Chapter 5
Elastic Half-Space
5.1 Volumetric Expansion on Surface
Let the loading application area be small in dimensions as compared to the body
dimensions, and we are interested in a stressed state in the vicinity of applied forces.
In this case, we can represent that the body is limited by only the plane tangential
to the body surface in the middle point of the load application area. This semiindefinite body whose material conforms to Hooke’s law is referred to as the [1]
elastic half-space.
Let us assume that external loads represent normal pressure p(x, y). Let us use
formulas (3.17) to find movements. By changing designations, let us write these
formulas as follows
u = ϕ + z
∂P
∂x
,
v = ψ + z
∂P
∂y
,
w = f +
∂P
∂z
,
(5.1)
where the harmonic functions ϕ, ψ, f and P are connected by the following
condition due to formula (3.17)
∂P
∂z
= −
1
3 − 4ν
∂ϕ
∂x
+
∂ψ
∂y
+
∂f
∂z
.
(5.2)
Four harmonic functions ϕ, ψ, f and P are found from the condition that they
disappear at the infinity boundary together with their derivatives, and there are
known stresses at the half-space boundary z = 0
© 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_5
45
Elastic Half-Space
5.1 Volumetric Expansion on Surface
Let the loading application area be small in dimensions as compared to the body
dimensions, and we are interested in a stressed state in the vicinity of applied forces.
In this case, we can represent that the body is limited by only the plane tangential
to the body surface in the middle point of the load application area. This semiindefinite body whose material conforms to Hooke’s law is referred to as the [1]
elastic half-space.
Let us assume that external loads represent normal pressure p(x, y). Let us use
formulas (3.17) to find movements. By changing designations, let us write these
formulas as follows
u = ϕ + z
∂P
∂x
,
v = ψ + z
∂P
∂y
,
w = f +
∂P
∂z
,
(5.1)
where the harmonic functions ϕ, ψ, f and P are connected by the following
condition due to formula (3.17)
∂P
∂z
= −
1
3 − 4ν
∂ϕ
∂x
+
∂ψ
∂y
+
∂f
∂z
.
(5.2)
Four harmonic functions ϕ, ψ, f and P are found from the condition that they
disappear at the infinity boundary together with their derivatives, and there are
known stresses at the half-space boundary z = 0
© 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_5
45