On the duration of shock loading and yield strength

N.F. Morozov and L.S. Shikhobalov / Physical Mesomechanics 13 1–2 (2010) 1–11

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On the duration of shock loading and yield strength N.F. Morozov and L.S. Shikhobalov* St. Petersburg State University, St. Petersburg, 199034, Russia Thermal atomic motion in a crystal is interpreted as a certain rapidly oscillating stress field called a fluctuation field. A fluctuation field theory is developed in the context of solid state physics and statistical physics. The theory is applied to the description of dislocation motion in a crystal at external stress lower than the threshold required for dislocation motion. The dislocation motion is thus due to the joint action of external and fluctuation stresses. The shock pulse duration at which stress fluctuations have no chance (with 0.99 probability) to reach the level required for dislocation motion is calculated. With this pulse duration, the material does not experience plastic deformation, whereas with a longer loading pulse at the same stress it does. The effect, i.e., the absence of plastic deformation with a short loading pulse, can be eliminated by increasing the stress in the pulse. This suggests that the material yield strength increases with decreasing the duration of shock loading. Keywords: dislocation, stress fluctuations, yield strength, shock loading

1. Introduction The basic mechanism of plastic deformation of crystalline materials is dislocation motion. The yield strength of crystalline materials, as a rule, displays a descending temperature dependence and hence it is thermal atomic oscillations that are conducive to dislocation motion. Thus, dislocation motion is due to the joint action of applied stress and thermal atomic oscillations. The effect of thermal atomic oscillations on each individual dislocation segment is discrete in time: the time intervals on which they assist and fail to assist the dislocation motion are alternating. According to the laws of statistical physics, the above time intervals alternate with a very high rate; therefore their timediscreteness escapes detection in ordinary experiments on plastic deformation of materials. However, intensive recent research in the mechanical properties of materials under supershort shock loading [1] brings up the reasonable question: How short must the duration of external loading be that thermal atomic oscillations have no time, to a high probability, to initiate dislocation motion? In this work, we calculate the loading pulse duration at which thermal atomic oscillations required for dislocation motion have no chance to occur (to a 0.99 probability) on * Corresponding author Dr. Lavrentii S. Shikhobalov, e-mail: [email protected] Copyright © 2010 ISPMS, Siberian Branch of the RAS. Published by Elsevier BV. All rights reserved. doi:10.1016/j.physme.2010.03.001

the basis of the Einstein–Debye fluctuation theory and data of [2]. With this loading pulse duration, the material does not experience plastic deformation, whereas with a longer pulse at the same stress it does. The effect, i.e., the absence of plastic deformation with a short loading pulse, can be eliminated by increasing the stress in the pulse and this suggests that the material yield strength increases with decreasing the duration of shock loading. 2. Stress field due to thermal atomic oscillations Let us consider a crystal as an elastic homogeneous isotropic solid that contains dislocations of one slip system. An immobile dislocation is set in motion providing that the tangential stress tensor component operating in its vicinity and along slip plane in the Burgers vector direction is greater in the absolute value than a certain threshold V 0 (representative of the resistance to dislocation motion). Let us analyze the case where the above external stress component Vext falls short of the threshold and is positive. Thus, 0 < V ext < V 0 , and the aid of thermal atomic oscillations is required to actuate the dislocations. Let us treat thermal atomic oscillations as a certain rapidly oscillating stress field — call it a fluctuation field — universally present in the solid and dependent on its temperature.

2

N.F. Morozov and L.S. Shikhobalov / Physical Mesomechanics 13 1–2 (2010) 1–11

The fluctuation stress field, like an ordinary stress field, is bound to obey the system of equations of elasticity. However, our further discussion makes clear that the dislocation motion is governed by only quite definite values of this field in small solid subregions. Hence most of the fluctuation stress field has no effect on dislocation motion and its detailed form is of no importance for the description of the process. In this context, formulation and solution of the elasticity problem for the fluctuation stress field is omitted in the discussion and we consider this field as a set of individual stress fluctuations (or “flashes” of stress fluctuations) arising in certain solid subregions. These terms are used to emphasize the time and space discreteness of the field. The field parameters are specified with resort to the known concepts of solid state physics and statistical physics. For the fluctuation field, we set the simplifying assumptions similar to those taken in [2]: – the fluctuations of all six independent stress tensor components are independent of each other and are alike in probability distribution; – the solid subregions involved in fluctuations are spherical; the subregion diameters D, unlike those taken in [2], are different and can assume any value from the lattice cell parameter a to the minimum linear dimension of the solid L 1; – the fluctuation stress field is homogeneously inside each subregion involved in fluctuation; – the duration Wf of a fluctuation “flash” in a subregion of diameter D is equal to the time it takes for an elastic wave to travel a distance D with a velocity of sound ct : D (1) ôf . ct By analogy with the known Einstein and Debye heat capacity models [3], we consider that the fluctuation stress field in the solid is produced by independent oscillators each of which has a certain frequency Q f , different for different oscillators and the number of which is equal to the number of internal degrees of freedom of the solid 3N – 6 | | 3N (N is the number of atoms in the solid). The frequency Q f is taken to be the reciprocal of the fluctuation duration Wf : ct 1 íf (2) . D ôf The quantity Q f is termed the oscillator frequency or the fluctuation frequency. From the above properties and from formula (2) it follows that to each oscillator corresponds a certain diameter D of subregions in which it generates a fluctuation stress, and to different oscillators correspond different diameters. 1

The minimum linear dimension of a solid in the case where the latter is a rectangular parallelepiped is the minimum length among the lengths of its three edges issuing from one vertex; if the solid is a cylinder, it is the minimum quantity among its height and diameter; if the solid is a plate, it is its thickness.

Of concern to us is the tangential fluctuation stress tensor component affecting the dislocation motion. We have 3N oscillators and equally distributed fluctuations of six independent stress tensor components; hence, the tangential fluctuation stress tensor component of interest is generated by N 2 oscillators. It is assumed that half of the oscillators produce stress of one sign, and the other produce stress of opposite sign. We consider only N 4 oscillators responsible for the tangential stress component codirectional with the tangential external stress component V ext . Note that the fluctuation stress is capable of moving dislocations not only in the direction of external stress, but in the opposite direction as well. However in the latter case, the fluctuation stress is bound to be greater (in the absolute value) than that in the former case due to the necessity to overcome the external stress along with the resistance to dislocation motion. Moreover, the probability of fluctuation decreases rapidly with increasing its value, and hence with a rather high external stress, the probability of fluctuations moving the dislocations in the direction of external stress is so much higher than the probability of fluctuations moving them in the opposite direction that the latter can be ignored. Let Q(D) be the diameter distribution density for solid subregions involved in fluctuations. Because the diameter range and the oscillator range have one-to-one correspondence, Q(D) is simultaneously the oscillator distribution density. By analogy with the Debye model [3], let the number of oscillators with frequencies in the range [ í f , í f  dí f ] be proportional to í f2 dí f (in the linear approximation in dí f ). Then in view of formula (2), we have Q ( D ) dD

const (c t3 D 4 ) dD.

Once the constant in this equality is found from the condition that the number of oscillators is N 4 and D  [a , L ] (where a is the lattice cell parameter and L is the minimum linear dimension of the solid), we conclude the following. The number of oscillators initiating fluctuations of the tangential stress component in subregions of diameter [ D, D  dD ], is

Q ( D ) dD

3 Na 3 L3 3 Na 3 d dD D | 4( L3  a 3 ) D 4 4D 4

(3)

(in the linear approximation in dD). In the above equation, a  L. Let the tangential stress tensor component in question reaches V f in a certain subregion of volume : (according to the assumption taken, it is homogeneous in this subregion, V f > 0). Hence, the corresponding elastic strain tensor component å ef óf (2G ) (G is the shear modulus). The elastic energy of this stress fluctuation is

Uf

óf

2 ³ dÙ ³ ó då e (Ù)

0

ó 2f Ù , 2G

(4)

where the subregion volume and the subregion itself are

N.F. Morozov and L.S. Shikhobalov / Physical Mesomechanics 13 1–2 ((2010)) 1–11 y

denoted by the same symbol :; factor 2 indicates that the stress tensor has a tangential component symmetric to the component in question. Let P ( V f ) be the fluctuation probability density of the stress V f . From the concept of statistical physics [4] it follows that the function P(Vf ) is proportional to exp(U f kT ), where k is Boltzmann’s constant and T is the absolute temperature of a solid. The total probability of all possible values of V f is equal to unity, and hence, in view of (4), we arrive at the following conclusion. Probability that the tangential component of the fluctuation stress field in the volume : takes on a value in the range [V f , V f  dV f ] (where 0  V f  f) is equal to

§ V 2Ù · 2Ù exp ¨  f ¸ dV f (5) ¨ 2GkT ¸ ðGkT © ¹ (in the linear approximation in dV f ). The factor of the exponent in formula (5) is found from the normalization condition: P (V f ) d V f

f

³ P (V f ) d V f

1.

(6)

0

At this point, the formulation of the fluctuation stress field model is completed. In a certain sense, this model is intermediate between the Einstein and Debye heat capacity models. In the Einstein model, oscillations of each atom are independent; therefore all oscillators cover subregions of atomic size. The Debye model uses standing waves in a solid with which atomic oscillations are fully correlated; therefore, each oscillator covers the entire solid. In our model, as distinguished from the above two, oscillators can cover subregions of any size from the lattice cell size to the size of the entire solid. It can be demonstrated that our model leads to the known Dulong–Petit law and this is an argument for its validity. 3. Dislocation motion due to external and fluctuation stress fields Assume that a “flash” of stress fluctuation occurs in a solid subregion intersected by a dislocation. We also assume that before the flash of stress fluctuation, a dislocation segment inside the subregion was rectilinear and immobile, and early in the flash it starts moving, while the remainder of the dislocation stays immobile. Because the dislocation is a continuous line and is immobile outside the subregion under study, the extreme points of this dislocation segment are immobile, too. This means that the segment is bent. Assume, for simplicity, that the segment takes the shape of a circular arc. In this case, we can restrict ourselves to analysis of the motion of its middle point. Let us represent the motion of the point, i.e., the inflection of the dislocation segment, through x. Let us resort to the equation of dislocation motion used in [5–7]. This equation, as applied to our case, can be written in the form:

ñb 16 Gb &x&  x c ð (1  í) l 2

V ext  V f  V 0 for x& ! 0,

3

(7)

where U is the matter density; b is the Burgers vector magnitude; c is a dimensionless parameter of order unity; G is the shear modulus; Q is Poisson’s ratio; l is the length of the dislocation segment in question; a dot above a symbol stands for a time derivative. In the left-hand side of the equation, the first term characterizes the dislocation inertia and the second term describes the dislocation self-action stress that tends to return the dislocation to the rectilinear position. The resistance to dislocation motion has the form of dry friction: V 0 sign x& (at V 0 const ! 0 and x& z 0), and hence with a dislocation velocity x& ! 0, it is equal to V 0 . Here it is assumed that the fluctuation stress V f meets the condition V f > V 0 – V ext . This condition ensures the positive range of the right-hand side of equation (7). Remind that we analyze the case where the external stress V ext satisfies the inequality 0 < V ext < V 0 . Previously, we assumed that the fluctuation stress V f is homogeneous inside the subregion involved in fluctuation. Assume that the external stress V ext is also homogeneous inside this subregion and that the both stresses V f and V ext remain invariant during the fluctuation Wf . In this case, the right-hand side of equation (7) is constant during the fluctuation. Solution of equation (7) for zero initial conditions ( x 0 and x& 0 at t = 0) gives

x(t )

ð (1  í) l 2 (V ext  V f  V 0 ) u 16 Gb ª §4 cG u «1  cos ¨ ¨ «¬ © l ñ ð (1  í)

·º t ¸» . ¸» ¹¼

Hence it follows that the maximum inflection of the dislocation segment in question is

ð (1  í) l 2 (V ext  V f  V 0 ). (8) 8 Gb It can be demonstrated that during the fluctuation Wf (determined by formula (1)), the dislocation segment will manage to bend to the indicated maximum value. Note that xmax ! 0 because V f > V 0 – V ext . Once the flash of fluctuation ends and the dislocation stops, two situations are possible according to the equation of dislocation motion. If the stress V f during the fluctuation is moderate, namely, satisfies the inequality: 3V  V ext Vf d 0 , (9) 2 the dislocation segment is kept in the bent position with an inflection equal to xmax because in this case, the dislocation self-action stress tending to rectify the dislocation is insufficient to overcome the resistance V 0 to backward dislocation motion. xmax

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If the fluctuation stress V f satisfies the inequality: 3V  V ext , Vf t 0 (10) 2 the dislocation segment starts to move backward on completion of the fluctuation. Its motion is described by the equation: ñb 16 Gb &x&  (11) x V ext  V 0 for x&  0, c ð (1  í) l 2 where it is taken into account that in backward dislocation motion, the resistance reverses sign and becomes equal to + V 0 instead of – V 0 . We put the sign of conditional, instead of strict, inequality in (10) to include the limiting case V f (3V 0  Vext ) 2 , where the dislocation is immobile, but is set in motion with the least increase in V f ; this case satisfies equation (11) at x xmax , x& 0, &x& 0. The solution of equation (11) for the initial conditions x xmax , x& 0 (at t = 0) is the function: ð (1  í ) l 2 ª «(V ext  2V f  3V 0 ) u x( t ) 16 Gb « ¬ º §4 · cG u cos ¨ t ¸  V ext  V 0 » . ¨ l ñ ð (1  í ) ¸ »¼ © ¹ By calculating the velocity x& from the above, we find that the dislocation stops when its inflection x takes on the following value — call it also the maximum value (and mark by an asterisk):

ð (1  í ) l 2 (2V 0  V f ). (12) 8 Gb Here, V f is the fluctuation stress during the fluctuation

preceding the flash. Note that xmax decreases with an increase in V f . This is because the bent dislocation, like a drawn and released bow string, is “rebound” as much farther as its bending is strong. Thus, the flash of stress fluctuation shifts one of the dislocation segments. In the same way, other flashes of stress fluctuations shift other dislocation segments. As a result, the whole dislocation structure is set in motion. The forgoing analysis of dislocation motion relates to the case where the solid under consideration is a continuum. However, the real crystal modeled by the solid has a discrete atomic structure. The discreteness, in particular, is that in energy terms, the dislocation benefits from taking quite definite positions in the crystal. These positions are d apart and this distance is of the order of the lattice cell parameter a. The dislocation energy in these positions is minimum, and in halfway between them it is maximum. Let the dislocation be in a position with minimum energy. If the stress displaces the dislocation from this position by a distance smaller than d 2 , the energy relief will return the dislocation to the initial position after stress removal. Hence, this dislocation motion makes no contribution

xmax

to the residual (plastic) strain of the crystal. In view of the above effect, let us analyze the dislocation motion with regard to only those stress fluctuations that initiate the maximum inflection of the dislocation segment: d xmax t . (13) 2

Substitution of xmax , xmax values found in (8) and (12) in (13) and consideration of inequalities (9) and (10) gives us the limitation on the fluctuation stress V f : 3V  V ext 4 Gbd or V 0  V ext  d Vf d 0 2 2 ð (1  í) l (14) 3V 0  V ext 4 Gbd . d V f d 2V 0  2 ð (1  í) l 2 For the range of V f satisfying limitation (14) not to be an empty set, it is necessary and sufficient that at least one of two-sided inequalities (14) has its left-hand side no greater than the right-hand side. The criterion for the condition to be met is V  V ext 4 Gbd . d 0 (15) 2 2 ð (1  í) l Condition (15) yields the limitation on the size of the solid subregion involved in fluctuation, since the dislocation segment length l found in this subregion and entered in (15) is proportional to its diameter D. Hereinafter the quantity l is taken to be the average dislocation segment length equal to 2 3 D1 ; thus, l 2 3 D. In this case, condition (15) is equivalent to the condition: D t D , (16) where 18 Gbd D . (17) ð (1  í)(V 0  V ext ) The foregoing allows the following conclusion. If stress fluctuation occurs in the solid subregion intersected by a dislocation, dislocation bending condition (13) is fulfilled in one, and only one, case where, first, the fluctuation satisfies inequality (16), i.e., covers a subregion of diameter D no less than D* , and, second, its value of V f is subject to limitation (14), i.e., to the condition: Ö d V f d Ø, where Ö V 0  V ext 

Ø

1

4 Gbd , ð (1  í) l 2 4 Gbd 2V 0  . ð (1  í ) l 2

(18)

(19)

The average length of a rectilinear segment confined in a sphere of diameter , with its extreme points on the sphere is understood as the height of a cylinder whose base coincides with the equatorial section of the sphere and whose volume is equal to the sphere volume. The thus found average length of the segment is 2 3 D.

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Note that in expressions (19), which specify ) and <, the last terms are determined by the parameter d from condition (13) and by the dislocation self-action toward the return of the bent dislocation segment to the rectilinear position. It is easy to verify that condition (16) is a necessary and sufficient condition for the inequality )d< (at l 2 3 D ) to hold and thus for the existence of V f values satisfying condition (18). With the stress V ext (0 < V ext < V 0 ) and fulfilled condition (16), the two-sided inequality < t ) > 0 holds true (the inequality < t ) follows from condition (16), and the inequality ) > 0 from first expression of (19) on the strength of V ext < V 0 ). The foregoing conclusions suggest that the stress fluctuation which fail to meet condition (16) or (18), i.e., those which cover solid subregions of diameter smaller than D* or have V f < ) or V f > <, fail to move dislocations on the macroscale. Below, we imply for brevity that these fluctuations are incapable of moving dislocations at all; accordingly, fluctuations which occur in solid subregions intersected by dislocations and satisfy (16) and (18) are capable of moving dislocations. 4. Probability of stress fluctuation in the vicinity of a dislocation Stress fluctuations may arise both in solid subregions containing dislocations and in those where they are absent. In the latter case, the fluctuations fail to initiate dislocation motion and, hence, plastic deformation of the solid. In this context, the question arises: What is the probability that a certain randomly chosen spherical solid subregion contains a dislocation if the dislocation structure of the solid is fixed?1 The dislocation structure of a solid, as a rule, is characterized by the so-called dislocation density — the total dislocation length per unit volume of a solid. Let us demonstrate with a simple example that knowledge of only this characteristic of the dislocation structure can be insufficient for determination of the above probability. Let there be parallel identical rectilinear dislocation segments in a solid. Assume that they are uniformly distributed in the solid volume. Then, the probability that a certain subregion contains a dislocation is proportional to the number of dislocations (with a subregion diameter smaller than the spacing of dislocations). However if these dislocations are spaced very closely and are nearly merging into a line, the probability in question is independent of the number of dislocations and is equal to the probability of intersection of a subregion with a dislocation. Thus in the general case, knowledge of the dislocation structure from the dislocation density only is really insuffi-

1

The notion “a subregion contains a dislocation” refers to the case where the subregion contains the entire dislocation as well as that where it contains only a certain dislocation segment.

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cient for determination of the probability of a certain solid subregion containing a dislocation. Further, we restrict our discussion to the case where the probability of interest depends only on the dislocation density in addition to the subregion diameter. For more accurate definition of this situation, we resort to the following considerations. Let us consider a set of all space points separated from dislocations by a distance smaller than D 2 , where D is a certain positive number much smaller than the minimum linear dimension of a solid L (0  D  L). Dislocations are normally continuous piecewise-smooth lines and therefore this set of points forms a set of curvilinear cylinders of radius D 2 with dislocations as their central lines. Let us choose a certain spherical subregion of diameter D in the solid. Clearly if the centre of the subregion is inside of any cylinder, the subregion is intersected by at least one dislocation, and if its centre is outside of the cylinders it escapes intersection with dislocations. We assume that with a randomly chosen subregion, its centre can fall, with equal probability, on any point of the solid (taking into account the condition D  L, we neglect the presence of a thin nearboundary layer of thickness D 2 , within which the centre of the subregion of diameter D can not fall). In this case, the probability that the chosen subregion contains a dislocation is apparently equal to the ratio of the solid volume occupied by the cylinders to the entire solid volume. Let the dislocations be no less than D 2 away from the boundary of the solid and no less than D apart. Thus, the cylinders are within the limits of the solid and are mutually nonintersecting. In this case, the solid volume occupied by the cylinders is equal to the sum of the cylinder volumes. Dislocations, as a rule, consist of fairly long near-rectilinear segments. Therefore, the sum of the cylinder volumes approximates the product of their cross-sectional area (the same for all cylinders) and the total length of their central lines equal to the total length of dislocations. Thus, the solid volume occupied by the cylinders is about 1 4 ðD 2 Ë, where / is the total length of dislocations in the solid. In the light of the foregoing, division of this quantity by the solid volume V gives us the probability of the chosen subregion containing a dislocation. The probability is 1 4 ðD 2 ë, where ë Ë V is the dislocation density. The obtained value of the probability can not be greater than unity. Hence, the diameter D of the solid subregion in question is bound to satisfy the condition D d 2 ðë . Thus in the case under consideration, the probability of interest depends only on the subregion diameter and dislocation density. Let us more accurately define the situation using the following statement. Let the solid contain a dislocation structure with a dislocation density O (O > 0). Let us randomly choose a spherical solid subregion whose diameter D satisfies the conditions:

N.F. Morozov and L.S. Shikhobalov / y Physical Mesomechanics 13 1–2 ( (2010) ) 1–11

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2 (20) and D  L, ðë where L is the minimum linear dimension of the solid. Considering that the dislocations are no less than D 2 away from the boundary of the solid and no less than D apart, the probability F that this subregion contains a dislocation is specified by the expression: ðD 2 (21) ÷|ë . 4 From the positive range of O and from the first condition of (20), it follows that 0 < F d 1.The quantity O1 2 entered in (20) is often interpreted as the average spacing of dislocations. In our further discussion, the subregion diameters are free of limitations (20). In this context, F specified by formula (21) can, by and large, be greater than unity. Therefore from this point on, the quantity F is a certain probability of a subregion containing a dislocation (hence for D > 0 and O > 0, it satisfies the inequality 0 < F d 1). Formula (21) is used at the end of the next section. In the final part of the work, it is demonstrated that with the model parameters corresponding to real crystals, the stress fluctuations actually affecting the dislocation motion are subject to conditions (20). For these fluctuations, the condition introduced before (21) for the relative position of dislocations and their position with respect to the solid boundary can be considered as formulated and fulfilled. All the foregoing means that for the stress fluctuations affecting the dislocation motion, the use of formula (21) is justified (and the inequality 0 < F d 1 holds true for them). Taking into account that to these fluctuations correspond subregions of diameters smaller than the characteristic length of rectilinear dislocation segments, we consider below, without going into details, that if a certain subregion contains a dislocation, this dislocation intersects the subregion all the way through (this assumption allows the use of the results of Section 3). 0Dd

5. Probability of the absence of stress fluctuations sufficient for dislocation motion Let external forces produce a rectangular V ext stress pulse of duration W in a solid. We assume as before that 0 < V ext < V 0 . Let us calculate the probability that during the pulse, no stress fluctuation sufficient for dislocation motion arises and the solid escapes plastic deformation. For this to occur, it is necessary and sufficient that during the pulse, each oscillator generating stress fluctuations fails to produce a fluctuation responsible for dislocation motion. Because the oscillators are independent, the desired probability is equal to the product of the probabilities of these events for all oscillators. Let us consider oscillators for which condition (16) is invalid, i.e., they produce stress fluctuations in solid subre-

gions of diameters smaller than D* . It follows from the results of Section 3 that the stresses produced by these oscillators are a fortiori insufficient for dislocation motion. This means that each of the oscillators contributes to the probability of interest in the form of a factor equal to unity (from this it follows that these oscillators have little or no effect on the probability). Now let us consider a certain oscillator for which condition (16) holds true. Let the oscillator produce a flash of stress fluctuation at a certain point in time. According to the results of Section 3, this flash provokes dislocation motion in one, and only one, case where, first, it covers the solid subregion intersected by a dislocation and, second has a value satisfying inequality (18). The first of these conditions is realized with a probability F, the second with a probability equal to the integral of the function P (V f ) over [), <]. Because these conditions are independent, the probability that the flash initiate dislocation motion is equal to the product of the function F and integral of P (V f ). Hence it follows that the probability of the reverse situation, i.e., where the flash fails to provoke dislocation motion, is equal to the difference between unity and above probability. Thus, the probability that the flash of stress fluctuation produced by the oscillator at a certain point in time fails to provoke dislocation motion is equal to Ø

1  ÷ ³ P (V f ) dV f ,

(22)

Ö

where the functions F and P (V f ) are determined respectively by expressions (21) and (5), and the limits of integration ) and < by expressions (19). The quantity (22) is positive and is no greater than unity due to 0 < F d 1, positive range of the function P (V f ), normalization condition (6) and inequality < t ) > 0 (valid with V ext < V 0 and condition (16)). Let us assume for a while that the ratio ô ô f is an integer (W is the duration of an external stress pulse, ô f is the duration of fluctuations produced by the oscillator). Hence in a time W,the oscillator with a frequency í f 1 ô f (see (2)) produces ô í f ô ô f flashes of stress fluctuations. Assuming these flashes independent, we conclude the following. The probability that in a time W, the oscillator fails to produce flashes of stress fluctuation responsible for dislocation motion is equal to the product of probabilities (22) for all flashes: Ø( D ) § ¨1  ÷(D ) P(V , D) dV f ³ f ¨ Ö(D) ©

ô

· ôf ( D ) ¸ , ¸ ¹

(23)

where F, ), <, P and ô f depend on the diameter D of subregions covered by the flashes (), < and P depend on D indirectly through l and :; remind that to each oscillator corresponds a quite definite value of the diameter D). In the calculations of the number of flashes, we assumed that the number ô ô f is an integer. Now we discard this

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assumption and take expression (23) valid at any W> ô f (it is assumed that W> ô f rather than W > 0 to fit the assumption of the V ext invariance during the fluctuation made in Section 3). Let us consider other oscillators for which condition (16) holds true. Before, we assumed that the solid subregions involved in fluctuations are D  [ a, L ] in diameter. We put D*  [a, L) in (16). Hence, the interval [ D* , L] is nonzero in length and falls within the diameter range [a, L]. Let us partition the interval [ D* , L] into n equal subintervals 'D and number the points of division with indices i (i = 1, 2, }, n, D* = D1 , L = Dn1 ). Let us take up oscillators generating fluctuations in subregions of diameters from a certain interval [ Di , Di1 ], where Di1 Di  'D. The number of these oscillators is specified by the quantity Q ( Di ) 'D expressed through formula (3) at D = Di and dD = 'D. Assuming that 'D is small, we assume that to all these oscillators corresponds the same diameter equal to Di . Hence, due to the independence of the oscillators, the probability that in a time W they fail to produce stress fluctuation responsible for dislocation motion is equal to the product of quantities (23) in amounts of Q ( Di ) 'D (with replacement of D by Di in (23)): ô

Q ( D ) 'D

Ø( Di ) § · ôf ( Di ) i ¨1  ÷(D ) P (V , D ) dV ¸ , (24) i ³ f i f¸ ¨ Ö ( Di ) © ¹ and the accuracy of this expression increases with a decrease in 'D, i.e., with an increase in the number of points of division n, since the quantity Q ( Di ) 'D specifies the number of oscillators in the linear approximation in 'D and these oscillators are assigned the same diameter equal to Di . It is likely that a similar probability for all oscillators satisfying condition (16) is expressed by the product of quantities (24) for all values of i from 1 to n with n o f in the expression derived. Remind that the oscillators for which condition (16) is invalid contribute to the probability of interest in the form of factors equal to unity. Taking into account the preceding, we arrive at the following. The probability p that in a time W (during the pulse of external loading), no stress fluctuations sufficient for dislocation motion arise is expressed as:

p

ô ª Q ( D ) 'D Ø( D1 ) · ôf ( D1 ) 1 «§¨ ¸ u ... lim « 1  ÷( D1 ) ³ P(V f , D1 ) dV f ¸ no  f «¨ D Ö ( ) © ¹ 1 «¬

Ø( Dn ) § ... u ¨1  ÷(Dn ) ³ P (V f , Dn ) dV f ¨ Ö ( Dn ) ©

ô

· ôf ( Dn ) ¸ ¸ ¹

Q ( Dn ) 'D º

» ». » ¼»

(25)

7

Each of the cofactors in the right-hand side of (25) is a certain value of quantity (22) raised to a positive power. Because quantity (22) is positive and is no greater than unity, p satisfies the inequality 0 < p d 1. Let us transform expression (25). We take a natural logarithm of both sides of equality (25). Using the properties of logarithmic functions, we obtain: n ô ln p lim ¦ Q( Di ) u no  f i 1 ô f ( Di ) Ø( Di ) § · u ln ¨1  ÷( Di ) ³ P (V f , Di ) dV f ¸ 'D. ¨ ¸ Ö ( Di ) © ¹ In this equality, the expression under the limit sign is the integral sum for a certain function defined on the interval [ D* , L ] of the diameter D. Hence, the right-hand side of the equality is equal to the corresponding integral; thus, we can write L ô ln p ³ Q( D ) u D ô f ( D ) Ø( D ) · § (26) u ln ¨1  ÷( D ) ³ P (ó f , D ) dó f ¸ dD. ¸ ¨ Ö( D ) ¹ © Substitution of the values of PdV f from formula (5) in (26) gives: L ô ln p ³ Q( D )u D ô f ( D )

where

§ 2 z1  z 2 ·¸ uln ¨1  ÷( D) ³ e dz ¸ dD , ¨ ð z0 © ¹ Ù óf ; z0 2GkT

z

Ù Ö; z1 2GkT

(27)

Ù Ø. (28) 2GkT

Note that z1 t z 0 > 0 on the interval of integration [ D* , L ], because for D t D* and 0 < V ext < V 0 we have < t ) > 0. Let us express the inner integral in (27) as the difference of integrals:

ln p

L

ô

Q( D ) u ³ D ô f ( D )

ª § 2 z1  z 2 2 z0  z 2 ·¸º uln «1  ÷(D)¨ e dz  ³ ³ e dz ¸» dD. (29) ¨ ð ð 0 «¬ 0 © ¹»¼ 2 2 Assume that z0 !! 1. Then, z1 !! 1, since z1 t z0 > 0. In this case, for calculations of the inner integrals in equality (29), we can use the known asymptotic series [8] whose form in our notations is z 2 0 z2 ³ e dz | ð 0 2

|1

º 1 1˜ 3 1˜ 3 ˜ 5 e z0 ª  2 4  3 6  K» . «1  2 2 z0 2 z0 2 z0 ð z0 ¬« ¼»

N.F. Morozov and L.S. Shikhobalov / y Physical Mesomechanics 13 1–2 1–11 ( (2010) )

8

In so doing, with a finite number of terms of the series, the error is less than the absolute value of the last term. Taking into account that this series is alternating and for z02 !! 1 each of its first terms is much greater than the next one, we consider only the first term in square brackets in this series. Then, the logarithmic function in (29) can be transformed as follows:

ª § 2 z1  z 2 2 z0  z 2 ·¸º ln «1  ÷(D )¨¨ ³ e dz  ³ e dz ¸» | ð 0 «¬ © ð 0 ¹»¼ 2 ª ÷( D ) § e  z02 e  z1 ·¸º» ¨ | ln «1   | z1 ¸» « 𠨩 z0 ¹¼ ¬ 2 2 z z   e 1 ·¸ ÷( D ) §¨ e 0 |  , z1 ¸ 𠨩 z 0 ¹ where the expansion of the logarithmic function into a power series includes only the first term on the strength of z02 !! 1, z12 !! 1 and 0 < F d 1. Let us substitute the approximate value of the logarithmic functions in equality (29); then, multiply the both sides of the equality by –1; take constant values outside the integral sign in the right-hand side; and finally, interchange the position of the right- and left-hand sides of the equality. Thus, we obtain 2 2 1 L Q( D ) ÷(D) ¨§ e  z0 e  z1 ¸· 1 ô ³ ô ( D ) ¨ z  z ¸ dD | ln p . (30) ð D f 1 ¹ © 0

Using the values of Wf , Q, ), <, F, z0 and z1 following from formulae (1), (3), (19), (21), (28) and considering that Ù 1 6 ðD 3 and l 2 3 D , we can express dependence (30) as:

3 3 ô ëNa 3c t GkT 8 ­ § 3 ° ¨ ðD u ®exp ¨  ¨ 12GkT °¯ ©

L

³

D

1 D

4

D

u

ª 9Gbd º «V 0  V ext  » ð (1  í ) D 2 ¼» ¬«

2

· ¸ ¸¸ u ¹

1

ª 9Gbd º u «V 0  V ext  »  ð (1  í) D 2 »¼ «¬ 2· § 3 9Gbd º ¸ ¨ ðD ª2  exp ¨  V  « 0 » u ¨ 12GkT ¬« ð (1  í ) D 2 ¼» ¸¸ © ¹ 1 ½ ª 9Gbd º ° 1 (31) u «2V 0  dD | ln . 2» ¾ p ð (1  í) D ¼» ° ¬« ¿ Let us express the model characteristics D, D* , b, d and L in terms of the lattice cell parameter a with the use of the dimensionless coefficients: D

D a , D

D a, b

â a, d

ã a, L

; a.

(32)

Because D  [ a, L ], D*  [a , L ), b | a, d | a, L !! a, the entered coefficients have the following values: 1 d D d ;, 1 d D* < ;, E | 1, J | 1, ; !! 1. Let us introduce the notations:

ðGa 3 9 âã , C . (33) 12 k ð (1  í) 3 ð ë N ct a Here W0 has the dimension of time, T0 has the dimension of temperature, C is a dimensionless quantity. By entering the quantities from (32) in (31) and using notations (17) and (33), we derive the final formula: 16

ô0

ô ô0

T T0

, T0

;

³

f (D, V 0 , V ext , T ) dD | ln

D ( V0 , Vext )

1 , p

(34)

where the integrand function: 2 § T ª V V 1 ­° º ·¸ ext ¨  0 D2 0 exp  f C ® » ¸u ¨ TD «¬ G D 2 D °¯ ¼ ¹ © 1

ª V  V ext º u «D 2 0  C»  G ¬ ¼ 2 1 § T ª 2V º · ª 2V º ½°  exp ¨  0 «D 2 0  C » ¸ «D 2 0  C » ¾ (35) ¨ TD ¬ G G ¼ ¸¹ ¬ ¼ °¿ © and the lower limit of integration:

D

2CG . V 0  V ext

(36)

Formula (34) establishes the relation between the duration W of an external stress pulse and the probability p that during the pulse, stress fluctuations responsible for dislocation motion have no chance to arise. Note that formula (34) allows no passage to the limit of the external stress V ext = 0, since its derivation neglects the fluctuations responsible for dislocation motion in opposition to the external stress and this is justified only with rather high V ext . 6. Effect of the loading pulse duration on the onset of plastic deformation Let us resort to formula (34) to determine the duration W of an external stress pulse within which the stress fluctuation fails to shift a dislocation with a certain prescribed probability p. The calculation is applied to an Al specimen. All quantities on which the left-hand side of formula (34) depends are uniquely expressed through the known characteristics of the material or the parameters measured directly in a test. The only exception is the resistance V 0 , since the external stress V ext rather than V 0 is measured in experiment. The resistance V 0 can be estimated as the limiting value of the material yield strength at T o 0 K (because in this case, no thermal atomic oscillations conductive to dislocation motion takes place and hence for the onset of dislocation motion, the external stress V ext is bound

N.F. Morozov and L.S. Shikhobalov / Physical Mesomechanics 13 1–2 ((2010)) 1–11 y

to exceed V 0 ). According to experimental data [9], it can be concluded that V 0 lies in the range (10 410 2 )G (it depends on the chemical composition, lattice type, impurity content, state of the defect structure and other characteristics of material). Let us consider an Al specimen in the shape of a disk of radius R = 1.5 cm and thickness L = 5 mm (which is close to the specimen shape and size used in impact tests). Then, the specimen volume V SR 2 L | 3.5 ˜ 10 6 m 3. Aluminum has a fcc lattice and hence has four atoms per lattice cell. The lattice cell parameter of aluminum a | 4 ˜ 1010 m. In view of this fact, we find the total number of atoms in the specimen: N 4V a 3 | 2.2 ˜ 10 23. We use the following characteristics of aluminum: G | | 2.65 ˜ 1010 Pa, Q | 0.35, ct | 3.15 ˜ 103 m s (the temperature dependence of these parameters is ignored). In the fcc lattice, the least magnitude of the Burgers vector b is a 2, and therefore we put â b a 1 2 . The quantity d entered in condition (13) is taken to be the spacing a 3 (2 2 )  between close-packed atomic rows. Then, ã d a 3 (2 2 ). (These values of b and d roughly agree with the characteristics of screw dislocations in aluminum.) Assume that the dislocation density is O = = 1012 m 2 , the resistance to dislocation motion is V 0 = = 0.2 GPa | 7.5 ˜ 10 3 G. Remind that k | 1.38 ˜ 10 23 J K . For the above parameters, we have from (33): C | 1.91, W 0 | 1.09 ˜ 10 29 s, T0 | 3.22 ˜ 10 4 K. Let us analyze the dependence of the integrand function f in formula (34) on the integration variable D. According to expression (35), this function vanishes at D = D * , since in view of (36) all quantities in square brackets in (35) take on the same value in this case. Evidently, the function f tends to zero at D o +f. Moreover, this function is nonnegative throughout the interval of integration [ D* , ; ]. This is easily verified with the use of formula (30) equivalent to (34) considering that z1 t z 0 ! 0 on the interval of integration [ D* , L ]. Figure 1 shows the function f at D t D* (for the above parameters and V ext = 0.13 GPa, T = 293 K). The function is a narrow peak positioned to the right of D = D* . A similar curve shape is found for other values of V ext and T used below. This fact allows the important conclusion. Actually, according to formula (34), if the function f vanished on the entire interval of integration [D* , ;], no dislocation motion would occur with a probability p = 1. This means that the dislocations are set in motion only by those stress fluctuations that provide a nonzero function f. It can be demonstrated that for all values of V ext and T considered below, the peak of the function f similar to that shown in Fig. 1 is within the interval [ D* , 27], where D*min | | 17, and the integral of the function f over the interval [27, ;] is negligible compared to the integral over [ D* , 27].

9

Fig. 1. Function B versus the diameter of subregions involved in stress fluctuations (D is the diameter in terms of the lattice cell parameter)

In view of D = Da, D* D *a and formulae (34), we conclude the following. The dislocation motion is determined mainly by those stress fluctuations that cover solid subregions of diameters D only little greater than D* , i.e., those with frequencies Q f close to í f ct D (here c t is the transverse wave velocity; D* is specified by expression (17)). In particular with the values of V ext and T considered below, the stress fluctuations covering subregions of diameter D  [17 a , 27 a ] play, according to the above, a dominant role in the dislocation motion. The function f thus reaches a maximum at D  [17.5, 19.5] (i.e., at D | 18.5a | | 7.4 ˜ 10 9 m), and the characteristic frequency í f c t D | c t (17 a ) | 4.6 ˜1011 Hz. The formulated conclusion proves several assumptions made in constructing the model under discussion. Actually, with the values of V ext and T considered, the dislocation motion owes mainly to the stress fluctuations in subregions of diameters D  [17a, 27a]. Because these diameters are much greater than the lattice cell parameter a, the proposed interpretation of thermal atomic oscillations as a stress fields in an elastic medium is justified. Using formula (3), it can be demonstrated that the total volume of these subregions is a0.7 of the solid volume and this justifies the idea of the fluctuation stress field as a set of individual flashes of fluctuations arising in different solid subregions. The conditions z02 !! 1 and W > Wf are also justified (because with the specified parameters, z02 ! 37, and the minimum duration of shock loading W | 10 8 s attained in experiments [1] is much greater than the duration of characteristic fluctuation ô f 1 í f D c t | 2.2 ˜10 12 s). Also valid with the specified diameters is the assumption D*  [a, L) (since D* | 17a, a  L) and conditions (20) (since for the dislocation density O 1012 m 2 in (20), 2 ðë | 1.1 ˜10 6 m | | 2 800 a ). Following (34), let us analyze the dependence of the external stress V ext on the pulse duration W and tempera-

N.F. Morozov and L.S. Shikhobalov / Physical Mesomechanics 13 1–2 (2010) 1–11

10

Fig. 2. External stress V ext in the loading pulse at which the specimen escapes plastic deformation with a probability F (Vext versus the pulse duration W)

Fig. 4. External stress V ext versus the absolute temperature 6

ture T for the previously specified values of the parameters and two values of the probability p = 0.99 and 0.01. This dependence is plotted in Figs. 2– 4. The presented data allow the following conclusion. If V ext , W and T are such that the point corresponding to them in Figs. 2– 4 is on or beneath the curve for p = 0.99, dislocations stay immobile with a probability of no less than 0.99, i.e., no plastic deformation takes place. If this point is on or above the curve for p = 0.01, the situation is the reverse and plastic deformation occurs with a probability of no less than 0.99. Noteworthy is that the curves corresponding to p = 0.99 and p = 0.01 are close to each other. Let us take the material yield strength as an external stress V ext , at which dislocations start to move. Then, the presented results allow us to state the following. The material yield strength is probabilistic; it can, by and large, assume different values even for identical specimens exposed to identical action. At the same time, the difference in yield strength for identical specimens is small enough. It can be concluded that the material yield strength is characterized by the stress V ext within the narrow zone between the curves V ext ( W) (or V ext (T )) for p = 0.99 and p = 0.01 (surely, other values of the probability, e.g., 0.999 and 0.001, can be used). Figures 2 and 3 show the functions V ext ( W) for two widely different ranges of the argument W: the duration W of the loading pulse in Fig. 2 varies about tenfold (from ~ 10 7 s to 10 6 s), and that in Fig. 3, about 15 times (from 10 8 s to 10 7 s). The foregoing data suggest that the ten-

fold variation of the loading pulse duration has little or no effect on the yield strength. In this case, it can be stated that the specimen displays so-called athermal plasticity. However, the variation in the loading pulse duration over several orders of magnitude does lead to wide variations in the yield strength. Moreover, as it follows from extrapolation of the plot in Fig. 3 to the range of higher W, even small external stress V ext can initiate plastic deformation of the specimen with time. Hence with any, even small, load the material undergoes creep.1 From the plots in Fig. 4 it follows that the yield strength of the material decreases with increasing its temperature, as does happen with the assistance of thermal atomic oscillations in dislocation motion. Let us consider the case where the characteristics of external loading — the stress V ext and the duration W — correspond to the point A above the upper curve in Fig. 3. In this case, the specimen experiences plastic deformation. Now we decrease the pulse duration W keeping the stress V ext constant (this variation fits in the motion of the image point along the segment AB). As soon as the pulse characteristics come into correspondence with the point B on the lower curve, the specimen escapes plastic deformation (with 0.99 probability). To the point B in Fig. 3 correspond V ext | 0.13 GPa and W | 10 6 s. Thus with a short loading pulse duration, the absence of plastic deformation of the specimen is possible, even though with a longer pulse at the same stress the specimen undergoes plastic deformation. The foregoing effect occurs because the stress fluctuations required for dislocation motion have no chance to arise during the loading pulse. This effect, i.e., the absence of plastic deformation with a short pulse, can be eliminated by increasing the stress in the pulse (because, according to

 

1

Fig. 3. Dependence V ext ( W) from Fig. 2 for a much wider range of W

In Egyptian pyramids, there is hardly any spacing between stone blocks. It is conceivable that this is due to the above effect: in thousands of years, the spacing of the blocks disappeared due to the creep phenomenon. The validity of this assumption can be verified by comparing the spacing in the upper and lower parts of a pyramid where the stresses produced by the gravity force of the overlying blocks are widely different.

N.F. Morozov and L.S. Shikhobalov / Physical Mesomechanics 13 1–2 (2010) 1–11

formulae (34) – (36), even a slight increase in stress V ext causes a considerable decrease in the probability p of the absence of plastic deformation of a solid; the same is directly evident from Figs. 2– 4). The result can be interpreted as the increase in material yield strength with decreasing the loading duration. Note that this conclusion refers not only to shock loading, but to any type of rectangular loading of solids. In summary, we would like to make several remarks. 1. According to the foregoing, for the pulse characteristics corresponding to the point B in Fig. 3, the time for which the stress fluctuations fail to shift dislocations is about 10 6 s. This means that the dislocation motion is jumping with an average frequency no greater than 10 6 Hz. Let us compare this frequency with the average frequency of stress fluctuations. According to formulae (2) and (3), the average frequency of fluctuations is  

 

L

íf

³ íf Q( D) dD a

L

³ Q( D ) dD

a

L

|

³ a

c t 3 Na 3 dD D 4D 4 3c | t | 6 ˜ 1012 Hz, N 4a 4

where we take into account that a  L and use the previously indicated values of a and c t . Hence it follows that the average frequency of stick-slip dislocation motion is many orders of magnitude lower than the average frequency of fluctuations. Note that this effect is comparable with Brownian motion of a particle suspended in a liquid, where the frequency of macroscopic jumps of the particle is also many orders of magnitude lower than the frequency with which liquid molecules impact on the particle. 2. Although the crystal in the work is a continuous medium, we have twice resorted to its discrete atomic structure. First, it is this property of a crystal that gave grounds to assume the existence of nonzero minimum size (equal to the lattice cell parameter) of solid subregions involved in fluctuations. Second, the discrete atomic structure was used to introduce condition (13) that gives limitations (16) and (18) on the diameters of solid subregions involved in fluctuations and on the fluctuation stress responsible for dislocation motion. 3. The revealed effect of the absence of plastic deformation with a short duration of external loading depends heavily on condition (13). We used this condition to emphasize that the dislocation motion contributes to the residual (plastic) strain of the crystal in one, and only one, case where

11

the maximum dislocation inflection xmax is no less than a certain value d 2 ; moreover, d > 0. If we assume that the inflection xmax can take on any nonnegative value, which corresponds to d = 0 in (13), the above effect does not occur. More precisely, formula (34) suggests that plastic deformation occurs with a probability of no less than 0.99 at any W> 10 29 s. This is because the contribution to plastic deformation in this case is by any arbitrary small dislocation motion. Note that at d = 0, C = 0 (on the strength of (32) and (33)). Therefore, the exponents for the function f in expression (35) become proportional to  D 3 and the function f is monotonically descending in D (at D > 0) rather than has the shape of the peak shown in Fig. 1. Moreover in this case, D* = 0 and D* = 0 (this follows from the equality C = 0 and formulae (36) and (32)); therefore the lower limit of integration in (34) should be taken equal to unity (because D [1, ;]). Thus, the results of the work allow the conclusion that decreasing the duration of loading of material increases its yield strength. This conclusion follows from formulae (34)– (36) and also directly from the data presented in Fig. 3. Some of the reported results were obtained earlier in [10]. The authors are thankful to S.S. Vallander and Ya.Yu. Nikitin for discussion of the paper and useful remarks. References [1] G.I. Kanel, V.E. Fortov and S.V. Razorenov, Shock waves in condensed-state physics, Physics-Uspekhi, 50 No. 8 (2007) 771. [2] A.A. Vakulenko and L.S. Shikhobalov, A new approach to the interpretation of the Bordoni internal friction peak in crystals, Dokl. AN SSSR, 259, No. 6 (1981) 1323 (in Russian). [3] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1972. [4] L.D. Landau and E.M. Lifshitz, Course of Theoretical Physics: Statistical Physics, Vol. 5, Pergamon Press, Oxford, 1980. [5] J.P. Hirth and J. Lothe, Theory of Dislocations, McGraw-Hill Book Company, New York, 1970. [6] L.S. Shikhobalov, Equation of Motion for a Dislocation in the Continuum Model, in Problems of Mechanics of Solids, Izd-vo LGU, Leningrad (1982) 73 (in Russian). [7] L.S. Shikhobalov, Study of the Relation of Micro- and Macroplasticity: Cand. Degree Thesis (Phys. & Math.), Izd-vo LGU, Leningrad, 1987 (in Russian). [8] H.B. Dwight, Tables of Integrals and Other Mathematical Data, The Macmillan Company, New York, 1947. [9] Physical Metallurgy, Ed. by R.W. Cahn, North-Holland, Amsterdam, 1965. [10] N.F. Morozov and L.S. Shikhobalov, Effect of impact loading duration on yield strength, Doklady Physics, 53, No. 10 (2008) 529.