmogorov normalized velocity spectra at all wavenumbers.A scale-by-scale budget for the second-order structure function $〈{(\delta u)}^2〉$ ($\delta u=u(x+r)-u(x)$, where u is the fluctuating streamwise velocity component and r is the longitudinal separation) is carried out to investigate the energy distribution amongst different scales in the boundary layer. It is found that while the small scales are controlled by the viscosity, intermediate scales over which the transfer of energy (or $〈{(\delta u)}^3〉$) is important are affected by mechanisms induced by the large-scale inhomogeneities in the flow, such as production, advection and turbulent diffusion. For example, there are non-negligible contributions from the large-scale inhomogeneity to the budget at scales of the order of $\lambda$, the Taylor microscale, in the region of the boundary layer extending from $y/\delta =0.2$ to 0.6 ($\delta$ is the boundary layer thickness).     

Stress concentration at an elliptic hole in transversely isotropic piezoelectric solids

Two-dimensional electroelastic analyses have been performed theoretically on a transversely isotropic piezoelectric material containing an elliptic hole, subjected to a uniform stress field and a uniform electric displacement field at infinity while the surface of the hole is free of traction and electrically open. Solutions are obtained by using the exact electric boundary condition based on the complex variation method. Explicit solutions for the distributions of the mechanical and electrical components on the rim of the elliptic hole are obtained. An interesting relationship between the stress concentration factor of an elliptic hole (K_t) and that of a circular hole (K_t∣_t=1), $K_t=1+\frac{K_t{\mid }_{t=1}-1}{t}$, is found in both elastic and piezoelectric materials. It is shown that the electromechanical coupling effect is helpful to reduce the stress concentration. And the influence of the dielectric parameter of the medium inside the hole on the stresses and the concerned stress concentration factor at the surface of the hole is weak in a wide range of the dielectric parameter. Comparisons with available results show good coincidence.