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Interdendritic Strain and Macrosegregation-Coupled Phenomena for Interdendritic Crack Formation in Direct-Chill Cast Sheet Ingots

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Abstract

In a study of the early stages of dendritic solidification in the direct-chill cast sheet ingots, the coupled effect of interdendritic strain and macrosegregation on the interdendritic cracks formation in dendritic equiaxed structure has been investigated by the metallographic study of ingot samples and by performing a set of mathematical analyses for AA-6061 and AA-1050 aluminum alloys. The metallographic investigation contains microstructure examinations and macrosegregation measurements of collected samples from plant trials. The mathematical analysis consists of a two-dimensional (2-D) fluid flow, heat flow, interdendritic strain, and macrosegregation-coupled model. Also, a simple approach to measure interdendritic crack has been developed based on the accumulative interdendritic strain criterion, local dendritic phases, and the crystal distortion correlation factor resulting from steep positive local segregation. The model predications have clarified the effect of high positive macrosegregation on the surface and subsurface interdendritic crack formation. It has been revealed that interdendritic strain starts to generate just below the liquidus temperature, resulting from shrinkage of liquid→solid phase transformation and contraction of dendritic solid in the incoherent mushy region. In this region, the coupled effect of the shrinkage/contraction mechanism increases the interdendritic distances between equiaxed crystals and the interdendritic crack begins to nucleate. Subsequently, in the coherent mushy region, the different interdendritic strain sources start to affect significantly the distances between equiaxed crystals in a diverse way, and therefore, the final morphology of interdendritic crack begins to form. The mechanism of interdendritic crack formation during dendritic equiaxed structure solidification and the possible solutions to this problem are discussed.

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Abbreviations

A :

elementary area (m2)

a, b:

constants in Eq. [10]

C p :

specific heat (kJ/kgK)

E η :

volumetric energy (W/s m3)

E :

modulus of elasticity (N/m2)

F c :

crystal distortion correlation factor

h(t):

heat transfer coefficient at time t (W/m2K)

H :

enthalpy (kJ/kg)

\( \bar{H} \) :

average enthalpy (kJ/kg)

k c :

coefficient in Eq. [8]

K e :

equilibrium partition coefficient

l coh :

coherent solid shell thickness (mm)

L s :

slab width (mm)

L :

latent heat of fusion (kJ/kg)

M :

bending moment (Nm)

n :

number of measured fields

P :

pressure (N/m2)

Q ϕ , Q x :

surface and x isotherm heat fluxes (kW/m2)

P w :

ferrostatic pressure (N/m2)

R d :

radius of equiaxed crystal (m)

S j :

macrosegregation ratio of element j

S meaj :

actual measured composition of j element (wt pct)

S oj :

nominal composition of j element (wt pct)

S pj :

predicted composition of element j (wt pct)

S v :

grain boundary surface area per unit volume (1/m)

t :

time (s)

t f :

solidification time (s)

T :

temperature (°C)

T ϕ , T w :

ingot surface and cooling fluid temperature (°C)

u :

local interdendritic liquid velocity (m/s)

V :

volume (m3)

V a :

dendritic growth rate (m/s)

x, y, z :

Cartesian coordinates (m)

\( \bar{X} \) :

mean measurements value (μm)

X i :

individual measurement value (μm)

α :

thermal expansion coefficient (1/°C)

\( \alpha^{*} \) :

instantaneous diffusion parameter (m2/s)

\( \varepsilon \) :

strain

\( \sum {\varepsilon_{i} } \) :

total strain per unit volume i

ϕ 1 :

grain size (μm)

K :

mushy permeability

η :

coordinate for local energy

λ :

thermal conductivity (W/m2K)

\( \rho \) :

density (kg/m3)

\( \bar{\rho } \) :

average density (kg/m3)

\( \sigma \) :

standard deviation (μm)

τ :

stress (N/m2)

ξ :

bulging (mm)

Δ :

infinitesimal element of length or time

acc:

accumulative

acc*:

interdendritic accumulative

c:

creep

eff:

effective

e:

elastic

l:

liquid

m:

mechanical

m k :

exponent in Eqs. [20] and [21] of solid phase k

T ph :

thermometallurgical

tot:

total

coh:

coherent

conv:

convection stream

eff:

effective

Int:

interdendritic liquid

l:

liquid

m:

melting

s:

solid

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Acknowledgments

The author wishes to express his sincere gratitude to Prof. Merton Flemings, former Head of the Department of Materials Science and Engineering, Massachusetts Institute of Technology (MIT), Cambridge, MA, for his early pioneering work in the fields of segregation and solidification cracks, which guided the author to several facts in these fields. Also, he offers his sincere gratitude to Prof. Hasse Fredriksson, Department of Materials Science and Engineering, Royal Institute of Technology, Institute of Materials Processing, Stockholm, Sweden, for his considerable supervision, guidance, helpful discussions throughout the work, and valuable assistance. The author aspirates the previous help and useful discussions from Prof. Michael Rappaz, Laboratoire de Metallurgie Physique, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland, and Prof. David Poirier, Department of Materials Science and Engineering, University of Arizona, Tucson, AZ. The author is also especially grateful for the financial support of Companies’ Chair of the Swedish Iron Masters Association, Stockholm, Sweden. He is also grateful to Egyptian Copper Works, Hagar El-Nawatia, Alexandria, Egypt, who kindly supplied him with ingot samples. The author’s sincere gratitude is due to the assistances in metallurgical laboratories, Faculty of Engineering, Ain Shams University, Cairo, Egypt.

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  1. Companies’ Chair of Swedish Iron Masters Association, S-100 44, Stockholm, Sweden

    Mostafa Omar EL-Bealy (Chair Professor of Materials Processing and Technology)

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  1. Mostafa Omar EL-Bealy
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Correspondence to Mostafa Omar EL-Bealy.

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Manuscript submitted August 29, 2011.

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EL-Bealy, M.O. Interdendritic Strain and Macrosegregation-Coupled Phenomena for Interdendritic Crack Formation in Direct-Chill Cast Sheet Ingots. Metall Mater Trans B 43, 635–656 (2012). https://doi.org/10.1007/s11663-011-9616-0

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