Temperature and strain-rate dependence of the flow stress of tantalum single crystals in cyclic deformation

(1)

HAL Id: jpa-00245821

https://hal.archives-ouvertes.fr/jpa-00245821

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Temperature and strain-rate dependence of the flow stress of tantalum single crystals in cyclic deformation

M. Werner

To cite this version:

M. Werner. Temperature and strain-rate dependence of the flow stress of tantalum single crystals in

cyclic deformation. Revue de Physique Appliquée, Société française de physique / EDP, 1988, 23 (4),

pp.672-672. �10.1051/rphysap:01988002304067200�. �jpa-00245821�

(2)

672 TEMPERATURE AND STRAIN-RATE DEPENDENCE OF THE FLOW STRESS OF TANTALUM SINGLE CRYSTALS IN CYCLIC DEFORMATION

M. WERNER

Max-Planck-Institut für Metallforschung, Institut für Physik,

7000 Stuttgart 80, ^FRG

Revue Phys. Appl. ²³ (1988) ⁶⁷² ^AVRIL ^1988,

In recent years progress has been made in our understanding of the strong dependence ^of ^the ^flow ^stress ^of b.c.c. metals

on temperature ^and strain rate below a critical temperature

Tk ⁽⁼ ^knee temperature). Ît îs ^now generally accepted that the mobility ôf ^screw dislocations represents ^the strain-rate-

controlling ^factor [1,2]. Because of their three-fold _sym- metry straight 111> screw dislocations in a b.c.c. crystal

are sessile. The high Peierls barrier _{may be} overcome with the aid of an applied ^stress ^or thermal activation making ^the

screw dislocations mobile.

The most appropriate theoretical model appears ^to be the formation and migration ôf ^kink pairs ôn the screw disloca- tions [3]. A recent theory ôf Seeger ^describes the "thermal"

component of ^the flow stress T* near and below the knee temperature by ^means of two different mechanical models and two different thermodynamical approximations. ^The analytical expressions ^are ^valid ⁱⁿ well-defined _ranges of stress and température.

As discussed elsewhere [4] the kink-pair formation may be calculated by ^the ^diffusion theory ^based ^on ^{the work} ^of

Kramers. At sufficiently ^low temperatures the kinks can move unhampered. ^This gives ^the ^same ^results ^as ^the ^trans- ition-state theory. By contrast, ^at higher temperatures the low mobilities correspond ^to viscous motion of the kinks.

The kinks are subject ^to Brownian forces due to their interaction with lattice vibrations. Thèse effects may be described in terms of either a kink mobility gk ^{or a} ^kink diffusivity Dk given by the Einstein-Nernst relationship.

For the calculation of the enthalpy of a pair ^of kinks,

Hkp(T*), ^two different mechanical models may be used [3]:

for small stresses we can use the elastic interaction approximation with a Coulomb potential. ^Then Hkp ^is ^given

by

where 2Hk denotes the enthalpy ôf â pair ôf ^two îsolated kinks, ând â ^can ^be expressed âs

In equation (2) 10 denotes the pre-logarithmic factor of the dislocation line-tension, ^{b the} Burgers ^vector ând â ^the height ôf ^the ^kinks.

At large ^stresses ^the kink-pair ^formation enthalpy may be calculated from the line-tension model [3,4]. ^{In the} special

case of a so-called Eshelby potential ^we ^have

where TP ^is ^an effective Peierls stress.

The experimentally determined kink properties ^can ^be ^com- pared to the theoretical predictions ^and provide ^a ^critical

test of the theoretical assumptions.

The cyclic deformation experiments ^were carried out in a

servo-hydraulic closed-loop control MTS-machine using plastic strain control. The advantage ^of ^the present experi-

mental procedure ^is ^{that all} measurements can be performed

on only ône single specimen ^which ^{has been} predeformed into cyclic saturation, î.e. ât nearly ^constant microstructure.

Measurements were carried out over a range of temperatures from 80 K to 450 K using ân isopentane ^bath ^cooled by liquid nitrogen ôr â heated silicone oil bath. Figure ¹ ^shows

the temperature dependence of T* for five different plastic shear-strain rates varying ^from 2·10-5 s-’ to 6·10-3 s-1.

Fig. ^1. Temperature dependence ôf ^the êffective ^stress The data lying ⁱⁿ ^the ^stress regime âbove âbout ^{80 MPa} âre describable in terms of the line-tension approximation. ^We find, using the effective Peierls stress Tp âs ân âdjustable

parameter, TP - ^{280 MPa.} The data in the stress regime

below 80 MPa ^were analyzed by â global least-squares ^fit ^to the complete expressions ôf ^the kink-pair ^formation ^rate ând in terms of the approximate relationships just ^mentioned.

The agreement of the values obtained by the different methods is excellent.

The formation enthalpy ² Hk ^of ^two isolated kinks was

obtained by determination of the knee temperatures for dif- ferent strain rates [5]. This procedure gives ^us 2Hk=0.98 ^eV.

From the parameter ^a ^we may calculate the kink height ^a.

The results are clearly incompatible ^with ^the existence of the shortest possible ^kinks only. ^Thus ^we ^have ^to ^conclude that ^a large fraction of the kinks are double kinks.

In the low-stress regime ^{the kink} diffusivity Dk appears in the expression for the temperature and the ^stress dependence of the strain rate. A temperature-independent Dk ^near ^and

above the knee temperature is fully compatible ^with ^our

measurements. For a simple picture ^{that the} ^kink ^diffusion proceeds by jumps ^of ^one ^lattice period ^we obtain from

Dk

⁼

b2yk ^a ^value ^for vk

⁼

3’1013 s-1 which is of the ^same order of magnitude ^as ^the Debye frequency.

Finally ^we obtain for the product ^of ^the density of screw

dislocations and the spacing between obstacles that are

impenetrable ^to ^kinks, the value 2.1·105 m-1.

Références :

[1] Hirsch, P.B., ^Proc. ^Conf. Strength ^of Metals and Alloys, Suppl. ^to ^Trans. Japan Inst. Metals 9 (1968) ^XXX.

[2] 0160esták, ^{B. and} Seeger, A., ^Z. Metallkde. 69 (1978) ^195.

[3] Seeger, ^A., ^Z. Metallkde. 72 (1981) ^369.

[4] Seeger,A. and Schiller,P., Physical Acoustics, ^Vol. ^{III A} (ed. by W.P.Mason) p. 361, ^Academic Press, 1966.

[5] Ackermann, F., Mughrabi, H., ^and Seeger, A., ^Acta metall. 31 (1983) ^1353.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304067200

Temperature and strain-rate dependence of the flow stress of tantalum single crystals in cyclic deformation

HAL Id: jpa-00245821

https://hal.archives-ouvertes.fr/jpa-00245821

Submitted on 1 Jan 1988

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Temperature and strain-rate dependence of the flow stress of tantalum single crystals in cyclic deformation

M. Werner

To cite this version:

M. Werner. Temperature and strain-rate dependence of the flow stress of tantalum single crystals in

cyclic deformation. Revue de Physique Appliquée, Société française de physique / EDP, 1988, 23 (4),

pp.672-672. �10.1051/rphysap:01988002304067200�. �jpa-00245821�

672

TEMPERATURE AND STRAIN-RATE DEPENDENCE OF THE FLOW STRESS OF TANTALUM SINGLE CRYSTALS IN CYCLIC DEFORMATION

M. WERNER

Max-Planck-Institut für Metallforschung, Institut für Physik,

7000 Stuttgart 80, FRG

Revue Phys. Appl. 23 (1988) 672 AVRIL 1988,

In recent years progress has been made in our understanding of the strong dependence of the flow stress of b.c.c. metals

on temperature and strain rate below a critical temperature

Tk (= knee temperature). It is now generally accepted that the mobility of screw dislocations represents the strain-rate-

controlling factor [1,2]. Because of their three-fold sym- metry straight 111&#x3E; screw dislocations in a b.c.c. crystal

are sessile. The high Peierls barrier may be overcome with the aid of an applied stress or thermal activation making the

screw dislocations mobile.

The most appropriate theoretical model appears to be the formation and migration of kink pairs on the screw disloca- tions [3]. A recent theory of Seeger describes the "thermal"

component of the flow stress T* near and below the knee temperature by means of two different mechanical models and two different thermodynamical approximations. The analytical expressions are valid in well-defined ranges of stress and température.

As discussed elsewhere [4] the kink-pair formation may be calculated by the diffusion theory based on the work of

Kramers. At sufficiently low temperatures the kinks can move unhampered. This gives the same results as the trans- ition-state theory. By contrast, at higher temperatures the low mobilities correspond to viscous motion of the kinks.

The kinks are subject to Brownian forces due to their interaction with lattice vibrations. Thèse effects may be described in terms of either a kink mobility gk or a kink diffusivity Dk given by the Einstein-Nernst relationship.

For the calculation of the enthalpy of a pair of kinks,

Hkp(T*), two different mechanical models may be used [3]:

for small stresses we can use the elastic interaction approximation with a Coulomb potential. Then Hkp is given

by

where 2Hk denotes the enthalpy of a pair of two isolated kinks, and a can be expressed as

In equation (2) 10 denotes the pre-logarithmic factor of the dislocation line-tension, b the Burgers vector and a the height of the kinks.

At large stresses the kink-pair formation enthalpy may be calculated from the line-tension model [3,4]. In the special

case of a so-called Eshelby potential we have

where TP is an effective Peierls stress.

The experimentally determined kink properties can be com- pared to the theoretical predictions and provide a critical

test of the theoretical assumptions.

The cyclic deformation experiments were carried out in a

servo-hydraulic closed-loop control MTS-machine using plastic strain control. The advantage of the present experi-

mental procedure is that all measurements can be performed

on only one single specimen which has been predeformed into cyclic saturation, i.e. at nearly constant microstructure.

Measurements were carried out over a range of temperatures from 80 K to 450 K using an isopentane bath cooled by liquid nitrogen or a heated silicone oil bath. Figure 1 shows

the temperature dependence of T* for five different plastic shear-strain rates varying from 2·10-5 s-’ to 6·10-3 s-1.

Fig. 1. Temperature dependence of the effective stress The data lying in the stress regime above about 80 MPa are describable in terms of the line-tension approximation. We find, using the effective Peierls stress Tp as an adjustable

parameter, TP - 280 MPa. The data in the stress regime

below 80 MPa were analyzed by a global least-squares fit to the complete expressions of the kink-pair formation rate and in terms of the approximate relationships just mentioned.

The agreement of the values obtained by the different methods is excellent.

The formation enthalpy 2 Hk of two isolated kinks was

obtained by determination of the knee temperatures for dif- ferent strain rates [5]. This procedure gives us 2Hk=0.98 eV.

From the parameter a we may calculate the kink height a.

The results are clearly incompatible with the existence of the shortest possible kinks only. Thus we have to conclude that a large fraction of the kinks are double kinks.

In the low-stress regime the kink diffusivity Dk appears in the expression for the temperature and the stress dependence of the strain rate. A temperature-independent Dk near and

above the knee temperature is fully compatible with our

measurements. For a simple picture that the kink diffusion proceeds by jumps of one lattice period we obtain from

Dk

b2yk a value for vk

3’1013 s-1 which is of the same order of magnitude as the Debye frequency.

Finally we obtain for the product of the density of screw

dislocations and the spacing between obstacles that are

impenetrable to kinks, the value 2.1·105 m-1.

Références :

[1] Hirsch, P.B., Proc. Conf. Strength of Metals and Alloys, Suppl. to Trans. Japan Inst. Metals 9 (1968) XXX.

[2] 0160esták, B. and Seeger, A., Z. Metallkde. 69 (1978) 195.

[3] Seeger, A., Z. Metallkde. 72 (1981) 369.

[4] Seeger,A. and Schiller,P., Physical Acoustics, Vol. III A (ed. by W.P.Mason) p. 361, Academic Press, 1966.

[5] Ackermann, F., Mughrabi, H., and Seeger, A., Acta metall. 31 (1983) 1353.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304067200

7000 Stuttgart 80, ^FRG

Revue Phys. Appl. ²³ (1988) ⁶⁷² ^AVRIL ^1988,

In recent years progress has been made in our understanding of the strong dependence ^of ^the ^flow ^stress ^of b.c.c. metals

on temperature ^and strain rate below a critical temperature

Tk ⁽⁼ ^knee temperature). Ît îs ^now generally accepted that the mobility ôf ^screw dislocations represents ^the strain-rate-

controlling ^factor [1,2]. Because of their three-fold _sym- metry straight 111> screw dislocations in a b.c.c. crystal

are sessile. The high Peierls barrier _{may be} overcome with the aid of an applied ^stress ^or thermal activation making ^the

The most appropriate theoretical model appears ^to be the formation and migration ôf ^kink pairs ôn the screw disloca- tions [3]. A recent theory ôf Seeger ^describes the "thermal"

component of ^the flow stress T* near and below the knee temperature by ^means of two different mechanical models and two different thermodynamical approximations. ^The analytical expressions ^are ^valid ⁱⁿ well-defined _ranges of stress and température.

As discussed elsewhere [4] the kink-pair formation may be calculated by ^the ^diffusion theory ^based ^on ^{the work} ^of

Kramers. At sufficiently ^low temperatures the kinks can move unhampered. ^This gives ^the ^same ^results ^as ^the ^trans- ition-state theory. By contrast, ^at higher temperatures the low mobilities correspond ^to viscous motion of the kinks.

The kinks are subject ^to Brownian forces due to their interaction with lattice vibrations. Thèse effects may be described in terms of either a kink mobility gk ^{or a} ^kink diffusivity Dk given by the Einstein-Nernst relationship.

For the calculation of the enthalpy of a pair ^of kinks,

Hkp(T*), ^two different mechanical models may be used [3]:

for small stresses we can use the elastic interaction approximation with a Coulomb potential. ^Then Hkp ^is ^given

where 2Hk denotes the enthalpy ôf â pair ôf ^two îsolated kinks, ând â ^can ^be expressed âs

In equation (2) 10 denotes the pre-logarithmic factor of the dislocation line-tension, ^{b the} Burgers ^vector ând â ^the height ôf ^the ^kinks.

At large ^stresses ^the kink-pair ^formation enthalpy may be calculated from the line-tension model [3,4]. ^{In the} special

case of a so-called Eshelby potential ^we ^have

where TP ^is ^an effective Peierls stress.

The experimentally determined kink properties ^can ^be ^com- pared to the theoretical predictions ^and provide ^a ^critical

The cyclic deformation experiments ^were carried out in a

servo-hydraulic closed-loop control MTS-machine using plastic strain control. The advantage ^of ^the present experi-

mental procedure ^is ^{that all} measurements can be performed

on only ône single specimen ^which ^{has been} predeformed into cyclic saturation, î.e. ât nearly ^constant microstructure.

Measurements were carried out over a range of temperatures from 80 K to 450 K using ân isopentane ^bath ^cooled by liquid nitrogen ôr â heated silicone oil bath. Figure ¹ ^shows

the temperature dependence of T* for five different plastic shear-strain rates varying ^from 2·10-5 s-’ to 6·10-3 s-1.

Fig. ^1. Temperature dependence ôf ^the êffective ^stress The data lying ⁱⁿ ^the ^stress regime âbove âbout ^{80 MPa} âre describable in terms of the line-tension approximation. ^We find, using the effective Peierls stress Tp âs ân âdjustable

parameter, TP - ^{280 MPa.} The data in the stress regime

below 80 MPa ^were analyzed by â global least-squares ^fit ^to the complete expressions ôf ^the kink-pair ^formation ^rate ând in terms of the approximate relationships just ^mentioned.

The formation enthalpy ² Hk ^of ^two isolated kinks was

obtained by determination of the knee temperatures for dif- ferent strain rates [5]. This procedure gives ^us 2Hk=0.98 ^eV.

From the parameter ^a ^we may calculate the kink height ^a.

The results are clearly incompatible ^with ^the existence of the shortest possible ^kinks only. ^Thus ^we ^have ^to ^conclude that ^a large fraction of the kinks are double kinks.

In the low-stress regime ^{the kink} diffusivity Dk appears in the expression for the temperature and the ^stress dependence of the strain rate. A temperature-independent Dk ^near ^and

above the knee temperature is fully compatible ^with ^our

measurements. For a simple picture ^{that the} ^kink ^diffusion proceeds by jumps ^of ^one ^lattice period ^we obtain from

b2yk ^a ^value ^for vk

3’1013 s-1 which is of the ^same order of magnitude ^as ^the Debye frequency.

Finally ^we obtain for the product ^of ^the density of screw

impenetrable ^to ^kinks, the value 2.1·105 m-1.

[1] Hirsch, P.B., ^Proc. ^Conf. Strength ^of Metals and Alloys, Suppl. ^to ^Trans. Japan Inst. Metals 9 (1968) ^XXX.

[2] 0160esták, ^{B. and} Seeger, A., ^Z. Metallkde. 69 (1978) ^195.

[3] Seeger, ^A., ^Z. Metallkde. 72 (1981) ^369.

[4] Seeger,A. and Schiller,P., Physical Acoustics, ^Vol. ^{III A} (ed. by W.P.Mason) p. 361, ^Academic Press, 1966.

[5] Ackermann, F., Mughrabi, H., ^and Seeger, A., ^Acta metall. 31 (1983) ^1353.