SHOCK COMPRESSION OF METALS

V. P. Shubin

1.

The problem of studying the physico-mechanical properties of metals under shock compression is inseparably connected with two phenomena: impact and large plastic deformations.

The question of impact has a fairly long history. Galileo Galilei (1564–1642) attempted to measure by balances the force of impact of a water jet, which later only Mariotte (1620–1684) succeeded in doing. The well-known Dutch physicist Gravesande (1688–1742) also attempted to measure the force of a falling load by means of lever balances specially constructed by him, but without success. Soon, through the works of Huygens (1629–1695), Mariotte, and others, as N. N. Davidenkov [1] points out, the nature of impact—specified and measured not by force of weight, but by change of mass—was clarified, and the dispute concerned only what should be taken as the measure of impact: the product of mass and velocity (according to Descartes), or the product of mass and the square of velocity (according to Leibniz).

Isaac Newton (1642–1727), in his remarkable book [2], indicates that the greatest geometers of that time—C. Wren (1632–1723), Wallis (1616–1703), and C. Huygens—were the first to derive the laws of impact and reflection of bodies, and that these laws were subsequently verified experimentally by Mariotte. Also known is Newton’s own work concerning impact; for example, the concept of the coefficient of restitution belongs to Newton. On the basis of his experiments he determined that the coefficient of restitution for the collision of glass with glass is equal to \(15/16\), and of iron with iron is equal to \(5/9\).

Theoretical investigations into questions of impact of elastic bodies began later. The first theoreticians in this field were Riccati, Thomas, Young, and the famous French mathematician, mechanician, and physicist Poisson (1781–1840). Poisson was the first to approach the solution of the problem of the collision of elastic rods, although his conclusions that two rods of one and the same material and with identical cross sections cannot separate from one another if only their lengths are not equal to each other are erroneous.

In 1835 Navier (1785–1836), and then, at the beginning of the eighties of the last century, Sebert and Hugoniot, Boussinesq [3], and especially Saint-Venant [4], considered certain questions concerning the impact of elastic bodies by falling loads and the collision of elastic rods. In studying the latter problem, they attempted to determine the post-impact velocities and thus shed light on the laws of impact.

However, only Soviet scientists—N. N. Davidenkov, M. A. Bolshanin, V. D. Kuznetsov, F. F. Vitman, N. V. Kunin, and others—developed the questions of shock compression of metals in considerable detail.

Corresponding Member of the Academy of Sciences of the USSR A. A. Ilyushin [5] writes: “The theory of plastic flow and dynamics is intended for calculations of significant plastic deformations of bodies under the possible presence of large pressures and velo-

...and in any case beginning with certain average velocities of the order of

\[ 10^{-2}\ \frac{1}{\mathrm{sec}}. \]

The initial shape of bodies is then substantially changed as a result of plastic deformation.” A. A. Ilyushin also outlined paths for developing this problem, namely:

a) the study and formulation of the laws of plasticity for finite deformations, high pressures, and velocities;

b) the investigation of the dynamic characteristics of metals by wave methods of the theory of plasticity;

c) the solution of the most important technical problems concerning the strength of bodies under the dynamic application of loads, and the development of the spatial problem of dynamic elastic-plastic deformations.

Kh. A. Rakhmatulin [6], in the field of the theory of dynamic plasticity, poses the problems:

a) determining the laws of momentum transfer in a plastic medium;

b) determining strength criteria for metals under dynamic loads.

Impact deformation of metals is very widespread and has a number of special features arising from the fact that the effect of recovery, or recrystallization, is here considerably weakened because of the short duration of the deformation process. Because of its prevalence, impact deformation deserves special consideration. Usually the work of deformation is determined, while the forces can be found by differentiating the work curve. In impact compression tests, two quantities are usually determined: the specific work required to fracture the specimen, called the “fracture factor,” and the specific work sufficient to produce a definite compression of the specimen (usually by 80%), called the “upsetting factor.”

2.

The basic regularities of dynamic compression were established by Prof. N. F. Kunin [7]. He shows that the law of static compression:

\[ P h^{m} = P_{0} h_{0}^{m} \]

is also valid for dynamic compression (lead, copper, aluminum, tin). Figure 1 gives a diagram of static and dynamic compression of lead. It is seen that the dynamic curve lies above the static one at equal compressions. The static curve was obtained by him by graphical integration of the “Force—deformation” curve. The polytropic exponent “\(m\)” in dynamic compression proved to be the same as in static compression. Table 1 gives the results of N. F. Kunin’s work.

The conventional yield point \(\sigma_{0}\) in dynamic compression is considerably greater than in static compression. The ratio

\[ \frac{\sigma_{0\,\mathrm{dyn}}}{\sigma_{0\,\mathrm{stat}}} = z \]

N. F. Kunin calls the dynamic coefficient.

Table 1

E. P. Unksov [8] points out that practitioners have observed that, of two hammers with equal impact energy, the one with the greater mass of falling parts gives the greater upsetting. In other words, there is an indication that, with increasing deformation rate, the resistance of the metal increases. Many theorists and experimentalists have arrived at analogous conclusions, in particular Ludwik [9]. He explained the increase in resistance to deformation with velocity by an increase in internal friction, assuming, in accordance with the hydrodynamic theory of viscous fluids, that the frictional forces are proportional to the deformation rate. Ludwik further notes that, with increasing deformation rate, the brittleness of the metal increases.

Fig. 1

M. Kodrom [10] presents the results of experiments on the deformation of lead cylinders at different velocities under room-temperature conditions. He found that, with increasing deformation rate, the resistance of lead increases sharply. The course of the dependence curves is analogous to the course of these curves at low velocities. For brass and steel he arrived at opposite conclusions.

Engineer Heller [11] points to an increase in resistance to deformation even with a very slight increase in velocities. Thus, increasing the deformation rate by a factor of 2—from 0.3 to 0.6 m/sec—Heller obtained an increase in energy consumption of 70%.

Engineer Riedel [12] gives a formula for the resistance of a metal to deformation as a function of velocity:

\[ \sigma = 3.8 + 0.1 v \ \text{kg/mm}^2 . \]

Thus, there exists an opinion that the expenditure of energy increases significantly with increasing deformation rate, although there are also contrary opinions. For example, Prof. Shaposhnikov [13], compressing copper dies, varied the velocity from 1.98 to 9.90 m/sec and did not notice any substantial difference in the results.

In the present work we set ourselves the task of considering certain particular cases of deformation of metals by impact and of elucidating the regularities observed in this process. In the laboratory of the Department of Strength of Materials of the Tomsk Polytechnic Institute, on an Amsler testing machine with the greatest—

...with a ram lift height equal to 4.5 m, a large number of experiments were carried out on specimens of steel 30, steel 40, brass, aluminum, copper, zinc, tin, and lead.

Before proceeding to the solution of the problem posed, we consider it necessary to dwell on certain theoretical considerations, apparently common to all metals and to any conditions of deformation of metals by impact.

Let us assume that a specimen of length \(h\) and cross-sectional area \(F\), deformed by some load \(Q\) falling from a height \(H_0\), performing the work \(A_0 = H_0 Q\), accumulates the potential energy of elastic deformation

\[ u_0=\frac{Q^2 h}{2EF}. \tag{1} \]

Let us now assume that a specimen having an initial length \(h_0 > h\) is deformed to the height \(h\) and accumulates, upon impact, the same potential energy. For this, obviously, it will be necessary for some load \(Q_1\) to be dropped from the height \((H+H_0)\). The work produced by the load in this case will be \(A_1 = Q_1(H+H_0)\). The expression for the potential energy for this case will have the form:

\[ u_1=\frac{Q_1^2 h_0}{2EF}. \tag{2} \]

From the equality, within the limits of elastic deformations, of the works of the potential energy it follows that the works \(A\) and \(A_1\) are equal to each other (by condition), that is,

\[ H_0Q=(H+H_0)Q_1, \tag{3} \]

whence

\[ Q_1=\frac{QH_0}{H+H_0}. \tag{4} \]

Equating the values of the potential energy in the first and second cases and replacing \(Q_1\) by its value according to formula (4), we obtain

\[ \frac{Q^2h}{2EF}=\frac{Q^2H_0^2h_0}{2EF(H+H_0)^2}, \]

whence it follows that

\[ (H+H_0)^2\cdot h=H_0^2\cdot h_0. \tag{5} \]

Carrying out some transformations of this expression, we obtain

\[ (H+H_0)=H_0\frac{1}{\sqrt{\dfrac{h}{h_0}}}. \]

Let us clarify what \(\sqrt{\dfrac{h}{h_0}}\) represents under the condition \(h_0>h\).

\[ \sqrt{\frac{h}{h_0}} = \sqrt{\frac{h}{h_0}+1-1} = \sqrt{\frac{h-h_0}{h_0}+1} = \sqrt{1-\frac{h_0-h}{h_0}} = \sqrt{1-\delta_{\text{comp}}}. \]

Thus,

\[ H+H_0=H_0\cdot\frac{1}{\sqrt{1-\delta_{\text{comp}}}}. \]

From this expression it is easy to obtain

\[ H=H_0\cdot \frac{1-\sqrt{1-\delta_{\text{comp}}}}{\sqrt{1-\delta_{\text{comp}}}} . \tag{6} \]

Let us determine the quantity \(H_0\); for this we start from the equality

\[ (H+H_0)^2\cdot h=H_0^2\cdot h_0 \]

\[ S=(H+H_0)^2\cdot h-H_0^2\cdot h_0=0. \]

\[ \frac{\partial S}{\partial H_0} = 2\cdot (H+H_0)h-2H_0h_0 = 2[H\cdot h-H_0(h_0-h)]=0. \]

Whence it follows that

\[ H_0=\frac{Hh}{h_0-h}. \tag{7} \]

In the case where we have a series of tests,

\[ H_0=\frac{\sum Hh}{\sum (h_0-h)} . \tag{8} \]

Obviously, for a definite lifting height of the load—\(H\)—for a given series of tests \(H_0=\text{const}\). If, moreover, the weight of the lifted load is equal to \(Q\), then the work of this load over the height \(H_0\) will be \(H_0\cdot Q=A=\text{const}\), whence

\[ H_0=\frac{A}{Q}. \]

Let us introduce a certain quantity \(B\), having the dimension of the cube of a linear quantity, such that the product of \(B\) by the modulus of normal elasticity \(E\) of the material under investigation will be equal to \(A\), i.e.

\[ B\cdot E=A. \]

In turn, the quantity \(B\) may be represented as the product of a certain dimensionless coefficient \(2\beta_c\) by the volume of the specimen \(V\)

\[ B=2\beta_c\cdot V. \]

With these assumptions formula (6) assumes the form:

\[ H\cdot Q = 2\beta_c\cdot V\cdot E\cdot \frac{1-\sqrt{1-\delta_{\text{comp}}}}{\sqrt{1-\delta_{\text{comp}}}} . \tag{9} \]

From formula (9) it follows that

\[ \beta_c = \frac{H\cdot Q\sqrt{1-\delta_{\text{comp}}}} {2\cdot V\cdot E\left(1-\sqrt{1-\delta_{\text{comp}}}\right)} . \tag{10} \]

This formula of ours for the general impact coefficient \(\beta_c\) makes it possible to pose a number of problems, namely:

1. Determination of the influence of the load weight \(Q\) and the lifting height of the load.
2. Determination of the influence of Young’s modulus, i.e., in fact, other conditions being equal, determination of the influence of the metal itself.
3. Determination of the influence of the specimen volume.
4. Determination of the influence of the relative geometric dimensions.

Let us introduce the concept of the average resistance to deformation of a metal under impact compression. By this resistance we shall mean the ratio of the difference of the works \(A_1-A_2\) to the displaced volume, i.e.

\[ K=\frac{A_1-A_2}{V_n}, \tag{11} \]

where \(A_1\) is the work of the falling load,

$A_2$ — the work of the load’s rebound after impact

\[ V_n = V \ln \left( \frac{1}{\delta_{\text{compr}}} \right), \tag{12} \]

$V$ — volume of the specimen before impact,

\[ \delta_{\text{compr}} = \frac{h_0 - h}{h_0} \]

— shock compression,

$V_n$ — displaced volume.

3.

Influence of Young’s modulus on the value of the coefficient $\beta_c$ and on the resistance of metal deformation to shock compression.

To clarify this question, we tested, under shock compression, specimens of steel 40, copper, brass, aluminum, zinc, tin, and lead. All specimens had constant identical dimensions: height $h_0 = 15\ \text{mm}$, diameter $d_0 = 10\ \text{mm}$. The impact was made from a height $H = 100\ \text{cm}$ with a load $Q = 31.2\ \text{kg}$. The number of revolutions of the copra drum was $n = 1500\ \text{rev/min}$. The average values obtained from these experiments have been compiled by us in Table 2.

Fig. 2

From the data of Table 2, a graph was constructed of the relationship between the values of the modulus of normal elasticity $E$, the impact coefficient $\beta_c$, and the coefficient of deformation resistance $K$ (Fig. 2).

Analysis of this graph allows the following conclusions to be drawn:

a) With increasing modulus of normal elasticity $E$, the impact coefficient $\beta_c$ increases, which indicates the validity of this coefficient as a quantity characterizing the impact process.

b) The coefficient $K$ decreases with increasing modulus of normal elasticity $E$, and very considerably (for lead $K = 20580$, while for steel $K = 1237$).

In Fig. 2, for copper, a deviation from the conclusion made above is noticeable; this may be explained by the specificity of copper deformation under impact loading (with comparatively small reductions, there is a considerable loss of elastic properties).

4.

Influence on the impact coefficient $\beta_c$ and the resistance coefficient $K$ of the height of lifting the load $H$.

To clarify this dependence, we tested a large number of specimens made of the materials indicated in § 3, having constant dimensions:

Table 2

Note: For lead, the height of fall of the load is $H=75$ cm.

\(h_0 = 15\) mm, \(d_0 = 10\) mm. The drop height \(H\) was varied within the following limits: for steel, copper, and aluminum from \(H = 150\) cm to \(H = 75\) cm; for brass and zinc from \(H = 125\) cm to \(H = 75\) cm; for tin from \(H = 100\) cm to \(H = 50\) cm, and for lead from \(H = 75\) cm to 25 cm. The different drop heights of the load were adopted because, at greater drop heights, the brass, zinc, tin, and lead specimens after impact either cracked apart (brass, zinc) or were strongly flattened (tin, lead).

Table 3 gives the basic data of these experiments.

Analysis of the data in Table 3 shows that increasing the drop height of the impacting load has a substantial effect on the values of \(\beta_c\) and \(K\).

For steel, for example, increasing the drop height of the impacting load causes the values of \(\beta_c\) and \(K\) to change in the following ratio:

\[ \frac{\beta_c(150)}{\beta_c(75)} = \frac{8150 \cdot 10^{-6}}{9440 \cdot 10^{-6}} = 0.865, \]

\[ \frac{K(150)}{K(75)} = \frac{2485}{842} = 2.95, \]

i.e., when the drop height of the impacting load is changed by a factor of two, the average resistance to impact changes by 295%, and the coefficient by 13.5%. It is natural to expect that increasing the drop height of the load causes an increase in the energy expenditure for deformation in the same ratio as the average resistances to impact change, i.e., the values of \(K\).

5.

The influence of relative geometrical dimensions on the values of \(\beta_c\) and \(K\). In the present section we consider the question of the influence of the coefficient

\[ \alpha = \frac{h_0}{d_0} \]

on the change in the values of \(\beta_c\), the average resistance to deformation under impact \(K\). To resolve this question, we carried out experiments with specimens having the following values of \(\alpha\)

\[ \alpha = 0.5;\quad 1.0;\quad 1.5;\quad 2.0;\quad 2.5;\quad 3.0 \]

and made of steel 30, copper, brass, zinc, and gray cast iron. The weight of the falling load was \(Q = 25\) kg. The drop heights were selected differently, as shown in Tables 4 and 5.

Graphs of the variation of \(K\) and \(\beta_c\) as functions of \(\alpha\) are presented in Figs. 3 and 4. Analysis of the tables and graphs makes it possible to draw the following conclusions:

a) For all the metals tested by us, as \(\alpha\) increases, the average resistance to deformation of the metals under impact increases.

b) The smallest value of \(K\) corresponds to the value \(\alpha = 3.0\), and the largest to the value \(\alpha = 0.5\).

c) As the value of \(\alpha\) decreases, the change in the values of the impact coefficient \(\beta\) for steel and copper follows a smoothly increasing curve.

For brass, cast iron, and zinc, the change in \(\beta\) as a function of \(\alpha\) has a complex character; moreover, all of them have two maxima at \(\alpha = 1.5\) and \(\alpha = 0.5\). At \(\alpha = 1.0\), for these metals the value of \(\beta_c\) drops. This is evidently explained by the fact that all these metals belong to the category of brittle metals.

6

Let us consider the question of the dependence of the impact work \(A_1 - A_2\) and the relative compressive deformation under impact

\[ \delta_{\text{comp}} = \frac{h_0 - h}{h_0} \]

on the initial impact velocity \(V\). For this purpose, the experimental data from the preceding sections were processed by us in the direction of determining the impact velocity \(V\) and are summarized in Table 6.

Table 3

Table 4

Table 5

Fig. 3

Fig. 4

Table 6

Using the data of this table, graph 5 was constructed for steel and lead. The graphs for other metals have a similar form. It is clearly seen that the regularity of the change in \(A_1 - A_2\) and \(\delta_{\text{compressed}}\) with the change in velocity obeys the law of a straight line.

An increase in the velocity of the striking load increases the value of the work of impact and, correspondingly, the deformation of the metal.

Fig. 5

An increase in the velocity of the striking load from \(384\ \text{cm/sec}\) to \(542\ \text{cm/sec}\) causes, respectively, the following increases in \((A_1 - A_2)\) and \(\delta_{\text{compressed}}\).

Table 7

We shall also pose the question of the influence of the rigidity of the support, at constant values of the load \(Q\) and the height of fall \(H\), and with constancy of the geometric dimensions of specimens of various materials, on the values of the impact coefficient \(\beta_c\) and the resistance coefficient \(K\). Different support conditions are created with the aid of the device shown in Fig. 6. It consists of a cylindrical vessel (the body of an artillery shell

Fig. 6

Labels in Fig. 6: sample; piston; cylinder; liquid.

\(d = 120\) mm), filled with various liquids: water, oil, molten paraffin, and kerosene. A steel piston was inserted into the vessel; on its surface it had four sealing rings, preventing the liquid from being expelled from the cylinder during impact. On the end surface of the piston, at the center of the circle, the test specimen was installed. The impact was produced by a weight \(Q = 31.2\) kg, of which 8.6 kg is accounted for by the striker of the special device.

Table 8 gives the density characteristics of the liquids.

Table 8

The experiments were carried out with specimens of steel 40, steel 30, copper, brass, aluminum, zinc, tin, and lead. The specimen dimensions were: \(h_0 = 15\) mm, \(d_0 = 10\) mm. The height of fall of the weight \(Q\) in all cases was \(H = 100\) cm. The mean values of the results from several experiments for each specimen are presented in Table 9.

Fig. 7

On the basis of the data of this table, in coordinate axes—the density of the supporting cushion and the values of \(K\) and \(\lambda\)—graphs 7 and 8 were plotted. These graphs also give data for a rigid support.

Analysis of the data in Table 9 and the graphs leads to the following conclusions:

a) The average resistance of materials to deformation under impact, \(K\), and the impact coefficient \(\beta_c\) depend substantially on the rigidity of the supporting cushion. The largest values of \(K\) correspond to a rigid support.

b) As the density of the supporting cushion decreases, the value of the average resistance \(K\) decreases.

c) The variation of the impact coefficient \(\beta_c\) as a function of the rigidity of the supporting cushion is rather complex. It does not appear possible to relate the change in \(\beta_c\) to the density of the supporting cushion as the latter is decreased.

Fig. 8

For example, the density of the oil supporting cushion differs from the density of the paraffin supporting cushion by only 0.02, whereas the change in the value of \(\beta_c\) is very considerable.

Thus, we are able only to establish the fact of a significant influence of the density of the supporting cushion on the results of shock compression of metals.

We introduced the assumption that, under impact, the deformation of the vessel walls \((\delta = 15\ \text{mm})\) is negligibly small and does not affect the experimental results.

Table 9

Continuation of Table 9

References Used

1. Davidenkov, N. N. Dynamic Tests of Metals. ONTI, 1936.
2. Newton, I. Mathematical Principles of Natural Philosophy.
3. Boussinesq. Application des Potentielles, 1885.
4. Saint-Venant. Addition to the translation of Clebsch’s book Theorie de l’elasticite, §§ 60–61.
5. Ilyushin, A. A. Izvestiya of the Academy of Sciences of the USSR, Department of Technical Sciences, No. 9, 1950.
6. Rakhmatulin, Kh. A. Izvestiya of the Academy of Sciences of the USSR, Department of Technical Sciences, No. 9, 1950.
7. Kunin, N. F. The Law of Plastic Tension and Compression of Metals. Chelyabinsk, 1943.
8. Unskov, E. P. On the deformation of metal by impact. NIITMASH, No. 8, 26–40, 1933.
9. Ludwik. Elemente der technologischen Mechanik. Berlin, 1909.
10. Kazron, K. Hot Working of Metals.
11. Feller. The influence of the number of revolutions on the temperature of the tool in cold presses, 1927.
12. Riedel. The Principle of Energy Expenditure in Forging under a Press, 1913.
13. Shaposhnikov, N. Questions of Metallurgy, p. 189, Kharkov, 1926.