The macro-flow stress (Moderator's Comment)

HAL Id: jpa-00245784

https://hal.archives-ouvertes.fr/jpa-00245784

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.

The macro-flow stress (Moderator’s Comment)

U.F. Kocks

To cite this version:

U.F. Kocks. The macro-flow stress (Moderator’s Comment). Revue de Physique Appliquée, Société

française de physique / EDP, 1988, 23 (4), pp.381-382. �10.1051/rphysap:01988002304038100�. �jpa-

00245784�

381

The macro-flow stress* (Moderator’s Comment)

U.F. Kocks

Center for Materials Science, ^{Los Alamos National} Laboratory,

Los Alamos, ^NM 87545, ^U.S.A.

Revue Phys. Appl. ²³ (1988) ^{381-382 AVRIL} 1988,

Micro-Flow versus Macro-Flow

Figure ^{1 shows}

^schematic stress/strain ^curve

at low strains: after

linear-elastic rise, ^there is

rather rapid transition toward

asymptote with very low slope. This is realistic behavior for many pure metals ^and solution-hardened alloys.

In other cases,

sharp transition is obtained in

plot ^{of o2 vs}

[1-3].

Fig. ¹

FRANCIOSI has addressed the regime ^of beginning plasticity ^{in detail.} The mechanisms are quite complex: ^the spatial ^{extent of the} plastic regions

increases drastically, and the mobile dislocation structure undergoes rapid rearrangements-toward

dynamic equilibrium ^or steady ^state. [Schôck pointed ^out that the statistics should be

more

complex, ^{due to} correlations between neighboring

dislocation segments.]

An experimental observation is that this region

of micro-flow depends ^{on the detailed} specimen history; ^{it is, for} example, sharper after pre-

straining ^{in the same} direction, less sharp ^after opposite straining ^or recovery; and the transition sometimes involves an "inverse transient", i.e.

yield drop. ^In the absence of the latter, ^{it is} difficult to define

"proportional limit": it

decreases, perhaps ^{to zero, when the} magnitude ^of

the considered offset is decreased.

*

Work supported by ^{the U.S.} Department ^of Energy.

Figure ^{1 shows two} idealizations by ^{an elastic} ideally-plastic model: the dotted line breaks at

presumed proportional limit; ^{it does not describe} the subsequent ^{behavior at all.} The dashed line matches the asymptotic, "(macro-)flow stress"; ^it ignores the transition regime, i.e. "micro-flow".

A significant advantage of the latter method is that the locus of the macro-flow stress

function of plastic pre-strain ^{is identical to the} stress/strain ^{curve at strains} large compared ^to the elastic ones: the "hardening curve".

The important point ^{to be made is that micro-} flow and macro-flow are different regimes ^of behavior, ^{due to different} micro-mechanisms, ^and following ^different macroscopic phenomenology [3].

"Ideal plasticity" may be ^an acceptable ^ideal-

ization for macro-flow, it is not for micro-flow.

Rate Effects: Thermal Activation versus Overstress The flow stress as defined above depends

^the

rate of straining-usually mildly, ^{but in} principle always. ^{It is} well established that this rate

dependence is, ^over

very wide range of the variables, ^{due to} thermal activation of disloc- ations over the (intrinsic ^or extrinsic) obstacles to their motion. The rate dependence ^is therefore

coupled ^{to the} temperature dependence of the flow stress, in the way shown by the abscissa of Fig. ^2.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:01988002304038100

382

Thermal activation vanishes at absolute zero

temperature, ^{or when the time} available for it

vanishes, i.e. when the strain rate demanded is infinite. Neither of these limits

be achieved in practice, ^and usually the mechanism of deform- ation changes ^before

gets ^near them,

^SCHWINK has shown in

particular ^{case. In} principle, however, ^{it is} easy ^to imagine dislocation motion under mechanical forces only:

^{the stress is} raised, more dislocations move farther and farther (a process described by Fig. ^{1 and} by FRANCIOSI);

but there comes

mechanical threshold above which

no

equilibrium position ^{can be} found for any mobile dislocation. This should correspond ^{to the flow}

stress in Fig. ² extrapolated ^{to absolute zero} temperature. Thermal activation allows further dislocation motion at

lower stress, ^at

^certain rate [3-5J.

Figure 3 shows the rate effects only, ^at

constant temperature. The mechanical threshold is shown as

vertical dashed line: it corresponds ^to

a "rate independent material", where the rate is indeterminate at the flow stress. The solid line shows the lowering of the flow stress by ^thermal

activation.

In both Figures ^{2 and} 3, ^{it is} possible ^to assign

"lower threshold" to the flow stress, below which the rate of deformation is "essentially zero"; it is shown dotted. ^In many ^cases, this lower limit is

elusive

the proportional ^limit

of elasticity, ⁱⁿ others it

be ascertained by supplemental experiments ^and meaningful ^models

[2,3]. ^{This is} typically ^{the case in} two-phase alloys. ^Note, however, ^{that even if}

independent lower limit for flow exists, it does ^not replace

the need for the upper limit of quasi-static flow, the mechanical threshold, ^{but is}

^second strength parameter.

All the above arguments ^{referred to scalar} strength ^{and rate} properties. ^{Be it} merely ^stated

that the mechanical threshold is an appropriate scaling parameter ^for yield ^and flow surfaces

[1,3], whereas the second parameter, ^{if it} exists, such as

lower limit for flow in two-phase alloys,

may well have ^tensor character and would then correspond ^to "kinematic hardening".

Conclusion

The mechanics notion of

limit surface in stress space above which equilibrium ^{is not} possible corresponds ^to

concept ^of crystal plasticity, the "mechanical threshold". It is, however, ^{not true that the} regime inside this limit

is elastic: _permanent plastic strains occur, and creep ^at low rates happens below this limit. A

strictly ^{elastic domain} may often ^not exist or if it does, is much harder to define. The macro-flow

stress

be measured by back-extrapolation of the

stress/strain ^{curve to zero} plastic strain, and the mechanical threshold

be measured by back-extra- polation of the flow stress to zero temperature.

All these considerations referred to

material in

given "state", ^{and this state}

^{assumed not}

to change through, e.g., mobile dislocation rearrangements. Evolution of the state itself is described by

change ^{of the} (macro-)flow ^stress with (plastic) pre-strain [3]. ^In fact, ^the mechanical threshold could serve

good ^first

state parameter [4]. ^{The rate} of evolution is itself dependent

^{the rate of} straining, ^{and this}

is the major ^{cause of rate} sensitivity ^{at elevated} temperatures; ^this regime

^not covered in the present discussion.

References

1. U.F. Kocks, "Constitutive Relations for Slip":

in Constitutive Equations ⁱⁿ Plasticity,

A.S. Argon, ^ed. (MIT ^Press 1975), pp. 81-115.

2. U. F. Kocks, "Superposition ^of Alloy Hardening,

Strain Hardening, ^and Dynamic Recovery": ⁱⁿ Strength ^{of Metals and} Alloys, ^P. Haasen, V. Gerold, ^G. Kostorz, eds. (Pergamon 1980)

pp. 1661-1680.

3. U. F. Kocks, "Constitutive Behavior Based

Crystal Plasticity", ⁱⁿ Unified Constitutive Equations for Plastic Deformation and Creep

in Engineering Alloys, ^{A. K.} Miller, ^ed.

(Elsevier Applied ^Science 1987), ⁱⁿ press.

4. U. F. Kocks, A. S. Argon, ^{and M. F.} Ashby,

"Thermodynamics ^and Kinetics of Slip":

Progr. Mater. Sci. 19 (1975).

5. P. S. Follansbee and U. F. Kocks, ^{"A Constitu-} tive Description of the Deformation of Copper

Based on the Use of Mechanical Threshold Stress

as an

Internal State Variable", ^Acta Metall.,

in press (1987).

6. G. Regazzoni, ^{U. F.} Kocks, ^{and P.S.} Follansbee,

"Dislocation Kinetics at High ^Strain Rates":

Acta Metall., ⁱⁿ press (1987).