Strengthening Mechanisms in Metallic Materials

The yield strength of metals and alloys reflects the amount of force or stress to move dislocations in them. Thus, to increase their yield strength, obstacles to the motion of dislocations are engineered or created in them. The four most basic ways to strengthen metals and alloys are:

1. solid solution strengthening2. refinement of grain size3. increase of dislocation density4. dispersed phases or particles

solid solution strengthening

The effects of the solute atoms are due to dilatational, distortional, and/or stiffness misfits (with the host atoms) that interact with and retard the motion of dislocations. Interstitial solutes anchor dislocations at rest whereas substitutional solutes retard moving dislocations. The anchoring or locking of the static dislocations is manifested during tensile test by a sharp yield point that gives rise to an upper and a lower yield stress in the stress strain curve ( e.g. in low carbon annealed steels and annealed or heat treated Al-Mg solid solution alloys). Accompanying the sharp yield point in mild steel is an initial inhomogeneous plastic deformation called lower yield point elongation (Fig.).

The effect of the substitutional solutes is analogous to frictional effects experienced by a body in motion. Extra energy or stress is exerted to overcome the friction. The frictional stress to overcome is proportional to the solute concentration. The strengthening depends also on the differences in atomic sizes and the crystal structures of the solute and host atoms. These differences lead to elastic misfit strains that strengthen the host lattice – the same effect as in particle strengthening. The following Figs. show the strengthening effects of carbon (interstitial) and substitutional solutes in pure iron.

Grain size strengthening

This strengthening is due to the large misorientation at the grain boundaries; the tilt boundaries that form subgrains may also contribute to strengthening. This effect is illustrated in Fig. The deformation in a grain cannot continue to adjoining grains because the motion of dislocations is blocked The finer the grain size i.e. more grain boundaries give more obstacles to dislocation motion. This is expressed by the following Hall-Petch equation.

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Hall-Petch Eq.

σys = yield stress σo and k are constants d = average grain size

Examples are given in Fig.

Fig. Crystals are strengthened by grain boundaries blocking dislocations creating a pile-up.

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Dislocation strengthening

The presence of other dislocations in the structure will interfere with the motion of a dislocation and thus increase the resistance or strength of the material. This is the basis for strain hardening or cold work hardening. The density of dislocations is estimated as the total length of dislocations in cm/cm3. This density ranges from about 106 in annealed metals to about 1012 – 1013 cm/cm3 in fully cold-worked structure. The increase in dislocation density raises the strength of the crystal according to the following Eq. Where ρ is the dislocation density.

The manner in which the dislocation density increases from 106 to 1013 is accompolished by a dislocation mechanism shown in Fig.

Dispersed phase or particle strengthening

The strengthening from the dispersed phases may be treated on a continuum basis, whereby, the multiphase alloy may be regarded as a composite. The properties of the aggregate or composite may be determined by the rule of mixtures:

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Where Pagg is the aggregate or composite property, Pi is the property of the ith constituent and fi is the volume fraction of the ith constituent in an aggregate containing n phases. The application of this Eq. to the structure-sensitive property of yield strength requires one of two hypotheses:

One hypothesis assumes that each of the phases experiences equal strain., In this case the average stress in the alloy for a given strain will increase linearly with the volume fraction of the strong phase. This is illustrated in the following Fig. which shows the effect of the volume fraction of martensite (hard phase) on yield strength of an aggregate of austenite and martensite in 304 stainless steel.

The alternative hypothesis is that the phases are subjected to equal stresses. The average strain of the alloy can be calculated from the above Eq.

The strengthening from dispersed particles, which may be coherent or non-coherent (Fig.) is treated as interaction with dislocations. The influence of these particles is to block or retard the motion of dislocations. For coherent particles, the dislocation moves or cuts through the particle, as it is part of the host lattice. However, this is accomplished at a much higher force than when it moves through the host lattice. Thus, we observe a very high increase in the strength of the material. For noncoherent particles, the particles block the dislocation motion by acting as anchor points as depicted in the Fig. above. The following Fig. depicts the shear stress pushing the dislocation anchored by hard particles with interparticle spacing of λ .the strengthening arises from these particles is inversely

proportional to λ,

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Thus, the closer the particles, the stronger is the material. The number of particles per unit volume controls the property. For a fixed weight percent of the particles, smaller-sized particles will exhibit greater strengthening than larger sized particles.

In addition to dislocation particle interactions, Mott and Nabarro suggested that a source of strengthening is the elastic strain mismatch of the particle and matrix. The increase of yield strength is given by: Where ε is the elastic strain field and f is the volume fraction of the dispersed particles. This Eq. is especially true for coherent particles.The overall yield strength of metals and alloys is the result of the four factors just described. Neglecting the interactions between them, the yield strength of a crystalline

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material based on dislocation mechanisms may be taken as the additive effects of the four factors. Thus,

o = inherent resistance of the lattice to dislocation motionss for solid solution, gs for grain size, sh for strain hardening (dislocations) and p is for dispersed phases and/or particles.

Example

The strength of a low carbon steel is 622 MPa for ASTM grain size #2 (d=180 μm) and 663 MPa for ASTM grain size #8 (d=22.5 μm). What will the strength be for ASTM grain size #10 (d=11.2 μm).?

and

Solving for k and σo gives ; k = 301 MPa√μm and σo = 599.5 MPaThus, the Hall-Petch Eq. for this steel is:

For an ASTM #10 sample with d=11.2 μm we find σys = 689.4 MPa

Example

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The flow stress of a copper alloy increases from 2 MPa to 55 MPa when the dislocation density increases from 107 cm-2 to 1010 cm-2. Calculate the flow stress for a similar heavily deformed copper alloy with a dislocation density of 1012 cm-2.

and Solving for k and τo gives k = 5.47x10-4 MPa-cm and τo = 0.27 MPa

For ρdisl. = 1012cm-2

= 547.3 MPa

Grain size measurement

ASTM : n = 2N-1 n = number of grains per in2 at 100X magnification and N is the ASTM grain size number.For ease of calculation, assume grains of square shape and edge length at 100 X of D100 , thus n = 1/( D100)2 and D = (2[(1-N)/2]/100) in.

The stress required to force the dislocation between the obstacles is

A simple expression for the linear mean free path of the dispersed particles is

Where f is the volume fraction of spherical particles of radius r.

Example

An aluminum-4% Cu alloy has a yield stress of 600 MPa. Estimate the particle spacing and particle size in this alloy.

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At this strength level we are dealing with a precipitation-hardened alloy that has been aged beyond the maximum strength. The strengthening mechanism is dislocation bypassing of particles. G = 27.6 GPa , b=2.5x10-8 cm, τo = 600/2 = 300 MPa

Wt% of α phase (Al) =

Wt% of θ phase (CuAl2) =

Volume of α phase =

Volume of θ phase =

Volume fraction of α phase = 0.96Volume fraction of θ phase = 0.04

Assuming spherical particles ,

Edge Dln

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•Edge dln Movement

Edge and Screw Dlns

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