Page 251 - Mechanics of Asphalt Microstructure and Micromechanics

CHAPTER 8

Finite Element Method and

Boundary Element Method

8.1 Introduction
This chapter will discuss the application of Finite Element Method (FEM) and Boundary
Element Method (BEM). The general philosophy is to present: 1) a brief review on the
very fundamentals of the two methods; 2) implementation of constitutive models using
implicit and semi-implicit methods for FEM; 3) implementation of interface models on
FEM; and 4) some literature in modeling asphalt concrete (AC) using BEM. There are
many references on using FEM for simulating AC behavior and the models discussed in
Chapter 7 are mainly implemented on FEM. Therefore, the literature revi ew will not
cover those using FEM. Several general works are suggested for more details, such as
Zienkiwicz (1977); Huebner et al. (1995); Scarpas (2004); Brebbia et al. (1984); and Gaul
(2003). Many commercial programs such as ABAQUS, ASYS, and ADINA are conve-
niently available and so no detailed discussions on general FEM are considered neces-
sary. However, the application techniques such as interface element, infinite element,
rigid element, which may find direct applications, are discussed instead.
A special FEM, CAPA-3D, dedicated to modeling and simulation of pavements and
paving materials developed by Scarpas and his colleagues (2004), is a unique reference
for most readers.

8.2 Numerical Solution Approaches to Elasticity Problems, FEM
8.2.1 Theory
In elasticity, two commonly used approaches are the force method and the displacement
method. FEM is typically a displacement-driven approach. This approach involves the
following procedure: 1) obtaining strains from displacements; 2) obtaining stresses from
strains; and 3) solving the equilibrium equations expressed in displacements and their
partial derivatives with boundary conditions. A brief review is given as follows:
u ∂ u ∂
Strain-displacement relationship: ε = ( i + j )/2 (8-1)
ij x ∂ x ∂
j i
Stress-strain relationship (Hooke’s Law): σ = D ε (8-2)
ij ijkl kl
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