Chapter 9
Plane Problem of Elasticity Theory
9.1 Functions of Stresses
Previously (p. 16), it was established that the dependency between relative elongations and normal stresses in the case of plane strain (ε z = 0) could be obtained
from the respective dependencies in the case of a plane stressed state (σ z = 0)
by substituting the Young modulus E and Poisson coefficient ν with E ∗ and ν ∗ ,
respectively, using formulas (1.21). Therefore, both these stress–strain states are
combined by the concept of “plane problem” not always mentioning what exact
state is considered.
Let us consider plane strain (ε z = 0). In this case, equilibrium equations:
∂σ x
∂x
+
∂τ xy
∂y
= 0,
∂σ y
∂y
+
∂ τ yx
∂x
= 0
(9.1)
can be met if we consider that
σ x =
∂ 2 U
∂y 2 , σ y =
∂ 2 U
∂x 2 , τ xy =
∂ 2 U
∂x∂y
,
(9.2)
where U is an arbitrary function differentiated for the sufficient number of times,
which is called (Airy, [5]) the stress function.
Let us also use the fact that volumetric expansion in an elastic body (1.13) is a
harmonic function (see p. 31). This means that in the case of a plane stressed state
(σ z = 0), the following equation is true:
∂ 2
∂x 2 +
∂ 2
∂y 2
(σ x + σ y ) = 0.
(9.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_9
87
Plane Problem of Elasticity Theory
9.1 Functions of Stresses
Previously (p. 16), it was established that the dependency between relative elongations and normal stresses in the case of plane strain (ε z = 0) could be obtained
from the respective dependencies in the case of a plane stressed state (σ z = 0)
by substituting the Young modulus E and Poisson coefficient ν with E ∗ and ν ∗ ,
respectively, using formulas (1.21). Therefore, both these stress–strain states are
combined by the concept of “plane problem” not always mentioning what exact
state is considered.
Let us consider plane strain (ε z = 0). In this case, equilibrium equations:
∂σ x
∂x
+
∂τ xy
∂y
= 0,
∂σ y
∂y
+
∂ τ yx
∂x
= 0
(9.1)
can be met if we consider that
σ x =
∂ 2 U
∂y 2 , σ y =
∂ 2 U
∂x 2 , τ xy =
∂ 2 U
∂x∂y
,
(9.2)
where U is an arbitrary function differentiated for the sufficient number of times,
which is called (Airy, [5]) the stress function.
Let us also use the fact that volumetric expansion in an elastic body (1.13) is a
harmonic function (see p. 31). This means that in the case of a plane stressed state
(σ z = 0), the following equation is true:
∂ 2
∂x 2 +
∂ 2
∂y 2
(σ x + σ y ) = 0.
(9.3)
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021
V. Molotnikov, A. Molotnikova, Theory of Elasticity and Plasticity,
https://doi.org/10.1007/978-3-030-66622-4_9
87