G22.3033-002: Topics in Computer Graphics:
Lecture #7
Geometric Modeling
New York University
Elasticity Theory Basics
Lecture #7:
20 October 2003
Lecturer:
Denis Zorin
Scribe:
Adrian Secord, Yotam Gingold
Introduction
This lecture is the summary of the basics of the elasticity theory. We will discuss the funda-
mental concepts of stress and strain, and the equations describing the deformation of an elastic
object in equilibrium. We will focus on materials following the linear material law.
The linear material law can be thought of as a generalization Hooke’s law f = −kx to 3D
deformable objects. The most straightforward generalization would be to state that stretching
or compressing an object in any direction results in an elastic force in the same direction which
is proportional to the extension. However, this would match well only a small class of mate-
rials. In three dimensions in most cases a stretch in one direction changes the dimensions in
the perpendicular directions. The two aspects of deformation are characterized using two con-
stants: the Young’s modulus, which corresponds to the constant in Hooke’s law, and is denoted
E or Y , and the Poisson’s ratio, which is denoted ν. The Poisson ratio is the ratio of stretch to
perpendicular change. To formulate the equations of balance for elastic objects and define the
linear material law more precisely, we start with a discussion of the counterparts of forces and
displacements in the equations of elasticity: strain and stress tensors.
Stress and Strain
There are two kinds of forces acting on a small volume of a body:
1. Volume forces
• f is the volume force density;
• dFvol = fdV is the differential volume force, where dV is the differential volume
element.
2. Surface forces
• t(n) the vector surface force density, where n is the normal to the differential sur-
face element dA;
• t(n) is also called the stress vector;
• dFsurf = t(n)dA is the differential surface force, acting on the differential surface
element dA.
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G22.3033-002: Lecture #7
Stress tensor
As