CRITERION FOR THE ACTION OF APPLIED STRESS [N THE
`MARTENSITIC TRANSFORMATION“
`
`J. R. PATEL and M. COHENl
`
`'i'hc ntartensitic reaction is treated as El strain translarmation with shear and (lilatatinnnl displace—
`ments». respectively parallel and normal to the habit plane. W'hcn external forms :irc acting.
`the
`resulting effect on the Air, temperature is calculated from the mechanical work clone on or by the
`transforming region as the resolved shear and normal components of the applied stress are carried
`through the corresponding transformation strains. This energy term is added algebraically to the
`chemical free energy change of the reaction. to compute the alteration in temperature at which the
`critical value of the thermodynamic driving force is attained to initiate the transformation. The
`transformation is aidcd by shear stresses. but may be aided or opposed by the normal stress corn—
`poncnt depending on whether the latter is tensile or compressive.
`The above criterion for the action of applied stress has successfully predicted the quantitative
`change. in the Mg temperature of iron-nickel and iron-nickcl-carlmn alloys under uniaxial tension.
`uninxial compression and hydrostatic pressure. M, is raised by tension. less so by compression. nntl is
`lowered by hydrostatic pressure.
`
`UN CRITERIUM DE L'ACTlON D‘UNE TENSION APPLIQUEE SUR LA
`TRANSFORMATION MARTENSI'I‘IQL'F.
`La réactiou martensitiquc est: considérée comma une transformation dc déformation avcc dos
`(iéplatements induits par un cisaillement et tine dilatation. respectivement parallels er normalc an
`plain limitc. Quand des efforts extérieurs agissent, leur eflet stir in. temperature M, est calculé tl‘aprcs
`lc travail cfféctué stir ou par la région en transformatiOn quaml les deux composantcs. do cisaillement
`et normale. de la. tension appliquée sont reliées aux deformations correspondantes dc la transforma-
`tion. Cettc énergie est ajoutéc algébriq uement an changement de l’énergie libre de la reaction, cc: qui
`permet dc calculcr la modification (le la temperature pour laquelle on atteint la valeur critique dc la
`“force motrice” thermorlynnmique, pour iitirier la. transformation. La transformation est airlée par
`les tensions tic cisaillemcnt, mais pent Etre aidée on contrariée par la composante normale de la.
`tension, stiivartt quc c’est une composantc cl'cxtensiou on dc compression.
`Ce criterium dc l'action d'unc tension appliquée a prédit quantitativemcnt. d'une manicre satis—
`faisantc.
`ic changemctlt
`tie in tempérnture M,
`ties alliages fermnickel et fer—nickel-cnrhonc sous
`extension et compression uniaxiales et sour; une pression hydrostatique. (if; est élevéc par l‘cxtcnsion.
`im pen moins par in compression, ct est abaissée par la pression hydrOstatiquc.
`
`KRITERION FUR DIE “TIRKUNG AUSSERER SPANNUNGEN AL'F DIE
`MARTENSITBILDUNG
`
`Die lla'lanCnsitrcaktion wird als cine \-’erzerrttngsumwandlung mit Scherungsi Lind Dehnungsver-
`schiehungEn parallel unCl scltkrccht znr Habitusebene hehztnrlelt'.
`Itl‘l Falle tier Einwirkung iiusscrer
`Kraft: wirrl rlcr Gesamteffckt auf die ME-Tempcrmur aus dcr geleistctcn mechanischcn Arbeit auf
`ode-r ciorch die transformicrten Bereiche herechnet; dahei werrlen die Scherungs uncl Normals
`komponentcn der atlsseren Spannung mit den entsprechenclen Umwandlungsverzcrrungen zusammen-
`gcfnsst. Dieser Energiefaktor wird algebraisch zu clcr iindcrung tier frcicn chemischen Reaktionscn—
`ergie addiert um die Antlerung tier 'l‘empcra tnr, bei der dcr kritische Wert dcr thcrnlodyllamir‘vchcn
`Kraft die die Transformation auslost erreicht
`ist, zu berechnen. Scherspannungen konnen die
`Transformation unterstu‘tzen; jL‘llDCl‘l kann die Normalkomponente entweder eine untersttitzcnde
`an e t.
`hdcrdeine hindernde Wirkung haben. jc nachdcm ob es sich tlm cine Zug— Oder Druckkomponente
`Dies Kriteriqn fiir dic Wirkung der husseren Spannungen hat zu eincr richtigen Vornussage der
`quantitativeu Anderung der M, 'l‘empcratur in Eisen—Nickel- und Eisen—Nickel—Kohlenstolf Legier-
`ungcn unter einuchsigem lug, einachsiger Kornpression und hydrostatischem Druck gefuhrt. Ms
`wird (lurch Zug erhijht,
`in fieringerm Masse auch dutch Kompressiun unrl (lurch hydrostatischcn
`Druck verminilerr.
`
`1. Martensite Formation as :1 Strain
`Transformation
`
`The martensitic transformation is characterized
`
`by its disiplacive shear—like nature. Consequently, it
`may be regarded as a strain transformation or as a
`mode of deformation which competes with slip when
`external stresses are applied to the parent phase.
`According to Scheil [1], the shear stress required to
`activate the martensitic transformation decreases
`
`*Receivcd April 1, 1953.
`TDepartment of Metallurgy, Massachusetts Institute of
`Technology, Cambridge, Masaachusotts, U.S.A.
`
`ACTA METALLURGICA, VOL. 1, SEPT. 1953
`
`with decreasing temperature (becoming nil at the
`Elf, temperature where the reaction starts spontan-
`eously on cooling), whereas the shear stress required
`to initiate yielding by slip increases with decreasing
`temperature. Thus, at temperatures near M,, applied
`stresses should induce plastic deformation by the
`martensitic mode rather than by slip. On the other
`hand, when stresses are applied at temperatures
`sufficiently high above it!” the slip mechanism
`should supersede the transformation in the deforma-
`tion process.
`In a general way,
`these postulated
`trends have been confirmed by experiment.
`Scheil's concept leads to a shear-stress criterion for
`
`LLombard Exhibit 1030, p. 1
`
`Edwards Exhibit 1013, p. 1
`
`Edwards Exhibit 1013, p. 1
`
`

`

`532
`
`ACTA
`
`M ETALLURGICA, VOL. 1, 1953
`
`formation had occurred exclusively in the portion of
`the bar where the tensile stresses existed. No trans-
`
`the bar
`formation took place in the portion of
`sustaining compressive stresses,
`the latter being
`symmetrically equal in magnitude, but opposite in
`sign. to the aforementioned tensile stresses.
`When the acting stresses in the elastically bent
`bar are resolved into their normal and shear com-
`
`in
`ponents, it is noted that they are symmetrical
`magnitude about the neutral axis. Since the sign of
`the shear component is immaterial from the stand-
`point of the martensitic reaction, because of the
`many potential habit orientations,
`the striking
`selectivity of the transformation in the bar can only
`be attributed to the difference in sign of the normal
`stresses.
`
`This finding invalidates the shearistress criterion.
`along with other possible criteria based on shear
`strain or shear-strain energy, because the demon-
`strated effect of the normal component is neglected
`in such considerations. Moreover,
`if an}! of these
`criteria were appropriate, hydrOstatic pressure should
`have no influence on the transformation since no
`
`shear stresses would then be applied. In a similar
`sense, uniaxial compression and tension should have
`the same elTect on MR when the resolved shear stresses
`
`are the same. Experimental evidence is presented in
`the next section to show that this is not the case.
`
`2. Experimental Results on the Efiect
`of Applied Stress
`
`Figure 1 summarizes the changes in M, tempera-
`ture due to applied uniaxial tension, uniaxial com-
`
`l
`’ TA
`3
`-
`l
`'
`
`the effectiveness of applied stress in promoting the
`martensitic transformation. In other words, if yield-
`ing by slip does not intervene, the applied stress will
`activate the transformation when the shear—stress
`
`Component, resolved along a potential habit plane,
`reaches a critical value. This critical shear stress will
`
`depend on temperature and on the nature of the
`alloy system. However, the validity of this criterion
`may be questioned because shear strain is not the
`only displacement accompanying the martensitic
`transformation.
`in steels and iron-base alloys, for
`example, the volume change is of the order of +4 per
`centian expansion which must certainly interact
`with the applied stresses.
`In fact, the macrostrains attending the formation
`of individual plates of martensite in single crystals of
`a 70 per cent iron --30 per cent nickel alloy have been
`measured directly by Machlin [2], using fiducial
`marks inscribed on various crystallographic faces of
`the specimens. His results showed a homogeneous
`displacement* for the transformation,
`involving a
`shear strain (w) of 0.20 along the habit plane and a
`dilational strain (to) of 0.05 normal
`to the habit
`plane. The magnitude of these transformation strains
`are well beyond the elastic range of the parent phase,
`and indicate that the martensitic reaction should be
`
`sensitive to both the shear and normal components
`of the applied stress. The measured strains do not
`actually account for the over-all conversion of the
`austenite lattice to that of martensite: further atomic
`movements are necessary in order to complete the
`change in structure. Whether the latter movements
`occur simultaneously with, or in sequence with, the
`observed macrodisplacement is still a controversial
`matter [3; 4; 2: 5], but for the present purposes, we
`are only concerned with the macrostrain of the trans-
`formation because this is what interacts with the
`
`applied stress in the treatment to be discussed here.
`More recently,
`the importance of the dilational
`component of the transformation strain was demon-
`strated by Kulin [6]. A bar of 0.5 per cent carbon+20
`per cent nickel steel in the austenitic state was sub-
`jected to uniform elastic bending, and was cooled
`until martensite
`started to form. Subsequent
`metallographic examination disclosed that the trans-
`
`
`‘This kind of displacement is termed a Bowlers—type strain
`14] or an invariant-plane strain [2]. It can be described by the
`motion of a plane of no rotation and no distortion (the in-
`variant plane) in a direction not necessarily contained in the
`plane. The invariant plane is found to lie parallel to the habit
`plane of the martensitic plate. In the case cited above,
`the
`invariant planes slide past one another to provide a com—
`ponent of shear strain (yo) parallel to the habit plane. and
`also move apart to provide a component of dilational strain
`(so) normal to the habit plane.
`
`
`
`
`
`
`
`CHANGElNMsTEMPERHTURE-'C
`
`30L
`
`20 ,.
`
`,0;
`
`’23I/I
`353:, c. 2m. N. STEEL
`lx’unmmt
`‘
`\\.
`TENSION
`
`ummmt
`
`COMPRESSlON
`
`
`HYDROSYnTlC
`
`COMPRESS:ON
`7|0 '
`
`70s., F2, 309;, n. ALLEY
`
`l
`|
`I
`"I'
`
`J
`‘
`
`
`
`-20
`
`~30 _
`
`o
`
`'
`4
`
`|
`|
`l
`'
`l
`24
`20
`l6
`I2
`8
`smcss on PRESSURE no" PSI
`
`I
`25
`
`52
`
`FIGURE 1. Change in My temperature as a function of
`loading condition.
`
`LLombard Exhibit 1030, p. 2
`
`Edwards Exhibit 1013, p. 2
`
`Edwards Exhibit 1013, p. 2
`
`

`

`PATEL AND COHEN:
`
`MARTENSITE TRANSFORMATION
`
`533
`
`free energy accompanying the transformation of one
`moi of austenite to one moi of martensitc at the M,
`
`temperature is F“ —- F‘ = -—29(} cal/moi.) In the
`iron—nickel system, tier, occurs at F” — F'“ = -200
`cal/moi over the range of compositions under cen-
`sideration here [91.
`In the present study, the thermodynamic work of
`Jones and Pumphrey [9] is used for the iron—nickel
`system, and this is extended to the iron-nickel-
`carbon case. The method of calculation is described
`
`in Appendix I. Figure 2 depicts the change in free
`
`-
`7‘
`_
`I
`i
`I
`500 :—
`II
`
`pression and hydrostatic pressure. The tension and
`compression experiments Were performed on T‘s-inch-
`and Tl—incli-diameter rods (respectively) of a 0.5 per
`cent carbon-‘20 per cent nickel steel, whereas the
`pressure experiments were carried out on {6—inch-
`diameter rods of a 70 per cent iron—3O per cent nickel
`alloy. Both materials were austenitizcd at 1095°C
`and oil quenched to room temperature. The é—inch—
`diameter specimens were stress relieved at 425°C:
`this treatment was not found to be essential for the
`
`smaller size specimens. The M, temperature for the
`nickel—carbon steel was about —40°C, and that of
`the ironenickel alloy was about -—20°C.
`The M, temperatures were determined by electri-
`cal
`resistance measurements while the austenitic
`
`specimens were being subcooled in the stress-free
`and variously loaded conditions. The procedures f0r
`the tensile and cornpressive runs have been described
`previously [6]. The hydrostatic pressure runs were
`made in Professor P. W. Bridgman’s laboratory at
`Harvard University:
`the details will be presented
`elsewhere.
`
`It is evident from Figure 1 that the three methods
`of stressing produced virtually linear changes in the
`NI, temperature, at least up to about 15,000 psi.
`The corresponding slopes are +1.0, +0.65 and
`—0.57°CI/ 1000 psi, respectively.
`for
`the cases of
`tension, nompreseion and pressure. Thus, there is an
`appreciable difference between the effects of uniaxial
`tension and uniaxial compression, despite the iden—
`tical shear components in these two instances. The
`influence of hydrostatic pressure is also noteworthy,
`not only because of its magnitude but because of its
`sign which differs from that of the other two methods
`of loading. Clearly, any criterion based on shear alone
`cannot account for these results.
`
`3. Criterion for the Efiectivenes: of
`
`Applied Stress
`
`3.1 Free Energy Change (in the flflzrtemiiic Reaction
`
`The criterion to be prOposed here is a thermo-
`dynamic one (at) based on the assumption that the
`M, temperature of a given alloy occurs at a certain
`value of the driving force (#AF), and (b) taking in
`account the mechanical work performed on or by the
`transforming region as the resolved normal and shear
`forces are carried through the respective transforma-
`tion displacements.
`It has been shown that iron-carbon alloys start to
`transform spontaneously to martensite at tempera—
`tures corresponding to a constant driving force of
`—AF : 290 cal/moi [7; 8]. (This means that for a
`wide ra nge of carbon contents the change in chemical
`
`
`
`400 .--
`
`_]|
`”205; mchL
`e Ms ranpeeaTuREs
`i
`
`‘
`400
`soo
`son
`1000
`race
`rswannrues — -«
`
`—600 L—L-
`o
`ace
`
`FIGURE 2. Change in free energy attending the austenite—
`to—martcnsitc reaction in iron-nickel alloys. Plotted from the
`equation of lanes and Pumphrey [9].
`
`energy as a function of temperature and composition,
`attending the martensitic transformation in iron-
`nickel alloys. Figure 3 gives similar information for
`the iron-nickel-carbon system. In both graphs, the
`particular compositions were chosen because fairly
`accurate data. were available for their iii, tempera-
`tures, and because they straddle the two alloys
`employed in this investigation. It will be noted that
`in each series the M, temperature occurs at a fixed
`value of the chemical free energy change, the latter
`being about #200 cal/mo] in the iron-nickel alloys
`and *370 cal /mol in the iron—nickel-carbon alloys.
`These values signify that the martensitic reaction
`does not
`take place at
`the temperature (Tn) of
`thermodynamic equilibrium between the austenite
`and martensite, but only after supercooling suffi—
`ciently to achieve the required driving force for the
`
`LLombard Exhibit 1030, p. 3
`
`Edwards Exhibit 1013, p. 3
`
`
`
`
`
`600 —
`
`.
`i
`400‘
`
`\
`
`339g NICKEL.\
`30s. Mic-(EL
`\
`27% wears.
`23’s NI-CKE..\\‘
`
`.
`
`.
`
`
`
`l
`
`.
`
`—=_
`*- -sF =zoc C-GWN-sl AT M5:
`_ ‘ist, NLCKEL
`|
`x
`!
`
`20C 1
`
`0
`
`-2oo
`
`EE
`
`BU
`
`g 5
`
`?LI.
`,-
`t.
`
`Edwards Exhibit 1013, p. 3
`
`

`

`53-1
`
`ACTA METALLURGICA, VOL. 1, 1953
`
`inasmuch as the free energy of
`transformation.
`nucleation decrmses as the driving force increases,
`this is tantamount to saying that .Mx occurs on cool-
`ing at the temperature where the free energy of
`nucleation drops to a critical level [I] ; 12].
`
`a
`I
`1
`
`
`300
`
`of U can readily be converted to calories per mol,
`to match the units of F‘” — F“.
`
`For uniaxial tension or compression, at considera-
`tion of Mohr's circle in Figure -l shows that the
`resolved shear and normal stresses are
`ll
`
`(2)
`
`(3)
`
`r
`
`cr
`
`% in sin 26,
`
`:i: %crl (1 + cos 26),
`
`[he applied stress
`where arl = absolute value of
`(tension or compreSSion) and 6 2 angle between the
`specimen axis and the normal to any potential habit
`plane.
`
`
`
`D'=%ll+co:‘.39}
`
`1' : % ism 231
`
`FIGURE 4. Mohr’s circle for tension showing the shear (r)
`and normal (:7) components of stress as a function of the applied
`stress in) and the orientation (9) between the stress axis and
`the normal to the potential habit plane.
`
`U may now be expressed as a function of the
`orientation of a transforming martcnsitic plate:
`
`(4)
`
`U = é'i’o or; sin 29 :I: %eu 01(1 + ens 28)
`
`Since we are concerned with the plates that form
`first (at £118) under the influence of applied stress, it
`is necessary to find the particular orientation which
`yields a maximum value of U:
`
`(5)
`
`-
`(0)
`
`d7}? = 7001 cos 29 :i: 6w1(— sin 26) = 0
`
`n
`. s
`sing? a ,
`cos 26— tan 26 — 1 £0
`
`Substituting the known components of the trans—
`formation strain into equation {6) and then into
`equation (4). it is now possible to obtain Um“. It is
`then assumed that UElm contributes to (or detracts
`from)
`the chemical Free energy change to aid (or
`oppose)
`the start of the transformation. As illus-
`trated in Figure 5, if Um“ is positive, the degree of
`
`LLombard Exhibit 1030, p. 4
`
`Edwards Exhibit 1013, p. 4
`
`
`
`
`son —
`
`400—
`
`|.O%C,20%NI—.._V
`owe 20°; Ni— "
`' °'
`"
`I.O%C, rz'r, Ni.“
`
`_
`°
`F zoo—
`3
`
`OU E
`
`a"
`LL.
`l
`I
`‘:
`
`0
`
`—eoo
`
`
`
`"AF = 3m Cal/Mel
`
`,/
`AT MS
`-400__
`_
`
`
`
`a MS TEMPERATURES
`
`-500l
`J__
`L
`i
`o
`200
`400
`500
`800
`Icon
`TEMPERATURE — ”K
`
`
`
`FIGURE _3. Change in free energy attending the austenite-
`to-martensitc reaction in iron-nickel-carbon alloys.
`
`3.2 Rafe of Applied Sires:
`
`The work (U) done on or by the transformation
`due to the action of applied stress is comprised of
`two terms:
`(mg) the shear stress resolved along a
`potential habit plane times the transformation shear
`strain. and (are) the normal stress resolved perpens
`dicular to the habit plane times the normal com-
`ponent of the transformation strain. Thus
`
`(1)
`
`U = T'Yn + can.
`
`0’ is numerically pesitive when the normal stress is
`tensile, and negative when this component is com-
`pressive. “r is always taken to be positive because the
`many habit permutations (:I:.{259} in these alloys)
`virtually permit shearing in either sense- Hence, in
`effect, shear stresses will stimulate the transforma-
`tion, but normal stresses may aid or oppose it
`depending upon whether cr is tensile or compressive.
`The units of U in equation (1) are stress times
`transformation strain or mechanical work per unit
`Volume of austenite reacted to martensite. Values
`
`Edwards Exhibit 1013, p. 4
`
`

`

`PATEL, AND COHEN:
`
`MARTENSiTE TRANSFORMATION
`
`r." 90v'l
`
`supercooling required to reach the appropriate
`driving force for initiating the reaction is reduced.
`and the MS temperature is thereby raised to AM; by
`
`
`
`= 1.3333! ,H'molfiC
`
`
`
`,/g MS WITHOUT APPLIED sraess
`o n's WITH APPLIED srness
`
`FIGURE 5. Schematic diagram showing how the mechani-
`cal energy Umsx due to the applied stress system changes the
`M, temperature by contributing to the thermodynamic driv-
`ing force of the martensitic transformation. In this case the
`driving forCc to start the transformation does not vary with
`the Ma temperature.
`
`is the temperature
`the applied stress. Thus M5’
`defined by the following relationship. and can be
`calculated therefrom through the known tempera-
`ture-dependence of F” — FA:
`
`This siopc holds for the linear portion of the curve.
`and prevails for the temperature range of interest.
`The increase in M, due to applied stress of 1000
`psi is
`
`(10)
`
`
`rid/f:
`do = $.32 = + 1.070(3/103 135i (for tension)
`
`In the case of uniaxial compression. :1 similar analy-
`sis gives
`
`(11) d3? = 'w = + 0.72DC/103 psi (for
`compression)
`
`the same treatment
`With hydrostatic pressure.
`may be employed.
`In this situation,
`there are no
`shear COmponcnts to aid the reaction. The applied
`pressure interacts only with the dilatational strain,
`and opposes the transformation. I f 0'1 = the magni-
`tude of
`the hydrostatic pressure. equation (4}
`becomes
`
`(12)
`
`U = — £061
`
`there being no orientation dependence in this case.
`For in : 1000 psi‘
`
`(7? (FM— F‘) at Mi = (FMH‘ FAMtMa-l- Um
`
`(13)
`
`U = -— 40 in-lbs/in3 = ~— 0.47 Cal/moi
`
`The calculation will now be carried through for
`the conditions of 1000 psi tension. in the iron-nickel
`alloys,
`
`From equation (6}, 26‘ = 79°, thus giving the orient-
`ation of the first martensitic plates to form on cools
`ing under the applied stress. Substituting in equa-
`tion (4):
`
`(8}
`
`Um“ = 122 in-lbs/in“ = 1.42 cal/Incl.
`
`Using the curve from the 0.5 per cent carbon—20
`per cent nickel steei in Figure 3:
`
`*Equation (7) is not strictly correct uniess the driving force
`to start the transformation is independent of the M. tempera—
`ture. as is the case for the alloy sttems under consideration
`here (Figures 2 and 3}. Appendix I presents a. corresponding
`treatment for the more general situation in which the driving
`force for starting the transformation varies with the M,
`temperature and therefore with the composition of the alloy.
`fAccording to Machlin's observations [21,
`so equals 0.05,
`but more precise dilatometric measurements indicate that the
`unit bulk expansion. is closer to 0.04. This represents a more
`exact value of so because the latter must correspond to the
`volume change.
`
`Using the curve for the T0 per cent iron~30 per
`cent nickel alloy in Figure 2 (since experimental
`results are available on this material for comparison
`purposes):
`
`(14) w = 1.23 caljmol-°C
`
`from which
`
`(15)
`
`.
`dM -- 0.47
`E1 ; —1§§ =' - 0.38°C/10a p51
`(for hydrostatic pressure):
`
`Table I summarizes these calcuiations and the
`
`corresponding experimental findings. The agreement
`is quite good. undoubtedly better than is justified.
`The criterion proposed here successfully predicts the
`lowering of M. by hydrostatic pressure and the
`raising of M5 by uniaxial tension and compression.
`
`IDr. J. C. Fisher kindly suggested a similar caiculation for
`the case of hydrostatic pressure (private communication).
`Note added in proof (August 24. 1953). The treatment of
`Dr. Fisher mentioned here has since appeared in Acta Met,
`-1 (1953) 310.
`
`LLombard Exhibit 1030, p. 5
`
`Edwards Exhibit 1013, p. 5
`
`Edwards Exhibit 1013, p. 5
`
`

`

`536
`
`ACTA METALLURGICA, VOL. 1, 1953
`
`TABLE I
`
`EFFECT or APPLIED Stanss on THE M, TEMPERATURE
`
`
`Stress System
`
`Uniaxial Tension
`
`Uniaxial Compression
`
`Hydrostatic Pressure
`
`Material
`0.5% C, 20% Ni, bal. Fe
`0.5% C, 20%. Ni, bal. Fe
`TOE/u Fe, 30% Ni
`
`dM,
`_
`Calculated
`+1.UY°C/103 psi
`+0.72"(,:/-l(la psi
`-—-0.38°C/1[ll‘ psi —
`
`;-
`Experimental
`+1 (PC/103 psi
`+0.65°C/103 psi
`—0.57°C/103 psi
`
`Change in Jail,
`Calculated
`.-'
`— +16°C
`+10,6°C
`I
`;5. 7°C.?—
`IEETOOO psi
`: Experimental
`+l5°C
`+10°C
`-8. 5°C
`
`
`
`
`The raising of M, by compression is quantitatively
`explained on the basis that the resolved shearcom-
`ponent of stress aids
`the transformation more
`effectively than the compressive normal component
`opposes it. This is a. consequence of the fact that the
`shear displacement is much larger than the dilata—
`tional displacement. Under uniaxial
`tension.
`the
`transformation is aided by both the shear and
`(positive) normal components of stress, and there-
`fore the M, is raised even more than in the case of
`uniaxial compression. Under hydroetatic pressure,
`the stress system only opposes the transformation,
`and the AI, is lowered.
`
`4. Conclusions
`
`-L1 The role of applied stress in the martensitic
`transforlrsition has been quantitatively analyzed
`for the conditionsof uniaxial tension, uniaxial com-
`
`pression and hydrostatic pressure.
`4.2 It
`is assumed that the reaction starts spon-
`taneously on cooling at
`the temperature (M,)
`where the free energy change FM — F" reaches a
`critical negative value. \Vhen external stresses are
`applied, the work done on or by the transformation
`[as the acting forces are carried through the trans-
`formation displacements] contributes algebraically
`to the free energy change, thus raising or lowering
`the M“, temperature.
`1.3 To calculate this mechanical contribution to
`
`the normal as
`the thermodynamic driving force,
`well as theshear components of the applied stresses
`are taken into account,
`thereby recognizing the
`interaction with both the dilatational and shear
`strains of the transformation.
`
`in
`least
`4,-1 This treatment predicts that, at
`iron—nickel and iron-nickel—carbon alloys, 114’, should
`
`be raised by uniaxial compressive stress, should be
`raised even more by tensile stress, and should be
`
`lowered by hydrostatic pressure. These changes in
`M, have been confirmed quantitatively by experi-
`ment.
`
`Acknowledgments
`
`This work is part of a research program sponsored
`at the Massachusetts Institute of Technology by the
`Office of Naval Research.
`The authors are indebted to the Office of Naval
`
`Reseach for sponsoring this program. Th cy also wish
`to express their appreciation to Professor P. W.
`Bridgman of Harvard University for the use of the
`high pressure facilities in his laboratory and for his
`helpful advice and interest in this work. \Villiam R.
`Yankee and R. Sudenfield rendered effective aid in
`
`various phases of the investigation. Stimulating dis-
`cussions concerning the significance of
`the results
`were held with E. S. Machlin, J. C. Fisher and B. L.
`Averbach.
`
`APPENDIX I
`
`Free Energy Changes Accompanying the
`Martensitie Transformation
`
`The thermodynamics of the iron-nickel system
`have been investigated by Jones and Pumphrcy [9],
`who obtained the following expression for the free
`energy change
`accompanying the austenite-to-
`martensite reaction:
`
`(Al)
`
`F” — F" = NMAHN, + lama/AFR
`
`where NNl -—— moi fraction of nickel
`N],e = moi fraction of iron
`AHN, = difference between the heat of solution
`of
`nickel
`in ferrite
`and that
`in
`austenite —-— 2500 cal/moi (adjusted to
`match the iron-nickel phase diagram).
`
`LLombard Exhibit 1030, p. 6
`
`Edwards Exhibit 1013, p. 6
`
`Edwards Exhibit 1013, p. 6
`
`

`

`PATEL AND COHEN:
`
`MARTENSITE TRANSFORMATION
`
`537
`
`AFFe = free energy change attending the trans—
`fer of one tool of iron from austenite to
`ferrite.
`
`Values of All?Fe have been calculated by Zencr [7]
`from the specific heat data of Austin [13]. For the
`iron-nickel system, Jones and Pumphrey found that
`AF“ had to be increased by 25 per cent over the
`values given by Zener to assure good correlation
`with the equilibrium diagram. This is the reason for
`the empirical factor of 1.25 in equation (Al).
`To consider the effect of carbon, the method of
`Fisher [1.0] was utilized:
`
`(A2) F” — F" = NMAHN, + 1.25NFenFn
`
`+ Nanosoo _ 3,425 T) + AF,
`
`where NC = mol fraction of carbon
`AF; = free energy change due to ordering of
`carbon atoms in tetragonal martensite
`The temperature-dependence indicated by third
`term on the right side of equation (A2) is obtained
`from the activities of carbon in austenlte and in
`
`[10].
`ferrite, and has been discussed by Fisher
`Vilhen no carbon is present. equation (AQ) reduces
`to equation (Al).
`The results of equations (Al) and (A2) are plotted
`in Figures 2 and 3.
`
`It is doubtful that AH)“ is actually constant over
`the entire temperature range extending down to the
`subaero levels where the martensitic reaction takes
`
`place in these alloys. However, various attempts to
`use extrapolated values of ARM did not seriously
`alter the final result. Allowing AI- N, to change with
`temperature caused the (FM e F“) VS. T curves to
`become steeper, but
`this was partially counter-
`balanced by the fact that the driving force at Mr,
`then varied with the M3 temperature (see Figure
`6]. The treatment of
`this case is presented in
`Appendix II, but in view of the limited data avail—
`able, the more straightforward calculations given in
`the text are considered to be sufficiently appropriate
`for the problem at hand.
`
`APPENDIX II
`
`Effect of Applied Stress When the
`
`Driving Force at Mf,2 Depends on M.
`
`Let the curves 1, 2 and 3 in Figure 6 represent the
`variation of FM — F" with temperature for a series
`of alloys in a given system. Suppose that the M,
`temperatures superimposed on these curves do not
`lie at a fixed value of FM — FA. This is shown by
`
`curve 4 which indicates how the driving force to
`start the transformation depends on the M,
`tem-
`perature (and therefore on composition}. 1 t is not
`necessary to assume linear relationships for these
`curves.
`
`+
`
`lF'-F"l-— O a MS wITHOUT APPLIED STRESS
`
`a M; woe APPLIED STRESS
`
`FIGURE 6. Schematic diagram showing how the mechani-
`cal energy Lima: due to the applied stress system changes the
`M9 temperature of alloy 2 by contributing to the thermo-
`dynamic driving farce. In this case the driving force tolstart
`the transformation varies with the M, temperature according
`to curve -I.
`
`let a stress be
`Taking alloy 2 as an example,
`applied to the specimen during cooling such that
`Um“ is the work (in 'calories per mol) done on the
`first-formed plates of martensite as a result of the
`components of
`the applied stress being carried
`through the transformation displacements.
`in this
`instance, consider that the transformation is aided
`by the acting stresses, and M, is raised. It is now
`required to find the temperature (M!) where the
`combined effects of Um“ and the chemical
`free
`
`energy change (PM e F“) achieve the necessary
`driving home to start the transformation at MU.
`The graphical solution to this problem is shown in
`Figure 6. The dashed curve is drawn parallel to the
`F“ — FA curve. but is displaced dowuward by an
`amount Um“. The intersection of the dashed curve
`with the M5 temperature line (which designates the
`driving force necessary to start the transformation
`as a function of
`temperature) gives the required
`temperature 1113'. It is evident that the change of the
`martensite—start temperature due to applied stress
`isgreater in the case at hand than if the driving
`force at M, were independent of AL, (compare with
`Figure 5).
`the dashed curve is
`\Vith hydrostatic pressure,
`displaced upwards because U is negative and opposes
`the transformation. This results in a corresponding
`lowering of the 1115’ temperature.
`
`LLombard Exhibit 1030, p. 7
`
`Edwards Exhibit 1013, p. 7
`
`Edwards Exhibit 1013, p. 7
`
`

`

`538
`
`1.
`
`9‘9"?
`
`.-\ CTA M [C'l‘.'\LL[.'R(1](‘.\.
`_,
`1‘
`
`References
`
`VOL.
`
`1.
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`in Physical R-Tetallurgy
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`
`j.
`
`10.
`11.
`
`12.
`13.
`
`1. Iron Steel Inst.
`
`LLombard Exhibit 1030, p. 8
`
`Edwards Exhibit 1013, p. 8
`
`Edwards Exhibit 1013, p. 8
`
`