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ASM Handbook, Volume 4B, Steel Heat Treating Technologies.
J. Dossett and G.E. Totten, editors
Distortion Engineering
B. Clausen, T. Lübben, and R. Rentsch, Stiftung Institut für Werkstofftechnik
AS MENTIONED in the article “Basics
of Distortion and Stress Generation during
Heat Treatment” in this Division, distortion
is a system property, and distortion control
during manufacturing processes must follow
a system-oriented approach. The fact that the
corresponding system is the complete manufacturing chain is very important.
One answer to the question of what must be
done to control size and shape changes under
these conditions was given by Zoch (Ref 1).
He reports about the development of a methodology called distortion engineering. Always
taking into account the entire process chain,
this methodology consists of three levels of
investigations (Fig. 1). On Level 1, the parameters and variables influencing distortion in
every manufacturing step must be identified.
In general, a large number of parameters may
be important. Therefore, design of experiment
(DOE) techniques were used, which allow the
investigation of larger numbers of parameters
by a limited number of samples, enabling the
identification of cross-influencing parameters
as well as interdependencies.
On the basis of the resulting knowledge,
Level 2 focuses on understanding the distortion
mechanisms by using the concept of distortion
potential and its carriers (see the article “Basics
of Distortion and Stress Generation during Heat
Treatment” in this Volume). Modeling and
simulation not only are helpful, but in many
cases are necessary tools to fully understand
the mechanisms governing the distortion
generation.
Distortion engineering aims to compensate
distortion using the so-called compensation
potential (Level 3). On one hand, this approach
uses the conventional method to increase the
homogeneity, and respectively the symmetry,
of the carriers of the distortion potential. On
the other hand, well-directed insertions of additional inhomogeneity/asymmetries in one or
more of the distributions of the carriers can be
used to compensate the resulting size and shape
changes of the existing asymmetries. For example, an inhomogeneous quenching process can
be used to compensate shape changes from the
previous manufacturing process. In principle, a
compensation of single components is possible.
On this level, in-process measurement and control techniques are very important (Ref 2, 3).
The Collaborative Research Center “Distortion Engineering” at the University of Bremen
analyzed different manufacturing chains according to the three levels of distortion engineering
and rated the results according to the success of
the underlying analyses. Details of the investigated
manufacturing chains and the corresponding
results can be found in Ref 2 (rings, SAE 52100),
Ref 4 (rings, SAE 4140), Ref 5 and 6 (shafts,
SAE 5120), Ref 7 and 8 (bevel gear, 17 CrNi6 6),
and Ref 9 and 10 (aircraft panel, aluminum alloy).
The investigations of disks of the steel grade
SAE 5120 are used in the following sections to
describe distortion engineering in detail. Therefore, it is important to know that the distortion of
gear teeth is strongly correlated to the size and
shape changes of the base body of the gear. This
is, in many cases, very similar to a disk with a
hole. More details are given in the section “Influence of the Carrier Distribution of Mass—
Geometry.”
In the next sections are many results for
disks, which behave very similarly to the base
body of a gear with the same dimensions. Furthermore, and of greater importance, the application of distortion engineering is explained in
detail. For rings, all details of distortion engineering are presented in Ref 2.
Distortion Engineering, Level 1
—Identification of DistortionRelevant Parameters and Variables
More than 200 parameters can affect distortion (Ref 11). These parameters not only influence the distortion directly but also interact
with each other, thereby enhancing or reducing
effects of the others. This knowledge necessitates careful selection of influencing parameters
in the investigated manufacturing process chain,
and a test procedure that also detects interactions
of these parameters. The design of experiments
(DOE) approach can be used and can be structured into four subtasks (Ref 12):
Fig. 1
Methodology for distortion engineering. Courtesy of Thomas Lübben. Source: Ref 1
System analysis
Test strategy
Test procedure
Test evaluation
This approach for the investigation of the
distortion occurring in a case-hardened disk
with a centered hole is described subsequently,
using the base body of a gear wheel. The results
were taken from Ref 13.
392 / Distortion and Residual Stresses
System Analysis
Machining
Continuous/block cast.
Quenching
Clamping cond.
Temperature gradient
Cuttting speed
Cast temperature
Chemical composition
Quenching velocity
Feed rate
Homogeneity
Disk with distortion
Steel
Uni-/multidirectional
Forces
Temperature
(a)
Forging
Density change
Loading tool
Internal stresses
Internal streses
Temperature
Temperature
Carburizing
Annealing
Melt
production
Continuous/block cast.
Cast temperature
Chemical composition
ngth
Stre
s
egate
ertie
prop
ility
segr
(b)
enab
n to
natio
Hard
Incli
In the first step, every possible influencing
factor in each manufacturing step must be identified. This can be done by literature survey,
brainstorming, or with the help of fishbone
diagrams (Ref 14). For the example of a casehardened disk, Fig. 2(a) shows a rough fishbone
diagram, dividing the manufacturing chain into
the manufacturing steps. The diagram is
detailed for one branch in Fig. 2(b). Such a
detailed rendering must be done for every
branch.
In the second step, the identified parameters
must be rated. This can be done by experiments
testing the sensitivity of distortion for the
change of single parameters. Alternatively, discussions with scientific and industrial experts
can be helpful.
In the rating process, two aspects must be
taken into account: the possible influence of
the parameter on distortion and the range of its
variation. For the second aspect, an unchanged
product quality—for example, hardness—must
be ensured.
Figure 3 shows the parameters for the investigated manufacturing steps rated to be of
influence and rated to be variable in distortion-relevant measures. Carrying out a complete test plan with these definite influencing
factors—altogether 23—would take decades
(223 = 8,388,608 experiments). A sensible
approach to reduce the number of necessary
experiments is to detect the main influencing
parameters in separate manufacturing-stepspecific plans. Figure 4 shows the main influencing factors that were determined in such
plans. Because most influencing factors of the
manufacturing steps before heat treatment
were expected not to interact with the influencing factors of the heat treatment, these two
main parts of the entire investigation were
separated. It is very important to keep every
other parameter constant and to record the production flow to be able to rate and relate the
occurring effects correctly.
Melt
production
Fig. 2
Influence parameters. (a) Process chain for the manufacture of case-hardened disks. (b) Detailed illustration
for one branch of diagram from (a)
Fig. 3
Parameters for the investigated manufacturing steps rated to be of influence. Source: Ref 13
Test Strategy and Test Procedure
For this investigation, disks with an outer
diameter of 120 mm (4.7 in.), a height of
15 mm (0.6 in.), and a center hole diameter of
45 mm (1.8 in.) were produced.
Part 1—Casting, Forming, and Machining.
This section covers constant test conditions;
hardenability; temperature of forging; preheat
treatment; feed rate; cutting strategy; heating,
carburizing, and quenching; and parameters of
DOE.
Constant Test Conditions. All disks were
taken from two melts of SAE 5120 steel with
the same rolling strain, but with different hardenabilities. Both melts were molten in an electric arc furnace. Their degree of purity was
adjusted by the use of the Ruhrstahl-Heraeus
(RH) procedure. The continuous caster produced
square blooms with a side length of 256 mm
(10.07 in.) and a weight of 2 tonnes (2 metric
tons, or 2.2 tons), which were first hot rolled to
a size of 85 mm2 (3.34 in.2) in several steps.
Subsequently, the bars were hot rolled on a
round rolling and several three-roll reducing
blocks with inductive temperature control to bars
with a diameter of 73 mm (2.87 in.). The total
logarithmic strain was j = 2.75. After the hot
rolling process the bars were air cooled to room
temperature.
The steel bars of both melts were numbered
consecutively and marked over their whole length
for identification of the angular orientation of the
finished parts in the bars. After they were sawed
into billets of approximately 2 kg (4.4 lb), the orientation mark of the bars was replaced by a wire
in a small bore hole in each billet. The billets
Distortion Engineering / 393
were forged in an upsetting device with a load of
100 t (110 tons). They were prepunched and sized
to disks with a final height of 22.5 mm (0.88 in.) at
a main load of approximately 750 t (826 tons).
Finally they were punched with a load of 20 to
30 t (22 to 33 tons). The forging process leads to
an additional logarithmic strain of 0.98.
Hardenability. The hardenability serves as an
integrating parameter to describe the effect of
the alloying elements on distortion. The cast
analysis of the steel heats used is given in
Table 1. The calculated Jominy curves of the
two heats are given in Fig. 5 and show a significant difference in the hardenability of the two
melts. However, it must be kept in mind that
Fig. 4
the variation of the alloying elements also will
affect other properties of the material such as
the mechanical behavior.
Temperature of Forging. As one of the major
parameters in hot metal forming, the forging
temperature has a strong influence on the flow
stress as well as the material flow. The standard
of 1250 C (2280 F) is commonly applied
in industry. In cooperation with an industrial
partner, a lower value of 1150 C (2100 F)
was selected for the variation of the forming
temperature.
Preheat Treatment. After forging, the disk
blanks were stored batchwise in boxes and were
air cooled to room temperature and sandblasted.
Main influence factors of the investigated manufacturing steps. Source: Ref 13
Table 1 Chemical composition of the used steel heats
Chemical element, %
Hardenability
Low
High
C
Si
Mn
P
S
Cr
Mo
Ni
Cu
Al
N
0.20
0.21
0.23
0.09
1.35
1.35
0.011
0.013
0.020
0.026
1.02
1.24
0.03
0.09
0.10
0.12
0.12
0.10
0.04
0.03
0.015
0.012
Martensite or bainite phases can be found in the
microstructure because of the comparatively
high hardenability. To enhance the machinability, 20MnCr5 is pre-heat-treated before cutting,
mostly to adjust to a ferrite-pearlite structure.
Hardening and tempering is an alternative
preheat treatment. The choice of the preheat
treatment can influence the distribution of
residual stresses in the component by affecting
the generation of forces and heat during machining. Additionally, the specific volume of
the material is influenced by the microstructure
(see the article “Basics of Distortion and Stress
Generation during Heat Treatment” in this
Volume) and therefore the dimensional alterations, as well. Annealing to a ferrite-pearlite
structure was done at 930 C (1705 F) for
1 h. The disks were cooled within 3 min to
650 C (1200 F) and held there for 2 h. After
heat treatment the disks were cooled in air.
The hardening treatment started with austenitizing at 930 C for 30 min. The disks were
quenched with 10 bar nitrogen. The tempering
process was carried out at 620 C (1150 F)
for 4 h.
Feed Rate. Distribution of the residual stresses can be influenced by the cutting parameters,
mainly the feed rate (Ref 15). Therefore, the
feed rate of the last cut at the bottom disk face
was varied to obtain a difference in the distribution of residual stresses at the two faces
(Fig. 6). Feed rates of 0.1 and 0.3 mm (4 and
12 mils) were selected. The resulting residual
stresses are summarized in Table 2. The top
disk face always was cut with a feed rate of
0.3 mm, generating residual tangential stresses
of stang = 630 MPa (91.3 ksi) and residual
radial stresses of srad = 300 MPa (43.5 ksi).
Cutting Strategy. The turning of disks was
conducted in two clamping setups on a computer numerical control (CNC) turning center.
In the first setup, the disks were clamped with
a form-locking clamping technique applying a
Fig. 6
Illustration of the residual surface stresses on
disks. Source: Ref 13
Table 2 Influence of feed rate on residual
surface stresses, measured by x-ray
diffractometry
Residual surface stress at bottom face
stang
Feed rate (f) at bottom face
Fig. 5
Calculated Jominy curves of the investigated heats. Source: Ref 13
srad
mm
mils
MPa
ksi
MPa
ksi
0.1
0.3
4
12
390
630
56.5
91.3
20
300
2.9
43.5
394 / Distortion and Residual Stresses
clamping pressure of 30 bar. The top face was
turned first, followed by an external turning of
the outer surface and an internal turning of the
center hole. The bottom face was turned in a
second setup using segment jaws and a clamping pressure of 40 bar.
Results from preliminary investigations suggested the cutting strategy had a strong effect
on distortion of the disks. The analysis showed
that the banded structure, which remains in the
disks after turning, is affected by the applied
cutting strategy due to the change of the local
material removal. Therefore two strategies were
chosen for the DOE (Fig. 7). In cutting strategy 1,
a material layer with a thickness of 1 and
6 mm (0.04 and 0.24 in.) was removed from
the top side and the bottom side, respectively.
In cutting strategy 2, the thickness of the layer
removed from the top side was 4 mm (0.16 in.),
and the layer from the bottom side was 3 mm
(0.12 in.).
Heating, Carburizing, and Quenching. The
case-hardening process of the part 1 investigation (Fig. 4) was kept constant: a low-pressure
carburization in a two-chamber vacuum furnace
with gas quenching was applied. Each batch
consisted of eight disks hanging in a single
layer. Carburization was carried out at 940 C
(1725 F) in a C2H2 atmosphere. The casehardening depth was adjusted to 0.8 mm
(0.031 in.). The surface carbon content
averages 0.7 wt%. The batch was gas quenched
after holding for 20 min at 840 C (1545 F)
with 10 bar nitrogen. A tempering process was
not considered.
Parameters of DOE. Table 3 gives an overview of the evaluated factors of the DOE and
the appropriate levels. It was assumed that the
distortion is caused by main effects or interactions between two main factors. For this reason
a 2V5-1 test plan was chosen. As generator for
this fractional factorial test, G1 = ABCD was
used. This kind of test matrix has the advantage
that main effects and second-order interactions
can be separated (Ref 16). Main effects are
superimposed with fourth-order interactions,
and second-order interactions are superimposed
with third-order interactions. In general, higherorder interactions are rare and often can be
neglected. Each of the 16 variants was repeated
8 times for a total of 128 experiments.
Part 2—Heating, Carburizing, and
Quenching. This section discusses casting,
forming, and machining; constant heat treatment conditions; loading tool; carburizing
depth; hardening temperature; gas flow rate
during quenching; influence of material position in the original strand; and parameters of
DOE.
Casting, Forming, and Machining. In this
part of the investigation all green disks were
produced in the same way. For the manufacture
the melt with the low hardenability was used
(Table 1). The disks were forged with the standard temperature of 1250 C (2280 F). They
were annealed to a ferrite-pearlite structure
and sandblasted before cutting. The disks were
finished on a turning center in two clamping
setups by use of cutting strategy 1 (Fig. 7) and
a feed rate of 0.3 mm (12 mils).
Constant Heat Treatment Conditions. All
disks were case hardened in a gas carburizing
process with subsequent quenching in a gas
nozzle field (Ref 13). Carburizing was done in
a multiprocess bell-type furnace. The disks
were batched horizontally. Depending on the
hardening temperature, one or two disks were
carburized simultaneously (further described
subsequently). The nozzle field quenching was
done for one disk at a time. A tempering process was not considered.
Loading Tool. Creep caused by gravity during heat treatment is one important reason for
distortion, and the loading tools used can have
a significant effect on it (Ref 2). Moreover,
Fig. 7
heating uniformity, flow resistance of furnace
gas during thermochemical heat treatment, and
flow state of the quenchant are related to the
loading pattern and the design of the loading
tools (Ref 17). By selecting two loading tools
(two-line and three-point support), different
stress states by dead load during the final heat
treatment were realized (Fig. 8). With the line
loading tool, disks were placed on two hightemperature steel bars with a width of 10 mm
(0.4 in.) and a length of 100 mm (4 in.). With
the three-point loading tool, the disks were
placed horizontally on a ring with an outer
diameter of 76 mm (3 in.) and a wall thickness
of 3 mm (0.12 in.), with three supporting areas
of an arc length of 10 mm. During carburizing,
the disks were aligned to the direction of the
positive x-axis.
Comparison of the applied cutting strategies. (a) Cutting strategy 1. (b) Cutting strategy 2. Source: Ref 13
Table 3 Levels of evaluated factors in the first part of investigation (casting, forming, and
machining)
Level
Factor code
Factor
–
+
A
B
C
D
E (ABCD)
Hardenability
Temperature of forging
Preheat treatment
Cutting strategy
Feed rate
High
1150 C (2100 F)
Hardened and tempered
Strategy 2
0.1 mm (4 mils)
Low
1250 C (2280 F)
Annealed to a ferrite-pearlite structure
Strategy 1
0.3 mm (12 mils)
Fig. 8
Different loading tools during case hardening. (a) Two-line loading tool. (b) Three-point loading tool. Source:
Ref 13
Distortion Engineering / 395
Carburizing Depth. A change of carbon content at the component surface influences the
phase transformation kinetics and stress states
during quenching. The disks were heated to
850 C (1560 F) in the preheated furnace.
After 20 min of temperature equalization, the
disks were heated to the carburizing temperature of 940 C (1725 F). Because the final carburizing depth depends on carburizing time, for
a depth of 0.6 mm (0.023 in.), the carburizing
time amounted to 65 min, and for 0.8 mm
(0.031 in.), 135 min.
Hardening Temperature. By increasing the
hardening temperature (from 840 to 940 C,
or 1545 to 1725 F), the thermal gradients in
the component were increased. This affects
the phase transformations and the stress states
during quenching. Cooling to hardening temperature was accomplished with 5 K/min to
840 C, followed by subsequent temperature
equalization. When the first disk was taken out
of the furnace for quenching in the gas nozzle
field (Fig. 9), a period of 10 min was required
for reheating the remaining disk to hardening
temperature before starting the second quenching process. At this temperature, no difference
in carburizing and hardness profiles between
the first and second disks was found. However,
in the case of the upper hardening temperature
of 940 C, large effects on both profiles were
expected due to the increasing carbon diffusivity at higher temperatures. To avoid this effect
on component distortion, only one disk at a
time was carburized at this level.
Gas Flow Rate during Quenching. By
increasing the flow rate of the quenching gas
from 8000 to 12,000 l/min (2115 to 3170 gal/
min), the cooling time from 800 to 500 C
(1470 to 930 F) was decreased from 24.8 to
20.0 s. Consequently, the thermal gradients in
the components were increased and the development of the phase transformations will be
modified. The gas nozzle field used consists of
64 nozzles (Fig. 9): 32 at the top and 32 at the
bottom. The nozzles are arranged in two concentric circles such that each nozzle feeds the
same surface fraction of the disks with gas.
Therefore, the distribution of the heat transfer
coefficients is symmetric over both faces.
Influence of Material Position in the Original
Strand. In another analysis of cylinders of
the same material (Ref 18), an explicit dependence on bending direction was identified for
cylinders produced from one bar. The reason
for this distortion behavior was explained by
different orientations of the banded structures
in different bars (Ref 19). Because the disks
had to be produced from different bars (bar
length was 3 m, or 10 ft), this possible barrelated effect on disk distortion was included
in this examination: the billets were sawed off
of two bars.
Parameters of DOE. Table 4 shows the
levels of the analyzed factors for a 2V5-2 test
plan (Ref 0). The following generators for this
fractional factorial test were used: G1 = BCDE,
G2 = ACE, and G1 G2 = ABD. This leads to
an aliasing of main effects and second-order
interactions (Ref 16). In this case it was
assumed that second-order interactions could
be neglected. Each of the 8 variants was
repeated 4 times for a total of 32 experiments.
How to Measure Disk Distortion. Before a
test can start, it must be decided which dimensions and shapes of the investigated sample will
be of major interest. These must be measured
before and after heat treatment. The uncertainty
of the measurement device must suit the size of
the effects that will be determined. The number
of measured points must suit the shapes and
dimensions of the investigated samples that will
be visualized. To reduce the measurement
effort in the main investigation, a few pretests
are useful.
In this example, the geometrical measurements were conducted on a coordinate measuring machine (CMM).
A rotary table with tilting and centering unit was
used for the measurement. The manufacturer’s
Fig. 9
specification of one-, two-, and three-dimensional
length measurement uncertainty is (length, L,
in mm):
U1 ¼ 1:2m þ L=500m
U2 ¼ 1:5m þ L=300m
U3 ¼ 2m þ L=300m
All disks were measured before and after
case hardening to evaluate the size and shape
changes due to heat treatment. At different
heights at the inner and outer surface of the
disks the measurement program included two
and three roundness plots, respectively. In addition, four flatness scans along circles were each
measured at different radii at the top and bottom surface (Fig. 10). According to the measurement program, the following size and
shape alterations were analyzed:
Change of radius of the inner and outer sur-
face at different z-values
Gas nozzle field for quench hardening of disks. (a) Photo of gas nozzle field used. (b) Diagram of distribution
of nozzles. Source: Ref 13
Table 4 Levels of evaluated factors in the second part of investigation (heating,
carburizing, and quenching)
Level
Factor code
A
B
C
D (AB)
E (AC)
Fig. 10
Factor
–
+
Loading tool
Volume flow rate
Hardening temperature
Carburizing depth
Steel bar
Two lines
8000 l/min (2115 gal/min)
840 C (1545 F)
0.8 mm (0.031 in.)
Bar A
Three points
12,000 l/min (3170 gal/min)
940 C (1725 F)
0.6 mm (0.23 in.)
Bar B
Positions of the measurements on the investigated disk and illustration of dishing determination
396 / Distortion and Residual Stresses
Height alterations at different radii (r)
Change in roundness deviations of the inner
and outer surface at different z-values
Change in flatness deviations of the top and
bottom surface
How to Analyze Disk Distortion. A characteristic distortion of disks is the formation
of a dish at the top and bottom surface due to
the final heat treatment (see the article “Basics
of Distortion and Stress Generation during Heat
Treatment” in this Volume). This distortion
phenomenon can be described by calculation
of its slope, m (Fig. 10). To exclude a characteristic and symmetric effect at the edges of the
disks due to case hardening, average z-coordinates can be calculated for at least four measurement points of the upper and lower
surface. The slope (m) is calculated by least
squares analysis in the r,z-plane for different
angles in circumferential direction (j).
Roundness deviations can be separated by
their respective forms, such as ovality and triangularity, using Fourier analysis. This method is
described in detail in the article “Basics of Distortion and Stress Generation during Heat Treatment” in this Volume. Fourier analysis also can
be applied to measurements of other dimensions
if they were done along a circular path. This
condition is fulfilled for the flatness scans and
for the dishing slope of the disk measurements.
How to Distinguish between Statistical
Scattering and Real Effects. To identify significant factors and interactions for the distortion behavior, the results of a DOE plan can
be evaluated by applying a t-test (Ref 20).
Three confidence intervals of 95, 99, and
99.9% can be defined, belonging to the probabilities for type I error: a1 = 0.05, a2 = 0.01,
and a3 = 0.001, respectively. By comparing
the effects with the confidence intervals, they
can be classified into four significance levels:
not significant (a), indifferent (b), significant
(c), and highly significant (d).
For further interpretation of results, it is
important to distinguish between significance
and relevance. If a DOE is carried out very
carefully, effects may turn up in the range of
0.05 mm (2 min.) for dimensions in the range
of 100 mm (4 in.) to be highly significant, but
in practice they are without relevance and can
be neglected in subsequent considerations.
Test Evaluation
This section includes discussion on size
changes and shape changes.
Size Changes, Part 1—Casting, Forming,
and Machining. Releasing the accumulated
distortion potential by case hardening leads to
characteristic size changes of the disks
(Table 5). The average values of the complete
DOE are shown in Fig. 11(a). In the axial direction, the size change results in an average
height increase of 1.71% (26 mm, or 1 mil).
There is a decrease of inner and outer radius.
The average inner radius after the final heat
treatment is –1.46% (–33 mm, or –1.3 mils) less
than after the cutting process, and the average
outer radius decreases approximately –0.50%
(–30 mm, or –1.2 mils). The average volume
change is 1.02%.
The so-called effects of a parameter variation
are defined as the difference between the average value of all measurements on the plus level
of a parameter and the mean of the minus level.
Figure 11(b) shows the situation for the parameter hardenability that leads to the largest size
changes of the samples. The dashed lines represent the average size changes, as shown in
Fig. 11(a). The dotted lines indicate the results
for the two levels of hardenability.
The forging temperature causes highly significant effects for the changes of height and
outer radius. However, these effects are comparably small and therefore not relevant. The
same is true for the interaction of hardenability
(A) and forging temperature (B). The preheat
treatment causes highly significant effects
for the changes of inner and outer diameter.
Cutting strategy and feed rate and all other
interactions have either no highly significant
effect or no relevant effect (Table 5).
Size Changes, Part 2—Heating, Carburizing, and Quenching. The average tendencies
of the size changes of disks in this second part
are equal to the first part, as seen in comparison
of Fig. 11(a) and 12(a). Due to the different
heat treatment facilities used in the various
parts of investigation, the average values differ.
The variation of volume flow rate (B), hardening temperature (C), and carburizing depth
(D) influence the size changes (Table 6). The
largest effects can be determined for the variation of the carburizing depth (Fig. 12b).
Shape Changes, Part 1—Casting, Forming,
and Machining. As relevant shape changes of
the analyzed disks, the ovality of the outer surface (Ro(2)) and the dish slope (m(0)) with average values of 4.1 mm and –0.11 mm/mm were
identified (Table 7).
The result of the DOE shows several highly
significant and significant main effects as well
as interactions. Their values at the outer surface, which are comparable to those at the inner
surface, are very small (approximately 1.0 mm,
or 0.04 mil). The main change in ovality actually was caused by hanging the disks in the heat
treatment process. Creep resulting from the
deadweight of the disks (1.2 kg, or 2.6 lb)
caused an elongation parallel to the gravitation
vector.
The change of the dish slope is influenced
mainly by the cutting strategy (D), as shown
in Fig. 13. For cutting strategy 4/3, the mean
change of the dish slope is –0.40 mm/mm,
which increases to 0.19 mm/mm for cutting
strategy 1/6.
Shape Changes, Part 2—Heating, Carburizing, and Quenching. In the second part of
the investigation the most important shape
changes are the second harmonic of the roundness plot of the outer surface (Ro(2), ovality),
the dish slope (m(0)), and its second harmonic
(m(2)) (Table 8).
The loading tool factor (A) has the strongest
influence on these shape changes. Figure 14
Table 5 Effects and significance levels of factors and interactions of the relative size changes of disks (casting, forming,
and machining)
A
B
C
D
E
AB
AC
BC
DE
AD
BD
CE
CD
BE
AE
Factors and interactions (as defined in Table 3)
CDE
BDE
ADE
ABC
BCE
ACE
ABD
ABE
ACD
BCD
Change of height, DH
Mean change: 1.71%; standard deviation of effect: 0.01%
Effect, %
–0.47
–0.06
. . .(d)
. . .(d)
Significance level
0.06
. . .(d)
–0.03
. . .(a)
0.00
. . .(a)
–0.01
. . .(a)
–0.03
. . .(a)
0.01
. . .(a)
0.04
. . .(b)
0.02
. . .(a)
0.02
. . .(a)
–0.01
. . .(a)
0.00
. . .(a)
0.01
. . .(a)
Change of inner radius, DRi(0)
Mean change: –1.46%; standard deviation of effect: 0.01%
Effect, %
0.13
0.01
. . .(d)
. . .(a)
Significance level
–0.01
. . .(a)
0.29
. . .(d)
–0.01
. . .(a)
0.03
. . .(b)
–0.01
. . .(a)
–0.06
. . .(d)
0.01
. . .(a)
0.00
. . .(a)
–0.03
. . .(c)
0.02
. . .(a)
–0.01
. . .(a)
0.01
. . .(a)
0.00
. . .(a)
Change of outer radius, DRo(0)
Mean change: –0.49%; standard deviation of effect: 0.01%
Effect, %
–0.19
–0.02
. . .(d)
. . .(d)
Significance level
0.00
. . .(a)
0.11
. . .(d)
–0.01
. . .(a)
0.00
. . .(a)
0.00
. . .(a)
0.00
. . .(a)
0.00
. . .(a)
0.00
. . .(a)
–0.01
. . .(a)
0.00
. . .(a)
–0.01
. . .(a)
0.00
. . .(a)
0.01
. . .(a)
(a) Not significant. (b) Indifferent. (c) Significant. (d) Highly significant
0.04
. . .(c)
Distortion Engineering / 397
After machining
After machining
After heat treatment
After final heat treatmemt
Mean ΔH/2 = 1.60‰
Mean ΔH/2 = 1.71‰
Mean Δri = –1.01‰
Mean Δra = – 0.19‰
(a)
Mean Δri = –1.46‰
Mean Δra = – 0.49‰
D – = 0.8 mm
D + = 0.6 mm
(a)
A–: high
A+: low
Effect ΔH/2 = –0.43‰
Effect ΔH/2 = –0.47‰
Effect Δri = 0.41‰
Effect Δra = 0.15‰
(b)
Fig. 12
Effect Δri = 0.13‰
Schematic of size changes in disk cross
section. (a) Mean size changes in part 2 of
investigation. (b) Effect of carburizing depth (D). Source:
Ref 13
Effect Δra = – 0.19‰
(b)
Fig. 11
Schematic of size changes in disk cross section. (a) Mean size changes in part 1 of investigation. (b) Effect of
the hardenability (A). Source: Ref 13
shows a detailed analysis of its influence on the
second harmonic of the Fourier analysis of the
outer roundness plots (this kind of data presentation is explained in the article “Basics of Distortion and Stress Generation during Heat
Treatment” in this Volume). The loading tool
influences the amplitude and the direction of
the change of ovality. The measured ovality
complies distinctly with the alignment of the
support lines of the loading tool. Still, the
change of ovality affected by the loading tool
differs only marginally from the mean change
of ovality caused by the case-hardening procedure described in part 1 of this investigation.
No change in amplitude of the second harmonic
of the outer radius was found for the three-point
loading tool (Fig. 14).
A similar result was found for the change of
the dish slope. The loading tool (A) is the only
factor that affects the averaged dish slope.
Considering the second harmonic of the dish
slope, a more complicated situation occurs
(Fig. 15). Graphically, this can be understood
as a deformation of the surface in the shape of
a saddle. Two preferred distortion directions
of the saddle are identified in the complex
plane. The slight angular shift between the
direction of the saddle and the orientation of
the loading tool could be explained by the distortion of the loading tools itself. Again no
influence was found for the three-point loading
tool.
Table 6 Effects of factors and interactions on size changes due to the final heat treatment
(heating, carburizing, and quenching)
A
B
D
BD
AD
AB
C
E
AC
CE
Factors and interactions
(as defined in Table 4)
CDE
BCE
BDE
BCD
BC
CD
DE
BE
ACD
ABC
ABE
ADE
Change of height, DH
Mean change: 1.60%; standard deviation of effect: 0.05%
Effect, %
0.03
0.18
. . .(a)
. . .(c)
Significance level
–0.43
. . .(d)
0.03
. . .(a)
–0.07
. . .(a)
–0.01
. . .(a)
0.06
. . .(a)
Change of inner radius, DRi(0)
Mean change: –1.01%; standard deviation of effect: 0.03%
Effect, %
–0.01
0.08
. . .(a)
. . .(c)
Significance level
0.41
. . .(d)
0.24
. . .(d)
0.04
. . .(a)
–0.03
. . .(a)
–0.05
. . .(a)
Change of outer radius, DRo(0)
Mean change: –0.19%; standard deviation of effect: 0.01%
Effect, %
–0.01
0.08
. . .(a)
. . .(d)
Significance level
0.15
. . .(d)
0.18
. . .(d)
0.00
. . .(a)
–0.02
. . .(a)
0.00
. . .(a)
(a) Not significant. (b) Indifferent. (c) Significant. (d) Highly significant
Distortion Engineering, Level 2—
Identification of the DistortionRelevant Mechanisms
Table 9 summarizes the results of the previous
section and shows the relationship of the carriers
of distortion potential, the parameters and variables, and the resulting effects concerning size
and volume changes, where only the significant
and highly significant results were chosen. The
parameters of casting, forming, and machining
are indicated by a subscript number 1; parameters
of part 2 of the investigation are indicated by a subscript number 2. Effects that are quite small (1 to
2 mm, or 0.04 to 0.08 mil), and therefore not relevant for industrial praxis, are parenthesized. In
the following sections the influence of the different
carriers on distortion generation and the
corresponding mechanisms are discussed. However, first the relationship between volume and size
changes is analyzed.
398 / Distortion and Residual Stresses
Table 7 Effects and significance levels of factors and interactions of ovality changes and dish slope changes of disks (casting, forming, and
machining)
A
B
C
D
E
AB
AC
BC
DE
AD
BD
CE
CD
BE
AE
Factors and interactions (as defined in Table 3)
CDE
BDE
ADE
ABC
BCE
ACE
ABD
ABE
ACD
BCD
Change of amplitude of outer surface ovality, DRo(2)
Mean change: 4.1 mm; standard deviation of effect: 0.3 mm
Effect, mm
–1.0
–1.0
. . .(d)
. . .(d)
Significance level
1.0
. . .(d)
–0.9
. . .(c)
0.2
. . .(a)
0.4
. . .(a)
0.5
. . .(a)
–0.7
. . .(b)
0.4
. . .(a)
0.9
. . .(c)
–0.5
. . .(a)
–0.1
. . .(a)
0.5
. . .(a)
0.2
. . .(a)
–0.1
. . .(a)
Change of dish slope, Dm(0)
Mean change: –0.11 mm/mm; standard deviation of effect: 0.01 mm/mm
Effect, mm/mm
0.03
–0.06
–0.04
. . .(a)
. . .(a)
. . .(a)
Significance level
0.00
. . .(a)
–0.08
. . .(b)
–0.02
. . .(a)
0.03
. . .(a)
0.59
. . .(d)
–0.12
. . .(c)
–0.02
. . .(a)
0.06
. . .(a)
0.00
. . .(a)
–0.05
. . .(a)
0.04
. . .(a)
0.01
. . .(a)
(a) Not significant. (b) Indifferent. (c) Significant. (d) Highly significant
Table 8 Effects of factors and interactions on shape changes due to the final heat’
treatment (heating, carburizing, and quenching)
A
B
D
BD
AD
AB
C
E
AC
CE
CDE
BCE
BDE
BCD
Factors and interactions (as defined in Table 4)
Change of amplitude of outer surface ovality, DRo(2)
Mean change: 3.1 mm; standard deviation of effect: 0.2 mm
Effect, mm
–3.9
0.2
. . .(d)
. . .(a)
Significance level
Fig. 13
Dependence of the change of the dish slope
on the cutting strategy
Volume versus Size Changes
CD
DE
BE
ACD
ABC
ABE
ADE
–0.2
. . .(a)
0.0
. . .(a)
0.2
. . .(a)
–0.4
. . .(b)
0.4
. . .(b)
Change of dish slope, Dm(0)
Mean change: 0.45 mm/mm; standard deviation of effect: 0.04 mm/mm
Effect, mm/mm
–0.30
0.02
–0.07
. . .(d)
. . .(a)
. . .(a)
Significance level
–0.05
. . .(a)
0.05
. . .(a)
–0.04
. . .(a)
–0.01
. . .(a)
Change of second-order dish slope, Dm(2)
Mean change: 0.09 mm/mm; standard deviation of effect: 0.01 mm/mm
Effect, mm/mm
–0.13
0.05
–0.05
. . .(d)
. . .(d)
. . .(d)
Significance level
–0.01
. . .(a)
0.00
. . .(a)
0.00
. . .(a)
–0.01
. . .(a)
(a) Not significant. (b) Indifferent. (c) Significant. (d) Highly significant
Table 9 shows that for different parameters
of the analysis, totally different relations
between the changes of the three dimensions—
height, bore radius, and outer radius—result. If
the data for carburizing depth and hardenability
are compared, very similar height changes but
totally different changes of the inner and outer
radii are found. Therefore, the question arises:
How is the additional volume distributed to the
three dimensions—height, bore radius, and outer
radius?
In general, the volume change of a case-hardened disk can be calculated from the equation:
DV DH 2 ðRo DRo Ri DRi Þ
¼
þ
V
H
R2o R2i
BC
(Eq 1)
where V is volume, H is height, Ro is outer
radius, and Ri is inner radius.
When applying this equation, it must be kept
in mind that, in general, the size changes can
depend on position. This is shown in Fig. 13
of the article “Basics of Distortion and Stress
Generation during Heat Treatment” in this Volume. Therefore, the averaged size changes must
be used for an estimation of the volume change.
If, instead of the inner and outer radii, the
average radius (R) and the wall thickness (W)
according to:
Ro þ Ri
;
R¼
2
Ring : 0 ¼
W ¼ Ro Ri
(Eq 2)
are used, Eq 1 becomes:
DV DH DR DW
þ
¼
þ
V
H
W
R
(Eq 3)
This is the only relation between volume and
size changes of disks that is known. The values
of the three-dimensional changes depend on a
huge number of parameters. Therefore, in general, only heat treatment simulation can predict
these data.
Some simplified cases are discussed in the
following paragraphs.
Transformation-Free Quenching. In the
article “Basics of Distortion and Stress Generation during Heat Treatment” in this Volume,
the calculated size changes of cylinders, plates,
and rings without phase transformations are
shown. In this case the volume change is zero
and the following equations are valid:
Cylinder : 0 ¼
DL
DR
þ2
L
R
(Eq 4)
DH DR DW
þ þ
H
W
R
(Eq 5)
where L is the length of the cylinder.
For a cylinder, the transformation-free case
gives a simple ratio between the size changes
of the two dimensions, length and radius:
DL DR
:
¼ 2
L R
(Eq 6)
This relation is independent from other parameters. However, if a body must be described
by three or more dimensions, only equations
similar to Eq 5 can be written without knowledge of the distribution of the volume change
to the different dimensions.
Through Hardening. If a hardening
process is analyzed where only martensite is
formed, then the volume change is constant
and does not depend on the cooling conditions.
The dimensional changes of a cylinder follow
Eq 7:
DL DV DR
:
¼ 2
L
V
R
(Eq 7)
Distortion Engineering / 399
where a is thermal diffusivity, and t is time.
For case hardening, the carbon profile Cð~
rÞ
will influence the volume change, too. Therefore, the influence of additional carbon on the
volume change also must be taken into account:
DV
DH DR DW
ðBi; Fo; Cð~
r ÞÞ ¼
þ þ
V
H
W
R
(Eq 11)
Influence of the Carriers Distribution
of Alloying Elements and
Microstructure
Fig. 14
Change in amplitude of the second harmonic of the outer radius dependent on the loading tools used.
(a) Two-line loading tool. (b) Three-point loading tool. Source: Ref 13
Fig. 15
Change in amplitude of the second harmonic of the dish slope dependent on the loading tools used.
(a) Two-line loading tool. (b) Three-point loading tool. Source: Ref 13
Figure 16 shows simulation results as function of the Biot number, which is defined as:
Bia ¼
V
a O
(Eq 8)
where a is the heat transfer coefficient, la is the
heat conductivity of the phase austenite, and O
is the surface.
From Fig. 16 it can be seen that the ratio
between radius and length change is not constant. As discussed in the article “Basics of Distortion and Stress Generation during Heat
Treatment” in this Volume, the final dimensions depend on the interactions of thermal
and transformation stresses. Therefore a variation of the Biot number in combination with
the martensite formation leads to different
ratios according to Eq 7.
Hardening Processes with Time-Dependent Phase Transformations. For these conditions no systematic simulations comparable to
Fig. 16 are known. Nevertheless, the dimensional changes result even in these more complicated cases from the interaction of thermal
and transformation stresses. Therefore similar
curves will exist. The main difference is that
the volume change is no longer constant but
also depends on the cooling conditions. By
use of the Fourier number (Fo), Eq 3 for disks
can be rewritten in the following form for
non-through-hardening conditions:
DV
DH DR DW
ðBi; FoÞ ¼
þ þ
V
H
W
R
Fo ¼
a
t
R2
(Eq 9)
(Eq 10)
In most cases, the distributions of alloying
elements and microstructure act in a coupled
way during the generation of distortion. Therefore, these two carriers are discussed together.
Table 9. shows that the analyzed parameters
that act on these carriers influence mainly the
size changes. Only the cutting strategy differs
from this and produces dishing. In the following subsections the corresponding mechanisms
are analyzed.
Carburizing Depth. Table 9 shows that the
decrease of the carburization depth from 0.8
to 0.6 mm (0.031 to 0.023 in.) leads to a volume change of –0.23%. There are principally
three reasons for this result:
First, the addition of carbon in the case layer
results in an increase of the local specific volume depending on the local amount of additional carbon (Fig. 17). Consequently, the
volume of the component also must increase.
Second, the additional carbon shifts the local
continuous cooling transformation (CCT) diagram to later transformation times and reduces
the local martensite start temperature. Therefore, additional martensite in the case layer
can be formed depending on the given cooling
conditions. This effect also will increase the
local specific volume.
Finally, the reduction of the local martensite
start temperature by the increased carbon content can lead to retained austenite. This effect
reduces the volume increase caused by the
two other mechanisms.
Assessing the effect of a carburizing depth
modification must take into account these three
effects. To receive an increase of carburizing
depth, additional carbon must be inserted into
the component (Fig. 18). Consequently, the
local specific volume will increase for all
phases in regions with enlarged carbon content.
The effect of the modified microstructure
also can be calculated. However, for this the
microstructure distributions of both conditions
must be known. Depending on the amount of
additional martensite and additional retained
austenite, the effect of the increased specific
volume by additional carbon can become larger
or smaller. To estimate it, the data given by
Lement (Ref 22) or Thelning (Ref 23) can be
used. But the complete distributions of the
microstructure must be known, and the equations given in these references must be
400 / Distortion and Residual Stresses
Table 9 Relationship between the carriers of distortion potential and the significant and highly significant effects of the investigated
parameters on size and shape changes.
Values in parentheses are too small to be relevant for industrial practice. The effects of volume change were calculated by use of the effects of size changes.
Size changes
Carriers of distortion potential
DH, %
Parameter and variables
Distributions of alloying elements and microstructure
Distributions of (residual) stresses and mechanical history
Distribution of temperature
D(a): carburizing depth
A(b): hardenability
C(b): preheat treatment
E(a): steel bar
D(b): cutting strategy
E(b): feed rate
A(a): loading tool
B(b): temperature of forging
B(a): cooling rate
C(a): hardening temperature
–0.43
–0.47
...
...
...
...
...
(–0.06)
0.18
...
DRi(0), %
0.41
0.13
0.29
...
(–0.06)
...
...
...
(0.08)
0.24
Shape changes
DRo(0), %
0.15
–0.19
0.11
...
...
...
...
(–0.02)
0.08
0.18
DV, %
DRo(2), mm
Dm(0), mm/mm
–0.23
–0.96
0.14
...
...
...
...
...
0.33
0.37
...
(–1.0)
(–0.9)
...
...
...
–3.9
(–1.0)
...
...
...
...
...
...
0.59
...
–0.3
...
...
...
Dm(2), mm/mm
(–0.05)
...
–0.13
(0.05)
...
(a) Factor defined in Table 4. (b) Factor defined in Table 3
Distance to surface, mils
0.132
4.5
0.8
Martensite
4.0
3.0
ΔD/D
2.5
2.0
ΔV/(3V)
1.5
1.0
0.126
Quenching
Austenite
0.124
Heating
ΔL/L
0.122
0.5
1.0
Bia
7.8
15.7
23.6
31.4
39.3
47.2
0.4
0.6
0.8
1.0
Distance to surface, mm
1.2
0.6
CD = 0.6 mm
0.4
CD = 0.4 mm
0.2
1.5
0
Carburizing
2.0
0.120
0
Fig. 16
0
0.0
0.5
0.0
0.0
Ferrite +
cementite
Δρ–1
0.128
Carbon content, mass %
0.130
3.5
Specific volume, cm3/g
Relative dimensional change, mm/mm
5.0
Diameter and length change of throughhardened cylinders of SAE 52100. Courtesy
of B.P. Maradit. Source: Ref 21
integrated over the volume of the sample/component. This is done during heat treatment simulation. The basics of this technique are
presented in the article “Modeling and Simulation of Quenching, Residual Stress, Distortion,
and Quench Crack Formation” in this Volume.
Hardenability. The decrease of the hardenability in the investigations of level 1 leads to
a volume change of –0.96%. This was the largest effect of all investigated parameters. The
reasons for this result are mainly the modified
CCT diagram and also the space requirements
of the additional alloying elements.
Greater hardenability means that the CCT
diagram will be shifted to later times. If a casehardened component made of SAE 5120 contains ferrite and pearlite in the core, an increased
hardenability can lead to the formation of bainite
instead of ferrite and pearlite. Furthermore, the
bainite portions below the case layer can be
reduced by additional martensite formation.
The larger specific volume of bainite compared
to ferrite and pearlite in the core leads to an
enlarged space requirement (Fig. 19). Below
the case-hardening layer the additional martensite leads to the same result because of its higher
Fig. 17
0.4
0.8
1.2
1.6
Carbon content, %
0.2
Fig. 18
Carbon profiles for carburizing depths (CD) of
0.4 and 0.6 mm (15.7 and 23.6 mils)
Fig. 19
Specific volume of different microstructures.
Source: Ref 22
2.0
Changes of local specific volume by case
hardening. Source: Ref 22
specific volume compared to bainite. Because
this mechanism can act nearly over the complete
volume—only in the case layer can the effect be
neglected—it works very effectively, which
explains the large volume increase observed
here. However, if the microstructure of a component consists of 100% martensite, then the use of
a melt with higher hardenability will produce
only a small effect that depends on the modification of the specific volume by the variation of the
alloying elements.
Systematic investigations from Mallener are
provided in Ref 24. He investigated the influence of hardenability on the bore diameter of
a gear. Altogether, 100 melts of SAE 5120
(20MnCr5) were characterized by the Jominy
hardness at J10 and the resulting change of bore
diameter of the gear. Figure 20 shows that an
increased hardenability tends to reduce the bore
diameter. However, the scatter of the measured
data is large. This means that not only the transformation behavior influences the size change
of the bore. Other parameters that were changed
by the chemical composition—for example,
heat conductivity, thermal strains, or yield
strength—also have an influence.
Preheat Treatment. The modification of the
preheat treatment from a hardened and tempered state to a ferritic-pearlitic microstructure
leads to a volume enlargement of 0.14%.
The reason for this can be found in the different