In this assignment, the Kalthoff problem was examined by apply a finite element model based on the Johnson-Cook relation. The material modeled was 4340 steel. Table 1 lists the material properties and material constants for the Johnson-Cook model for 4340 steel.
 

Material	A (MPa)	B (MPa)	C	n	m	density (kg/m3)	m (degrees C)	K (GPa)	µ (GPa)	c (J/kg C)
4340 Steel	792.19	509.51	0.014	0.26	0.55	7,840	1,520	157	76	477

Table 1 Material properties and material constants for the Johnson-Cook model. Source: Batra, R.C., et. al., "Critical Strain Ranking of 12 Materials in Deformations Involving Adiabatic Shear Bands", Transactions of the ASME, Vol. 62, p. 252-255, March 1995.
 

    Once the finite element model mesh created with the proper constants, the mesh was submitted to a program called Dyna2D for evaluation. With the output from Dyna2D, a visualization package called Orion was used to interpret these results. Orion created a plot of the time history of the maximum principal stresses and a plot of the time history of the Von-Moses stress. The smallest value of time in the time history plot of the maximum principal stresses, for any of the elements examined, where the maximum prinicpal stress is equal to 1.5 A is called t_brittle. An example of how this "measurement" is given in Figure 1.

Figure 1 A time history plot of the maximum principle stresses.  The horizontal grey line is from 1.5 A ( 1.19 GPa)
and the grey vertical line intercepts the time axis at  t_brittle.

    The smallest value of time in the time history plot of the Von Mises stress, for any of the elements examined, where the Von Mises stress has dropped to 90% of its maximum value is called t_ductile. All of the times and the specific mesh elements which failed are listed in Table 2.
 

	Brittle Behavior		Ductile Behavior
Vo (m/s)	tbrittle (s)	Number of Failed Mesh Element	tductile (s)	Number of Failed Mesh Element
10	11	43,49,55	23	112
20	10	55	16	112
30	10	43,49,55,78	14	112
40	10	43,49,55,78	14	112, 162
50	10	43,49,55,78	13	180,174,168,162
60	11	43,49,55,78	13	109

Table 2 A table of results from the Johnson-Cook finite element model for the Kalthoff problem.
 

    The values for t_ductile and t_brittle are plotted versus the impact velocity, V_o, in Figure 2. Please note that the two curves do not intersect. Therefore, no critical time, t_critical, can be found.

Figure 2 A graph of the time to failure versus impact velocity.

    From the number of simulations run, no t_critical could be found. But if the graph is examined, the slope of t_brittle seems to be increasing and the slope of t_ductile seems to be leveling off or decreasing. Therefore, I would suggest to run several more simulations with the impact velocity greater than 60 m/s. Then one should be able to find a t_critical.