Numerical Model of Bond Strength Measurements
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NUMERICAL MODEL OF BOND STRENGTH MEASUREMENTS Gerald L. Nutt, Chemistry and Materials Science Department, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA. 94550
ABSTRACT We have reported bond strength measurements of metal/ceramic interfaces using shock waves to separate the bond by spallation[1,2]. The technique relies on the interpretation of the free surface velocity of the metal film as it is spalled from the substrate. We answer several questions relating to the details of the interaction of the shock with the interface. Specifically, we examine the role of sound speeds in the measurements. We also calculate the plastic strain in the the bond region and verify the theory relating to the jumpoff velocity of the scab to the bond strength. INTRODUCTION The bond strength measurements are a simple extension of a well known method of spall measurement used for homogeneous materials. A plain stress wave is generated in the substrate, propagating in a direction normal to the plane of the interface, as shown schematically in Fig. 1. The compressive stress wave is reflected at the free surface of the metal overlayer as a tensile wave incident on the interface. We model the experiment numerically using the two-dimensional contiuum mechanics computer code DYNA2D[3]. Thin metal foil Ni
Sapphire
From laser To VISAR
Compressive wave
Interface
launched into sapphire
Figure 1. Schematic of exfoliation experiment. The reflected tensile stress separates the bond, and the newly created free surface at the bond plane communicates the event to the outer free surface. Measurements of the surface velocity history, made with a laser interferometer, are then used to determine the maximum stress at bond separation. Consider the z,t-diagram shown in Fig. 2. The wave reflected from the surface is a centered rarefaction, propagating along C_ characteristics. The characterisic equations relate the sound speed, flow velocity and axial stress in the region between the free surface and the bond plane: dz -- uc;dtr-pcdu=O (1) dt along C+, and
Mat. Res. Soc. Symp. Proc. Vol. 159. 01990 Materials Research Society
466
dz T=u--c ; do,+pcdu=O
(2)
along C_. u and c are the flow velocity and sound speed respectively. a, is the axial stress, and p, the density. Path of Ni free surface created when
--c
Figure 2. x,t-plot of boundaries and characteristic trajectories in specimen.
One generally assumes the sound speed and density are constant along C+ and C_. Integrating along the C_ characteristic from the free surface boundary to the spall event, and then along the C+ characteristic back to the free surface with constant density and sound speed we get, -o
poc(u,
-
Uo)
0,= poc(u, - uk),
(4)
(5)
where the subscript, s, indicates the quantity evaluated on the bond plane at the instant of spall. p0 is the normal density of the material. Solving for o-
0,,=
IPoc(Uo - uk).
(6)
Identifying u0 with the free surface velocity at the instant of reflection of the incident shock wave it is just the "jump off" velocity c
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