Edge crack due to circular rigid punch in incomplete contact

A circular rigid punch with friction is assumed to be in contact with a half-plane and the punch has one end sliding on the half-plane and another end with a sharp corner. The contact length is determined by satisfying the finite stress condition at the sliding end of the punch. An oblique edge crack is assumed to be initiated near the end with the sharp corner where the infinite stresses exist. Coulomb's frictional force is assumed to act on the contact region. The cracked half-plane is mapped into a unit circle by using a rational mapping function and the problem is transformed into a standard Riemann–Hilbert problem, which is solved by introducing a Plemelj function. The contact length, the stress intensity factors of the crack and the resultant moment about the origin of the coordinates are calculated with different frictional coefficients and oblique angles of the crack. The stress distributions on the contact region are also shown.     

Thermal relaxation effects in rapid sliding contact with friction

Summary. A rigid die slides at constant sub-critical speed on a homogeneous, isotropic linear coupled thermoelastic half-space. Friction exists, and a dynamic steady state of plane strain is considered. An exact integral transform solution for the related problem of moving surface traction is obtained, and asymptotic expressions valid when thermal relaxation is prominent are extracted.These are used to derive an analytic solution for the sliding problem, and formulas for contact zone size and location, and unilateral constraints imposed by non-tensile contact and non-positive frictional work rate. Expressions for three body wave speeds and a Rayleigh wave speed show, save for the rotational wave case, clear dependence on thermoelastic coupling and thermal relaxation.Calculations for 4340 steel show that the problem eigenvalue is similar to its isothermal counterpart for high sliding speeds, but that the average contact zone temperature increase is less pronounced than when classical Fourier heat conduction effects dominate. Calculations for a hypothetical material similar to steel show that increasing the thermal relaxation time can in effect suppress both the Rayleigh wave and second sound body wave.     

Eigenvalue analysis for a frictional punch problem of a concave notch with a fixed edge

Summary A numerical solution for the eigenvalue in a frictional punch problem is presented. One edge of a concave notch is assumed to be fixed, and the other edge is in contact with the frictional punch (Fig. 1). In addition to the normal pressure, the friction force is also applied along the contact edge of punch. The friction coefficient is assumed within the range (−0.3, 0.3). The notch angle between two edges is varying within the range 180 to 360 degrees. The target function method is used to evaluate the eigenvalue which is an exponent in the stress expression near the crack tip. Finally, the obtained eigenvalues are expressed in the form of a function of the friction coefficient and the angle of the concave notch. The pressure under the punch and the stresses at the corner points are singular, in general.     

Dynamic crack extension along the interface of materials that differ in their thermal properties: with and without thermal relaxation

Summary Moving surface stresses cause crack extension along the interface of perfectly bonded thermoelastic materials at a constant sub-critical speed. The materials differ only in their thermal properties, and are governed by coupled thermoelastic equations that admit as special cases Fourier heat conduction as well as thermal relaxation with one or two relaxation times. A dynamic steady state of plane strain is assumed. The exact transform solution for a propagating displacement and temperature discontinuity is used to find solutions to the interface crack valid away from the crack edge for low extension speeds and solutions valid at the crack edge for high speeds. Results show that Fourier heat conduction dominates the former case, but solution behavior in the latter is dependent upon the particular thermal model. Thermal mismatch is seen to by itself cause a solution behavior similar to that for bonded dissimilar isothermal elastic solids. In particular, the two-relaxation time solution exhibits both oscillatory and non-oscillatory terms, and the interface temperature at the crack edge is finite.     

Eigenvalue Analysis for a Frictional Punch Problem of Concave Notch

A numerical solution for the eigenvalue in a frictional punch problem is presented, One edge of a concave notch is assumed to be traction free, and the other edge is in contact with the frictional punch ( Fig, 1), In addition to the normal pressure the friction force is also applied along the contact edge of punch. The friction coefficient δ is assumed within the range – 0.3 ⩾ 6 ⩾ 0,3. The notch angle β between two edges is varying within the range π ⩾ β ⩾ 2π. The target function method is used to evaluate the eigenvalue λ₁ in the stress expression σ_ij ≈ O(1/r^λ1). Finally, the obtained eigenvalues are expressed in the form of λ₁ = s(δ, β). The pressure under punch and the stresses at the corner points are singular in general.     

Steady state thermoelastic contact problem in a functionally graded material

This paper is concerned with the stationary plane contact of a functionally graded heat conducting punch and a rigid insulated half-space. The frictional heat generation inside the contact region due to sliding of the punch over the half-space surface and the heat radiation outside the contact region are taken into account. Elastic coefficient μ, thermal expansion coefficient α_t and coefficient of thermal conductivity k are assumed to vary along the normal to the plane of contact. With the help of Fourier integral transform the problem is reduced to a system of two singular integral equations. The equations are solved numerically. The effects of nonhomogeneity parameters in FGMs and thermal effect are discussed and shown graphically.     

Aerodynamic structure of a flat plate laminar boundary layer with diffusion flame

An aerodynamic structure of a laminar boundary layer over a flat plate with uniform fuel injection from the flat plate and with diffusion flame is investigated numerically. Elliptic type conservation equations are used to take into account the pressure variation within the boundary layer. Velocities and the pressure are solved numerically by SIMPLER algorithm. One step irreversible chemical reaction of methane is assumed. An Arrhenius type chemical reaction rate model is assumed and the pre-exponential factor is varied from 1.0 × 10¹² to 1.0 × 10³⁰ m³/(kg s) as a parameter of the reactivity in order to elucidate the effect of the reactivity on the structure of the boundary layer. When the chemical reaction is very fast, the leading edge of the reaction zone reaches the flat plate. As the chemical reaction rate is decreased with a decrease in the pre- exponential factor, the leading edge of the reaction zone parts from the flat plate and it shifts downstream. The fuel is injected in front of the leading edge of the reaction zone, where the air is dominant, and the oxygen penetrates into the fuel dominant zone through the region between the leading edge and the flat plate. As a consequence, a premixed gas is formed around the leading edge of the reaction zone. The premixed gas seems to react in the region apart from the main visible flame.Part of this work was presented at ICCM 86, Tokyo, Japan, May 25–29, 1986     

Analysis of functionally graded nanodisks under thermoelastic loading based on the strain gradient theory

In this paper, the thermoelastic behavior of a functionally graded nanodisk is studied based on the strain gradient theory. It is assumed that the nanodisk thickness is constant, and a power-law model is adopted to describe the variation of functionally graded material properties. Furthermore, the nanodisk angular acceleration is taken to be zero while it is subjected to an axisymmetric loading. Also, it is assumed that any variation in temperature occurs only in the radial direction. The equilibrium equation and the boundary conditions are deduced from Hamilton’s principle. The obtained results are compared with those of classical theory. These results show that both theories predict the same trend for the variation in radial displacements. The differences between the stresses obtained from classical and strain gradient theories are clearly highlighted. Increasing the value of the material inhomogeneity parameter, n, considerably affects the magnitudes and the corresponding peak values of the high-order stress \(\bar{\tau }_{rrr}\). Any rise in temperature at the outside radius has a direct effect on the total stresses and radial displacements in the nanodisk. Also, the effects of external load at the inner and outer radii on radial displacement as well as stress components are fully investigated.     

Numerical investigations of flat punch molding using a higher order strain gradient plasticity theory

In this work we have revisited the problem of molding a deformable substrate with a rigid flat punch. The work is motivated by the recent experiments by Chen et al. (Acta Mater 59:1112?1120, 2011) where it was shown that systematically determined characteristic molding pressure H increased significantly with decrease in punch width, for widths less than \(\sim 25 \; \mu m\) . This size effect, akin to the indentation size effect observed in nano-indentation of metals, assumes importance in applications involving molding of metallic microstructures. Numerical simulations have been conducted within the framework of a finite deformation higher order strain gradient model. While classical plasticity predicts almost uniform stress with severe plastic strain concentration at the sharp corners to prevail just beneath the punch, our simulations present a significantly different picture. Very narrow punches have fairly uniform plastic strain with severe concentration of strain gradients and large contact stresses close to the edges. Wider punches however, behave in a manner closely resembling the predictions of classical plasticity.     

Contact problem of a functionally graded layer resting on a Winkler foundation

The contact problem for a functionally graded layer supported by a Winkler foundation is considered using linear elasticity theory in this study. The layer is loaded by means of a rigid cylindrical punch that applies a concentrated force in the normal direction. Poisson’s ratio is taken as constant, and the elasticity modulus is assumed to vary exponentially through the thickness of the layer. The problem is reduced to a Cauchy-type singular integral equation with the use of Fourier integral transform technique and the boundary conditions of the problem. The numerical solution of the integral equation is performed by using Gauss–Chebyshev integration formulas. The effect of the material inhomogeneity, stiffness of the Winkler foundation and punch radius on the contact stress, the contact area and the normal stresses are given.