Temperature rise and stress induced by microcracks in accelerating structures

The temperature rise and induced stress due to Ohmic heating in the vicinity of microcracks on the walls of high-gradient accelerating structures are considered. The temperature rise and induced stress depend on the orientation of the crack with respect to the rf magnetic field, the shape of the crack, and the power and duration of the rf pulse. Under certain conditions the presence of cracks can double the temperature rise over that of a smooth surface. Stress at the bottom of the cracks can be several times larger than that of the case when there are no cracks. We study these effects both analytically and by computer simulation. It is shown that the stress in cracks is maximal when the crack depth is on the order of the thermal penetration depth.

Figure 1

Transgranular view of cracks in the material of the accelerating structures (reproduced from Ref. 6).

Figure 2

CuAg5 pulse heating sample, $T=110°\mathrm{C}$ (courtesy of Lisa Laurent, SLAC).

Figure 3

Elevated temperature profiles at five time intervals, for a smooth, uncracked copper surface and heat flux $q=2.7\times {10}^9\mathrm{W}/{\mathrm{m}}^2$.

Figure 4

Part (a) is the diagram of a crack of depth $d$ on an infinite metallic plane, with zero flux on the plane surface ($y=0$), and a flux $q$ injected into both side walls of the crack ($|y|<d$). Part (b) is the diagram of a heat source of width $2d$ placed symmetrically with respect to $x$ axis on the surface of a half plane $x>0$, with zero flux at $x=0$ for $|y|>d$ and flux $q$ injected at $x=0$, $|y|<d$.

Figure 5

Temperature rise vs time from calculation and simulation results: the solid curve with closed symbols represents the temperature rise at the infinite surface of a plane, the dashed curve twice that value, and the solid curve with open symbols represents the temperature rise at the crack upper corners. Cross marks along both solid curves are simulation results from ANSYS [16].

Figure 6

Part (a) is the temperature contour plot for the trapezoid-shaped crack at $2.1\mu \mathrm{s}$. Part (b) is the von Mises stress contour plot for the trapezoid-shaped crack, the square-bottom crack, and the round-bottom crack at $2.1\mu \mathrm{s}$.

Figure 7

Average and maximum stress at the bottom of a round-bottomed crack (with and without sidewall heating) versus crack depth, compared with plane surface stress (straight line at lower part of the plot). Pulse duration $2.1\mu \mathrm{s}$, heat flux $q=2.7\times {10}^9\mathrm{W}/{\mathrm{m}}^2$ is applied. Both solid curves represent the case with sidewall heating along the crack with the closed symbol for the maximum stress at the crack bottom and the open symbol for the average stress over the semicircled crack bottom. The dashed curves denote no sidewall heating cases, with the closed symbol for the maximum stress at the crack bottom, and the open symbol for the average over the crack bottom.

Figure 8

Stresses at the bottom of the round-bottomed crack with and without sidewall heating, compared with plane surface stress with pulse duration $0.21\mu \mathrm{s}$, heat flux $q=2.7\times {10}^{10}\mathrm{W}/{\mathrm{m}}^2$. Similar to Fig. 7, solid curves represent the case with sidewall heating with the closed symbol for the maximum stress at the crack bottom and the open symbol for the average stress over the crack bottom. The two dashed curves denote the no sidewall heating cases.