Chapter 2. Literature review
2.5 Previous research on impulse loading on honeycomb protected structures
2.5.1 Impact dynamics and contact law
When two granular particles are in contact with each other under no external loading, the initial contact established between their surfaces is the point of contact (or likely a line contact). Hertz (1881) published the first article on contact mechanics between solid elastic bodies that describes the local elastic deformation relation between two spherical particles. Hertz's law is widely used in diversified fields of contact mechanics, such as for bearings, metal forming, wheel-rail contact, gasket seals, etc.
Hertz (1881) has studied two identical spherical particles of radius ‘R’ assuming that both particles are topographically smooth on macro and micro scales, such that the elastic wave's motion in the particle can be ignored and the tangential surface interaction between the particles is frictionless.
A schematic view of two particles in contact under compressive force ‘F’ is shown in Figure 2.18 a. The typical force-displacement curve during interaction of granular chain is shown in Figure 2.18 b. When these particles are compressed against each other towards the contact point ‘O’, the relative approach between the distant points of two particles is given as 1 2, where 1, 2 represents the displacement of distant point for particles 1 and 2, respectively. Here,
0for compression and
0 for tension and the particles deform locally at the point of contact ‘O’ and form a circular contact area. Since the material has a characteristic property to regain its original shape, a resisting force is developed between the two particles. It is assumed that the radii of curvatures of the contacting particles are large as compared to radius of the contact circle, which results in less displacement. The problem is also considered as elasto-static, i.e. the material interaction is linear elastic, and the two particles are in static equilibrium such that the displacement is independent of time.Based on these assumptions, the generalized interaction force derived by Hertz for two particles is given as (Johnson and Johnson 1987).
3/ 2 ; 0 0 ; 0 F k
(2.8)
Figure 2.18 Schematic diagram: a) two spherical bodies in contact, and b) force-displacement plot with Hertzian law
2 /
3A F
No tension, compression only
F
d
a) b)
The compressive force (F) between the two granular particles is considered zero in tension
0
and positive in compression
0
. During the compression process, the force is initially weak as compared to linear or harmonic spring force for small values of ‘’, and it rises sharply when the ‘ ’ value is increased as compared to the linear spring force. Therefore, the interaction force in the Hertz model is proportional to the3/2, which is a nonlinear contact interaction behavior between two spherical particles that results from the consideration of geometrical effects only. Further, its proportionality relation depends on the two particles contact geometry. Spence (1975) developed a detailed generalization of the Hertz contact law that describes the interaction of two particles of arbitrary axisymmetric contact surface. They calculated the stress distribution and the contact area of overlap between the two particles to determine the interaction force between the particles.
i) For spherical bodies
The Hertzian contact stiffness between two spherical interacting bodies is given as
2
2 3(1 ) k E r
(2.9)
where, = poisons ratio of spherical striker and ris the equivalent radius.
1 2
1 1 1
r r r (2.10)
ii) For cylindrical bodies in contact
The contact area generated by compressing two cylindrical surfaces against each other depends on the orientation angle
between the axes of the two cylinders. For small deformations, this could result in a circular contact area
90o
.Let the total displacement of two cylindrical particles under compression is denoted by ‘ ’. The relation between the contact force ‘F’and the displacement ‘ ’ is given by Khatri et al. (2012)
*
3/ 2 3/ 2
3/ 2 2
4 R 3
e cyl
F k E
F
(2.11)
where,
‘Re’ is an equivalent radius
R R
e sin (2.12)
‘E*’ is an effective elastic modulus
2
*
2(1 )
E E
(2.13)
iii) Spherical striker impacting rigid surface
The interaction behavior of spherical impactor on the rigid plate is similar to the interaction of two spherical impactors with an exception that for one surface, the radius of curvature is infinite.
The effective stiffness
ko between the spherical striker and rigid surface is given by Szuladzinski (2009)2
2
1
1 3 2
o 2
k E r
(2.14)
The maximum deflection (m) of the spherical striker is a function of mass and velocity of striker and contact stiffness.
2 0.4
0
5 2
o m
Mv
k
(2.15)
The contact time
to of the impact interaction is given as0.4 0.2
3.214 (2 ) 2
o
o o
t M
v k
(2.16)
The contact radius
a during interaction of striker with the rigid surface is given as2 0.2 2
5 2
o o
a Mv r
k
(2.17)
a = contact radius in mm
M =mass of the spherical striker in gm
E =modulus of elasticity of spherical striker in MPa
= poisons ration of spherical striker v0=velocity of spherical striker m/s r2= radius of spherical striker mm
iv) Spherical striker impacting an elastic surface
Figure 2.19 shows the spherical striker impacting on an elastic surface considered the plane surface as semi-infinite mass. The thickness of plate doesn’t affect the interaction between the spherical impactor and plate. When the thick plate was used,
and its thickness reduces below a certain range, it significantly affects the interaction of the impactor and the plate.
Figure 2.19 a) Spherical ball impacting on elastic surface, and b) Deformed contour of honeycomb panel subjected to spherical ball impact
The contact stiffness
ko , between the spherical striker and elastic surface is given as2 2
1 2 2
1 3 1 1 1
o 4
k E E r
(2.18)
The contact time between interaction of striker and elastic surface is given as
0.4
0 0.2
3.214
o o
t M
v k
(2.19)
Peak surface contact stress
cm0.2
0.8 0.4 2
0.499
cm o o
M k v
r (2.20)
Peak surface tension
tm1 2
tm 3 cm
v (2.21)
where,
1,
2 are the poisons ratio of spherical bodies and elastic surface.E1, E2 are the modulus of elasticity in MPa.
a) b)
The above fundamental concepts are employed to understand the interaction of impacting body on elastic substrate. Based, on the impact dynamics, literature review of theoretical aspect of low-velocity and high-velocity impact on composite material is conducted.