A Brief Review of Elasticity and Viscoelasticity for Solids 1 Introduction

H. T. Banks, S. H. Hu and Z. R. Kenz / Adv. Appl. Math. Mech., 3 (2011), pp. 1-51 9
Infinitesimal strain theory
Infinitesimal strain theory is also called small deformation theory, small displacement theory
or small displacement-gradient theory. In infinitesimal strain theory, it is assumed that
the components of displacement u i are such that their first derivatives are so small that
higher order terms such as the squares and the products of the partial derivatives of u i
are negligible compared with the first-order terms. In this case e ij reduces to Cauchy’s
infinitesimal strain tensor
ε ij = 1 [ ∂ui
+ ∂u ]
j
, (2.8)
2 ∂x j ∂x i
(hence the infinitesimal strain tensor is also symmetric). Thus, ε ij = ε ji . We note that
in this case the distinction between the Lagrangian and Eulerian strain tensors disappears
(i.e., E ij ≈ e ij ≈ ε ij ), since it is immaterial whether the derivatives of the displacement
are calculated at the position of a point before or after deformation. Hence,
the necessity of specifying whether the strains are measured with respect to the initial
configuration (Lagrangian description) or with respect to the deformed configuration
(Eulerian description) is characteristic of a finite strain analysis and the two different
formulations are typically not encountered in the infinitesimal theory.
2.1.3 Stress
Stress is a measure of the average amount of force exerted per unit area (in units N/m 2
or Pa), and it is a reaction to external forces on a surface of a body. Stress was introduced
into the theory of elasticity by Cauchy almost two hundred years ago.
Definition 2.4. The stress vector (traction) is defined by
T (n) = dF
dΓ = lim ∆F
∆Γ→0 ∆Γ ,
where the superscript (n) is introduced to denote the direction of the normal vector n of the
surface Γ and F is the force on the surface.
To further elaborate on this concept, consider a small cube in the body as depicted
in Fig. 2 (left). Let the surface of the cube normal (perpendicular) to the axis z be
donated by ∇Γ z . Let the stress vector that acts on the surface ∇Γ z be T (e 3) , where
e 3 = (0, 0, 1) T . Resolve T (e 3) into three components in the direction of the coordinate
axes and denote them by σ zx , σ zy and σ zz . Similarly we may consider surface ∇Γ x and
∇Γ y perpendicular to x and y, the stress vectors acting on them, and their components
in the x, y and z directions. The components σ xx , σ yy and σ zz are called normal stresses,
and σ xy , σ xz , σ yx , σ yz , σ zx and σ zy are called shear stresses. A stress component is positive
if it acts in the positive direction of the coordinate axes. We remark that the notation
σ face, direction is consistently used in elasticity theory.