7.1 Local deformation of structures due to impact .1 Kinetics of a direct central collision between
7.1.2 Indentation caused by contact force
The analysis of the collision between two deformable bodies given above indicates that the behaviour of the system in both the compression and the restitution phases of the collision is dominated by the relationship between the contact force and the local deformations (i.e. indentations) of the two bodies in contact (as sketched in Fig. 7.2(a)). Obviously, this relationship depends not only on the elastic-plastic properties of the deformable bodies, but also on the local geometry of the contact surfaces.
Normal contact of elastic bodies: Hertz theory
When two deformable solid bodies are brought into contact they touch ini- tially at a single point or along a line. Under the action of a very light load, they deform in the vicinity of the point of first contact so that they touch over a finite area, which is small compared with the dimension of the two bodies.
D K K p p
m
p e m
e m v
o f
c r c
= - = - o
* = ( - )
* =( - )
* *( )
2 2 2 2 2
2
2
1 2
1 2
D K K K p
o f m
= ∫ - = c
loss *
2
2
K K K p
f c o m
= = - c
*
2
2
In the following, it is assumed that the contact surfaces of the two bodies are smooth and the contact areas will develop axisymmetrically. Therefore, we can take the initial contact point as the origin of a cylindrical coordi- nate (r,q,z) and let the z-axis be along the normal direction of the contact surfaces, so that (r,q) forms the common tangential plane of the two bodies in contact, as shown in Fig. 7.3(a).
From the geometry, the profiles of the surfaces of the two bodies in contact can be approximately expressed by
[7.30]
where Rdenotes the radius of curvature of the surface at the origin and subscripts 1 and 2 pertain to bodies 1 and 2, respectively.
z r
R z r
1 R
2 1
2 2
2 2 2
= =
R1
sz so
R2 P
P
a a
z
r
a r
P
P R1
R1
R2
P a a
0
(a) (b)
(d) (c)
7.3 (a) Normal contact between two elastic spheres; (b) distribution of the normal pressure within the contact region; (c) contact between a sphere and an elastic half space; (d) contact between two cylinders.
In order to solve the distributions of the stress and displacement created by the contact force, the first step is the determination of the size and shape of the contact area as well as the distribution of normal pressure acting on it. In Hertz theory, the following assumptions are made:
• the contacting bodies are isotropic and elastic;
• the contact areas are essentially flat and small in comparison with the radii of curvature of the undeformed bodies in the vicinity of the interface;
• the contacting surfaces are perfectly smooth and frictionless, so only normal pressure needs to be taken into account.
The foregoing set of assumptions enable an elastic analysis to be conducted (refer to e.g. K.L. Johnson, 1985). Without going into the derivations, some of the major results of Hertz theory are summarised in what follows (1) For two spherical surfaces in contact under force P(Fig. 7.3(a)), the
contact pressure is distributed over a small circle of radius agiven by [7.31]
Here the equivalent Young’s modulus E* and the equivalent radius R are defined by
[7.32]
where E,vand R with a subscript are the Young’s modulus, Poisson’s ratio and radius of the spheres, respectively; subscripts 1 and 2 pertain to spheres 1 and 2, respectively.
The maximum contact pressure is found to be
[7.33]
which acts at the centre of the contact circle. The pressure distribution in the contact circle of radius ais given by
[7.34]
which is depicted in Fig. 7.3(b).
The total contact force P causes a relative displacement of the centres of the two elastic spheres,d, owing to the local deformation, and they are related to each other by
sz r so r ( )= -Êa
Ë ˆ
¯ È
ÎÍ ˘
˚˙ 1
2 1 2
so p p
P a
PE
= =Ê R*
ËÁ ˆ
¯˜ 3
2
6
2
2 3 2
1 3
E v
E
v
E R
R R
* ∫ -
+ - Ê
ËÁ ˆ
¯˜ ∫Ê +
Ë
ˆ
¯
- -
1 12 1 1 1
1
2 2 2
1
1 2
1
a PR
=Ê E Ë
ˆ
¯ 3 4
1 3
*
[7.35]
which can be rearranged as
[7.36]
with contact stiffness k* =4E*R1–2/3.
(2) For a spherical surface in contact with a flat surface under force P(Fig.
7.3(c)), we can take R2= •(hence R =R1) as a special case of (1). If we further assume that both bodies possess the same Young’s modulus Eand v=0.3, then E*=0.55Eand
[7.37]
(3) If a rigid punch (or projectile) is in contact with a flat elastic surface under force P, then as a special case of (1) we can take that R2= • (hence R =R1) with E1= •(hence E* =1.10E2∫1.10Eifv=0.3) and obtain
[7.38]
(4) In the case of line contact between two cylinders under load P(per unit length), see Fig. 7.3(d), the semi-contact-width is given by
[7.39]
where E* and Rare as defined in Eq. [7.32]. The maximum contact pressure is
[7.40]
where szmis the mean normal pressure in the contact region.
Normal contact of elastic bodies: Winkler foundation model The difficulties of elastic contact stress theory arise because the displace- ment at any point on the contact surfaces depends on the distribution of
s p ps
o zm p
P a
PE
= = =Ê R*
Ë ˆ
¯
2 4
1 2
a PR
=Ê E*
Ë ˆ
¯ 4
1 2
p
a PR
E
PE R
P
o E R
= Ê
Ë ˆ
¯ = Ê
ËÁ ˆ
¯˜ = Ê
ËÁ ˆ
¯˜
0 880 0 616 0 775
1
3 2
2 1
3 2
2 1 3
. s . d .
a PR
E
PE R
P
o E R
= Ê
Ë ˆ
¯ = Ê
ËÁ ˆ
¯˜ = Ê
ËÁ ˆ
¯˜
1 089 0 388 1 230
1
3 2
2 1
3 2
2 1 3
. s . d .
P= *dk
3 2
d = =Ê *
ËÁ ˆ
¯˜ a
R
P E R
2 2
2 1
9 3
16
pressure throughout the whole contact. This difficulty can be avoided if the solids are modelled by a simple Winkler elastic foundation rather than an elastic half space. As shown in Fig. 7.4, an elastic foundation of depth hand elastic modulus k is supposed to rest on a rigid base and be compressed by an axisymmetric rigid indenter. The profile of the indenter is taken as the sum of the profiles of the two bodies (with radius of curvature R1and R2, respectively) being modelled. By recalling Eqs [7.30] and [7.32], the profile of the indenter is given as
[7.41]
For the axisymmetric case, under compression by force P, the contact will be developed into a circular area of radius a, and it can be proven (see K.L.
Johnson, 1985) that
[7.42]
while, for the two-dimensional contact of a long cylinder on a Winkler foundation
[7.43]
Equations [7.42] and [7.43] provide the relationships between the applied load and the size of the contact region. Comparing them with those obtained in Hertz theory (e.g. Eq. [7.35]), agreement can be reached if we choose k/h =1.70E*/afor the axisymmetric case and k/h =1.18E*/afor the two-dimensional case. If the depth of the foundation,h, is fixed, then we have to make the elastic modulus k inversely proportional to a, which increases with the indentation. In other words, the elastic modulus of the Winkler foundation in this model has to be reduced with the increase in P or in a.
P ka h
a R
a
= Ê R Ë
ˆ
¯ =
2
3 2
2 2
d P ka
h a
R
a
= Ê R Ë ˆ
¯ =
p d
4 2
3 2
z r z z r
R R r
( )= + = Ê + R
Ë ˆ
¯=
1 2
2
1 2
2
2
1 1
2
R P
sz
h 7.4 A rigid sphere is pressed on a Winkler foundation.