Elastic-plastic impact model

Prediction of the restitution coefficient has been a challenging research topic for decades. Unfortunately, no reliable and accurate prediction method has been found so far. However, some useful simplified models with certain limits have been developed. One of them is the elastic-plastic impact model in which the compression process is assumed to be plastic with part of the kinetic energy stored for later elastic rebounding, with the rebound process considered to be completely elastic [Johnson, 1985]. In this model, it is postulated that (1) during the plastic compression process, a — r3/2a (2) during the compression process, the averaged contact pressure pm is constant and is equal to 3 Y and (3) the elastic rebound process starts when maximum deformation is reached. Therefore, the compressional force is... 

The elastic-plastic model reveals that the restitution coefficient depends not only on the material properties but also on the relative impact velocity. Equation (2.166) also indicates that the restitution coefficient decreases with increasing impact velocity by an exponent of 1/4, which is supported by experimental findings, as shown in Fig. 2.18. For high relative impact velocities, the model prediction is reasonably good. However, for low relative impact velocities, the prediction may be poor because the deformation may not be in a fully plastic range as presumed. 

Example 2.4 A copper ball of 1 cm diameter normally collides with a stainless steel wall with an impact velocity of 0.5 m/s. Estimate the restitution coefficient using the elastic-plastic model. What is the rebound velocity of the ball The yield strength of copper is 2.5 x 108 N/m2. It can be assumed that the yield strength of the stainles steel is higher than that of copper. 

In a study of a vehicle rollover, simulations were performed to duplicate experimental rollover test of a vehicle at 30 mph on a FMVSS 208 rollover cart impacting a water-filled decelerator system, thereby throwing the vehicle off the cart. The initial roll angle was 23 degrees. The vehicle, at the time of departure, had a velocity of approximately 25 mph in the translational direction and 1.5 rad/s in the roll direction as shown in Figure 9. A number of different analyses with rigid and flexible bodies have been performed on this vehicle [20, 21]. Two additional simulations, related to the topic of this paper, were also performed and the results were compared to those from the experiment. It must be kept in mind that for these two simulations, the elasticity/plasticity of the roll bars have been neglected, however, friction is modeled in both the piecewise and continuous analyses. 

Plastics are very often used in temperature ranges lower than their Tgs, so that then-constitutive relations used for analyses should be linear elastic. However, strictly speaking, then-moduli have a weak dependency to loading rates. If the dependency is a problem for impact analyses, the variation of moduli should be compensated. For this purpose, viscoelastic models are sometimes used to express the loading rate dependency (Sato and Ikegami 2000). 

The enhanced plasticity and toughness of modem adhesives have become an important property in applications where dissipation of energy in the case of impact, the ability to compensate thermal movement, or the reduction of vibration lead to added value and improved service performance. Viscoelastic behavior can theoretically be predicted and analyzed by means of mechanical models including combinations of elastic and viscous elements to simulate the time-dependent viscoelastic stress-strain response to mechanical loads. 

The eontact properties of the two polymers were disclosed in a previous paper, determined by impact experiments [9]. In that work, it was found that PS displayed nearly pure elastic behavior throughout a collision, whereas HDPE exhibited a complex fransition from elastic to viscoelastic to purely plastic dissipative losses under the same conditions. Due to its non-ideahty, we anticipate describing HDPE accurately will be a much greater challenge since the viscoelastic losses are ovo-looked by the Thornton model. Unfortunately, no CFDM exists to date that properly fits the contact behaviour of HDPE. 

---