Time-harmonic elastic singularities and oscillating indentation of layered solids

A new approach is proposed for obtaining the dynamic elastic response of a multilayered elastic solid caused by axisymmetric, time-harmonic elastic singularities. The method for obtaining the elastodynamic Green’s functions of the point force, double forces and center of dilatation is presented. For this purpose, the boundary conditions in an infinite solid at the plane passing through the singularity are derived first by using Helmholtz potentials. Then the Green’s function solution for layered solids is obtained by solving a set of simultaneous linear algebraic equations using the boundary conditions for both the singularities and for the layer interfaces. The application of the point force solution for the oscillating normal indentation problem is also given. The solution of the forced normal oscillation is formulated by integrating the point force Green’s function over the contact area with unknown surface traction. The dual integral equations of the unknown surface traction are established by considering the boundary conditions on the contact surface of the multilayered solid, which can be converted into a Fredholm integral equation of the second kind and solved numerically.

Flamant solution of a half-plane with surface flexural resistibility and its applications to nanocontact mechanics

This article presents a semianalytical solution to a half-plane contact problem subjected to an arbitrarily distributed surface traction. The half-plane boundary is treated as a material surface of the Steigmann–Ogden type. Under the assumption of plane strain condition, the problem is formulated by coupling the methods of an Airy stress function and Fourier integral transforms. Stresses and displacements in the form of semi-infinite integrals are derived. A non-classical Flamant solution that is able to simultaneously account for the surface tension, membrane stiffness, and bending rigidity of the half-plane boundary is derived through limit analysis on the half-plane contact problem owing to a uniform surface traction. The fundamental Flamant solution is further integrated for tackling two half-plane contact problems owing to classical contact pressures corresponding to a rigid cylindrical roller and a rigid flat-ended punch. The resultant semi-infinite integrals are integrated by the joint use of the Gauss–Legendre numerical quadrature and the Euler transformation algorithm. Extensive parametric studies are conducted for comparing and contrasting the effects of Gurtin–Murdoch and Steigmann–Ogden surface mechanical models. The major observations and conclusions are two-fold. First, the introduction of either surface mechanical model results in size-dependent elastic fields. Second, the incorporation of the curvature-dependent nature of the half-plane boundary leads to bounded stresses and displacements in the fundamental Flamant solution. This is in contrast to the otherwise singular classical and Gurtin–Murdoch solutions. For all four case studies, the Steigmann–Ogden surface model also results in much smoother displacement and stress variations, indicating the significance of surface bending rigidity in nanoscale contact problems.

Influence of substrate base on sports field covered with bermuda grass

The sports field consists of three layers (sub-base, base (substrate) and grass). The base is responsible for the radicular development of the grass, directly influencing the characteristics that provide quality, water drainage and lawn durability and allows athlete performance. The construction of sports field base generally follows USGA (United States Golf Association) recommendations for golf course greens. Sand is the main component based on its high capacity for drainage. This study aimed to define the best composition of the base for sports fields covered with Bermuda grass. The following treatments were evaluated: T1: sand; T2: sand (80%) + peat (20%); T3: sand (90%) + clay soil (10%); T4: sand (70%) + sandy soil (30%) during 12 months. The experimental design consisted of a randomized block design, with a 4 x 12 factorial design (treatments x months), with 3 replicates and each plot measuring 3 x 4 m. During one full year, the following parameters were evaluated: surface traction of the turf, mechanical resistance of the base to penetration, humidity of the base, concentration of nutrients in grass clippings, and chemical analysis of the base. The tallest grass occurred in substrate of sand mixed with peat. This mixture also promoted the highest N and P levels in the leaves of Bermuda grass. Playability of the sports field, as determined by mechanical strength and surface traction, was unaffected by the type of substrate. The presence of peat, sand and clay soil mixed with sand promoted greater water retention at the base.

Numerical Analysis of Inverse Elasticity Problemwith Signorini's Condition

An optimal control problem is considered to find a stable surface traction, which minimizes the discrepancy between a given displacement field and its estimation. Firstly, the inverse elastic problem is constructed by variational inequalities, and a stable approximation of surface traction is obtained with Tikhonov regularization. Then a finite element discretization of the inverse elastic problem is analyzed. Moreover, the error estimation of the numerical solutions is deduced. Finally, a numerical algorithm is detailed and three examples in two-dimensional case illustrate the efficiency of the algorithm.

Bypassing slip velocity: rotational and translational velocities of autophoretic colloids in terms of surface flux

A standard approach to propulsion velocities of autophoretic colloids with thin interaction layers uses a reciprocity relation applied to the slip velocity although the surface flux (chemical, electrical, thermal, etc.), which is the source of the field driving the slip, is often more accessible. We show how, under conditions of low Reynolds number and a field obeying the Laplace equation in the outer region, the slip velocity can be bypassed in velocity calculations. In a sense, the actual slip velocity and a normal field proportional to the flux density are equivalent for this type of calculation. Using known results for surface traction induced by rotating or translating an inert particle in a quiescent fluid, we derive simple and explicit integral formulas for translational and rotational velocities of arbitrary spheroidal and slender-body autophoretic colloids.