ON THE DEPENDENCE OF STRENGTH ON EXPOSURE AND OF THE RATE OF FRACTURE ON THE STRENGTH OF A BODY

A. A. VOROB'EV

Introduction

The concept of the strength and stability of solid bodies is usually associated with the notion of a catastrophic process of fracture occurring when the destructive factor reaches some critical value.

The idea of the catastrophic nature of fracture is based on the observation that a specimen subjected to a load below the limiting one does not fail for a long time. On visual observation it seems that, when the load reaches or exceeds the limiting value, the specimen fails immediately. Thus there arose technical characteristics of a body: tensile strength, electrical strength, melting temperature, which are practically necessary and satisfactorily describe the behavior of a material under the corresponding load.

An analogy is observed among the regularities describing the mechanical, thermal, and electrical fracture of solid bodies.

The time to fracture of a specimen under tension (lifetime) \(t_{\text{fract}}\) increases exponentially, or even more rapidly, as the magnitude of the stress decreases and the temperature is lowered. The same dependence of the breakdown time on overvoltage is observed in the electrical breakdown of solid dielectrics.

In the electrical form of breakdown, the electrical strength increases with increasing temperature of a solid dielectric. At exposures of less than \(10^{-4}\) sec, the mechanical tensile strength also increases with increasing temperature. In the region of large exposures and high temperatures, the tensile strength and the strength of dielectrics under electrothermal breakdown decrease as the temperature rises.

Straight lines representing the dependence \(\ln t_{\text{fract}} = f(\sigma)\) at different temperatures \(T\) for specimens of a given substance, or straight lines \(\ln t_{\text{fract}} = f(T^{-1})\), constructed for different stresses \(\sigma\), intersect at one point.

The dependences \(\ln t_{\text{fract}} = f(\beta)\), where \(\beta\) is the magnitude of the overvoltage in electrical breakdown, obtained at different temperatures, also intersect approximately at one point. Therefore, for the given ma-

there is such a quantity \(t_{\mathrm{fract}}\) and a corresponding mechanical stress \(\sigma\), or electrical overvoltage \(\beta\), which do not depend on temperature. The strength of the body in this case proves to be independent of exposure. In the breakdown of crystals of alkali-halide salts, the strength does not depend on the composition of the crystal in the range of durations \(10^{-9}\) sec. At an exposure of \(10^{-8}\) sec, the strength does not depend on temperature [1].

In the case of thermal destruction (evaporation) of a body, an exponential increase in the evaporation time is observed as the temperature of the body is lowered.

Consequently, in mechanical, electrical, and thermal destruction of a solid body, the time to fracture increases as the magnitude of the destructive factor decreases, i.e., the stress, the electric-field strength, or the temperature.

The mechanical strength of thin threads increases as the cross-sectional area of the thread decreases; the electrical strength and breakdown time of thin dielectric layers increase as the thickness of the layer being broken down decreases. The time to rupture of thin threads has not been studied.

The identical course of the dependence of strength on temperature and on the duration of loading in mechanical, electrical, and thermal destruction of solid bodies makes it possible, within certain limits, to model any one of the listed modes of destruction by means of another.

The two possible modes of mechanical destruction of solids—“catastrophic” and with preparatory processes—and the several possible modes of breakdown are opposed to one another in the literature and regarded as mutually exclusive. It seems to me, however, that depending on the conditions of destruction, it is possible to bring about destruction either by one mode or by the other. In this connection it is of interest to consider the laws of destruction of a solid body under the condition either of catastrophic destruction or as a result of preparatory processes.

In practice there are more possibilities for better regulating the process of electrical destruction of a solid body during its breakdown. The possible physical processes in breakdown have also been worked out in sufficient detail. The numerous physical theories of electrical breakdown are divided into two groups: some theories describe breakdown as a catastrophic phenomenon that occurs when a field strength equal to its electrical strength \(E_{\mathrm{pr}}\) is reached in the dielectric; other theories proceed from the assumption that, when \(E_{\mathrm{pr}}\) is reached, processes develop in the dielectric—for example, ionization by impact—which lead to its destruction.

Catastrophic destruction is possible only under a short-time load, whose duration must be less than the duration of the elementary acts that prepare destruction, i.e., in the limiting case, less than the duration of processes in excited molecules. In the other case, destruction is preceded by preparatory processes, for example: in mechanical destruction, diffusion, dislocation, or other processes (creep); and in electrical breakdown, ionization preparatory processes.

Let us consider, using ionic crystals [1, 2, 3] as an example, the expressions for strength obtained when it is calculated in each of the cases of destruction considered.

Calculation of the Theoretical Strength of Solids under Their Mechanical or Electrical Fracture

The force of interaction between two ions varies according to the known curve shown in Fig. 1. The calculation of the theoretical strength can be carried out using various approximate expressions for this curve. The curve in the region of interest to us, which includes the maximum attractive force \(F_{\max}\) that must be overcome in order to fracture the body, may approximately be represented by a segment of a sinusoid with the indicated parameters [4, 5].

Fig. 1.

The work expended in fracture and calculated per unit area corresponds to the shaded area (Fig. 1) and is approximately expressed as

\[ \int_{r_0}^{\lambda/2} F_{\max}\sin \frac{2\pi r}{\lambda}\,dr = \frac{\lambda}{\pi}F_{\max} = 2U_s, \tag{1} \]

since in fracture this work is expended in forming surfaces and appears as the surface energy of two new surfaces, equal to \(2U_s\), where \(U_s\) is the surface energy. The quantity \(F_{\max}\) is the maximum force withstood by the system under its stretching

\[ F_{\max}=\frac{2\pi}{\lambda}U_s. \tag{2} \]

Hooke’s law for the system has the form

\[ F=E\frac{r}{r_0}, \tag{3} \]

where \(E\) is the modulus of elasticity, \(r_0\) is the equilibrium distance between ions, when the resultant attractive and repulsive forces are equal to zero. From (3) and Fig. 1 we have

\[ \frac{dF}{dr}=\frac{E}{r_0}=\frac{2\pi}{\lambda}. \]

The magnitude of the theoretical mechanical strength under elastic stretching and brittle fracture is determined by the condition

\[ F_{\max}=\left(\frac{EU_s}{r_0}\right)^{1/2}. \tag{4} \]

If fracture also occurs without preparatory processes, by the forces of the electric field \(E\) acting on the charges, then the theoretical strength is determined from the condition

\[ F_{\max}=eE_{\mathrm{pr}}, \]

where \(E_{\mathrm{pr}}\) is the electric strength

\[ E_{\mathrm{pr}}=e^{-1}F_{\max}=e^{-1}\left(r_0^{-1}EU_s\right)^{1/2} \tag{5} \]

increases with a decrease in the lattice parameter \(r_0^{-1/2}\), with an increase in the modulus of elasticity \(E\), in the mechanical tensile strength \(F_{\max}\), and in the surface energy of the body \(U_s\).

A. A. Vorob’ev and E. K. Zavadovskaya [6] proposed a criterion for crystal breakdown under the condition that fracture occurs when the energy transmitted by an electron to the lattice reaches the magnitude of the lattice energy, and, if successive accumulation of energy by particles is possible in the process of its excitation, a fraction of this magnitude.

The energy accumulated by an electron per unit time in a field \(E\) is

\[ \Delta W=eE\cdot v, \tag{6} \]

where \(v\) is its group velocity.

The mobility \(b\) of electrons in the conduction band is determined by

\[ b=\frac{l}{m}\,t, \tag{7} \]

where \(l\), \(m\), and \(t\) are, respectively, the charge, mass, and time of free motion of the electron. Since the probability of collision of an electron is \(t^{-1}\), the drift velocity of electrons in the field \(E\) is determined by

\[ v=bE=\frac{e}{m}\cdot t\cdot E. \tag{8} \]

The increase in energy of an electron in the field is

\[ \Delta W=\frac{e^2E^2}{m}\cdot t. \tag{9} \]

The energy accumulated by an electron over the time \(t\) is equal to

\[ W=t\Delta W=\frac{(e\cdot E\cdot t)^2}{m}. \tag{10} \]

At breakdown, the energy accumulated by the electron over the mean free path during the time \(t\) is transferred to the lattice and is expended in breaking the bonds between particles; this will occur when the energy transmitted by the electrons to the lattice sites is equal to the lattice energy, i.e. \(W\simeq U\). If accumulation of lattice excitations from individual interactions with electrons is possible, one should write the proportionality relation, i.e.

\[ W=cU, \tag{11} \]

where \(c=\mathrm{const}\) is a quantity depending on the number of acts of interaction of conduction electrons with lattice sites after which its destruction will occur. From conditions (10) and (11) we obtain

\[ cU=\frac{(e\cdot E_{\mathrm{pr}}\cdot t)^2}{m}, \tag{12} \]

whence it follows that

\[ E_{\mathrm{pr}}=(c\cdot m\cdot U)^{1/2}\cdot (et)^{-1} \tag{13} \]

the electric strength of solid dielectrics increases with an increase in the lattice energy proportionally to \(U^{1/2}\) and with a decrease in the time of free ...

of the mean free path \(t\) of the electrons, i.e., the time of development of breakdown \(t_{\mathrm{br}}\). From (13) it follows that

\[ E_{\mathrm{br}}=\operatorname{const}\, t_{\mathrm{br}}^{-1}. \tag{14} \]

It has been shown experimentally that the electrical strength of crystals increases with increasing mechanical strength, surface energy, and lattice energy, and with decreasing lattice parameter [6], i.e., (13) is confirmed.

In crystals of alkali-halide salts, at a field intensity equal to the electrical strength, internal mechanical stresses arise which reach the technical strength of the dielectric [7].

Thus, from the condition of rupture of the lattice by the forces of the electrostatic field, or of destruction of the lattice by electrons during impact ionization, as well as destruction of the lattice by ponderomotive forces in an electric field, concordant results are obtained concerning the theoretical strength of solid dielectrics under electrical breakdown and the dependence of the electrical strength of crystals on their composition and structure.

On the dependence of the theoretical electrical or mechanical strength of bodies on exposure

From the description of breakdown as a consequence of energy transfer by moving electrons, there follows a hyperbolic dependence between the magnitude of the electrical strength \(E_{\mathrm{br}}\) and the breakdown time \(t_{\mathrm{fract}}\). Such a dependence for crystals of alkali-halide salts is observed experimentally down to a breakdown time of \(10^{-9}\) sec [8]. At a breakdown time on the order of nanoseconds, the electrical strength proves to be the same for alkali-halide crystals of different chemical composition.

From condition (14) it follows that if breakdown develops in a dielectric over a time \(t_{\mathrm{fract}}\), and to complete it it is necessary to expend some amount of energy constant for the given dielectric, then there is a hyperbolic dependence between the electrical strength \(E_{\mathrm{br}}\) and the breakdown time \(t_{\mathrm{fract}}\) (14).

Dependence (14) also follows from the following general physical considerations. With an increase in the electrical strength of dielectrics \(E_{\mathrm{br}}\), in stronger prebreakdown fields, electrons will move faster and accumulate the energy needed to destroy the structure in a shorter prebreakdown time \(t_{\mathrm{fract}}\). Therefore, in the simplest case one may write condition (14), or a more complex function \(t_{\mathrm{fract}}=f(E^{-1})\).

If destruction occurs after plastic deformation, then the mobility of dislocations increases rapidly with increasing mechanical stress for bodies of high strength [11]. In this case the time for the preparatory processes of destruction is shortened. The increase in the tensile strength of a body and in its electrical strength with decreasing exposure follows from the condition that a constant amount of energy is expended on its destruction. A decrease in exposure must be accompanied by an increase in the power and amplitude of the destructive load [12].

The values of the mechanical strength determined by condition (2), and of the electrical strength (5), for cases of destruction of a body without preparatory physical processes, do not depend on time.

In brittle fracture of a body, or in its breakdown as a result of rupture in an electric field, the time of fracture does not enter into formulas (2) and (5). These formulas describe destruction without preliminary preparation

with the velocity of propagation of an elastic wave in the body, which is a characteristic of the body and not of the conditions of its loading.

The foregoing also implies the converse conclusion. If there is a dependence of the time to fracture on the magnitude of the strength of the body being fractured, then its fracture is preceded by certain preparatory processes. In the electrical breakdown of solid dielectrics it is practically possible to realize both of the described types of fracture. Down to exposures of \(10^{-9}\), an increase in electrical strength was observed with decreasing breakdown time.

Conclusion

From the formulas of theories describing fracture as a catastrophic phenomenon occurring when the destructive factor reaches a value equal to the strength of the body, no dependence follows either of strength on exposure or of the rate of fracture on the strength of bodies. When the kinetics of development of the fracture process is taken into account, a hyperbolic or exponential dependence between the time to fracture and the magnitude of the destructive factor appears in the formulas of the theory. A decrease in the resistance of bodies with increasing time to fracture is observed experimentally in the mechanical, electrical (breakdown), and thermal (evaporation) fracture of solids.

This regularity is a consequence of the need to expend, for fracture, a constant amount of energy, which for ionic compounds is equal to the lattice energy [13].

In an energetic description of the fracture of a solid body, three stages may be conventionally distinguished: the formation of fracture, its progress, and the completion of fracture. Formation occurs on the rising part of the load pulse. In the process of dynamic fracture, the rate of energy supply is equal to or exceeds the rate of its absorption. If fracture begins on the leading front of the pulse, then it occurs under conditions in which the supplied power exceeds the consumed power.

In the second stage, fracture may also proceed under conditions in which the supplied power is equal to the expended power. In this case fracture occurs at an unchanged value of the destructive factor, for example, of the electric voltage. Under the condition that the consumed power required for fracture exceeds the power of the source, fracture then proceeds under nonstationary conditions with a decreasing amplitude. In the final stage, a large part of the energy required for fracture is expended.

For fracture of a body it is necessary to expend a constant amount of energy \(U\), equal to the product of the exposure \(t_{\text{fract}}\) and the power \(W\) released in the volume being fractured, \(U = W t_{\text{fract}}\). With decreasing exposure \(t_{\text{fract}}\), the power \(W\), which is proportional to the square of the amplitude of the destructive factor \(A^2\), increases:

\[ U = W \cdot t_{\text{fract}} \approx A^2 t_{\text{fract}} = \text{const}. \tag{15} \]

Hence the magnitude of the amplitude of the destructive factor and the exposure are related by the condition

\[ A \approx t_{\text{fract}}^{-0.5}. \]

Consequently, from general energetic considerations it also follows that, as the time to fracture \(t_{\text{fract}}\) decreases, the magnitude of the pulsed mechanical or electrical strength, or the thermal strength in fracture (evaporation), must increase.

The amount of energy necessary for the destruction of a body and its strength increase with an increase in the binding energy between the particles of the body. Therefore the curves described by equation (14) will not coincide for bodies of different chemical composition. From the condition of constancy in (2) it follows that, with an increase in the strength of the body, the time to its fracture \(t_{\text{fract}}\) decreases, while the rate of the fracture process increases.

Conditions (14) and (15) also formally describe catastrophic fracture of bodies. In this case the rate of fracture is determined by the propagation of elastic waves in the body, which is a quantity constant for the given body. It increases with an increase in the strength of the bond between the particles of the body.

The increase of the destructive factor and the fracture of the body may end at a value considerably exceeding the true strength of the body. The magnitude of the binding energy between the particles of the body and the energy of its fracture are constant. The effect under consideration—the increase in the measured pulse strength—should be regarded as an “instrumental effect,” connected with the conditions of the experiment, and not with the essence of the phenomenon of fracture in the case considered [14].

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