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Dynamic Strength of Materials
Ronald W. Armstrong
Center for Energetic Concepts Development
Department of Mechanical Engineering
University of Maryland, College Park, MD 20742
Presentations for Masters Course in Shock Physics,
Institute of Shock Physics, Imperial College, LondonMarch 24,25, 2011
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Crystal grains, dislocations, slip, polycrystal plasticity/ fracturing
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TOPICSI. Dislocation velocity-dependent mechanics
1. Thermal activation
2. Dislocation pile-ups
3. Combined consequences: σ = σ{(dε/dt), T, ℓ-1/2}
(i) Ductile-brittle transition/Charpy impact results
(ii) Plastic instability/shear banding
(iii) Taylor cylinder impact results
II. Dislocation generation vs. velocity mechanics
1. Shock front dislocation generations
2. Copper, ARMCO iron and tantalum results
3. Shockless isentropic compression experiments (ICEs)
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Constitutive Equation RelationsThe total range for creep, slip, twinning and cleavage dynamics:
ε =ε{∆t, σ, D, T} → σ = σ{(dε/dt), T, ℓ-1/2}
1. Thermal activation - strain rate analysis, TASRA, (dε/dt) = (dε/dt){T, τTh}:
thus (∂τth/∂T)ln[dε/dt](∂T/∂ln[dε/dt])τTh (∂ln[dε/dt]/∂τTh)T = -1.0
and (dε/dt) = (dε/dt)0exp{-(G0 - ∫v*dτTh)/kBT}, with v* = A*b,
and v* = W0/τTh and τTh = τ - (τG + kSεℓ-1/2).
2. The Hall-Petch microstructural stress intensities, “k”s:
For a circular pile-up; n(τ – τ0ε) = m*τC and n = 2α(τ – τ0ε)ℓ/πGb
thus σ = mT[(τG + τTh) + (πm*GbτC/2α)1/2ℓ-1/2] = σ0ε + kεℓ-1/2
and kAl < kCu < kMg << kα-Fe with kε < ky.p. << kT ~ kC << KIC = σ(πc)1/2
with c and ℓ being analogous in comparison with the fracture mechanics KIC
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“Dynamic Strength of Materials” Bibliography1. A. Seeger, in “Dislocations and Mechanical Properties of Crystals”,
edited by J.C. Fisher, W.G. Johnston, R. Thomson, and T. Vreeland, Jr. (John
Wiley & Sons, Inc., N.Y., 1956) p. 243.
2. “Dislocation Dynamics”, edited by A.R. Rosenfield, G.T. Hahn, A.L.
Bement, Jr., and R.I. Jaffee (McGraw-Hill Book Co., N.Y., 1968).
3. R.W. Armstrong, in “Yield, Flow and Fracture of Polycrystals”, edited by
T.N. Baker (Appl. Sci. Publ., London, U.K., 1983) p. 1.
4. “Mechanics of Materials”, edited by M.A. Meyers, R.W. Armstrong, and
H.O.K. Kirchner (John Wiley & Sons, Inc., N.Y., 1999).
5. R.W. Armstrong and S.M. Walley, “High Strain Rate Properties of Metals and
Alloys”, Intern. Mater. Rev. 53, [3], 105-128 (2008).
6. R.W. Armstrong, W. Arnold, and F.J. Zerilli, “Dislocation mechanics of shock-
induced plasticity”, J. Appl. Phys., 105, 023511 (2009), 7 pp.
7. R.W. Armstrong, “Dislocation Viscoplasticity Aspects of Material Fracturing”, Eng.
Fract. Mech., 77, 1348-1359 (2010).
8. R.W. Armstrong, in “Mechanical Properties of Nanocrystalline Materials”,
edited by J.C.M. Li (Pan Stanford Publ., Singapore, 2011) Chap. 3, pp. 61-91, in print.
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Zerilli-Armstrong Constitutive Equations
(dε/dt) = (1/m)ρbν
ν = ν0exp[-(G0 - ∫A*bdτTh)/kBT] and A*b = W0/τTh
Computational (Z-A) equations:
σ = σG + Bexp[-βT] +
B0[εr(1 – exp{-ε/εr})]1/2exp[-αT] +kεℓ-1/2
in which
(β,α) = (β0,α0) – (β1,α1)ln(dε/dt)
bcc case: α =α0 = α1= 0 fcc case: B = β = β0 = β1 = 0
F.J. Zerilli and R.W. Armstrong, J. Appl. Phys. 61, 1816-1825 (1987)
F.J. Zerilli and R.W. Armstrong, J. Appl. Phys. 68, 1580-1591 (1990)
F.J. Zerilli, Metall. Mater. Trans. A, 35A, 2547-2555 (2004)
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Tensile ductile-brittle transition
σy = σC at T = TC = (1/β)[lnB – ln{(kC – ky) + (σC- σ0G)ℓ1/2} - lnℓ-1/2]R.W. Armstrong, Metall. Mater. Trans., 1, 1169-1176 (1970).
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Impact
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Charpy impact transition temperature
α’σy = σC at TC = (1/β)[lnα’B – ln{(kC – α’ky) + (σ0C – α’σ0G)ℓ1/2} - lnℓ-1/2]
R.W. Armstrong, Eng. Fract. Mech., 28, 529-538 (1987)
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Tensile plastic instability: σ = dσ/dε
R.W. Armstrong and F.J. Zerilli, Mech. Mater., 17, 319-327 (1994)
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BCC vs. FCC plastic instability dependences
R.W. Armstrong and F.J. Zerilli, Mech. Mater., 17, 319-327 (1994)
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Tantalum: σ = σ{(dε/dt), T, ℓ-1/2}
F.J. Zerilli and R.W. Armstrong. J. Appl. Phys., 68, 1580-1591 (1990)
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Shear banding based on σ = Kεn and σ = dσ/dε;
from (dP/dℓ0) = 0 and raised to (d2P/dℓ02) = 0
Measurements from M.R. Staker, Acta Metall., 29, [4], 683-689 (1981)
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Shear banding from a dislocation pile-up avalanche
R.W. Armstrong, C.S. Coffey, and W.L. Elban, Acta Metall., 30, 2111-2118 (1982)
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Shear band susceptibility from the pile-up avalanche model
R.W. Armstrong and F.J. Zerilli, Mech. Mater., 17, 319-327 (1994)
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Titanium: σ = σ{(dε/dt), T, ℓ-1/2}
F.J. Zerilli and R.W. Armstrong, Shock Compression of Condensed Matter – 1995,
edited by S.C. Schmidt and W.C. Tao (Amer. Inst. Phys., N.Y., 1996) CP 370, pp. 315-318
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Isothermal and adiabatic stress strain curves for Ti6Al4V material
F.J. Zerilli and R.W. Armstrong, Shock Compression of Condensed Matter – 1995, edited by S.C. Schmidt and W.C. Tao (Amer. Inst. Phys., N.Y., 1996) pp. 315-318
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Copper: σ = σ{(dε/dt), T, ℓ-1/2}
F.J. Zerilli and R.W. Armstrong, J. Appl. Phys., 61, 1816-1825 (1987);
see P.S. Follansbee and U.F. Kocks, Acta Metall., 36, 81-93 (1988).
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Copper Taylor cylinder impact result
F.J. Zerilli and R.W. Armstrong, J. Appl. Phys., 61, 1816-1825 (1987)
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Original Taylor cylinder test result on mild steel
W.E. Carrington and M.L.V. Gaylor, Proc. Roy. Soc. London A, 194A, 323-331 (1948)
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SHPB twinning measurements compared to Z-A slip calculations
R.W. Armstrong and F.J. Zerilli, J. Phys. Fr. Colloq., 49, [C3], 529-534 (1988)
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Armco iron Taylor test involving twinning and slip
F.J. Zerilli and R.W. Armstrong, Shock Compression of Condensed Matter, edited by
S.C. Schmidt and N.C. Holmes (Elsevier Sci. Publ. B.V., N.Y., 1988) pp. 273-277.
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Sequential twinning and slip in the Taylor cylinder impact test
J. B. McKirgan, “Microstructurally-based EPIC simulations of Taylor impact tests”, M.Sc. Thesis, University of Maryland, 1990
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Model for a propagating shock frontσ = P{(1 – ν)/(2 – ν)} for one-dimensional strain
F.A. Bandak, D.H. Tsai, R.W. Armstrong and A.S. Douglas, Phys. Rev. B, 47, 11681-11687 (1993)
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MD model of shock-induced dislocation dipole structures
F.A. Bandak, R.W. Armstrong, and A.S. Douglas, Phys. Rev. B, 46, 3228-3235 (1992).
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Model of 3-D post-shock deformation
F.A. Bandak, R.W. Armstrong, and A.S. Douglas, Phys. Rev. B, 46, 3228-3235 (1992)Analogy with “channeling” of deformation in post-deformation of neutron-irradiated materials
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Transition from strain rate control by dislocation
velocity to control by dislocation generationFor dislocation velocity control:
(dε/dt) = (1/m)ρbν
σ = σG + Bexp[-βT] + B0[εr(1 – exp{-ε/εr})]1/2exp[-αT] +kεℓ-1/2
in which
(β,α) = (β0,α0) – (β1,α1)ln(dε/dt)
For dislocation generation control:
(dε/dt) = (1/m)b(dρ/dt)∆xd
for which, at limiting small value of v* ~ b3, and m = 2
σTh = (2G0G/v*) – (2kBT/v*) [ln{(dε/dt)0/(dε/dt)}]
R.W. Armstrong, W. Arnold, and F.J. Zerilli, Metall. Mater. Trans. A, 38A, 2605-2610 (2007)
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SHPB indication of dislocation generation
R.W. Armstrong, W. Arnold and F.J. Zerilli, Metall. Mater. Trans. A, 38A, 2605-2610 (2007)
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Connection of Follansbee et al results with
Swegle and Grady shock measurements
R.W. Armstrong, W. Arnold, and F.J. Zerilli, Metall. Mater. Trans. A, 38A, 2605-2610 (2007).
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R.W. Armstrong, W. Arnold, and F. J. Zerilli, Metall. Mater. Trans. A, 38A, 2605-2610 (2007);
W. Arnold, Dynamisches Werkstoffverhalten von Armco-eisen bei Stosswellenbelastung, Fortschrittberichte VDI, 5, 247,VDI-Verlag GmbH, Duesseldorf, DE, 1992
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Shocked Armco iron at different grain sizes and plastic strain rates
R.W. Armstrong, W. Arnold and F.J. Zerilli, J. appl. Phys., 105, 023511 (2009)
-Fe
STRAIN RATE (s-1
)
10-3 10-2 10-1 100 101 102 103 104 105 106 107 108
ST
RE
SS
(G
Pa)
0
1
2
3
4
5
6
7
8Swegle and Grady
Arnold - yield, 20
Arnold - flow, 20
Arnold - yield, 80
Arnold - flow, 80
Arnold - yield, 400
Arnold - flow, 400
Twin (20 )
Twin (80 )
Twin (400 )
ZA (20 )
ZA (80 )
ZA (400 )
DISLOCATION GEN
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Tantalum
F.J. Zerilli and R.W. Armstrong, J. Appl. Phys., 68, 1580-1591 (1990)
TANTALUM
STRAIN RATE (s-1
)
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105
TR
UE
ST
RE
SS
(M
Pa
)
0
200
400
600
800
1000
HOGE & MUKHERJEE
ZERILLI & ARMSTRONG
TWIN, MURR, ET AL.
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Tantalum Conventional and Shock Results
R.W. Armstrong, W. Arnold and F.J. Zerilli, Metall. Mater. Trans. A, 38A, 2605-2610 (2007)
TANTALUM
STRAIN RATE (s-1
)
10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106 107
TR
UE
ST
RE
SS
(M
Pa)
0
1000
2000
3000
4000
5000
6000
7000
HOGE & MUKHERJEE
ZERILLI & ARMSTRONG
TWIN, MURR, ET AL.
MEYERS
= 2U0/V0 + (1400 MPa)ln( / 0)
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Ta single crystal/polycrystal SHPB results
D. Rittel, M.L. Silva, B. Poon and G. Ravichandran, Mech. Mater., 41, 1323-1329
(2009); R.W. Armstrong and F.J. Zerilli, J. Phys. D: Appl. Phys., 43, 492002 (2010)
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Critical activation volumes, v*, under shock loading
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Laser-shocked deformation of copper
W.J. Murphy et al., J. Phys.: Condens. Matter, 22, 065404 (2010); R.W.
Armstrong and F.J. Zerilli. J. Phys. D: Appl. Phys., 43, 492002 (2010)
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The strain and strain rate from combined
dislocation displacements and generations
The strain from displaced and generated dislocations
∆ε = (1/m)[ρNb∆xN + ∆ρGb∆xd +ρGb∆xd]
The strain rate is obtained then for the presumed time-
dependent parameters as
(dε/dt) = (1/m)[ρNbνN +(dρG/dt)b∆xd + ρGbνG]
with neglect of (dρN/dt). If it were assumed that νN = νG = ν*,
and ρN and ρG could be combined as ρT then
(dε/dt) = (1/m)[(dρG/dt)b∆xd + ρTbν*]
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Shockless isentropic compression experiments (ICEs)
The resident dislocation density is required to “carry the load”,
and because ρN is low, νN is so high as to be controlled by “drag”!
σTh = {1 – [c(dε/dt)/β1σTh]-β1T }[Bexp(-βT)]
in which
c = c0m2β1/ρb2 and bτTH = c0ν.
At high (dε/dt):
σTh = (c0m2/ρb2)(dε/dt)
F.J. Zerilli and R.W. Armstrong, Acta Mater., 40, 1803-1808 (1992);
R.W. Armstrong, W. Arnold and F.J. Zerilli, J. Appl. Phys. 105, 023511 (2009)
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Copper SHPB, Shock, and ICE results
H. Jarmakani et al., Mater. Sci. Eng. A, 463, 249 (2007);
R.W. Armstrong, W. Arnold and F.J. Zerilli, J. Appl. Phys., 105, 023511 (2009)
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Drag-controlled shockless ICE results for copper
R.W. Armstrong, W. Arnold and F.J. Zerilli, J. Appl..Phys., 105, 023511 (2009)
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Strength of Material DynamicsSUMMARY
1. An introduction has been given on a dislocation mechanics basis to coupling of
model thermal activation – strain rate analysis (TASRA) and grain size related,
dislocation pile-up, constitutive relations.
2. Attention was directed to the strain rate, temperature, and grain size dependencies
that are established for evaluations of such material strength properties as: (i)
conventional stress – strain; (ii) ductile-brittle transition; (iii) Charpy v-notch impact;
(iv) plastic instability; (v) shear banding; (vi) Taylor cylinder impact; (vii) shock; and,
(viii) shockless isentropic compression experiments (ICEs).
3. Beyond giving emphasis to the dislocation velocity and generation being thermally
activated processes at the crystal lattice scale, the fuller description involves key
features of (i) dislocation pile-up associated internal stress concentrations, (ii) relief
needed for dislocation generations at all lattice points along a propagating shock front;
and, (iii) trade-offs between dislocation density and velocity in distinguishing between
very different strength properties at comparable shock and shockless deformation rates.