Open access peer-reviewed chapter

Study on Dislocation-Dopant Ions Interaction during Plastic Deformation by Combination Method of Strain-Rate Cycling Tests and Application of Ultrasonic Oscillations

Written By

Yohichi Kohzuki

Submitted: February 19th, 2020 Reviewed: April 20th, 2020 Published: May 20th, 2020

DOI: 10.5772/intechopen.92607

From the Edited Volume

Electron Crystallography

Edited by Devinder Singh and Simona Condurache-Bota

Chapter metrics overview

695 Chapter Downloads

View Full Metrics


Strain-rate cycling tests associated with the ultrasonic oscillation were conducted for the purpose of investigation on the interaction between dislocation and dopant ions during plastic deformation of seven kinds of single crystals: NaCl doped with Li+, K+, Rb+, Cs+, F−, Br− or I− ions separately. Relative curves between the stress decrement (Δτ) due to ultrasonic oscillatory stress and strain-rate sensitivity (λ) of flow stress under superposition of the oscillation are obtained by the original method (combination method of strain-rate cycling tests and application of ultrasonic oscillations) at 77 K to room temperature and have stair-like shapes for the specimens at low temperatures. The Gibbs free energy for overcoming of the dopant ion by dislocation at absolute zero is calculated from the data analyzed in terms of Δτ vs. λ. As a result, the obtained energies are found to be varied linearly with the isotropic defect around it in the each specimen.


  • dislocation
  • ultrasonic oscillatory stress
  • activation energy
  • monovalent ion
  • isotropic strain

1. Introduction

Dislocation (linear defects in crystal) motions are related to the plasticity of crystal in a microscopic viewpoint. It is well known that the solution hardening depends on dislocation motion hindered by the atomic defects around impurities in crystals and is namely influenced by the dislocation-point defects interaction, which has been widely investigated by various methods. For instance, measurements of yield stress (e.g., [1, 2, 3, 4, 5, 6, 7]) and proof stress (e.g., [8, 9]), micro-hardness tests (e.g., [10, 11, 12, 13, 14]), direct observations of dislocation (e.g., [15, 16, 17, 18, 19, 20, 21]), internal friction measurements (e.g., [22, 23, 24, 25, 26, 27]), or stress relaxation tests (e.g., [28, 29]) have been carried out so far. Nevertheless, it is difficult to obtain such information on the motion of the dislocation which moves by overcoming the forest dislocations and the weak obstacles such as impurities during plastic deformation of bulk. A large number of investigations have been conducted by the separation of the flow stress into effective and internal stresses on the basis of the temperature dependence of yield stress, the strain rate dependence of flow stress, and the stress relaxation. Yield stress depends on dislocation velocity, dislocation density, and multiplication of dislocations [30]. On the other hand, the effect of heat treatment on the micro-hardness is almost insensitive to the change of atomic order of point defects in a specimen. As for direct observations, electron microscopy provides the information on dislocation motion for a thin specimen but not for bulk, and also light scattering method is useful only for a transparent specimen. X-ray topography is the lack of resolution in the photograph, so that the specimen is limited to the low dislocation density below 104 cm−2. Internal friction measurements concern the motion of the dislocation which breaks away from the weak obstacles between two forest dislocations by vibration [31]. Stress relaxation tests are generally assumed that internal structure of crystals does not change, i.e., dislocation density and internal stress are constant. Above-mentioned methods cannot provide the information on dislocation-obstacles interaction in bulk during plastic deformation.

In this chapter, the study on interaction between a dislocation and dopant ions is made by the strain-rate cycling tests during the Blaha effect measurement. The original method (strain-rate cycling tests associated with the Blaha effect measurement) is different from above-mentioned ones and would be possible to clear up it. The Blaha effect is the phenomenon that static flow stress decreases when an ultrasonic oscillatory stress is superimposed during plastic deformation [32]. Ohgaku and Takeuchi [33, 34] reported that the strain-rate cycling under the application of oscillation can separate the contributions arising from the interaction between a dislocation and dopant ions and from the dislocations themselves during plastic deformation at room temperature. Using ionic single crystals of KCl doped with Br (0.5, 1.0, and 2.0 mol%) or I (0.2, 0.5, and 1.0 mol%) [35] and of NaCl doped with Br (0.1, 0.5, and 1.0 mol%) [36], they discussed temperature dependence of the effective stress due to monovalent dopants (i.e., Br or I) and found that the measurement of strain-rate sensitivity under the ultrasonic oscillatory stress provides useful information on a mobile dislocation-the dopant ions interaction [35, 36]. The information on the dislocation motion breaking-away from dopant ions [37, 38, 39, 40] and also X-irradiation induced defects [41] with the ultrasonic oscillatory stress has been successively provided by the original method, which seemed to separate the contributions arising from the dislocation-the point defects interaction and from dislocations themselves during plastic deformation of crystals.

The Blaha effect was found by Blaha and Langenecker when the ultrasonic oscillatory stress of 800 kHz was superimposed during plastic deformation of Zn single crystals. The same phenomenon as Zn crystals has been also observed in many metals (e.g., [42, 43, 44]). Since this phenomenon has a significance as an industrial purpose, it has been widely made to apply to the plastic working technique: wire drawing, deep drawing, rolling, and another metal forming techniques (e.g., [45, 46, 47, 48, 49, 50, 51, 52, 53]).

The strain-rate cycling tests associated with ultrasonic oscillation were carried out here for NaCl single crystals doped with various monovalent ions separately. The monovalent ion is considered to have isotropic strain in the alkali halide crystal because its size is different from the substituted ion of the host crystal. Dopant ions are expected to cause the hardening due to the dislocation motion hindered by the defects around them at low temperature. Its force-distance profile between a dislocation and an atomic defect is expressed by Cottrell and Bilby [54]. This chapter refers to the energy supplied by the thermal fluctuations, when the dopant ions are overcome by a dislocation with the help of thermal activation during plastic deformation of crystals. This is estimated from the dependence of the effective stress due to impurities on activation volume, which reveals the force-distance profile, given by the measurement of the stress decrement due to application of ultrasonic oscillatory stress and strain-rate sensitivity of flow stress under superimposition of ultrasonic oscillation. And further, it is presented that the difference in size of isotropic strain around the various dopants different from host ion has an influential factor of the energy for overcoming the dopant ion by a dislocation in several kinds of alkali-halide single crystals.


2. Combination method of strain-rate cycling tests and the Blaha effect measurement

Specimens used in this work were seven kinds of single crystals: NaCl doped with Li+, K+, Rb+, Cs+, F, Br or I ions separately. Each concentration of the dopants was 0.5 mol% in the melt. The specimens were prepared by cleaving the single crystalline ingots, which were grown by the Kyropoulos method [55] in air, to the size of 5 × 5 × 15 mm3. Furthermore, they were kept immediately below the melting point for 24 h and were gradually cooled to room temperature at a rate of 40 K h−1. This heat treatment was carried out for the purpose of reducing dislocation density as much as possible.

The schematic illustration of apparatus is shown in Figure 1. A resonator composed of a vibrator and a horn was attached to the testing machine, INSTRON Type 4465. The specimens were lightly fixed on a piezoelectric transducer and then cooled down to a test temperature. Each specimen was held at the test temperature for 30 min prior to the following test. The specimens were deformed by compression along the <100> axis at 77 K up to the room temperature, and the ultrasonic oscillatory stress was intermittently superimposed for 1 or 2 min by the resonator in the same direction as the compression. The temperature measurements of specimens were conducted by heater controlled using thermocouples of Ni-55%Cu vs. Cu. As for the tests at 77 K, the specimen was immersed in the liquid nitrogen. The stability of temperature during the test was kept within 2 K. The resonant frequency was 20 kHz from a multifunction synthesizer and the amplitude of the oscillatory stress was monitored by the output voltage from the piezoelectric transducer set between a specimen and the support rod, which was observed by an a.c. voltmeter or an oscilloscope. Since the wavelength, which is 226 mm on the basis of calculating from the data of ref. [56], is 15 times as long as the length of specimen, the strain of specimen is supposed to be homogeneous.

Figure 1.

Schematic block diagram of apparatus system.

Strain-rate cycling tests made between the crosshead speeds of 10 and 50 μm min−1 were performed within the temperatures. The strain-rate cycling test associated with the ultrasonic oscillation is illustrated in Figure 2. Superposition of oscillatory stress (τv) causes a stress drop (Δτ) during plastic deformation. When the strain-rate cycling between strain-rates of ε̇1 (2.2 × 10−5 s−1) and ε̇2 (1.1 × 10−4 s−1) was carried out keeping the stress amplitude of τv constant, the variation of stress due to the strain-rate cycling is Δτ’. The strain-rate sensitivity (Δτ’/Δlnε̇) of the flow stress, which is given by Δτ’/1.609, was used as a measurement of the strain-rate sensitivity (λ = Δτ’/Δlnε̇). Slip system for rock-salt structure such as NaCl crystal is {110} <11_0> so that shear stress (τ) and shear strain (ε) calculated for the slip system were used in this study.

Figure 2.

Explanatory diagram of a change in applied shear stress, τa, for the strain-rate cycling test between the strain rates, ε̇1 (2.2 × 10−5 s−1) and ε̇2 (1.1 × 10−4 s−1), under superposition of ultrasonic oscillatory shear stress, τv .


3. Relation between stress decrement (Δτ) and strain-rate sensitivity (λ)

Figure 3 shows the influence of temperature on Δτ vs. λ curve for the NaCl:Rb+ (0.5 mol%) single crystals at strain 6%. The variation of λ with Δτ has stair-like shape: two bending points and two plateau regions are on the each curve. That is to say, the first plateau region ranges below the first bending point at low Δτ and the second one extends from the second bending point at high Δτ. λ gradually decreases with increasing Δτ between the two bending points. The length of Δτ within the first plateau region is named τp as denoted in Figure 3. τp tends to be lower at higher temperature. Similar phenomena as Figure 3 are observed for all the other NaCl single crystals contained with the monovalent impurities (i.e. Li+, K+, Cs+, F, Br or I ions).

Figure 3.

Relation between the stress decrement (Δτ) and the strain-rate sensitivity (λ) for NaCl:Rb+ (0.5 mol%) at strain 6% and various temperatures. The numbers besides each symbol represent the temperature (reproduced from Ref. [58] with permission from the publisher).

The relation between Δτ and λ reflects the effect of ultrasonic oscillation on the dislocation motion on the slip plane containing many weak obstacles such as impurities and a few forest dislocations during plastic deformation [40]. Δτ vs. λ curve is divided into three regions as shown in Figure 3. Within the first plateau region of relative curve (i.e. region 1 in Figure 3), the application of oscillation with low stress amplitude cannot influence the average length of dislocation segments (l¯) and l¯ is considered to remain constant. All weak obstacles act as impedimenta to the dislocation motion there. In region 2, the dislocation begins to break-away from the weak ones between the forest dislocations by applying oscillation with high stress amplitude. As a result, l¯ begins to increase and the λ of flow stress starts to decrease at the stress decrement Δτ of τp. This is because λ is inversely proportional to l¯ [57]. Some weak obstacles stop acting as impedimenta in the region. The weak obstacles are supposed to be monovalent dopants (Li+, K+, Rb+, Cs+, F, Br or I ions) and not to be vacancies here, since the vacancies have low density as against the dopants in the specimen. When the specimens were plastically deformed, it is imagined that a dislocation begins to overcome from the dopants which lie on the dislocation with the help of thermal activation. Then, τp is considered to represent the effective stress due to the ions. Accordingly, τp is expected to decrease with increasing temperature. Δτ vs. λ curves shown in Figure 3 correspond to this. As the temperature becomes larger, τp shifts in the direction of lower Δτ. τp depends on type and density of the weak obstacle [36, 38]. Applying still larger stress amplitude during plastic deformation of the specimens, the second plateau region within stage 3 becomes to appear on the relative curves in Figure 3. In stage 3, the dopants are no longer act as the impedimenta to mobile dislocations and the dislocations are hindered only by forest dislocations. Then, l¯ becomes constant again. This leads to the constant λ of flow stress. λp denoted in Figure 3 is introduced later.


4. Model overcoming the thermal obstacle by a dislocation

A dislocation will encounter a stress field illustrated schematically in Figure 4 as it moves through on the slip plane containing many weak obstacles and a few strong ones. In the figure, the positive stress concerning axis of the ordinate opposes the flow stress (applied stress τ) and the negative stress assists it. Extrinsic resistance to the dislocation motion has two types: one is long-range obstacle (the order of 10 atomic diameters or greater) and the other short-range obstacle (less than about 10 atomic diameters). The former is considered to be forest dislocations, large precipitates or second-phase particles, and grain boundary, for instance, and the latter impurity atoms, isolated and clustered point defects, small precipitates, intersecting dislocations, etc. Overcoming the latter type of obstacles (byname, thermal obstacles) by a dislocation, thermal fluctuations play an important role in aid of the flow stress above the temperature of 0 K. Then the aid energy, ΔG, supplied by the thermal fluctuations is given by the shaded part in Figure 4. Thus the dislocation can move through below τ0 (i.e. effective stress τ* due to short-range obstacles and internal stress τidue to long-range ones in Figure 4). τ0 is the value of τ at 0 K. As for the long-range obstacles (byname, athermal obstacles), the energy barrier is so large that the thermal fluctuations play no role in overcoming them within the temperature range.

Figure 4.

Stress fields encountered by a dislocation moving through the crystal lattice [57].

The representation of Figure 5 is concerned with a common type of thermal activation barrier. The free energy (G) varies with the distance (x) between a dislocation and the obstacle as given in Figure 5(a). When a dislocation overcomes the short-range obstacles, the free energy becomes high on account of the work (ΔW) done by the applied stress. Then the resistance (F), where it can be defined by the differentiation of free energy with respect to x (i.e. G/x), to the dislocation motion is revealed as Figure 5(b) in accord with the abscissa of Figure 5(a). F value is maximum at position x1. Figure 5(b) corresponds to typical force-distance curve for short-range obstacle among those in Figure 4. Shape of this curve represented by F(x) means the model overcoming the obstacle by a dislocation. G0, which is taken as the shaded area under F(x) between saddle-point positions x0 and x2 in Figure 5(b), is the Gibbs free energy of activation for the breakaway of the dislocation from the obstacle in the absence of an applied stress (in this case it is equivalent to the Helmholtz free energy for the dislocation motion).

Figure 5.

The process for thermal activated overcoming of the short-range obstacle by a dislocation. Variation in (a) the Gibbs free energy of activation and (b) the force acted on the dislocation with the distance for a dislocation motion [57].


5. Relation between the effective stress due to impurities on activation volume

When the dislocation breaks-away from the defects on a slip plane with the aid of thermal activation during plastic deformation, observations of τp and λp would provide information on the dislocation-defect interaction in the specimen. λp is the difference between λ at first plateau place and at second one on Δτ vs. λ curve as presented in Figure 3, which has been regarded as the component of strain-rate sensitivity due to dopant ions when a dislocation moves forward with the help of oscillation [40].

Figure 6 shows the relation between τp and activation volume (V) for NaCl:Rb+ (0.5 mol%). The activation volume has been expressed as [57].

Figure 6.

Relation between τp and activation volume (V) for NaCl:Rb+ (0.5 mol%) (reproduced from Ref. [58] with permission from the publisher).


where k is the Boltzmann constant and T is the absolute temperature. Here, the lnε̇τ in Eq. (1) is obtained from λp. Eq. (1) is namely replaced by


This dependence (τp vs. V) also represents the force-distance profile between dislocation and Rb+ ion. The τp vs. V curve gives the value of G0 for the specimen. The G0 values for the other specimens (i.e. NaCl: Li+, K+, Cs+, F, Br or I) are similarly estimated and are listed in Table 1.

SpecimenG0 (eV)
NaCl:Li+ (0.5 mol%)0.55
NaCl:K+ (0.5 mol%)0.60
NaCl:Rb+ (0.5 mol%)0.61
NaCl:Cs+ (0.5 mol%)0.82
NaCl:F (0.5 mol%)0.69
NaCl:Br (0.5 mol%)0.47
NaCl:I (0.5 mol%)0.53

Table 1.

Values of energy G0.

Figure 7 shows the obtained energies G0 with the isotropic defect size (Δε), which is estimated from the difference between the lattice constants of host crystal and dopant, around ion doped in the each specimen. The ions beside each plot represent the dopants in NaCl single crystals. G0 values vary linearly with Δε in the specimens. The intercept of the straight line is 0.36 eV, which is considered to be the interaction energy between dislocation and inherent obstacle of the host crystal because Δε is zero.

Figure 7.

Variation of the interaction energy (G0) between dislocation and the dopant ion with the defect size (reproduced from Ref. [58] with permission from the publisher).


6. Conclusions

The following conclusions were derived from the data analyzed in terms of the Δτ vs. λ curves for NaCl: Li+, K+, Rb+, Cs+, Br, I, F, Br or I single crystals.

  1. The relation between Δτ and λ has stair-like shape for the specimens at a given temperature and strain. There are two bending points and two plateau regions. λ decreases with Δτ between the two bending points. The measurement of τp and V calculated with λp provides information on the interaction between mobile dislocation and the dopant ion in the specimens during plastic deformation.

  2. The Gibbs free energy G0 for the overcoming of dislocation from the dopant is obtained for each of the specimens and increases linearly with increasing the defect size Δε. This result leads to the conclusion that the dopant ion as weak obstacle to dislocation motion becomes slightly stronger with larger defect size around the dopant in NaCl single crystal.



Dr. T. Ohgaku, as well as S. Yamaguchi, M. Azuma, H. Teraji, E. Ogawa and Y. Yamanaka are acknowledged for his collaboration in the analysis on Δτ and λ data, as well as for their experimental assistance.


Conflict of interest

The author declares no conflict of interest.


  1. 1. Chin GY, Van Uitert LG, Green ML, Zydzik GJ, Kometani TY. Strengthening of alkali halides by divalent-ion additions. Journal of the American Ceramic Society. 1973;56:369-372. DOI: 10.1111/j.1151-2916.1973.tb12688.x
  2. 2. Suszyńska M. Effect of impurity concentration and plastic deformation on dislocation density of KCl crystals. Kristall und Technik. 1974;9:1199-1207. DOI: 10.1002/crat.19740091015
  3. 3. Kataoka T, Yamada T. Yield strength and dislocation mobility of KCl-KBr solid solution single crystals. Japanese Journal of Applied Physics. 1977;16:1119-1126. DOI: 10.1143/JJAP.16.1119
  4. 4. Boyarskaya YS, Zhitaru RP, Palistrant NA. The anomalous behaviour of the doped NaCl crystals compressed at low temperatures. Crystal Research and Technology. 1990;25:1469-1473. DOI: 10.1002/crat.2170251219
  5. 5. Boyarskaya YS, Zhitaru RP, Palistrant NA. Influence of the state of the impurity on the deformation-rate dependence of the yield stress of NaCl:Ca single crystals. Soviet Physics - Solid State. 1990;32:1989-1990
  6. 6. Okazaki K. Solid-solution hardening and softening in binary iron alloys. Journal of Materials Science. 1996;31:1087-1099. DOI: 10.1007/BF00352911
  7. 7. Tabachnikova ED, Podolskiy AV, Smirnov SN, Psaruk IA, Liao PK. Temperature dependent mechanical properties and thermal activation plasticity of nanocrystalline and coarse grained Ni-18.75 at.% Fe alloy. IOP Conference Series: Materials Science and Engineering. 2014;63:012105. DOI: 10.1088/1757-899X/63/1/012105
  8. 8. Pratt PL, Harrison RP, Newey CWA. Dislocation mobility in ionic crystals. Discussions of the Faraday Society. 1964;38:211-217. DOI: 10.1039/DF9643800211
  9. 9. Newey CWA, Harrison RP, Pratt PL. Precipitation hardening and dislocation locking in doped NaCl. Proceedings of the British Ceramic Society. 1966;6:305-316
  10. 10. Chin GY, Van Uitert LG, Green ML, Zydzik G. Hardness, yield strength and Young’s modulus in halide crystals. Scripta Metallurgica. 1972;6:475-480. DOI: 10.1016/0036-9748(72)90031-2
  11. 11. Green ML, Zydzik G. Effect of heat treatment on the microhardness of some mixed and doped alkali halides. Scripta Metallurgica. 1972;6:991-994. DOI: 10.1016/0036-9748(72)90159-7
  12. 12. Andreev GA, Klimov VA. Influence of the state of an impurity on the microhardness of NaCl:Sr single crystals. Soviet Physics - Solid State. 1980;22:2042-2043
  13. 13. Buravleva MG, Rozenberg GK, Soifer LM, Chaikovskii EF. Changes in the flow stress of LiF:Mg2+ and LiF:Co2+ crystals during precipitation of solid solutions. Soviet Physics - Solid State. 1980;22:150-152
  14. 14. Narasimha Reddy K, Subba Rao UV. Influence of gadolonium impurity on microhardness of host alkali halide crystal. Crystal Research and Technology. 1984;19:K73-K76. DOI: 10.1002/crat.2170190730
  15. 15. Strunk H. Investigation of cross-slip events in NaCl crystals by transmission electron microscopy. Physica Status Solidi (A). 1975;28:119-126. DOI: 10.1002/pssa.2210280111
  16. 16. Appel F, Messerschmidt U. The interaction between dislocations and point obstacles: A comparison of the interaction parameter distributions obtained from computer simulation and from in situ high voltage electron microscopy straining experiments. Materials Science and Engineering: A. 1982;52:69-74. DOI: 10.1016/0025-5416(82)90070-2
  17. 17. Messerschmidt U, Appel F, Schmid H. The radius of curvature of dislocation segments in MgO crystals stressed in the high-voltage electron microscope. Philosophical Magazine A. 1985;51:781-796. DOI: 10.1080/01418618508237587
  18. 18. Kataoka T, Ohji H, Morishita H, Kishida K, Azuma K, Yamada T. In-situ observation of moving dislocations in KCl crystal by laser-light topography. Japanese Journal of Applied Physics. 1989;28:L697-L700. DOI: 10.1143/JJAP.28.L697
  19. 19. Kataoka T, Ohji H, Kishida K, Azuma K, Yamada T. Direct observation of glide dislocations in a KCl crystal by the light scattering method. Applied Physics Letters. 1990;56:1317-1319. DOI: 10.1063/1.102504
  20. 20. Kataoka T. The light scattering topography method: Direct observation of moving dislocations. Butsuri. 1992;47:713-716. (in Japanese)
  21. 21. Messerschmidt U. Dislocation Dynamics during Plastic Deformation. Berlin Heidelberg: Springer; 2010. DOI: 10.1007/978-3-642-03177-9
  22. 22. Indenbom VL, Chernov VM. Determination of characteristics for the interaction between point defects and dislocations from internal friction experiments. Physica Status Solidi A: Applications and Material Science. 1972;14:347-354. DOI: 10.1002/pssa.2210140142
  23. 23. Schwarz RB, Granato AV. Measurement of the force-distance profile for the interaction between a dislocation and a point defect. Physical Review Letters. 1975;34:1174-1177. DOI: 10.1103/PhysRevLett.34.1174
  24. 24. Ivanov VI, Lebedev AB, Kardashev BK, Nikanorov SP. Interaction of dislocations with pinning centers in magnesium at temperatures 295-4.2K. Soviet Physics - Solid State. 1986;28:867-868
  25. 25. Kosugi T, Kino T. Experimental determination of the force-distance relation for the interaction between a dislocation and a solute atom. Journal of the Physical Society of Japan. 1987;56:999-1009. DOI: 10.1143/JPSJ.56.999
  26. 26. Kosugi T. Temperature dependence of amplitude-dependent internal friction due to simultaneous breakaway of a dislocation from several pinning points. Materials Science and Engineering: A. 2001;309-310:203-206. DOI: 10.1016/S0921-5093(00)01792-5
  27. 27. Gremaud G. Dislocation-point defect interactions. Materials Science Forum. 2001;366-368:178-246. DOI: 10.4028/
  28. 28. Dotsenko VI. Stress relaxation in crystals. Physica Status Solidi B. 1979;93:11-43. DOI: 10.1002/pssb.2220930102
  29. 29. Urusovskaya AA, Petchenko AM, Mozgovoi VI. The influence of strain rate on stress relaxation. Physica Status Solidi (A). 1991;125:155-160. DOI: 10.1002/pssa.2211250112
  30. 30. Johnston WG, Gilman JJ. Dislocation velocities, dislocation densities, and plastic flow in lithium fluoride crystals. Journal of Applied Physics. 1959;30:129-144. DOI: 10.1063/1.1735121
  31. 31. Granato AV, Lücke K. Theory of mechanical damping due to dislocations. Journal of Applied Physics. 1956;27:583-593. DOI: 10.1063/1.1722436
  32. 32. Blaha F, Langenecker B. Dehnung von Zink-kristallen unter ultraschalleinwirkung. Naturwissenschaften. 1955;42:556. DOI: 10.1007/BF00623773
  33. 33. Ohgaku T, Takeuchi N. The relation of the Blaha effect with internal friction for alkali halide crystals. Physica Status Solidi A: Applications and Material Science. 1988;105:153-159. DOI: 10.1002/pssa.2211050115
  34. 34. Ohgaku T, Takeuchi N. Relation between plastic deformation and the Blaha effect for alkali halide crystals. Physica Status Solidi A: Applications and Material Science. 1989;111:165-172. DOI: 10.1002/pssa.2211110117
  35. 35. Ohgaku T, Takeuchi N. Interaction between a dislocation and monovalent impurities in KCl single crystals. Physica Status Solidi A: Applications and Material Science. 1992;134:397-404. DOI: 10.1002/pssa.2211340210
  36. 36. Ohgaku T, Teraji H. Investigation of interaction between a dislocation and a Br ion in NaCl:Br single crystals. Physica Status Solidi A: Applications and Material Science. 2001;187:407-413
  37. 37. Ohgaku T, Hashimoto K. Strain rate sensitivity of flow stress under superimposition of ultrasonic oscillatory stress during plastic deformation of RbCl doped with Br or I. Materials Science and Engineering A. 2005;400-401:401-404. DOI: 10.1016/j.msea.2005.03.058
  38. 38. Ohgaku T, Matsunaga T. Interaction between dislocation and divalent impurity in KBr single crystals. IOP Conference Series: Materials Science and Engineering. 2009;3:012021. DOI: 10.1088/1757-899X/3/1/012021
  39. 39. Kohzuki Y. Study on the interaction between a dislocation and impurities in KCl:Sr2+ single crystals by the Blaha effect-part IV influence of heat treatment on dislocation density. Journal of Materials Science. 2009;44:379-384. DOI: 10.1007/s10853-008-3150-8
  40. 40. Kohzuki Y. Bending angle of dislocation pinned by an obstacle and the Friedel relation. Philosophical Magazine. 2010;90:2273-2287. DOI: 10.1080/14786431003636089
  41. 41. Ohgaku T, Migiuma S, Nagahira D. Interaction between dislocation and defects induced by X-irradiation in alkali halide crystals. Radiation Measurements. 2011;46:1385-1388. DOI: 10.1016/j.radmeas.2011.05.067
  42. 42. Nevill GE, Brotzen FR. The effect of vibrations on the static yield strength of low-carbon steel. Journal of ASTM International. 1957;57:751-758
  43. 43. Langenecker B. Effects of ultrasound on deformation characteristics of metals. IEEE Transactions on Sonics and Ultrasonics. 1966;SU-13:1-8. DOI: 10.1109/T-SU.1966.29367
  44. 44. Izumi O, Oyama K, Suzuki Y. Effects of superimposed ultrasonic vibration on compressive deformation of metals. Materials transactions, JIM. 1966;7:162-167. DOI: 10.2320/matertrans1960.7.162
  45. 45. Evans AE, Smith AW, Waterhouse WJ, Sansome DH. Review of the application of ultrasonic vibrations to deforming metals. Ultrasonics. 1975;13:162-170
  46. 46. Jimma T, Kasuga Y, Iwaki N, Miyazawa O, Mori E, Ito K, et al. An application of ultrasonic vibration to the deep drawing process. Journal of Materials Processing Technology. 1998;80-81:406-412. DOI: 10.1016/S0924-0136(98)00195-2
  47. 47. Susan M, Bujoreanu LG. The metal-tool contact friction at the ultrasonic vibration drawing of ball-bearing steel wires. Revista de Metalurgia (Madrid). 1999;35:379-383. DOI: 10.3989/revmetalm.1999.v35.i6.646
  48. 48. Murakawa M, Jin M. The utility of radially and ultrasonically vibrated dies in the wire drawing process. Journal of Materials Processing Technology. 2001;113:81-86. DOI: 10.1016/S0924-0136(01)00635-5
  49. 49. Susan M, Bujoreanu LG, Găluşcă DG, Munteanu C, Mantu M. On the drawing in ultrasonic field of metallic wires with high mechanical resistance. Journal of Optoelectronics and Advanced Materials. 2005;7:637-645
  50. 50. Lucas M, Gachagan A, Cardoni A. Research applications and opportunities in power ultrasonics. Proceedings of the Institution of Mechanical Engineers, Part C. 2009;223:2949-2965. DOI: 10.1243/09544062JMES1671
  51. 51. Siddiq A, El Sayed T. Ultrasonic-assisted manufacturing processes: Variational model and numerical simulations. Ultrasonics. 2012;52:521-529. DOI: 10.1016/j.ultras.2011.11.004
  52. 52. Makhdum F, Phadnis VA, Roy A, Silberschmidt VV. Effect of ultrasonically-assisted drilling on carbon-fibre-reinforced plastics. Journal of Sound and Vibration. 2014;333:5939-5952. DOI: 10.1016/j.jsv.2014.05.042
  53. 53. Graff KF. Ultrasonic metal forming: Processing. In: Gallego-Juarez JA, Graff KF, editors. Power Ultrasonics: Applications of High-Intensity Ultrasound. Cambridge UK: Elsevier; 2015. pp. 377-438
  54. 54. Cottrell AH, Bilby BA. Dislocation theory of yielding and strain ageing of iron. Proceedings of the Physical Society of London. 1949;62:49-62. DOI: 10.1088/0370-1298/62/1/308
  55. 55. Pamplin BR, editor. Introdction to crystal growth methods. In: Crystal Growth. Pergamon Press: Oxford; 1975. pp. 1-11
  56. 56. Sirdeshmukh DB, Sirdeshmukh L, Subhadra KG. In: Hull R, Osgood RM, Sakaki H Jr, Zunger A, editors. Alkali Halides: A Handbook of Physical Properties. Berlin Heidelberg: Springer-Verlag; 2001. p. 41
  57. 57. Conrad H. Thermally activated deformation of metals. Journal of Metals. 1964;16:582-588. DOI: 10.1007/BF03378292
  58. 58. Kohzuki Y, Ohgaku T. Activation energy for dislocation breakaway from various monovalent ions in NaCl single crystals. Philosophical Magazine. 2016;(96):3109-3119. DOI: 10.1080/14786435.2016.1227099

Written By

Yohichi Kohzuki

Submitted: February 19th, 2020 Reviewed: April 20th, 2020 Published: May 20th, 2020