材料应力应变曲线出现单峰或多峰的理论解释

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1、材料应力应变曲线出现单峰或多峰的理论解释具有低或者中等层错能的金属，他们的回复过程比较慢，热加工过程中，动态回复未能同步抵消 加工过程中的位错的增殖积累，即出现加工硬化现象，流变应力增加，在某一临界形变条件下， 会发生动态再结晶在再结晶时,大量的位错被再结晶核心的大角度截面推移而消除，当这样 的软化过程占主导地位时，流变应力下降,应力应变曲线出现峰值由于在再结晶形核和长大的 同时材料继续形变，再结晶形成新的晶粒也经受形变，即硬化因素又重新增加当新晶粒内形变 达到一定程度后，又可能开始第二轮再结晶这样复杂的硬化和软化的叠加情况下，应力应变曲 线有可能周期地出现峰值材料发生动态再结晶时，应力应变曲

2、线可能出现单峰也可能出现多峰 当应变速率高或者形变 温度低时，应力应变曲线出现一个宽阔的单峰当应变速率减小或者形变温度增加时，应力应 变曲线会从单峰过渡到多峰形状其原因在于：在高应变速率或者低形变温度下，再结晶的速度慢，当第一轮再结晶未完成时就开始第二轮再 结晶,所以在应力应变曲线出现第一个峰后，材料始终保持部分再结晶状态，应力应变曲线就平 滑了;而在低应变速率或者高形变温度下，往往在第一轮再结晶完成后才开始第二轮再结晶，这 些过程重复,应力应变曲线就出现多峰摆动的状态。10 Basic Concepts1.4. STRESS AND STRAIN RELATIONS 141. Stress-

3、Strain and Stress- Strain- Rate LawsIn the previous sections the stresses and the strains were treated separately. In this section the relations between stress and strain will be treated.In uniaxial loading(1.17a)and(1.17b)The engineering strain in the xrdirection isLoading of this nature is perform

4、ed on a tensile testing machine.A typical stress-strain curve is given in Fig. 1.6. As the load rises from point A to point B. the elastic range. Hookes law prevails, i.e.,j (b)where the proportionality factor E is called Hookers or Youngs modulus of elasticity. As the load is removed, the strain wi

5、ll return to zero. One can repeal loading again along lhe line ABC. With relief of the load, line CD will be followed At point D the load is fully removed, but the specimen does not return to its original length The elastic limit has been exceeded and permanent plastic deformation has occurred. Poin

6、t B may be recognized as the place at which the elastic limit was reached and plastic flow was first observed Point B is hard to detect, and with better and more sensitive instrumentation deviation from elastic deformations is detected at lower values of stress and strain On subsequent loading (afte

7、r point D), a hysteresis loop will form. However, this hysteresis will disappear if loading and unloading are performed slowly. Proceeding with the loading beyond point C. the characteristic line for plasticFIGURE 1.6. Stress-strain curve with cycling. 12flow after the detour to the path CD is pract

8、ically identical with the characteristic line obtained by continuous loading from A to B to C and beyond So also for unloading from point E to point F and reloading to point E and beyond. Although the differences in the characteristics caused by interruptions are the topics of many studies, they are

9、 minute and can be neglected when only the overall picture is studied. Upon loading from point F to point E as described, the specimen behaves elastically; but the stress at point E is at the flow value for the material. If loading is continued beyond point Et the material will behave plastically, b

10、ut higher loads are required to cause further elongation. On the other hand, on unloading from point E the iiiaicrial behaves elastically. The straight line EF is parallel to AB. One notices the hardening of the material This phenomenon is called strain hardening. One can consider the stress-strain

11、diagram as the geometrical assembly of all the yield points of a strain-hardenable material On loading the workpiece behaves plastically, while on unloading it behaves elastically. Note that the definition of the yield point here is quite different from the value of stress at (say) 0.2% offset. Unde

12、r some conditions (at temperatures above the recrystallization temperature) the material is sensitive to strain rates and the stress-strain curve varies with the rate of loading. If the loading is performed at different rates, the stress-strain curve will differ as shown in Fig. 1.7. The plastic ran

13、ge of the characteristic curve gets higher with higher loading rates and strain rates.For each specific speed of loading, the stress-strain curve can be approximated by two straight lines as shown in Fig. 1& The slope of the steqxjr line is given by the elasticity E. and its inclination isa = arctan

14、 EThe upper inciination p expresses the ability of the material to strain-harden. If this ability is negligible, fl becomes zero as shown in Fig. 19. In all metabforming operations the elastic range is very small relative to the plastic range. Therefore, the elastic range can be ignored. In that cas

15、e tana is effectively infinite in the elastic range and the diagram of Fig. LIO resultsIn Fig. l0 lhe only characteristic property of the material is its yield strength a0, which can be determined from a tensile test. A typical tensile-testFIGURE 1.7. Stress vs. strain at various strain rates.FIGURE

16、 1.8. Modulus of elasiicily and modulus of plasticityFIGURE 1.9. Stress vs. strain with no strain hardeningFIGURE 1.10. Stress vs. strain with zero clastic range.specimen is shown in Fig. 11. Before neckings the stress field in a tensile test(d)From the diagram of load as a function of displacementt

17、 one can compute the values of the axial stress The axial stress at yield is thus the yield strengthPlacing Eqs. (e) and (d) in the Mises yield criterion (Eq(1.7) will gives an identity, and therefore the tensile test determines the value of %The tensile test provides. experimentallyt the relationsh

18、ip between one component of the stress (o() and its corresponding strain component (). A mathematical model is expected to provide relationships among each component of the stress field (ab alt. and am if the principal directions serve as the axes of the coordinate system) and all componenu of the d

19、eformation field. As observed in Section l.l and Eqs. (1.4) and (1.5)t the stress components can be replaced by the average and deviator components of the stress The deformation field can likewise be presented through strain or strain-rate components.The relation between the stress field and the str

20、ain field, in the plastic range, as presented by Mises, relates the components of the stress deviator to the strain-rate components as follows:(l8)FIGURE 1.11. A lensulatest specimenStress and Strain Rehtiom 13where a0 is a characteristic property of the material, expressing here the tensile strengt

21、h in uniaxial loadingMaterial obeying Mises* stress-deviator-strain-rate relations Eq. (18) is called Mises material. For Mises material the stress-deviator components relate to the strain-rate components. The hydrostatic component of stress (mean or average stress) is not determined uniquely from t

22、he strain rates alone.Evaluation of the sum of the squares of the individual components of the stress deviator of Eq. (L18) leads toSf+SZ附士詈 (券豐宀弼(.7) which is the previously presented Mises yield criterion. Thus Mises constitutive equation given by Eq. (1.18) leads to the determination of the yield

23、 criterion given by Eq. (1.7).The relation of the effective stress deviator (Sctt) to the yield in tension (a0) is given by Eqs. (b) and (1.7) of Section 1.1 as follows:尙*%(119)Thus, for Mises materialwhere and Sctt are defined respectively by Eq. (1.16a) and by Eq. (b) of Section LI as follows: 鬲H疗

24、W +爲+ di /r /sf+弗+s 金*Historically, however an early expression for the constitutive equations was presented, followed by the suggestion of the Mises yield criterion and leading to Eq. (1.18). While the basic concq)ts can be studied without the following historical review, it may be of interestPropo

25、sing that the strain rate components are each proportional to its respective component of the stress deviator, Sl Vcnant* suggested the following relation:i u *iu -(0where p is a scalar function of the strain-rate components and cin. ThisBasic Conceptsfunction p can be determined by first identifyin

26、g a quantity I defined by2/ - c? + 垢1(g)Substitution of Eq. (f) into Eq. (g) leads to2一屮(Sf+ S召+ S缶)(h)Inserting the yield criterion of Eq. (1.7), Sf +into Eq. (h)gives27 环冷(i)and4 事范冷甲0)a0 1 %St- Venanfs1 description for the characteristic relation becomes the following Mises relation:如疗%九打鄴k豐九wher

27、eI 1(1 + !I *!)Equation (k) can be derived directly from Eq. (1.18).1.4.2. The Extended Stress-Strain RelationsA closer look at the stress-strain relations for uniaxial tension as provided by the tensile testing of real material is justified because tensile testing is a most common procedure to dete

28、rmine strength characteristics for metals. In Figs 1.6 and 1.7 straining was carried out only to a limited extent. When straining is continued up to failure by fracture, a more complete picture is observed. In Fig. 12 the abscissa is the engineering strain, defined as the elongation perss3KASozo:33N

29、eN3MAXIMUM LOADENGINEERING STRAIN(1-21)FIGURE 1.12. Tensile-tcsl characteristics.unit original length /0 of Fig. ll. If. at any instant during the deformation, the length of the narrow section of the tensile bar is h then the engineering strain is defined asThus when the axial direction of the speci

30、men, which coincides with the loading direction, is designated as a principal direction I9 then the strain is actually one component of the three principal strains. The other components not plotted in Fig. aren * m - -1!(122)Checking for incompressibility one finds that Cj + cn + cIH s 0.The ordinat

31、e in Fig. 12 is the engineering stress in the axial direction, defined as the load per unit original area of the specimen:p恳厂-s- J-(1.23)Note that the engineering stress in the axial direction is only one of the three principal components. However, the stress components normal to the loading directi

32、on are zero.When the specimen is not loaded, both S and e are zero. With the initiation of loading the machine starts to pull on the specimen, forcing it to elongate, so that 0 increases Since the specimen resists the elongation, the load P and engineering stress S increase. At low values of load, t

33、he engineering stress S is proportional to the engineering strain c:16 Bask Conceptswhere the proportionality factor E is Hookes (Youngs) modulus of elasticity. The stresses and strains are elastic, which means recoverable. If the load is removed and strain drops to zero, the specimen is restored to

34、 its onginal dimensions.After yield, the proportionality between stress and strain no longer prevails, and if the load is removed the deformed rod does not recover its original shape or length. Permanent nonrecoverable plastic deformations have occurred during loading. With further increase in loadi

35、ng the rod elongates further but the slope of the engineering stress-strain curve decreases The slope of the curve in the plastic range represents the strain- hardening characteristics of the material tested Finally the characteristic line flattens, strain-hardening capability decreases with further

36、 straining, and the stress reaches a peak beyond which further extension requires lower and lower loads.Careful measurement of the rod diameter will reveal that up to the peak load, the diameter decreases homogeneously with increasing strain, so that volume constancy is retained. At the peak, and be

37、yondt the deformation ceases to be uniform. The plastic deformations become localized and necking commences at the locality of the plastic deformations while the rest of the specimen returns to behaving elastically. The reduced cross section of the necked region is the cause of the lower loads. When

38、 the load decreases, the stress on the larger cross section of the specimen on both sides of the necked region (See Fig 13) drops below the plastic limit and reenters the elastic region. On further extension, the necked region elongates while the rest of the specimen contracts. The engineering strai

39、n as defined by Eq. (1.21), where the measured A/ is the net elongation of the plastic necked region minusPPFIGURE L13 NeckingNECKED PLASTIC REGION Stress and Strain ReUtions 17contraction of the elastic region, loses all meaning beyond necking. Furthermore. if one defines the true stress as the loa

40、d per unit instantaneous area, then the true stress cannot be defined accurately beyond necking Presently the translation from engineering stress-strain curves to true stress-strain curves will be performed. When elongation continues far enough beyond necking, fracture will occur. The interpretation

41、 of the behavior of the stress-strain curve beyond necking becomes so unreliable that if (for example) short and long billets of the same material are loaded, their characteristics beyond necking nevertheless are not identical(See Fig 14) Thus, the first precaution to be taken in order to be able to

42、 compare results of sq)arate experiments is standardization of the design of tensile bars.The true stress is defined as the force (P) per unit instantaneous area (V):where the cross-sectional area of the specimen (A) and the load (P) are constantly changingTrue strain is defined as the elongation pe

43、r unit instantaneous length and has been shown in Eqs. (1 .ll)v (1.13)t and (i4) to be300:04 K3AStrew and Strain Relariom 1918 Basic Concepts(1.25b)Thus the relation of true strain to engineering strain is expressed by0 ln(4 + c)Note that for small strainswhen c c 1, then c(126)Strew and Strain Rela

44、riom 19Strew and Strain Relariom 19Bui during gross plastic deformations both and are large(127)The relation of true stress a to engineering stress will be found next. The load P is the product of true stress a and instantaneous cross-sectional area and it is also equal to the engineering stress S m

45、ultiplied by the original cross-sectional area Ao:P SAa oAorBy volume constancy.(b)and therefore0 28)Substitution of Eq. (128) into Eq. (a) gives(1.29)The graphical presentation of the tensile test in terms of the engineering stress-strain relations can be replaced, through Eqs. (1.25b) and (l.29)t

46、by a true-stress-true-strain curve. Fig. l5.The characteristic line in Fig. 15. as reproduced from the recording of the hypothetical tensile test of Fig. 1.12t terminates at necking. As mentioned previously, because of the localization of the strain at the necked region and the relief of strain else

47、where, the readings of c and lherefore those of and a are not meaningful beyond the point of necking. Necking occurs at small strains, while for the purpose of metalworking the stress-strain relations at much higher strains are of great importance Several techniques to extend theTRUE STRAIN $ !(!) I

48、n()FIGURE 1.15. True-stress-tne-strain curve.b SS3JUS wnorlstress-strain characteristic curves have been constructed. Among them are:1. Extrapolation beyond necking.2. Extending the stress-strain curve by multiple tensile testing with displaced origin 3. Determination of the flow strength by a compr

49、ession test. (See Section 2.4.6 of Ref. 1 of Chapter 2.)L4.2.1. Extrapolation beyond Necking. The point of necking is unique in that the load P from Eq. (1.27) and the engineenng stress S are both at their maximum points. Thus, differentiatingo of Eq. (1.29) with respect to leads todo d.(c)乔乔(S)Y乔+S

50、eso thatAnd because S is at its maximum point, therefore(d)da I乔I20 Basic ConceptsSubstituting again from Eq. (1.29)t the result is貂 i(5IncckingEquation (1.30) indicates that al the point of necking the slope of the true-stress-true-strain characteristic curve is equal to the stress at that point. B

51、y extrapolation, then, the curve in Fig. 15 can be extended farther. The farther the straight extrapolation is taken, the more inaccuracy is encountered.1.4.2.2. The Extended Stress-Strain Curve. The slope at necking can serve as an estimate of the flow strength beyond necking, but not very reliably

52、 nor too far away from the point of neckingIn his work on wire drawing, Wistreich2 developed extended stress-strain curves for the electrolytic copper with which he experimented. A heavy-gauge wire was drawn through a succession of dies with reduction of about 5 to 10% through each die. After drawin

53、g through each die. a specimen was taken for tensile testing As the wire was reduced through the dies it became harder and harder because the strain hardening raised the yield and flow stresses. Individual t rue-stress-st rain curves for each specimen were constructed. The point of zero strain was s

54、hifted to the right by the amount indicated by the total reduction achieved through the drawing prior to machining of the tensile rod.The effective or true strain for the origin isD -2 In(】31)where RQ is the original radius of the wire before the first draw on the machine The radius of the wire afte

55、r the instantaneous draw, before the specimen was machined, is Rf. The abscissa of Fig. I6 (which is analogous to Fig. 22 of Wistreich) is the true strain For example, when the ratio of the3匚。3吐3FIGURE 1.16. Flow stress vs. effective strain.Power and Work of Deformation 21TRUE STRAINFIGURE 1.17. Ext

56、ended stress-strain curve.SS3S mowdiameter of the original annealed material (2R0) to the final diameter (2Rf)of the rod after several reductions is 2:1 (/?0/7?z 2). lhe total reduction in area is 75%. equivalent to a irue strain of e = 1.3863. When a specimen of the rod is then made after 75% reduc

57、tion in area, its stress-strain curve starts with the origin al 0 h 13863 The envelope of all these curves is called the extended stress-st rain curve This curve can. in turn, serve for the determination of lhe average flow stress for any rod with any amount of cold work if the amount of cold work c

58、an be defined by a value of its true or effective strain.Some materials may behave differently when drawn than when extruded In Chapter 7. entitled u Hydrostatic Extrusion/1 the effect of environmental pressure on ductility and strength is discussed in more detail. In Fig. 1.17 the effect of differe

59、nt modes of deformation on the extended stress-strain curve is shown. This effect is real and should be accounted for. From curves C to B lo A in this figure, strengthening becomes more effective with increasing environmental pressure 1.5. POWER AND WORK OF DEFORMATION 1.5J. PowerThe power (work per unit time) spent per unit volume for Mises material is(132)Power and Work of Deformation 21(132)Power and Work of Deformation 21(132)