Chapter 6
Herz’s Task
6.1 Deformation of Adjoining Bodies
Let us consider two elastic bodies contacting before deformation in some point O
(Fig. 6.1). Assume this point as the origin of a rectangular system of Cartesian
coordinates Oxyz and direct the axis Oz along the common normal line to the
surfaces of the adjoining bodies in the point O, and assume the direction of the
internal normal line to the surface of body 1 as a positive direction of this axis.
Represent the surfaces of the adjoining bodies 1 and 2 using the functions:
z 1 = F 1 (x, y), z 2 = F 2 (x, y), (F 1 (x, y) F 2 (x, y)) .
(6.1)
Hereinafter, the indexes 1 and 2 mean the first and the second body, respectively. In
what follows, it is also suggested that the functions F 1 and F 2 are sufficiently smooth
in the point O and can be represented by a formal power series in its vicinity.
After compressing, the equations of deformed surfaces will be
˜
z 1 = F 1 (x, y) + w
o
1 , ˜
z 2 = F 2 (x, y) + w
o
2 ,
(6.2)
where w o
1 and w o
2 designate the movement components of body surface points in the
direction of the axis Oz.
Let S mean the contact area of the adjoining bodies after deformation. We have
˜
z 1 = ˜
z 2 for z 1 , z 2 ∈ S; ˜
z 1 > ˜
z 2 for z 1 , z 2 /
∈ S.
(6.3)
Using the representations of the functions z 1 and z 2 under formulas (6.2), the
conditions (6.3) can be written as follows
w
o
1 − w
o
2 = F 2 − F 1 in the region S,
(6.4)
© 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_6
55
Herz’s Task
6.1 Deformation of Adjoining Bodies
Let us consider two elastic bodies contacting before deformation in some point O
(Fig. 6.1). Assume this point as the origin of a rectangular system of Cartesian
coordinates Oxyz and direct the axis Oz along the common normal line to the
surfaces of the adjoining bodies in the point O, and assume the direction of the
internal normal line to the surface of body 1 as a positive direction of this axis.
Represent the surfaces of the adjoining bodies 1 and 2 using the functions:
z 1 = F 1 (x, y), z 2 = F 2 (x, y), (F 1 (x, y) F 2 (x, y)) .
(6.1)
Hereinafter, the indexes 1 and 2 mean the first and the second body, respectively. In
what follows, it is also suggested that the functions F 1 and F 2 are sufficiently smooth
in the point O and can be represented by a formal power series in its vicinity.
After compressing, the equations of deformed surfaces will be
˜
z 1 = F 1 (x, y) + w
o
1 , ˜
z 2 = F 2 (x, y) + w
o
2 ,
(6.2)
where w o
1 and w o
2 designate the movement components of body surface points in the
direction of the axis Oz.
Let S mean the contact area of the adjoining bodies after deformation. We have
˜
z 1 = ˜
z 2 for z 1 , z 2 ∈ S; ˜
z 1 > ˜
z 2 for z 1 , z 2 /
∈ S.
(6.3)
Using the representations of the functions z 1 and z 2 under formulas (6.2), the
conditions (6.3) can be written as follows
w
o
1 − w
o
2 = F 2 − F 1 in the region S,
(6.4)
© 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_6
55