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Precipitation hardening

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Abstract

The topic of precipitation hardening is critically reviewed, emphasizing the influence of precipitates on the CRSS or yield strength of aged alloys. Recent progress in understanding the statistics of dislocation-precipitate interactions is highlighted. It is shown that Pythagorean superposition for strengthening by random mixtures of localized obstacles of different strengths is rigorously obeyed in the limit of very weak obstacles; this had been known previously as a result of computer simulation experiments. Some experimental data are discussed in light of this prediction. All of the currently viable mechanisms of precipitation hardening are reviewed. It is demonstrated that all versions of the theory of coherency hardening are woefully inadequate, while the theory of order hardening is capable of accurately predicting the contribution of γ′ precipitates to the CRSS of aged Ni-Al alloys. It is also convincingly shown that a new theory based on computer simulation experiments of the motion of dislocations through arrays of obstacles having a finite range of interaction cannot explain these same data, and is of doubtful validity in other instances for which its success has been proclaimed. A new theory of hardening by spinodal decomposition is proposed. It is based on the statistics of interaction between dislocations and diffuse attractive obstacles, and is shown to be in very good quantitative agreement with much of the limited data available. Some of the problems that remain to be addressed and solved are discussed.

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Abbreviations

A :

Amplitude of a composition modulation during spinodal decomposition.

a(ap) a(ap):

Lattice parameter of the matrix (precipitate).

B :

Dimensionless variable in the order strengthening theory of Ardell, Munjal, and Chellman.

b bp:

Burgers vectors of total (partial) dislocations.

C C0:

Solute concentration (initial value) in atom fraction.

C G :

A constant in an expression for the maximum interaction force in modulus hardening.

Csl, C’sl:

Empirical constants in the theory of Schwarz and Labusch for energy conserving and energy storing interactions, respectively.

c ij :

Single crystal elastic moduli.

D :

Spacing of the dislocation pair in order strengthening.

dI dII:

Effective particle diameters for the leading (I) and trailing (II) dislocations of a pair at the critical configuration during order strengthening.

F m :

Maximum force of interaction that an obstacle can withstand.

F mz :

Maximum force of interaction between a precipitate and a dislocation on a slip plane at distance z from the particle center.

f :

Volume fraction of precipitate.

G, G111:

Shear modulus on the slip plane in the slip direction of the matrix of an fcc crystal.

Giso :

Shear modulus of the matrix of an isotropic crystal.

G p :

Shear modulus of the precipitate.

ΔG GP :

G

g :

Distribution function of the variable contained within the parentheses.

h :

Twice the amplitude of a zig-zag dislocation in the presence of attractive obstacles.

J c :

A constant in Cahn’s theory of spinodal de composition.

j :

The number of dislocations in a procession during the shearing of ordered precipitates.

K :

Force of interaction between two partial dislocations of Burgers vector bp.

k :

Index representing successive obstacles encountered

L :

Effective spacing of obstacles along a dislocation at the critical configuration.

L F :

The Friedel spacing.

L M :

The Mott spacing.

L s :

The square lattice spacing.

LI,LII:

Effective spacings of obstacles along the leading and trailing dislocations of a pair at the critical configuration in order strengthening.

ℒ :

Edge length of a cube containing one obstacle on average.

l :

Effective particle diameter for the trailing partial at the critical configuration in stackingfault strengthening.

m :

An exponent appearing in equations for the maximum force and CRSS due to modulus hardening.

n :

Number of obstacles contained within a dimensionless search area generated by circle rolling.

n s :

Number of obstacles per unit area in the glide plane.

n v :

Number of obstacles or precipitates per unit volume.

p :

An adjustable parameter in the Hüther and Reppich theory of order strengthening by strong pair coupling.

q :

An exponent in an empirical addition rule.

R(R *):

Radius (dimensionless radius) of a curved dislocation.

Rc(R *c ):

As above, at the critical breaking stress.

r :

Radius of a spherical precipitate.

r 0 :

Inner cut-off distance in the expression for the dislocation line energy.

r s :

Planar radius of a spherical precipitate.

S(S *):

Area (dimensionless area) swept out by a dislocation in circle rolling.

S *c :

Dimensionless critical area swept out by a dislocation in circle rolling, containing one obstacle.

S F :

Area swept out by a dislocation in Friedel statistics.

s 0 :

Dimensionless, radius-independent search area at the critical configuration in circle rolling.

s 0i :

As above, but containing on average one obstacle of typei in a random mixture of distinct obstacles.

U 0 :

Energy of interaction between an obstacle and a dislocation.

u :

A dimensionless variable in the order hardening theory of Ardell, Munjal, and Chellman, equal to 2〈rs〉/LF.

V :

A parameter related to the maximum force of interaction between a flexible edge dislocation and a spherical coherent precipitate.

wm(wp:

Ribbon width of a stacking-fault in the matrix (precipitate).

X i :

Fraction of obstacles of typei in the slip plane; an areal concentration.

x :

Spatial coordinate parallel to an initially straight dislocation line.

Y :

An elastic modulus resisting lattice deformation during spinodal decomposition.

y :

Spatial coordinate measuring the displacements of a dislocation from a straight line.

z:

Distance between the center of a spherical precipitate and the slip plane of a dislocation,

α:

A coefficient in the expression for the dislocation line tension.

βc :

Dimensionless critical force exerted by a dislocation on an obstacle.

Γ:

Line tension of a dislocation.

Γe, Γs :

Line tension of a pure edge, screw dislocation.

γs :

Energy of a matrix-precipitate interface created by slip.

γapb :

Antiphase boundary energy on the slip plane of an ordered precipitate.

γsfm :

Stacking-fault energy of the matrix.

γsfp :

Stacking-fault energy of the precipitate.

〈γsf〉:

Average stacking-fault energy of the matrix and precipitate phases.

Δγ γsfm− γ:

sfP

Δ :

Fractional misfit between the lattice parameters of the matrix and precipitate phases,

ε:

Constrained strain; the fractional misfit between anin situ coherent precipitate and the matrix.ξ Z/r.

η:

A measure of the lattice strains produced during spinodal decomposition.

η0 :

Parameter measuring the ratio of the obstacle range and the square root of its breaking strength.

θc :

Critical angle through which the dislocation turns at an obstacle in,e.g., circle rolling.

Λ:

Outer cut-off distance in the expression for the dislocation line energy.

λ:

Wavelength of a composition modulation during spinodal decomposition.

v(vp:

Poisson’s ratio of the matrix (precipitate).

ξ:

Angle between the dislocation line and its Burgers vector.

τc(τ *c ):

CRSS (dimensionless CRSS) predicted theoretically

gT **c :

Reduced theoretical CRSS; gTc/β 3/2c

gT **o :

Experimentally determined value of gT **c relevant to data on order hardening.

gTci (gT *ci ):

CRSS (dimensionless CRSS) predicted theoretically for obstacles of typei in a random mixture of distinct obstacles.

gTcc:

Theoretically predicted CRSS due to chemical hardening.

gTcG:

Theoretically predicted CRSS due to modulus hardening.

gTcO:

Theoretically predicted CRSS due to order hardening.

gTcS:

Theoretically predicted by CRSS due to hardening by spinodal decomposition.

gTce :

Theoretically predicted CRSS due to coherency hardening.

gTcγ :

Theoretically predicted CRSS due to stackingfault strengthening.

gTss :

Contribution of the solid solution matrix to the CRSS.

gT01, τ02 :

Contributions of Class 1 and Class 2 precipitates to the CRSS in alloys aged to contain bimodal γ′ particle size distributions.

gTt :

Experimentally measured CRSS of a crystal.

Δτ:

Contribution of precipitates to the CRSS determined experimentally.

ΔgT0:

As above, in order strengthening,

Δτs:

As above, in hardening by spinodal decomposition.

Δτε:

As above, in coherency hardening.

ΔgTγ:

As above, in stacking-fault strengthening.

θ:

Angle between tangential directions of the dislocation line at successive obstacles in circle rolling.

θ i :

Similar to θ, but for obstacles of typei in a random mixture of distinct obstacles.

θo :

Lower limit on θ which defines a boundary of the search area in circle rolling that contains one obstacle on average.

χ :

A constant in the theory of coherency strengthening.

ψc :

Critical breaking angle (cusp angle) included between adjacent arms of the dislocation at an obstacle.

ψw :

The difference between the maximum and minimum values of ψc.

Ω(Ω*):

Range (dimensionless range) of interaction between an obstacle and a dislocation.

〈〉:

Symbols denoting the average value of the quantity contained within. max, min, Subscripts denoting maximum, minimum, exp, experimental, and/or theoretical values of a theor parameter or variable.

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  1. Department of Materials Science and Engineering, University of California, 90024, Los Angeles, CA

    A. J. Ardell (Professor and Chairman)

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This paper is based on a presentation made at the symposium “50th Anniversary of the Introduction of Dislocations” held at the fall meeting of the TMS-AIME in Detroit, Michigan in October 1984 under the TMS-AIME Mechanical Metallurgy and Physical Metallurgy Committees.

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Ardell, A.J. Precipitation hardening. Metall Trans A 16, 2131–2165 (1985). https://doi.org/10.1007/BF02670416

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