Chapter 1 Introduction
3.4 Finite-Element Simulations
Finite-element simulations were performed to complement every experiment listed in Table 3.4. In addition, three more numerical runs were performed at CJ speeds that were of special interest. They are designated the S1, S2, and S3 runs, and were performed to reveal which CJ speed caused the greatest transverse shear stresses. S2 was computed with the CJ speed equal to the second critical speed ofTang(1965), S3 was computed with the CJ speed equal to the shear speed of aluminum, and S1 was
computed after a trial process to approximately locate the CJ speed for maximum shear amplification.
Figure 3.3: Comparison of boundary conditions.
3.4.1 Tube Model
The specimen tube was modeled with axisymmetric solid elements using the com- mercial explicit finite-element code LS-Dyna V.960. Full integration was used. To decrease computational costs, only 35.56 cm of the tube was modeled. The left end of the tube was simply supported radially but not axially (Fig. 3.3a). Twenty elements were used in the radial direction and 5039 elements in the axial direction. The whole model had 100780 elements. The choice of the large number of elements was made after a trial and error process at resolving the high-frequency shear stresses.
The location for recording stresses was at 12.7 cm from the left end and in the mid- dle of the wall thickness. The location for recording radial displacements was at 12.7
cm from the left end and on the outside diameter of the wall. The choice of 12.7 cm avoided contamination by reflected dilatational waves and eliminated the influence of boundary conditions from the right end throughout the duration of the simula- tion. A comparison between the simulated and experimental boundary conditions is illustrated in Figs. 3.3a and3.3c, respectively.
3.4.2 Pressure Distribution
The traveling pressure load model followed the one used in the study by Beltman and Shepherd (2002). The load was prescribed as a function of time at each node.
The force history was a discrete version of the exponential approximation to the Taylor-Zeldovich model:
P(x, t) =
P1 0< t < tcj,
(P2−P3) exp(−(t−tcj)/T) +P3 tcj < t <∞,
(3.1)
wheretcj is the arrival time of the detonation front,P1 is the initial pressure,P2 is the CJ pressure, T is the decay time, and P3 is the pressure at the end of the expansion wave. Similar to Beltman and Shepherd (2002), P3 and T were approximated as constants by curve-fitting the pressure trace from the second transducer. These values can be found in Table3.4. The exponential decay from the CJ point was approximated by 5039 linear segments over 0.11 ms, which was the duration of the simulations. Both the tube mesh and the traveling load were constructed using the pre-processor LS- Ingrid. A comparison between a typical experimental pressure trace and a curve-fit for the FEM is shown in Fig. 3.4.
3.4.3 Mesh Size and Temporal Resolution Considerations
There is currently no model in the literature that can estimate the frequency content of transverse shear stresses for the current experiments. The numerical study first began with coarse meshes and low sampling rates. They were refined until the shear stresses were resolved reasonably well.
Time (ms)
Pressure(MPa)
3.75 4 4.25 4.5
0 0.5 1 1.5 2
Measured
Exponential Decay Approximation
Figure 3.4: Comparison of measured (pressure transducer 2) and approximated pres- sure profiles for Shot 41. The apparent oscillations in the measured pressure trace were an artifact caused by the structural vibrations of the pressure transducer mounts.
3.4.3.1 Flexural Wavelength Considerations
To begin with, the simulations must be good enough to at least simulate the flexural (breathing mode) waves (in the form of hoop strains) so that they can be compared with the experimental strain signals. An upper bound of the mesh size could be estimated using flexural wavelength. Flexural waves for these experiments ranged from about 37 to 40 kHz, with wavelengths from 55 to 97 mm. Since this was much larger than the wall thickness of 1.5 mm, as long as the mesh size was smaller than the wall thickness, flexural waves could be more than adequately resolved. Note that since the flexural wavelengths were also much larger than the gage length of strain gages (0.81 mm), the experimental hoop strains were also adequately resolved.
3.4.3.2 Detonation Cell Size Considerations
Another way to think about the mesh size is to consider the detonation wave cell size.
The detonation wave was modeled in this numerical study as a traveling load. This was a numerically planar front. However, real detonation wave fronts have three-
dimensional transverse disturbances with wavelengths known as the detonation cell size (see Fig.1.4). For the current mixtures, the smallest cell size was 1.3 mm. This corresponded to a stoichiometric H2+O2 mixture with no dilution. Although these complicated three-dimensional features were not modeled in the loading, the mesh size must be chosen so that in terms of the detonation cell size, the structure was better resolved than the loading.
3.4.3.3 Chosen Resolution
It was found after several trials that considerations of flexural wavelengths and detona- tion cell sizes were insufficient in resolving transverse shear because higher-frequency components were being discovered as the mesh size went down and sampling rate went up. Down to a mesh size of 71 micron (axial) by 74 micron (radial) and up to a sampling rate of 91 megasample per second, the shear stress traces seemed to have converged, although a rigorous convergence study had not been attempted. At this spatial resolution, the mesh was capable of resolving transverse shear waves with frequency components up to 3.3 MHz traveling at a speed of 3576 m/s (highest in this study) with 15 elements per wavelength. The time step, automatically imposed by LS-Dyna V.960, was 10 ns.
3.4.4 Shear Stress Amplification Factor Scaling
One way to study shear resonance is to use a dynamic amplification factor similar to that used in flexural wave resonance studies. For this purpose, the dynamic shear amplification factor, Φs, is defined as:
Φs = τdynamic
τstatic , (3.2)
whereτdynamicis the maximum dynamic transverse shear stress (acting perpendicular to thez-axis in Fig. 3.3) andτstaticis the static equivalent. τstatic had to be computed using linearly elastic shell theory. An approximate analytical static model is shown in Fig. 3.3b. It is a semi-infinite tube simply supported at the right end and internally
pressurized fromz = 0 toz = 12.7 cm, where the data for FE analysis were recorded.
It is well-known that the form of the equations for this model are the same as those for a semi-infinite beam on an elastic foundation. From the handbook by Young (1989, p.149, case 2), the transverse shear Vs is
Vs = −P
2λ[(sinλB−cosλB) exp(−λB)−(sinλA−cosλA) exp(−λA)]
×cosh (λ(B −z)) cos (λ(B −z)) +P
2λ[(sinλB+ cosλB) exp(−λB)−(sinλA+ cosλA) exp(−λA)]
×sinh (λ(B −z)) sin (λ(B−z)) +P
2λ[coshhB−A−zisinhB−A−zi + sinhhB−A−zicoshB−A−zi
−coshhB −A−zisinhB−A−zi
−sinhhB −A−zicoshB−A−zi], (3.3)
where
hB−A−zi=
0, z > B−A B−A−z, z < B−A
(3.4)
and
λ =
3(1−ν2) R2h2
1/4
, A= 0.127 m, B = 1.524 m. (3.5) The average shear stress τstatic is then
τ = Vs
h. (3.6)
The shear stressτ near the loading discontinuity is plotted in Fig. 3.5for an example case where the internal pressure P was equal to aPcj of 2.0 MPa. Theτstaticused for normalization was taken to be the maximumτ, which was located at the loading dis-
continuity. This value (recalculated for each Pcj) was used to normalize the dynamic shear stresses in Fig. 3.9 (to be discussed in Section 3.5.2).
Axial Location (cm)
Stress(MPa)
11 12 13 14 15
-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
Figure 3.5: Static shear stress for the analytical model (pressure load discontinuity is at 12.7 cm).