Spall strength

Spall strength The dynamic tensile strength of a material associated with tension that results from the wave interaction of rarefaction waves. When the spall strength is exceeded, the material separates, or spalls. ... 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

A proportionality between the theoretical spall strength and the bulk modulus is obtained when a two-parameter model is chosen to represent the intermolecular potential. Other two-parameter representations of the intermolecular potential, such as the Lennard-Jones 6-12 potential, will yield a similar proportionality although the numerical coefficients will differ slightly. 

With a three-parameter model of the intermolecular potential, the theoretical spall strength is not simply a constant times the bulk modulus. Although the slightly greater accuracy obtained is not critical to the present investigation, an energy balance is revealed in the analysis which is not immediately transparent in the Orowan approach. 

To derive an explicit expression for the theoretical spall strength, an ana-... 

Figure 8.1. The cold compression-tension behavior of condensed matter. The volume dependence of energy and pressure or tension are illustrated. The cohesive energy and maximum tension (theoretical spall strength) are properties of the material.

Within the present analysis, the theoretical spall strength is determined from the minimum of (8.3)... 

Table 8.1. Theoretical Spall Strength of Selected Materials.

Note that the theoretical spall strength now depends upon the cohesive energy as well as the bulk modulus. Representative values for selected metals are shown in Table 8.1. These can be compared with experimental spall strengths in later sections. 

Impact of a thin plate on a sample of interest which is, in turn, backed by a lower impedance window material leads to an interaction of waves which will carry an interior planar region into tension. Spall will ensue if tension exceeds the transient strength of the test sample. A velocity or stress history monitored at the interface indicated in Fig. 8.4 may look as indicated in Fig. 8.5. The velocity (stress) pull-back or undershoot carries information concerning the ability of the test material to support transient tensile stress and, with appropriate interpretation, can provide a reasonable measure of the spall strength of the material. 

Generally the material response stress versus particle velocity curves in Fig. 8.6 are nonlinear and either a graphical or more complicated analytic method is needed to extract a spall strength, Oj, from the velocity or stress profile. When behavior is nominally linear in the region of interest a characteristic impedance (Z for the window and for the sample) specify material... 

Figure 8.7. Propagation of wave profile in an elastic-plastic material from the spall plane to the monitoring interface. The wave front propagates at a plastic wave speed whereas the wave release propagates at an elastic wave speed and complicates the analysis of the material spall strength.

Equations (8.11) through (8.14) are reasonable estimates of the spall strength... 

Romanchenko and Stepanov (1981) recognized that the impulse imparted at the spall plane to material downstream, because of the elastic-plastic nature of this material, led to an attenuating tensile stress pulse propagating toward the sample-window interface as is illustrated in Fig. 8.9. Thus, the maximum tension inferred from the measured spall signal should be adjusted for this attenuation in estimating a material spall strength at the spall plane. 

Figure 8.9. The effect of attenuation of the pullback wave signal in an elastic-plastic material. Amplitude of the pullback signal at the recording interface will be diminished due to wave attenuation and will not provide an accurate measure of the material spall strength.

Figure 8.10. Approximation of the pullback signal amplitude and shape used to estimate correction to the spall strength due to elastic-plastic attenuation.

Novikov, S.A., Shear Stress and Spall Strength of Materials Under Shock Loads (Review), J. Appl. Mech. Tech. Phys. 22 (3), 385-394 (1981). 

Some specimens have withstood over 500 cycles. Untreated specimens of these concretes commonly failed the test at approximately 80 cycles. The elastic modulus criterion (less than 60% of original by resonant frequency analysis) was used to determine failure. Specimens often failed the test with no visible cracking or spalling. Strength retention in these specimens was usually high (50% or more). 

In the P-u plane we see that a tension is formed at the center of symmetry. If the tension exceeds the dynamic tensile strength (spall strength) of the material, then the material will fail and part at this plane. This is called spalling. The two rarefactions are dropping the shock pressure before they interact from Pi to zero. The Hugoniots of these two waves are the positive pressure Hugoniots. 

We do not have much data on the spall strength of many materials. Even when we do have data, the local conditions and history of that one particular specimen come heavily into play. Table 19.1, however, lists approximate values that can be expected for the spall behavior of several materials of common interest. The values for spall strength shown in Table 19.1 are considerably higher than the equivalent static tensile strengths of these same materials, ranging from a factor of 2.5 to as high as 10. 

Table 19.1 Spall Strength Estimates for Several Materials...

From Table 19.1 we see that the strongest of the steels has a spall strength of only 3 GPa therefore the slab of stainless steel in this example should spall. 

From these equations it is possible to calculate the reaction on the fibre. Dividing the reaction by the fibre diameter is the pressure which the fibre exerts on the matrix. It was assumed that matrix spalling occurs when this stress exceeds the spalling strength of the matrix which is taken as om = SmEm-The spalling length should be added to the free length of the fibre. 

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