2.4 Effects of dynamic loading
2.4.1 Stress wave propagation and its effects on energy absorption
Elastic stress wave
Upon the application of dynamic loading, the suddenly gained stress or the suddenly gained particle velocity at the points on the loaded surface will be propagated away from the surface in the form of stress waves.
If the normal stress applied is smaller than the yield stress of the mater- ial,Y, then the stress wave is elastic and propagates at a speed of
[2.53]
where cLdenotes the speed of the longitudinal elastic stress wave,Eand rare the Young’s modulus and the density of the material, respectively.
For longitudinal elastic stress wave, there exists a relation between the stress sand particle velocity v
[2.54]
where the quantity s/vis termed stress wave impedance. Therefore, a yield velocity can be defined as a material’s property as follows
[2.55]
Only when the particle velocity gained from dynamic loading (e.g. from impact) is below this yield velocity will the stress wave be purely elastic.
For mild steels,cLª5100 m/s and typically if E/Y=500, then the yield veloc- ity vyis about 10 m/s.
Plastic stress wave
If the stress created by the dynamic loading is beyond the yield stress of the material, or the particle velocity gained from impact is beyond the yield velocity, then in addition to the elastic stress wave propagating with veloc- ity cL, plastic stress waves will be initiated and propagate away from the loaded region. Readers are advised to consult Johnson (1972) or Johnson
v Y E
Y c
Y Ec
y
L
= = = L
r r s = E vr =rc v= E
c v
L L
c E
L = r
and Yu (1989) to acquire more information on plastic wave propagation.
Only the most fundamental results are summarised herein to facilitate the discussion on energy-absorbing structures in the following chapters.
(1) If the material displays linear strain-hardening as shown in Fig. 2.2(b) or Fig. 2.3(b), then the longitudinal plastic wave will propagate with a speed of
[2.56]
where Epdenotes the strain-hardening modulus of the material. For structural metals, typically Epis usually 2 or 3 orders smaller than E, so that propagation of the plastic wave is at least an order slower than that of the elastic precursor.
(2) If the material displays non-linear strain-hardening and obeys a stress–strain relation s=s(e) in its plastic range, then the longitudi- nal plastic wave will propagate with a speed of
[2.57]
where (ds/de) represents the tangential modulus of the material, which in general varies with stress/strain which is brought about by the stress wave.
(3) For decreasingly strain-hardening material, whose tangential modulus (ds/de) reduces with increasing strain as shown in Fig. 2.14(a), the lon- gitudinal plastic wave which brings about a higher stress (or a higher strain) will propagate with a lower speed. Consequently, the plastic wave is scattered, as sketched in Fig. 2.14(b).
cp= ds de r
c E
p
= p
r
(a) 1 s
e
(b)
< <
s
t
t1 t2 t3
de ds
2.14 Stress waves in decreasingly strain-hardening materials: (a) the tangential modulus reduces with increasing strain; (b) change in wave shape during propagation.
(4) For increasingly strain-hardening material, whose tangential modulus (ds/de) increases with increasing strain as shown in Fig. 2.15(a), the longitudinal plastic wave which brings about a higher stress (or a higher strain) will propagate with a higher speed. Consequently, the plastic wave is convergent and will eventually form a shock wave, as sketched in Fig. 2.15(b), which is characterised by strong discontinu- ity in stress and particle velocity at the shock front.
If the applied dynamic load brings about a stress marked by point C in Fig. 2.15(a), then the shock wave will propagate at a speed deter- mined by the slope of the straight line (shown by the broken line in Fig. 2.15(a)) between the initial yield point A and point C, and can be expressed as
[2.58]
where [s] and [e] denote the jump (discontinuity) in stress and strain, respectively. Examples of applying shock wave theory in structures will be found in Sections 4.6 and 10.4.
Effects of stress waves on deformation mechanisms and energy absorption
Elastic and plastic waves may affect the energy absorption of materials and structures in various complex ways, depending on the dynamic loads, the structure’s configuration and the material’s properties. Herein we illustrate only a few of those effects that are frequently encountered in the analyses of energy-absorbing structures.
cshock =
[ ] [ ]
s er
(a) A
C s
[e ]
[s ]
e
(b)
< <
s
t
t1 t2 t3
2.15 Stress waves in increasingly strain-hardening materials: (a) the stress–strain curve; (b) change in wave shape during propagation and formation of a shock wave.
(1) At the dynamically loaded region (e.g. the region where a structure is impinged by a rigid projectile), the high stress brought about by the strong plastic compression waves may cause a local plastic collapse – for example (see Section 10.4 for details), a layer of honeycomb cells may be plastically collapsed at the loading end. After some energy has been dissipated in this localised zone, the rest of honeycomb may experience only elastic deformation.
(2) When a compressive elastic wave produced by a dynamic load or impact reaches the distal surface (e.g. the distal end of a long bar), it will reflect from that surface. If the surface is free (i.e. with no con- straint to its displacement), the reflected wave will be a tensile one which propagates back from the distal surface. For brittle materials such as concrete and rocks, whose tensile strength is low, this reflected tensile wave may cause fracture of material some distance away from the free surface; consequently a portion of material will be separated and fly away. This kind of failure is termed spallingand the flying layer will bring a notable portion of the input energy away by its kinetic energy.
(3) When a compressive elastic wave produced by a dynamic load or impact reaches the distal surface where no displacement is allowed (e.g. a clamped distal end of a long bar), the reflected wave will also be compressive and it will result in the magnitude of the compressive stress being doubled. The increase in the stress magnitude may create a plastic compressive wave, so plastic deformation and energy dissi- pation would first occur in the region close to the distal fixed surface rather than in the loading region. In the case of cellular materials (e.g.
honeycombs), cells may first be plastically collapsed at this clamped distal end.
(4) All the above cases usually occur when the dynamic loading is along the longitudinal direction of the structural components, such as a long bar subjected to a compressive force pulse or impact along its axial direction. If a slender structural component (e.g. a beam or a thin plate) is subjected to a dynamic loading in its transverse direction, then the stress waves along that direction will disappear in a brief time period as a result of the frequent wave reflections between two close surfaces of the beam (or the plate) along its thickness direction.
However, elastic flexural waves (i.e. bending waves) will propagate away from the loading region.
As is well known, the elastic flexural wave has a frequency-dependent speed when it propagates along a beam, so it is dispersive although there is no plastic dissipation. A few numerical studies on the dynamic response of a cantilever beam to a tip impact (Symonds and Fleming,
1984; Reid and Gui, 1987; Hou et al., 1995) have found that the dynamic behaviour of an elastic-plastic cantilever under impact is dis- tinctly different from that predicted by a rigid-plastic analysis (Parkes, 1955).A thorough investigation made by Yu et al. (1997) has confirmed that these differences can be attributed to the interaction between reflected elastic flexural waves from the distal clamped end and the travelling plastic ‘hinge’. Thus, it is evident that, although the elastic flexural wave does not directly dissipate energy, it does affect the energy dissipation in the cantilever beam by altering the deformation pattern and influencing the evolution of plastic regions.