Chapter 2
The First Basic Problem of Elasticity
Theory
2.1 Equilibrium Equations
Assume that an equilibrated, arbitrarily distributed system of forces acts on the solid
body surface. Strains and stresses caused by this system of forces are continuous
unrestrictedly differentiated functions of coordinates of all internal points of the
body. On the surface confining the body, strains and stresses can be diverse. When
approaching boundary surfaces from inside the body, strains and stresses must
satisfy some boundary conditions that will be considered separately.
Refer to Fig. 2.1. It shows an infinitely small parallelepiped with ribs dx, dy, dz
separated from the vicinity of an arbitrary point inside a body. Internal forces act on
the face of the separated element. The figure shows only components of these forces
in the direction of the coordinate axis Ox. By zeroing their geometric sum, we
obtain
∂σ x
∂x
+
∂τ xy
∂y
+
∂τ xz
∂z
dxdydz = 0.
By reducing by dxdydz and making a circular permutation of indexes of
coordinate axes, we obtain
∂σ x
∂x
+
∂τ xy
∂y
+
∂τ xz
∂z
= 0,
∂σ y
∂y
+
∂τ yz
∂x
+
∂τ yz
∂z
= 0,
∂σ z
∂z
+
∂τ zx
∂x
+
∂τ zy
∂y
= 0.
(2.1)
© 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_2
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