Chapter 6
Two-Dimensional Problems in Elasticity
(in Polar Coordinate System)
6.1 Introduction
In any elasticity problem, the proper choice of the co-ordinate system is extremely
important since this choice establishes the complexity of the mathematical expressions employed to satisfy the field equations and the boundary conditions.
In order to solve two-dimensional elasticity problems by employing a polar coordinate reference frame, the equations of equilibrium, the definition of Airy’s stress
function and one of the stress equations of compatibility must be established in terms
of polar co-ordinates.
6.2 Strain–Displacement Relations
Case 1: For Two-Dimensional State of Stress.
Consider the deformation of the infinitesimal element ABCD, denoting r and θ
displacements by u and v respectively. The general deformation experienced by
an element may be regarded as composed of (1) a change in the length of the sides,
and (2) rotation of the sides as shown in Fig. 6.1.
Referring to the figure, it is observed that a displacement “u” of side AB results
in both radial and tangential strains.
Therefore,
radial strain = ε r =
∂u
∂r
(6.1)
and tangential strain due to u per unit length of AB is
(ε θ ) u =
(r + u)dθ − r dθ
r dθ
=
u
r
(6.2)
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_6
163
Two-Dimensional Problems in Elasticity
(in Polar Coordinate System)
6.1 Introduction
In any elasticity problem, the proper choice of the co-ordinate system is extremely
important since this choice establishes the complexity of the mathematical expressions employed to satisfy the field equations and the boundary conditions.
In order to solve two-dimensional elasticity problems by employing a polar coordinate reference frame, the equations of equilibrium, the definition of Airy’s stress
function and one of the stress equations of compatibility must be established in terms
of polar co-ordinates.
6.2 Strain–Displacement Relations
Case 1: For Two-Dimensional State of Stress.
Consider the deformation of the infinitesimal element ABCD, denoting r and θ
displacements by u and v respectively. The general deformation experienced by
an element may be regarded as composed of (1) a change in the length of the sides,
and (2) rotation of the sides as shown in Fig. 6.1.
Referring to the figure, it is observed that a displacement “u” of side AB results
in both radial and tangential strains.
Therefore,
radial strain = ε r =
∂u
∂r
(6.1)
and tangential strain due to u per unit length of AB is
(ε θ ) u =
(r + u)dθ − r dθ
r dθ
=
u
r
(6.2)
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
T. G. Sitharam and L. Govindaraju, Theory of Elasticity,
https://doi.org/10.1007/978-981-33-4650-5_6
163