678 F.Yan et al.:J.Mater.Sci.Technol.,2011,27(8),673-679 Table 2 Parameters used for flow stress calculation for the three types of mi- crostructures Structure dr/nm Sv/nm-I 0LAB/deg·fLAB/% fvol/%ccal/MPa DS-region 140 0.014 1.3 21 22 518 NT-region 47 0.043 63 666 SB-region 62 0.032 1.3 8 15 713 of high amount of geometrically necessary dislocations be assumed to be a linear additivity of disloca- (GNDs).The localized shear strain that results in the tion strengthening (o(p))from low angle dislocation formation of SBs in the T/M lamellae has been de- boundaries and isolated dislocations and boundary termined to be 2-412).This strain amplitude present strengthening(o(b))from high angle boundariesliol. in a width of 1 um gives a strain gradient around 2- For the NT-region,only boundary strengthening will 4 um-1,which is two orders of magnitude higher than be involved.Here the high angle boundaries refer to that of HPT.The low bound for the GND density to the boundaries larger than the critical angle that de- accommodate such strain gradient (z)can be esti- termines the transition of strengthening from disloca- matod=.By inserting x=2-4 um tion strengthening to boundary strengthening.Such and b0.256 nm,the GND density is approximated critical angle is generally very small,being 2-3341. as (1.8-3.6)x1016 m-2,which is close to the estimate In the present investigation,the critical angle was set according to Eq.(1):1.6x1016 m-2.This value is to be 2.By taking o(p)is proportional to square root higher than the value of DS-region by a factor of 2-3 of dislocation density and (o(b))is inversely propor- However,further investigation is required to clarify tional to the square root of boundary spacing(Hall- such point. Petch),the flow stress can be related to the structural It is seen that the well-developed shear bands are parameters: mainly composed of low angle dislocation boundaries =00+aMGbyp+K/Vd (4) This seems contrary to the previous investigations where shear bands in LN-DPD Cu have been be- =0o +aMGV1.5Sb0LAB fLAB+ lieved to be composed of randomly-oriented grains, implying the formation of high density of high angle KV(I-f九AB)Sv2 (5) boundaries.However.Fig.9 in literature1 revealed where oo is the flow stress (20 MPa),a is a con- that only thin SBs gives continuous diffraction circles, stant (0.24),M is the Taylor factor (3.0),G is whereas for the well-developed SBs the diffraction cir- the shear modulus (45 GPa),b is the Burgers vec- cles are discontinuous,which implies that high angle tor (0.256 nm),K is the Hall-Petch slope of coarse- boundaries are present in thin SBs but disappear in grained Cu(140 MPa-um1/2),S is the surface area wide ones.Concerning the development of SBs in- per unit volume ()0LAB and fLAB are the volves bending and necking of deformation twins,de- average misorientation angle and the fraction of low twinning and the evolution of dislocation structures in angle dislocation boundaries that are misoriented less the detwinned bands21,the thin SBs should consist of than critical angle,respectively.Equation (2)can twin boundaries and high angle boundaries due to the thus be rewritten: incomplete detwinning.This is consistent with the present observation,as high angle boundaries around 50 were detected.Accompanied by the complete de- Gtotal fais(0o +aMGV/1.5SdisbodaB fuB+ twinning,dislocation boundaries are formed,which is mainly composed of low angle dislocation boundaries. KV(1-fuiB)Stis/2)+ftwin(o+K/vd+fsb(co+ This means the evolution of dislocation boundaries after the detwinning as in literature [2]. aMG√/1.5Sb9兜BfB+KV/(1-fB)S-/2) 3.4 Strengthening mechanism (6) By inserting the parameters of the three different The above quantitative characterizations of the microstructures given in Table 2,Eq.(6)gives a flow three types of microstructures in LN-DPD Cu allow stress of 640 MPa,which is very close to the measured a quantitative analysis of the strengthening mecha- tensile yield strength of 620 MPal3). nisms in the LN-DPD Cu.Assuming a linear additiv- ity of strengthening from DS-(odis),NT-(otwin)and 4.Conclusions SB-(osb)regions weighted by their respective volume fraction fdis,ftwin and fsb,the flow stress(ototal)can A pure Cu (99.995 wt%)has been subjected to dy- be expressed: namic plastic deformation at cryogenic temperature to a strain of 2.1.Three types of microstructures(DS- Ototal =fdisOdis ftwinOtwin fsbOsb (2) region,NT-region,and SB-region)have been quanti- fied and the strengthening mechanism has been dis- For the DS-and SB-regions,strengthening can cussed.The following conclusions are drawn:678 F. Yan et al.: J. Mater. Sci. Technol., 2011, 27(8), 673–679 Table 2 Parameters used for flow stress calculation for the three types of mi￾crostructures Structure dr/nm Sv/nm−1 θLAB/deg. fLAB/% fvol/% σcal/MPa DS-region 140 0.014 1.3 21 22 518 NT-region 47 0.043 – – 63 666 SB-region 62 0.032 1.3 8 15 713 of high amount of geometrically necessary dislocations (GNDs). The localized shear strain that results in the formation of SBs in the T/M lamellae has been de￾termined to be 2–4[2]. This strain amplitude present in a width of 1 μm gives a strain gradient around 2– 4 μm−1, which is two orders of magnitude higher than that of HPT. The low bound for the GND density to accommodate such strain gradient (x) can be esti￾mated by: ρGND= √ 4x 3b [33]. By inserting x=2–4 μm−1 and b≈0.256 nm, the GND density is approximated as (1.8–3.6)×1016 m−2, which is close to the estimate according to Eq. (1): 1.6×1016 m−2. This value is higher than the value of DS-region by a factor of 2–3. However, further investigation is required to clarify such point. It is seen that the well-developed shear bands are mainly composed of low angle dislocation boundaries. This seems contrary to the previous investigations[2,4], where shear bands in LN-DPD Cu have been be￾lieved to be composed of randomly-oriented grains, implying the formation of high density of high angle boundaries. However, Fig. 9 in literature [1] revealed that only thin SBs gives continuous diffraction circles, whereas for the well-developed SBs the diffraction cir￾cles are discontinuous, which implies that high angle boundaries are present in thin SBs but disappear in wide ones. Concerning the development of SBs in￾volves bending and necking of deformation twins, de￾twinning and the evolution of dislocation structures in the detwinned bands[2], the thin SBs should consist of twin boundaries and high angle boundaries due to the incomplete detwinning. This is consistent with the present observation, as high angle boundaries around 50◦ were detected. Accompanied by the complete de￾twinning, dislocation boundaries are formed, which is mainly composed of low angle dislocation boundaries. This means the evolution of dislocation boundaries after the detwinning as in literature [2]. 3.4 Strengthening mechanism The above quantitative characterizations of the three types of microstructures in LN-DPD Cu allow a quantitative analysis of the strengthening mecha￾nisms in the LN-DPD Cu. Assuming a linear additiv￾ity of strengthening from DS-(σdis), NT-(σtwin) and SB-(σsb) regions weighted by their respective volume fraction fdis, ftwin and fsb , the flow stress (σtotal) can be expressed: σtotal = fdisσdis + ftwinσtwin + fsbσsb (2) For the DS- and SB-regions, strengthening can be assumed to be a linear additivity of disloca￾tion strengthening (σ(ρ)) from low angle dislocation boundaries and isolated dislocations and boundary strengthening (σ(b)) from high angle boundaries[10]. For the NT-region, only boundary strengthening will be involved. Here the high angle boundaries refer to the boundaries larger than the critical angle that de￾termines the transition of strengthening from disloca￾tion strengthening to boundary strengthening. Such critical angle is generally very small, being 2◦–3◦[34]. In the present investigation, the critical angle was set to be 2◦. By taking σ(ρ) is proportional to square root of dislocation density and (σ(b)) is inversely propor￾tional to the square root of boundary spacing (Hall￾Petch), the flow stress can be related to the structural parameters: σ = σ0 + αMGb√ρ + K/√ d (4) σ = σ0 + αMG1.5SvbθLABfLAB + K(1 − fLAB)Sv/2 (5) where σ0 is the flow stress (20 MPa), α is a con￾stant (0.24), M is the Taylor factor (3.0[9]), G is the shear modulus (45 GPa), b is the Burgers vec￾tor (0.256 nm), K is the Hall-Petch slope of coarse￾grained Cu (140 MPa·μm1/2), Sv is the surface area per unit volume (Sv=2 d [11]), θLAB and fLAB are the average misorientation angle and the fraction of low angle dislocation boundaries that are misoriented less than critical angle, respectively. Equation (2) can thus be rewritten: σtotal = fdis(σ0 + αMG 1.5Sdis v bθdis LABfdis LAB + K (1 − fdis LAB)Sdis v /2) + ftwin(σ0 + K/√ d + fsb(σ0+ αMG 1.5Ssb v bθsb LABfsb LAB + K (1 − fsb LAB)Ssb v /2) (6) By inserting the parameters of the three different microstructures given in Table 2, Eq. (6) gives a flow stress of 640 MPa, which is very close to the measured tensile yield strength of 620 MPa[3]. 4. Conclusions A pure Cu (99.995 wt%) has been subjected to dy￾namic plastic deformation at cryogenic temperature to a strain of 2.1. Three types of microstructures (DS￾region, NT-region, and SB-region) have been quanti- fied and the strengthening mechanism has been dis￾cussed. The following conclusions are drawn: