Trefftz Displacement Potential Function Method for Solving Elastic Half-Space Problems

Charles Chinwuba Ike,Hyginus Onah ,Benjamin Mama ,Cu Nwoji

doi:10.13189/cea.2021.090301

Abstract

The elastic half-space problem has been solved previously using Boussinesq, Papkovich, Love, and Green and Zerna, potential function methods. In this work, the Trefftz displacement potential function method is used to obtain the stress and displacement fields in an elastic half-space subjected to boundary loads. Point load and various distributed loads are considered. The problem is presented using displacement formulation as Navier–Lame equations. It is proved that the Trefftz functions are solutions of the Navier–Lame displacement equations. Strain fields are derived in terms of the Trefftz function using the strain-displacement relations. The stress fields are similarly derived. The Trefftz function for the case of a point load acting at the origin of the elastic half-space is derived using the double exponential Fourier transformation technique. Stress equilibrium boundary conditions are used to fully determine the Trefftz function. Stress and displacement fields for the point load are then determined. The solutions to stress and displacement fields for point load are then used as Green functions to obtain stress and displacement fields for uniformly distributed load over a finite line, circular area and rectangular area. It is found that the solutions obtained for the stress and displacement fields in the elastic half-space due to point and distributed loads are identical with previously obtained expressions, thus validating this work.

Highlights

The conclusions of the present work are as follows: The Trefftz displacement potential function method has been successfully used to find stress fields, and displacement fields in an elastic half-space (−∞ ≤ x ≤ ∞, −∞ ≤ y ≤ ∞, 0 ≤ z ≤ ∞) due to point load acting at the origin (0,0,0), uniformly distributed line load of finite length, and uniformly distributed loads over circular areas and over rectangular areas on the half-space
It was proved that the Trefftz displacement potential functions are solutions to the Navier–Lamé displacement formulation of elastostatic problems of the elastic half-space, the Trefftz functions satisfying the fundamental equations of 3D elasticity theory namely kinematic, generalised Hooke’s law and the differential equations of equilibrium
Equilibrating strain, stress and displacement fields were derived in terms of the Trefftz displacement potential functions

Summary

Introduction

Specific types of the elastic half-space problems are the Boussinesq, Kelvin, Mindlin and Cerrutti problems. Boussinesq type problem of the elastic half-space considers load applied normally to the boundary surface of the elastic half-space and such that the boundary is free of shear stresses. Kelvin problem consists of finding stress and displacement fields in an elastic space −∞ ≤ x ≤ ∞, −∞ ≤ y ≤ ∞, −∞ ≤ z ≤ ∞ due to a point load, where the space region is considered linearly elastic and isotropic [15]. Kelvin problem is the point load in an elastic (full) space problem. In Mindlin problem, an axisymmetric point load acts at the interior of a homogeneous, isotropic, elastic half-space. The objective of elastic half-space problems under known loads is to determine the stress and displacement fields

Objectives

Methods

Results

Discussion

Conclusion