Strengthening Mechanisms The mechanical properties of a material are controlled by the microstructure. Tensile strength is controlled by the work‐hardening rate. The work‐hardening rate controls the amount of uniform deformation (elongation). The higher the elongation, the tougher the material and the greater deformability. The ability of a metal to plastically deform depends on the ability of dislocations to move. Reducing or inhibiting mobility of dislocations enhances mechanical strength. Four main ways of controlling the strength: (a) Solid Solution Hardening; (b) Grain Boundaries; (c) Precipitation hardening and (d) Work Hardening (Dislocation Hardening) Strengthening in crystals results from the restriction of dislocation motion. We can restrict dislocation motion by altering, promoting, or adding: Bond type: Choice of basic material Dislocation‐ dislocation interactions: Work hardening Grain boundaries : Hall‐Petch relationship Solute atoms: Solid solution hardening Precipitates or dispersed particles : Precipitation hardening or dispersion hardening Phase changes: Transformation hardening or toughening.

How can we do to increase strength? • GENERAL One simple method is to place obstacles in the path of dislocations that will either slow them down or stop them completely until the stress is high enough to move them further. This works for crystals! In non‐crystalline materials we do different things. • SPECIFIC Dislocations distort the crystal lattice. Various obstacles also distort the crystal lattice. Stress/strain fields from both will interact with each other, which reduces v (the dislocation velocity). This in effect increases the stress required to cause the material to “flow” (i.e., it increases the flow stress) and thus the “strength” of the material. General model for strengthening (1) REFERENCE: L.M. Brown and R.K. Ham, in Strengthening Mechanisms in Crystals, edited by A. Kelly and R.B. Nicholson, Wiley, New York, 1971, pp. 9‐70. Consider a slip plane that contains a random array of obstacles. We don’t care what the obstacles are at this point.

As the dislocations are being anchor by the Gb ⎛φ ⎞ τ ≅ Cos⎜ c ⎟ obstacles, extra “work” is required to move the L' ⎝ 2 ⎠ dislocation through the array of obstacles. This results in a higher stress to cause “flow”. Gb τ Max = SOLID SOLUTION STRENGTHENING L •Impurity atoms that go into solid solution impose lattice strains on surrounding host atoms •Lattice strain field interactions between dislocations and impurity atoms result in restriction of dislocation movement •This is one of the most powerful reasons to make alloys, which have higher strength than pure metals.

•Example: 24k gold is too soft. If we put in 16% silver and 9% copper, we get an alloy that looks just like pure gold, but is much more strong and durable. We call this 18k gold. (18/24 = 75% gold) There are two types of solid solutions: Substitutional Solid Solution: Solute and solvent atoms are roughly of the same size and the solute atoms will replace the solvent atoms in its position in the crystal structure. Interstitial Solids Solution: The solute atoms are smaller than the solvent atoms and they will occupy interstitial positions in the solvent lattice.

Substitutional Ni/Cu

Interstitial

C/Fe The factors that control the tendency for the formation of substitutional solid solutions are given by the Hume‐Rothery Rules.

Factors for high solubility in Substitutional alloys (Hume‐Rothery Solubility Rules) •Similar atomic size (to within 15%) •Similar crystal structure •Similar electronegativity (otherwise a compound is formed) •Similar valence Composition can be expressed in weight percent, useful when making the solution, and in atomic percent, useful when trying to understand the material at the atomic level. Example Ni is completely miscible in Cu (all rules apply) Zn is partially miscible in Cu (different valence, different crystal structure) Ni ‐ Cu binary isomorphous Limited solubility alloy (eutectic) alloys Ni Cu Pb Cu crystal structure FCC FCC FCC FCC atomic radius 0.125 0.128 0.175 0.128 2.4% 36.7% electronegativities 1.8 1.8 1.6 1.8 valence 2 +2 + 2+, 4+ 2 +

Solubility Cu in Ni 100% Solubility Cu in Pb 0.1%

Solid Solution: homogeneous maintain crystal structure contain randomly dispersed impurities (substitutional or interstitial) Second Phase: as solute atoms are added, new compounds / structures are formed, or solute forms local precipitates Why it works??

Small atoms like to live here. (they reduce lattice strain caused by the dislocation).

Big atoms (or interstitials) like to live here (there is more space.)

Atoms of either type diffuse to dislocations during high temperature processing, then exert forces on the dislocation later to keep them stuck. The Principle •Any inhomogeneity in a crystal lattice will create a strain field within a lattice. • Dislocations will interact with those strain fields. •The type of interaction will determine the degree of hardening or softening. •In general, solute atoms increase the strengths of crystals. However, under the right conditions they can decrease the strength. •The increase arises due to: –Short range ┴ ‐ solute interactions –Long range ┴ ‐ solute interactions • Dislocations will interact with solutes that lie on, above, and below slip planes. The most intense interactions will occur in close proximity to the slip plane. •Small impurity atoms exert tensile strains (see figure below) •Large impurity atoms exert compressive strains

•Solute atoms tend to diffuse and segregate around dislocations to reduce overall strain energy –cancel some of the strain in the lattice due to the dislocations.

Compressive Strains Imposed by Larger Substitutional Atoms

Add nickel to copper, strength goes up, ductility goes down ‐ for the same reason: dislocation mobility is decreased. Note: trade‐off in properties The usual effect of solute addition is to raise the yield stress and the level of stress‐strain curve as a whole.

Since solute atoms affect the entire stress‐strain curve, these atoms have more influence on the frictional resistance to the dislocation motion than on the static locking of the dislocations. According to their strengthening effect solute atoms fall into two broad categories: •Solute atoms that produce dilatational strains or spherical or symmetrical distortions, such as substitute atoms. Their relative contribution to the strengthening is low G/10. •Solute atoms that produce distortional (shear) strains or non‐ spherical or non‐symmetrical distortions, such as the interstitial atoms. Their relative contribution to the strengthening is high 3G. Elements in solid solution usually strengthen crystals. For substitutional solutions, the yield strength increases Δτ 4 = CGε 3 in proportion to their concentration of solutes. Δc Where c is the concentration of solute expressed as an δa atomic fraction, ε is the misfit parameter, C is a constant ε = a and G is the shear modulus. δc The effect of substitutional solutes is mainly attributable to interaction of the solutes with the dilatational stress field around edge dislocations. Substitutional solutes have little interaction with screw dislocations. Hence, substitutional and interstitial solutes are often consider to function as local elastic distortions within alloys. The interaction of the solutes with the dislocations depends on the character of the elastic strain field around the dislocation. The elastic strain fields of edge and screw dislocations are profoundly different. For a screw dislocation with a z‐axis For an edge dislocation with a z‐axis parallel to the dislocation line. parallel to the dislocation line.

⎡ 00 ε13 ⎤ ⎡ εε1211 0 ⎤ ε = ⎢ 00 ε ⎥ ε = ⎢ εε 0 ⎥ ⎢ 23 ⎥ ⎢ 2221 ⎥ ⎣⎢ εε3231 0 ⎦⎥ ⎣⎢ 00 ε33 ⎦⎥ The mutually exclusive character of the two strain fields provides that ideal screw dislocations should interact only with other defects that cause shear distortions along the dislocation core. Edge dislocations strain fields interact with defects that cause either volumetric strain components or shear strain components. Substitutional solutes produce primarily volumetric changes, while interstitial solutes (insertion of atoms into octahedral and tetrahedral positions) will produce either volumetric and distortional (shear) strains. Hence, interstitial atoms are anticipated as more efficient in strengthening.

The graph below shows how different alloying additions affect the yield strength of a ferrite + pearlite structural steel. Strengthening due to different defects (Typical Values)

The more potent strengtheners are the defects that produce non‐symmetrical strain fields (i.e., those with large shear components) The strengthening potency of interstitials is usually much greater Δτ than that of substitutional solutes. The solubility of interstitials is normally very small. The strengthening potency is expressed as: c

Where c is the fraction of interstitial solute and Δτ = τ −τ ini The strengthening potency for interstitial solutes is at least an order of magnitude greater than that for substitutional strengthening. The magnitude of the misfit strain also plays a role gf Δa += εξττcG ε = initial misfit a

τinitial= shear yield strength of a pure material; ξ is a constant ; G is the shear modulus; εmisfit is the misfit strain; c is the solute fraction. The misfit strain exponent is f=3/2 and the concentration exponent g varies from 0.5 to 1. Solute interactions with dislocations. 1. Elastic interaction 2. Modulus interaction 3. Stacking fault interaction 4. Electrical (valence) interaction 5. Short‐range order interaction 6. Long‐range order interaction

•Solid solution strengthening is dominated by type 1 and 2 interactions. •Type 1, 2, and 6 interactions are “long range,” which means that they are relatively insensitive to temperature and are strong up to

~0.6 –0.7TMP. •Type 3, 4, and 5 interactions are “short range,” which means that they will contribute strongly to flow stress at low temperatures. The Size Effect & Interstitial Strain Field Shape

•The shape of the strain fields caused by defects and those around Big dislocations have a significant Substitutional impact on the strength of a solid. • Substitutional solutes generally stretch the lattice uniformly producing hydrostatic (spherical) Small strain fields around the solutes. Substitutional •These hydrostatic strain fields are relative “weak” obstacles to further dislocation motion in comparison to shear strain fields. Edge dislocations have both dilatational and shear (i.e., distortional) strain fields. Screw dislocations only have shear strain fields. •There is an interaction between the strain fields around solute atoms and the strain fields around dislocations. This interaction is based on reducing the strain energy associated with dislocations and solute atoms. •Using an edge dislocation in this example, the region above an edge dislocation is in compression. The region below the core is in tension. Solute atoms with dilatational strain fields will interact with these regions to cancel out strain and thus reduce the elastic strain energy of the system. •Both attractive and repulsive forces between solutes and dislocations will inhibit the motion of dislocations, thus increasing strength. Strain fields around solutes • FCC lattice : Substitutional solute: dilatational strain. Interstitial solute: dilatational strain. • BCC lattice : Substitutional solute: dilatational strain. Interstitial solute: distortional (shear) strain. This component is asymmetric! •What interactions might be expected between solutes and dislocations in different lattices? • FCC Lattice: Screw dislocations –little or no interactions with solutes. Edge dislocations – strong interactions with both types of solutes • BCC Lattice: Edge dislocations – strong interactions with both types of solutes. Screw dislocations – strong interactions with interstitial solutes • Considering these interactions, what do you think might happen in an HCP lattice? Which solutes will cause the most potent hardening? Modulus interaction Since solute atoms generally have different shear moduli than the solvent atoms, they impose additional strain fields on the lattice of the surrounding matrix. Modulus interactions occur if the presence of a solute atom changes the local modulus of the crystal.

When solutes with smaller shear moduli than the solvent (i.e., Gsolute < Gsolvent), the energy of the strain fields around dislocations will be reduced (i.e., elastic strain energy is reduced) which causes an attraction between the solutes and the dislocations. Both edge and screw dislocations are subject to this interaction. Size and modulus effects are detailed in some of Fleischer’s classic papers: (a) R.L. Fleischer, “Solution Hardening,” Acta Metallurgica, 9 (1961) pp. 996‐1000. (b) R.L. Fleischer, “Solution Hardening by Tetragonal Distortions: Application to Irradiation Hardening in Face Centered Cubic Crystals,” Acta Metallurgica, 10 (1962) pp. 835‐842.(c) R.L. Fleischer, “Substitutional Solution Hardening,” Acta Metallurgica, 11 (1963) pp. 203‐210. Ultimately, the effectiveness of solid solution strengthening depends upon the size mismatch and the modulus mismatch between foreign atoms and parent atoms. Strains induced by lattice and modulus mismatch Lattice misfit strains are proportional to the local 1 δa change in lattice parameter per unit concentration of ε Lattice = solute. This can be expressed as (where a is the lattice a δc parameter of the solute and c is the concentration of solute). 1 δG ε = The equation describing the modulus interaction is Modulus G δc similar: The modulus interaction energy can be either positive l or negative depending upon the sign of εmodulus. S mod ulus −= βεεεLattice

The total strain caused by both lattice and l ε mod ulus ε mod ulus = modulus mismatch (εs) has been shown by ⎛ 1 ⎞ ⎜1+ ε mod ulus ⎟ Fleischer to be: ⎝ 2 ⎠

In this equation, εlattice and β are always positive. ε’modulus is negative for “soft” atoms (size and modulus effects reinforce) and positive for “hard” atoms. β is an empirical parameter that is related to the importance of screw and edge dislocations during plastic flow. We can relate solid solution strengthening to our F general equation for strengthening. If we let L’ equal τ = Max the effective obstacle spacing, then the increase in bLl flow strength for solute atoms is: 2 2 2 Gb Gb In this equation, Fmax is proportional to Gb . For FMax = to...... “strong” obstacles (i.e., things that cause 5 10 tetragonal lattice distortions): Gb2 F = For “weak” obstacles (i.e., things that cause Max 130 spherical lattice distortions): ⎛ c ⎞ For tetragonal defects, solid solution ≅ γτGb⎜ ⎟ = γ cG TET ⎜ ⎟ strengthening is given as: ⎝ b ⎠

Common examples of tetragonal defects include interstitial solutes in BCC metals and interstitial‐vacancy pairs in FCC metals. Hardening by interstitials is about 50 times more efficient than hardening by substitutional atoms. The general strengthening equation for “conventional” 3 Gε 2 substitutional solid solution strengthening has been τ = S c estimated by Fleischer. It is: y 700

Solid‐Solution Strengthening: Size‐ and Modulus‐ effect Cu modulus effect ε s ≅| εG − 3εb | Tetragonal distortion for Cu 1 dG 1 da εG = εb = G dc a dc

4πr 3 δV ~ (3εb ) 3

R. Flescher, Acta Metall. 11, 203 (1963) e.g. Tetragonal distortion: τTET = αGb(√c/b)= αG √c In local‐force model, spacing between solute is L’ = b/√(2c) due to solute atoms in the 2 planes immediately adjacent to slip plane giving 2(c/b2) atoms/area. In most materials, elastic interaction energy concerns (i.e. size and modulus effects) dominate strengthening. However, in some materials chemical and electrical factors are also significant. Electrical interaction arises from the fact that some of the charge associated with solute atoms of dissimilar valence remains localized around the solute atom. For example, in ionic solids the addition of solute atoms with different valence than the solvent will alter the electronic charge distribution and energy. This can lead to interactions with dislocations that can produce substantial strengthening. The addition of a divalent ion to a monovalent crystal produces a tetragonal distortion and can lead to significant electrical interactions between the impurity and the ions that comprise the dislocation. The tetragonal distortion generally produces the largest component of strengthening. Tetragonal Distortion ‐ Examples

Tetragonal defects are “strong” obstacles to dislocation glide. Substitutional atoms produce spherical/uniform lattice strains and are “weak” obstacles. Other types of solid solution strengthening/hardening • Stacking Fault Energy (SFE) ; In materials containing stacking faults, solute atoms can preferentially segregate to the stacking fault (e.g., Suzuki atmospheres) or may be repelled away from them. This lowers the SFE for the solid which can lead to hardening. •Order hardening ; Atomic ordering can also produce significant strengthening. Short range order interaction arises from the tendency of solute atoms to arrange themselves so that they have more than the equilibrium number of dissimilar neighbors. The opposite of short‐range‐order is clustering where the solute atoms tend to group together in regions of the lattice. Long range order arises from alloys that form superlattices (long range periodic arrangement of dissimilar atoms). The movement of dislocations through the superlattice creates regions of disorder called anti‐phase boundaries (APB) because the atoms across the slip plane have become out‐of‐phase. Mechanical Effects Associated with Solid Solutions Well defined Yield Point in the stress‐strain curve Curve characteristic for annealed low‐carbon steel. There are two main theories (a) Cottrell‐Bilby: Dislocations in annealed steels (~107cm‐2) are locked by solute atoms (C and N have a greater mobility). When the stress is increased it reaches a point when the dislocation is unlock. The stress required to move an unlocked dislocation is less than the stress necessary to free them causing a yield drop. Without yield point With yield point

A: Yield stress (offset) E: Rupture stress* B: Upper yield point F: Uniform strain C: Lower yield point* G: Strain‐to‐failure* D: Yield stress (proportional) H: Lüders strain D’: Ultimate tensile Strength Area under curve:* Work of fracture, Toughness (not same as Fracture Toughness) Yield Point Phenomenon Metals, particularly low‐carbon steel, show a localized heterogeneous transition from elastic to plastic deformation. Yield point elongation. The load after the upper yield point suddenly drop to approximately constant value (lower yield point) and then rises with further strain. The elongation which occurs at constant load is called the yield‐point elongation, which are heterogeneous deformation.

Lüder bands or stretcher strains are formed at approximately 45o to the tensile axis during yield point elongation and propagate over the specimen. Plateau in the stress‐strain curve after the well‐defined Yield Point or Luders Bands

In the stress‐strain curve a plateau region in which the load fluctuates around a certain value follows the load drop. The elongation that occurs in this plateau is called the yield‐point elongation. It corresponds to a region of non‐homogeneous deformation. A deformation band appears and it propagates through the test sample. The deformation is restricted to the interface. This deformation band is known as the Luders Band.

Luders band formation during the stamping of low carbon steels lead to irregularities in the final sheet thickness. It can be overcome by (a) Changing the alloy composition to eliminate the yield point (steel with small additions of Ti, Al, V, Nb, etc); (b) pre‐stressing the sheet above its yield point. The upper yield point The onset of general yielding occurs at a stress where tha average dislocation sources can create slip bands through a good volume of the material. += σσσ isy

σs stress to operate the disloca sourcestion

σi friction stress (combining effect of all the obstacles to the motion of the dislocations) The upper yield point is associated with small amounts of interstitial or substitutional impurities. The solute atoms (C or N) in low carbon steel, lock the dislocations, raise the initial yield stress. The breakaway stress required to pull a dislocation line away from a line of solute atoms is A 3 σ ≈ 22 = 4GbaA ε rb o a is the atomic radius; ro is the distance from the line of the dislocation to the solute atom; ε is the misfit strain and b the Burger’s vector ; When the dislocation is pulled free from the solute atoms, slip can occur at lower stress. The lower yield point. The magnitude of the yield‐point effect depends on interaction energy, concentration of solute atoms.

Strain Ageing A steel subjected to a tensile test was stopped unloaded and reloaded immediately. Upon reloading the sample did not show a well‐defined yield‐point. However, on reloading after waiting for certain time (3 hours) the stress‐strain curve showed a well defined yield point and a plateau. This phenomenon is known as strain ageing. It is explained by the migration of interstitial atoms to the dislocation during the test stoppage, leading to dislocation. Strain ageing is a phenomenon in which the metal increase in strength while losing ductility after being heated at relatively low temperature or cold‐working.

This reappearance of the yield point is due to the diffusion of C and N atoms to anchor the dislocations. N has more strain ageing effect in iron than C due to a higher solubility and diffusion coefficient. Stretcher strains Strain ageing should be eliminated in deep drawing steel since it leads to surface marking or stretcher strains. To solve the problem, the amount of C and N should be lowered by adding elements such as Al, V, Ti, B to form carbides or nitrides. H = yield‐point elongation Serrated Stress‐Strain Curve This are manifestations of the Portevin‐Le Chatelier Effect. These irregularities can be caused by the interaction of solute atoms with dislocations, mechanical twinning, stress assisted martensitic transformation. The first type is known as the P‐LeCh effect. It occurs within a specific temperature and strain rate range. Solute atoms being able to diffuse in the sample at a speed greater than the displacement speed of the dislocations. Blue Brittleness Carbon steels heated in the range 230 and 370oC show a notable reduction in elongation. This phenomenon is due to the interaction of dislocations in motion with the solute atoms (C and N). When the speed of the interstitial atoms is more than that of the dislocations, the latter are continually capture by the former.

Blue brittleness occurs in plain carbon steel in which discontinuous yielding appears in the temperature range 500 to 650 K. During this blue brittleness region, steels show •Decreased tensile ductility. •Decreased notched‐impact resistance. •Minimum strain rate sensitivity. Note: This is just an accelerated strain aging by temperature. STRENGTHENING BY GRAIN SIZE REDUCTION The strengthening in polycrystals due to grain boundaries has been experimentally established ever since Hall (Hall 1951; Petch Before 1953) proposed his relation between the deformation grain size and the yield stress. After deformation All of the stretching is away from the grain boundaries. Dislocation motion takes place when a critical resolved shear stress is reached. Dislocations cannot penetrate grain boundaries, because the crystal planes are discontinuous at the g.b.. Therefore, making a smaller grain size increases strength (more obstacles and shorter mean slip distance). Adapted from Fig. 7.12, Callister 6e. (Fig. 7.12 is from A Textbook of Materials Technology, by Van •Grain boundaries are barriers to slip. Vlack, Pearson Education, Inc., Upper Saddle River, NJ.) •Barrier "strength“ increases with misorientation. slip plane B •Smaller grain size: more barriers to in ra g slip. g grain A r •Hall‐Petch Equation: a in b o −1/2 u σ =σ + k d n yield o y d Why? a r y Dislocation‐G.B. Interactions G.B. impede motion of dislocations along entire length of disl. Line. •Expect G.B. to be more efficient at pinning than dislocation obstacles. Stress to activate dislocation motion for a given grain increases with number of dislocations pile‐up at the GB due to stress concentration. For infinite grain, YS is intrinsic value of single crystal slip.

For finite and decreasing grain size, YS increases. From Ian Robertson, UIUC Influence of grain size on yield strength (brass) Strength triples as grain size goes from 100 μm to 5 μm.

0.75mm

Fig. 4.11(c), Callister 6e.

As d⇓, σys⇑ and the ductility or ⇑ or it is constant − 2/1 y σσ0 )( += ydkT d = grain size

Note how intercept depends on temperature. Slope doesn’t.

N.J. Petch, Fracture, Proceedings of Swampscott Conference 1959, p. 54. Of the different models to explain the Hall‐Petch behaviour, three fundamentally different approaches can be identified, namely pile‐ up models, dislocation density models and composite models.

Pile‐Up Model (Hall‐1951, with subsequent modifications by Petch‐1953 and Cottrell‐1964). The basic idea is that dislocations are assumed to pile‐up against a grain boundary, thereby causing a stress concentration. When the stress concentration equals a critical stress, assumed to activate new dislocation sources, yielding starts in the next grain. The simplest pile‐up we can imagine is a single‐layer pile‐up, as illustrated in the figure below. The number of dislocations in a single‐layer pile‐up, as a function of the applied stress and pile‐up length, has been derived by Eshelby et al. (Eshelby, Frank et al. 1951). The pile‐up length is then proportional to the grain size and going through the algebra we can write the tensile shear stress as:

Attractive features of this theory. (a)It gives an explanation for the sharp yield point behavior in low‐ carbon steels. (b)it is consistent with the inhomogeneous nature of plastic yielding in these steels. Major drawbacks is that it is not really applicable to all systems (e.g. fcc‐metals) and there are no direct observations of pile‐ups reported in the literature. Dislocation Pileup d Grain boundary

Force on lead dislocation = Nb2τ Frank‐Read source

τ Frank‐Read source stops N ……. 3 2 1 emitting dislocations when back‐stress caused by source dislocation loops becomes sufficiently high to shut τ down source.

)(

*2 r met, shuts NbG

is π

τ = source

* ≈ determines

source.

τ r condition

dislocations by

τ + of

this I

Nb dislocation τ condition

pileupn dislocatio pileupn down: the When

generated This number with d/4

Nb Nb

rGb =

π 2/~ 3 2 1 vector

dislocation, ‐ ……. N

by caused stressback stressback caused σ by Burgers Super

source τ d Pileup dislocation

stress τ ) )( 42 NbG (d/ around π friction * friction

field = = ≈

I τ τ Dislocation Stress Physical picture # 2: yielding begins when stress from pileup activates source in neighboring grain

Number of dislocations in pileup (superdislocation) : Physical picture: πd τ −τ*)( f*: force on leading dislocation N = necessary to allow it to escape 2Gb At yielding, f* = Nbf τ −τ*)(* d −ττπ*)( 2 = 2G

− 2/1 y ττ+= π )/*2(* dGf − 2/1 τ *+= ydk Physical picture # 2: yielding begins when stress from pileup activates source in neighboring grain

s*: source

s* Stress at source in neighboring r* grain due to pileup: d 2/ −≈ τττ*)( r r *

Yielding begins once stress at source reaches threshold: y 2/1 y += τττr dr )/*2(* − 2/1 τ *+= ydk d = grain size

At low T, yielding results in cracks Effect of grain size: Both yield & fracture stress are increased; Ductile‐brittle transition sent to lower T.

Fracture stress at T2

Yield stress at T2

Yield stress at T1

brittle ductile

d − 2/1 Dislocation Density Models They are all based on Ashby's original model (Ashby 1970). Assume that the strengthening due to dislocations can be separated into two different contributions, namely that from statistically stored dislocations ρS, and that from geometrically necessary dislocations ρG. ρS is grain‐size independent while ρG depends on the grain size.

Where m is the average Taylor factor and d is the grain size. The S dislocation density ρS is governed by the geometrical slip distance L , in the interior of the grains where the deformation is assumed to be uniform. The non‐uniform deformation in the grain boundary region is accommodated by the introduction of geometrically necessary dislocations. These can be seen as the strain bearers needed to account for the plastic incompatibilities in‐between grains (Ashby 1970).

Deformation of polycrystal grains in an uniform manner, causing voids and overlaps (top right), this are corrected by the introduction of geometrically necessary dislocations (bottom right), taken from Ashby (Ashby 1970). The flow stress can then, in the usual fashion, be expressed as proportional to the square root of the total dislocation density, which leads to:

In the case where grain boundary strengthening S dominates, L >>C1b, i.e. the deformation is inhomogeneous, (i.e. ρG > ρS) the above equation reduces to, Composite Flow Stress Models A third type of approach is the idea of describing the flow stress as the sum of the contribution from grain boundaries and the contribution from grain interiors. A number of different variants have been proposed (Hirth 1972; Thompson, Baskes et al. 1973; Meyers and Ashworth 1982). Thompson et al. (Thompson and Baskes 1973; Thompson, Baskes et al. 1973; Thompson 1975; Thompson 1975) developed a model to describe the Hall‐Petch behavior of fcc‐metals by combining concepts from Ashby's model with a composite‐type model (Hirth 1972). They assumed the dislocation density in the grain boundary region ρG, to be inversely proportional to the grain size but independent of strain. In their expression, the statistical density of dislocations was estimated to be inversely proportional to the geometrical slip distance, LS. The contributions to the flow stress from the different area fractions were then added, using a rule of mixtures. Assuming the area of the grain boundary region as LS/d, this leads to the following expression for the flow stress: When LS approaches d, the grain size, i.e. at very small strains, the above expression reduces to the traditional form, with K2 equal to k. The physical significance of K2 is not very clear, but it should basically have the same meaning as C2 in Ashby's model although the interpretation is not as straightforward.