2. Introduction to plastic deformation of crystals

Transcription

1 .1. Micrscpic mdels f plastic defrmatin f crystalline slids. Intrductin t plastic defrmatin f crystals As we knw frm fundamentals f plastic defrmatin f crystalline slids (e.g. [36]), Hke s law is valid nly in the elastic regin f defrmatin. With increasing stress, the prprtinality between the stress and strain gradually vanishes, and a range f plasticity ccurs. Unlike the elastic range where defrmatin is fully reversible, applying a stress in the anelastic regin results in a permanent plastic strain..1. Micrscpic mdels f plastic defrmatin f crystalline slids.1.1. Critical reslved shear stress In crystalline materials, plastic defrmatin usually ccurs by glide n slip planes alng certain slip directins. A slip plane and a slip directin cnstitute a slip system. Glide can thus be understd as sliding r successive displacement f ne plane f f c n A atms ver anther ne in a certain directin. That a dislcatin can start mving n its glide plane, a characteristic shear stress is required. Cnsider a cubic crystal illustrated in Fig d A sl. Let it be defrmed in cmpressin, H f c being a cmpressive frce applied nrmal t the face A b f the crystal. This prduces a stress σ = f / A. Assume a dislcatin n a slip plane A sl. with c the Burgers vectr H b and a frce H f b acting alng Fig Definitin f the rientatin factr its directin. The shear stress reslved n the slip plane A sl. due t the applied stress equals H H f f H H L b c L τ = = cs(b,d) cs(n,d) (.1.1) A A sl. where d H is the directin f cmpressin ais, n H is the nrmal t the slip plane. This equatin is Schmid s law [37], and the prduct f csines is called the rientatin factr m s. Cnsequently, eq. (.1.1.) becmes τ = σ m s (.1.) 3

2 .1. Micrscpic mdels f plastic defrmatin f crystalline slids with m s H H H H = cs(b,d) cs(n,d) (.1.3) It is pssible t define a critical reslved shear stress (CRSS), abve which the plastic defrmatin sets in as slip n a given slip plane. Micrscpically, slip is realized by the mtin f line defects in the crystal structure, that is dislcatins. While mving thrugh a crystal, dislcatins bring abut the plastic strain pl.. If mving dislcatins f a density ρ m are displaced by a distance each, the plastic strain will be = m ρ b. The derivatin f this equatin with pl. s m respect t time yields the Orwan equatin = m ρ bv s m (.1.4) pl. where v is the velcity f the mving dislcatins. The Orwan equatin establishes a direct cnnectin between the macrscpic parameter pl. and the micrscpic ne v..1.. Thermal activatin f dislcatin mtin Cnsider a dislcatin gliding in the directin under an applied reslved shear stress τ which gives rise t a frce τ b per unit length f the line (fr a review, see [38]). Assume the dislcatin encunters bstacles, each f which prduces a resisting frce f r, as sketched in Fig..1.. The frces depend n the psitin f the dislcatin with respect t the bstacle. The bstacles can be f different nature. Let the spacing f the bstacles alng the dislcatin line be l, s that the applied frward frce n the line per bstacle is τ bl. At the temperature f 0 K, glide will cease if τ bl is less than the bstacle strength f ma, and the dislcatin line will stp at the psitin 1. In rder t vercme the barrier, the line must mve t, which can ccur n accunt f thermal atmic fluctuatins at temperatures abve 0 K. In this case, an energy has t be supplied where G = (f τ* bl)d (.1.5) 1 r G is the change in the Gibbs free energy between the tw states 1 and. 4

3 .1. Micrscpic mdels f plastic defrmatin f crystalline slids On the ther hand, when a dislcatin scillates with an attempt frequency χ ( an atmic vibratin frequency) at a given temperature, it successfully vercmes χ ep( G/kT) barriers per secnd. Hence, the dislcatin velcity becmes f r f ma τbl 1 G τ V= W Fig..1.. The resisting frce f r versus distance fr the thermal barriers that ppse dislcatin mtin G v = κν ep- kt (.1.6) where κ is the frward distance f the dislcatin after a successful activatin. Substituting this dislcatin velcity int the Orwan equatin f the macrscpic plastic strain (.1.4) yields an Arrhenius rate equatin fr the plastic strain rate G = ep( ) (.1.7) pl. kt where is the pre-epnential factr, which is suppsed t be cnstant. Accrding t [39, 41-43] (fr a review, see [40]), the Gibbs free energy f activatin (lightly shaded area in the Fig..1..) is given by G = U T S τ V = U T S W (.1.8) where U is the change f the internal energy, S, the change f the activatin entrpy, V, the activatin vlume, and W is the s-called wrk term (dark shaded area in Fig..1.) that reflects the wrk dne n the system by the shear stress τ during thermal activatin W = τ lb (.1.9) with = 1. Besides, U T S is the Helmhltz free energy F= U T S (.1.10) Cnsequently, frmula (.1.8) becmes G = F τ lb (.1.11) The quantity lb is the activatin vlume intrduced abve V = lb (.1.1) 5

4 .1. Micrscpic mdels f plastic defrmatin f crystalline slids The activatin vlume can be understd as the activatin area swept by a dislcatin segment f length l ver the activatin distance multiplied by the Burgers vectr f the dislcatin. This quantity is als given by the thermdynamical relatin G V = ( ) T (.1.13) τ The activatin vlume can be determined frm macrscpical defrmatin eperiments by lgarithmic differentiatin f the Arrhenius equatin (.1.7), taking eq. (.1.13) int accunt, t Assuming ln V = kt τ T (.1.14) = cnst, this equatin can be cnverted, s that it cntains nly eperimentally measurable quantities [40] ln 1 V = kt (.1.15) σ T m s where m s is the rientatin factr intrduced abve. The term in parentheses is the inverse strain rate sensitivity r. It will be described in chapter.3. Lgarithmic differentiatin f eq. (.1.7) with respect t the temperature yields the thermdynamical activatin enthalpy ln kt T τ G = G T T τ * = H (.1.16) G where = S is the entrpy change mentined abve. Schöck has shwn T τ that under the assumptin that the main cntributin t the activatin entrpy results frm the temperature dependence f the shear mdulus µ, the Gibbs free energy f activatin can be calculated [39] T dµ H + msσv µ dt G = (.1.17) T dµ 1 µ dt 6

5 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals This frmula cntains nly measurable quantities, as d the equatins (.1.15) and (.1.16).. Dislcatin mechanisms cntrlling plastic defrmatin f crystals Fundamentally, the mtin f a dislcatin is ppsed by tw different types f barriers: shrt-range interactins with dislcatins that can be vercme by thermal activatin described in the chapter.1. and lng-range interactins due t, fr instance, dislcatins n parallel slip planes that cannt be surmunted with the help f thermal activatin. Thus, the (shear) flw stress cmprises tw cntributins: the thermal cmpnent τ * and the athermal cmpnent τ i (e.g. [38]). Hence, τ = τ * + τ i (..1) The lng-range cmpnent τ i leads t a lcal decrease in the applied stress τ, s that nly the cmpnent τ * acts t vercme the shrt-range bstacles. τ * is therefre called the effective stress. Dislcatin mechanisms gverning the plastic defrmatin f slids will be hereafter cnsidered with respect t their cntributins t the thermal and athermal parts f the flw stress...1. Thermally activated cntributin Lattice frictin (Peierls mechanism) Because f the peridicity f the crystal structure, a mving dislcatin in a crystal eperiences a ptential energy, r mre eactly a free energy, f displacement that varies with the lattice peridicity. This ptential energy is called the Peierls ptential. The stress necessary fr the dislcatin t surmunt it is named the Peierls stress (fr reviews, see [44-48]). In the case f a screw dislcatin at 0 K, this stress is given by τ.fr. µ πa = ep( ) (..) (1-ν ) b where µ is the shear mdulus, ν, Pissn s rati, and a, the interplanar distance between neighbring glide planes. Twice the amplitude f the peridic part f the Peierls ptential is called the Peierls energy, and it is related t the Peierls stress by U fr. = τ fr. ab/π (..3) 7

6 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals Cnsider the influence f temperature n the dislcatin mvement. At a finite temperature, a dislcatin cannt be cmpletely straight, and it des nt vercme the Peierls barrier simultaneusly all alng its line but cntains s-called kinks due t thermal fluctuatins. Tw kinks f ppsite sign (a kink pair) place shrt dislcatin segments int the adjacent lattice energy valleys (energy minima), the distance between which equals the distance between tw neighbring rws atms, i.e. the kink height h. h Fig...1. A kink mving with a velcity υ k under an applied stress..1.) The velcity f the kink is then Under a small applied stress, the kinks underg a diffusive drift, thereby bringing abut the glide mtin f the entire dislcatin. Cnsider a kink in a screw y dislcatin, and let τ be the shear stress cmpnent in the glide plane, acting in the directin f H b (Fig. τbh υ = D (..4) k k kt where D k is the diffusin cefficient f the kinks. In the limit f small eternal stress, the cncentratin f the kinks is almst equal t their thermal equilibrium cncentratin c k = d ep k Fk kt b (..5) where d is the shrtest repeat distance alng the dislcatin line, thus being f the magnitude rder f b. All these kinks have a drift velcity given by the equatin (..4), with the psitive and negative kinks drifting in ppsite directins. The net velcity f the dislcatin nrmal t itself is then υ = hc k υk (..6) Cmbining the three equatins (..4) t (..5) yields the dislcatin velcity τbh Fk υ = D k ep (..7) d kt kt 8

7 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals It can be shwn that this relatin is valid in a brad range f stresses [44]. The kink frmatin energy can be epressed by the Peierls energy F = (h/ π) (U Γ) (..8) where Γ is the line tensin. k fr. Chemical hardening: slutin and precipitatin hardening Slutin hardening is defined as an increase f the flw stress f a crystal cntaining nn-diffusing freign atms disslved in its lattice wing t interactins between the atms, s-called slute atms, and dislcatins in the crystal. The dislcatins are hindered by these interactins. At zer temperature, their mvement is pssible nly if the applied shear stress is higher than a critical shear stress τ c [49, 50]. At the critical stress, the frce a dislcatin eerts n the bstacles τ cbl is just high enugh t vercme the interactin frce f the bstacles. In additin t electrical interactins in inic crystals and semicnductrs, the rigin f the lcal interactin frces f int may be due t the paraelastic interactin n accunt f a size misfit between the slute atms and the matri as well as due t the dielastic interactin because f a mdulus misfit between them. There may als be interactins wing t a pssible nn-spherical symmetry f the stress fields f the pint defects. Real crystalline slids frequently cntain a certain amunt f precipitates, that is etrinsic particles. Dislcatin interactin with these defects causes the s-called precipitatin hardening. Tw different cases f such interactins may ccur. The particles may be either impenetrable r penetrable fr dislcatins [51]. In the frmer case, a dislcatin is frced by the applied stress t bw arund the particle and bypass it. The by-passing dislcatin leaves a lp arund the particle. This mechanism was suggested by Orwan and is therefre called the Orwan mechanism. In the latter case, the particle is sheared by the dislcatin as the latter mves thrugh the crystal. This can nly ccur if the interface between the particle and the matri is cherent. The cherent interface des nt shw any gemetric discntinuity in the atmic arrangement. Bth the Orwan and the cutting mechanisms can be discussed in terms f the interactin f a single dislcatin with a linear array f particles f diameter D and a 9

8 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals center t center distance l between them. This discussin leads t an epressin fr the critical reslved shear stress needed fr the dislcatin t vercme a rw f equidistant bstacles in the slip plane as shwn in Fig.... The dislcatin line is pressed against the rw by the applied stress and bends between the particles with a bending angle ψ. D l ψ Fig.... Interactin f a dislcatin with a rw f bstacles The angle ψ depends n inter-particle distance l, their diameter D, the increase in the applied shear stress τ due t the interactin with the particles, the magnitude f the Burgers vectr b, and the line tensin Γ accrding t the equatin τb(l - D) = Γsin ψ (..9) The right part f this equatin describes the pinning frce eerted by each particle n the dislcatin fpin = Γsinψ (..10) Depending n the rigin f the interactin between the particle and the dislcatin, there eists a maimum frce f m, which the particle can sustain. Its value depends n the distance f the slip plane with respect t the particle center. If this maimum frce is reached befre the bending angle becmes 90 C, the particle will be cut by the dislcatin. Hwever, in the case that the bending angle becmes 90 C befre f m is reached, the dislcatin by-passes the particle by the Orwan mechanism. Assuming this is the beginning f plastic defrmatin at 0 K, the fllwing equatins indicate the increase in CRSS, τ : Cutting mechanism τ = f m /(b(l - D)) fr f m < Γ (..11) Orwan mechanism τ = Γ /(b(l - D)) fr f m Γ (..1) 10

9 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals If the precipitates are small, they can be treated similarly t the slute atms. In this case, the bstacles interact with the dislcatins nly alng a shrt part f their ttal length. The bstacles can then be called lcalized bstacles r pint bstacles (D=0) in eq. (..11). Up t this pint in ur apprach, the bstacles were cnsidered t be arranged in a regular array. In practice, they are distributed irregularly, which can be treated in many cases by a randm arrangement, as shwn in the Fig...3. In this case, the average bstacle spacing becmes dependent n the stress, and eq. (..11) thus reads f = τ bl( τ ) (..13) yields m c c Fig...3. Dislcatin in a field f pint bstacles l The statistical prblem was treated fr the first time by Friedel [5]. The average spacing between bstacles is 1/3 l=(γ b / τ c) (..14) c where c is the atmic fractin f freign atms. Cmbining the last tw equatins 3/ 1/ 1/ τ c = f m c / b ( Γ ) (..15) This frmula describes the cntributin f a randm array f lcalized bstacles t the flw stress at zer temperature. At finite temperatures, this thery has t be cmbined with the thery f thermal activatin, described in chapter.1.. The Orwan prcess is always f athermal nature.... Athermal cntributin t the flw stress 0 y 1, y Fig...4. Interactin between tw screw dislcatins Cnsider tw screw dislcatins 1 and lying parallel t the z ais (Fig...4.) [38, 53]. Assume that y is cnstant. The interactin frce per length between the tw screw dislcatins with respect t the mtin in directin is f µ b = π + y (..16) The maimum interactin frce is btained 11

10 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals by differentiatin f this equatin with respect t the crdinate f dislcatin mtin with subsequently equating the result t zer, which yields and hence f µ b y = π ( + y ) = 0 (..17) f ma µ b = ± (..18) 4πy Accrding t Taylr s thery, the critical shear stress t mve ne dislcatin in the array f ther parallel dislcatins can be identified as the stress required t frce tw dislcatins lying n parallel slip planes f a minimum spacing y past each ther against their elastic interactin just defined. Hence, µ b τ p = (..19) 4πy Shuld the stress eceed the interactin frce between the tw dislcatins, they can pass against each ther. The minimum slip plane distance is cnsidered t be sme fractin f the average mutual dislcatin distance. In an array f parallel dislcatins f a density ρ, the average distance is 1/ ρ, s that the cntributin f the interactin between parallel dislcatins t the flw stress can be rewritten as where µ b ρ π 1/ τ p = α (..0) α / π is a numerical cnstant f abut Wrk-hardening and recvery During defrmatin, the dislcatin density increases, giving rise t an increase in the athermal cmpnent f the flw stress, in accrdance with eq. (..0). The increase f the flw stress is called wrk-hardening. Recvery is a thermal diffusinal rearrangement f crystal defects where internal strains present in a crystal are relieved. Such a rearrangement may result in dislcatin migratin and annihilatin leading t energetically favrable dislcatin arrays like subgrain bundaries, with the subsequent grwth f the subgrains [54, 55]. This phenmenn is named plygnizatin. When it takes place, the flw stress f a crystal decreases, and the latter becmes mre ductile. Apart frm the plignizatin, the decrease f the flw 1

11 .. Dislcatin mechanisms cntrlling plastic defrmatin f crystals stress is als assciated with mutual annihilatin f dislcatins (e.g. [36]). There are several recvery cntrlled creep mdels f crystals knwn up t date, and we will briefly cnsider the mst essential pints f these. Firstly, we assume that the applied stress f creep is a functin f time and strain [56] σ dσ = t σ dt + t d = Υ rec dt + Θd (..1) where Υ rec and Θ are defined as the σ t σ Y = rec t σ 0 + σ σ Θ = σ 0 Fig...5. Schematic display fr estimating recvery and wrk-hardening rates recvery rate and the wrk-hardening rate, respectively (Fig...5.). During creep at cnstant stress, the last frmula turns int the Bailey-Orwan equatin. Cnsequently, the steady state strain rate can be epressed st ( σ / t) = (..) ( σ / ) and it is reduced t st rec = Υ Θ t (..3).3. Macrscpic cmpressin eperiments The previus chapters shw that the flw stress measured during a defrmatin eperiment depends n the temperature and the strain rate applied wing t the thermal part f the flw stress. Furthermre, the defrmatin behavir is a functin f the crystal rientatin in accrdance with Schmid s law. Since the micrstructure f the sample is changed during the defrmatin as a result f dislcatin prductin and annihilatin, the defrmatin behavir depends als n the degree f the sample defrmatin. Therefre, the crystal micrstructure shuld be cnsidered a functin f the strain [57, 58] The stress-strain curve One f the macrscpic methds t study plastic defrmatin cnsists in cmpressin tests alng a defined defrmatin ais. The material is defrmed at a cnstant strain 13

12 .3. Macrscpic cmpressin eperiments up. l. rate and temperature, and the stressstrain r defrmatin curve, i.e. a plt f the stress applied t the sample versus its strain, is measured (e.g. [59]). It can have several different regins, tw f them being the yield pint and wrk-hardening ranges, as shwn in Fig The frmer Fig Schematic representatin f the yield pint effect. σ up. and σ l. are the cnstitutes a phenmenn where after upper and lwer yield stresses, respectively the stress reaches a maimum (the upper yield pint ), a finite amunt f plastic defrmatin ccurs at decreasing stress, s that the flw stress drps t the level f the lwer yield stress, after which the plastic defrmatin can ccur withut adding t the stress. This interval is called the steady state. After that, wrk-hardening starts taking place, that is the flw stress increases cnstantly as the strain rises. The wrk-hardening may be f varius rigins [36]. Mainly, the increase f the flw stress is a cnsequence f the increasing dislcatin density. This can be due t the mutual interactin between dislcatins mving n parallel slip planes, as was described by eq. (..0) f chapter... Besides, dislcatins mving n nn-parallel slip planes intersect each ther. Hence, the prduced elastic interactins give rise t the flw stress, with the latter depending n the dislcatin density in a similar way as eq. (..0) says. In additin, dislcatin intersectins result in the frmatin f jgs, the mtin f which cntributes t the flw stress, t. The wrk-hardening cefficient is inferred frm eq. (..1), and accrding t [60], becmes Θ= σ, T (.3.1) The ttal strain during a cmpressin eperiment cnsists f elastic and plastic parts = + = σ / E + (.3.) el. pl. pl. where el. is a functin f cmpressive stress accrding t Hke s law. E is Yung's mdulus. As a result, the ttal strain rate cntains elastic and plastic parts = pl. + σ / E (.3.3) 14

13 .3. Macrscpic cmpressin eperiments By means f special tests within the macrscpic defrmatin eperiments, ne can identify the parameters f thermal activatin described in chapter.1.. Their numerical values allw ne t cnclude n the mechanisms cntrlling the plastic defrmatin, which were discussed in chapter The strain rate sensitivity The dependence f the flw stress n the plastic strain rate can be epressed using the s-called strain rate sensitivity: σ r =. ln pl. r the stress epnent ln m* = lnσ (.3.4) (.3.5) The strain rate sensitivity can be measured by a strain rate cycling eperiment. It cnsists in an instantaneus change f the strain rate f the cmpressin eperiment, s that ne can identify the resulting stress increment σ and hence r accrding t eq. (.3.4) Anther way t btain the strain rate sensitivity is t d stress relaatin tests. These are a sudden stp f the lading prcess where the sample cntinues t defrm at a diminishing strain-rate and under the actin f a decreasing stress, while the ttal strain remains cnstant [61, 6]. Therefre, the plastic part f the strain rate becmes prprtinal t the negative stress rate accrding t equatin (.3.3) pl = σ / E (.3.6.) If ne plts the lgarithm f the negative stress rate versus the decreasing stress, ne btains the s-called relaatin curve f the test, the inverse slpe f which is the strain rate sensitivity r The temperature sensitivity The temperature sensitivity f the flw stress σ is measured by means f T temperature cycling tests. During such a test, the sample is unladed, the specimen temperature is increased r decreased, and the sample is reladed. Hence, an 15

14 .3. Macrscpic cmpressin eperiments increment f the flw stress σ appears. Using eq. (.1.16) and keeping in mind that ln / T τ ln / = τ T τ * T [53], the activatin enthalpy can be calculated σ H = - kt T (.3.7) 1 T r + r 1 where k is the Bltzman cnstant, T 1 and T are temperatures befre and after the change takes place, and r 1 and r are the strain rate sensitivities befre and after the cycle. This equatin deals with the prper averages f the temperatures and strain rate sensitivities befre and after the temperature change. 16