# Modeling of Flow Stress of 2026 Al Alloy under Hot Compression

Authors: Zheng-bing Xiao; Yuan-chun Huang; Yu Liu
Journal: Advances in Materials Science and Engineering
 (2016)

---

## Abstract

In order to investigate the workability and to optimize the hot forming parameters for 2026 Al alloy, hot compression tests were performed in the temperature range of 350~450°C with strain rates of 0.01~10 s−1 and 60% deformation degree on a Gleeble-1500 thermosimulation machine. The true stress-strain curves obtained exhibit that the stress increases dramatically at small strains and then moves forward to a steady state, showing dynamic flow softening. Meanwhile, on the basis of Arrhenius equation, a constitutive equation on the flow stress, temperature, and strain rate was proposed. Yet, the values of the predicated stress from the equation and the true stress differ by as much as 50.10%. Given the intricate impact of precipitation of the second phases on the strength of 2026 Al alloy, the introduction of a revised equation with the reinforcement of temperature was carried out, fitting well with the experiment data at peak stresses. What is more, both pictures obtained by scanning electron microscopy (SEM) and transmission electron microscopy (TEM) were compatible with all the inferences.


---

## Body

## 1. Introduction

Understanding the materials flow behavior during hot deformation is important for materials processing designers, especially for those dealing with metal processes like hot forging, rolling, and extrusion [1–3]. Usually, constitutive equations describing relationships on the strain rate, temperature, and flow stress, which can be used in the finite element simulation, are helpful in selecting the appropriate working parameters for a given metal to process. A great deal of investigation had attempted to set up constitutive equations to predict the flow stress, especially for the frequently used alloys like aluminum alloys, steels, and nickel based super alloys [4–10]. Generally, constitutive models now used can be classified into phenomenological type, physical-based type, and artificial neural network (ANN) type [11]. The Arrhenius equation of phenomenological type was the main one for its simplicity and accuracy [6, 12–16]. Numerous papers have been published based on the equations proposed by Sellars and Zener [17–19], either the constitutive equations or the modified ones, especially in the hot working of steels, all agreeing well with experiment results, assisting in selecting the hot deformation parameters [12, 20–27].But, unlike steels, microstructures of aluminum alloys will significantly change due to precipitation in the hot deformation processing, not to mention that those can be strengthened by heat treatment, 2xxx and 7xxx alloys, for example, which usually serve as skins and frames of airplanes. Second phases, like Al2Cu in 2xxx alloys and MgZn2 in 7xxx alloys, usually appear in the hot working and heat treatment, which need substantial increase of the stress to continue the hot deformation [15, 28–32]. Prediction of the peak stress for those alloys is hard for their temperature-dependent nature, but the stress-strain relationship, especially the peak stress, is crucial for processing, since it determines the selection of the machines to fulfill the upper compressive stress needed.As an improved version of 2024 Al alloy, 2026 Al alloy with lower Fe and Si content but minor addition of Zr to inhibit recrystallization during hot working was established by Alcoa in 1999, satisfying demand for better safety and durability without sacrifice of strength and toughness [33]. As a new alloy of high strength and high damage tolerance, 2026 Al alloy is used as the upholstery skin material for lower aero foil in airplanes like A380 from hot rolling.Unlike the traditional materials, works about 2026 Al alloy are rare to find, and the most existent investigations on 2026 Al alloy mainly focus on the microstructure evolution during hot processing or the precipitation of the second phases [33, 34]. To the authors’ knowledge, there is little information available in the literature about flow behavior of 2026 Al alloy; further study is needed to perform numerical simulation and to select proper process parameters. This paper will put forward and verify a new constitutive equation about the flow behavior of 2026 Al alloy under hot deformation.


## 2. Experiments

A commercial 2026 Al alloy of chemical composition (wt.%) 1.33Mg-4.20Cu-0.55Mn-0.15Zr-(bal.)Al, with minor elements like Si and Fe below 0.05 and 0.07 (wt.%), respectively, was used in the investigation. Specimens were prepared with diameter of 10 mm and height of 15 mm from the direct cast ingot homogenized for 490°C × 12 h + 520°C × 12 h. Friction between the specimens and upsetting dies was minimized by entrapping the lubricant with composition of 75% graphite + 20% machine oil + 5% nitric acid trimethyl benzene grease in a depth of 0.1 mm recessed into both ends of the specimens.As depicted in Figure1, the specimens were heated to 480°C at a heating rate of 2°C/s and held for 6 min and then cooled to the deformation temperature at 10°C/s and held for 4 min before test. The hot compression tests were carried out on a Gleeble-1500 thermosimulation machine at four different strain rates (0.01 s−1, 0.1 s−1, 1 s−1, and 10 s−1) and four different temperatures (300°C, 350°C, 400°C, and 450°C), with 60% final reduction in height. After the hot compression, specimens were quenched in water immediately for micrographic observation.Figure 1
Experimental procedure for hot compression tests.


## 3. Results and Discussion

### 3.1. True Stress and Strain

The true stress-strain curves obtained from hot compression tests for 2026 Al alloy are shown in Figure2. For aluminum alloys of high stacking fault energy, dynamic recovery is the main reason for the softening phenomenon [35], which is confirmed by the occurrence of dynamic softening on the curves in this work, for the flow stresses increase monotonically up to an almost steady state (e.g., samples deformed at strain rate of 0.01 s−1). As can be seen from the figures, the strain rate and temperature significantly affect the flow stresses. The true stress level will rise once deformation temperature decreases and strain rate increases.Figure 2
Typical true stress-strain curves for 2026 Al alloy under different deformation temperatures with strain rates of (a) 0.01 s−1; (b) 0.1 s−1; (c) 1 s−1; and (d) 10 s−1.
(a)
(b)
(c)
(d)Under low temperatures, restraint of the number of slip systems reduces the mobility at grain boundaries, thus increasing the flow stress. This was approved further by the dislocation walls in Figure3, for the TEM observation of the sample at 300°C/10 s−1, which shows the highest flow stress during compression.Figure 3
TEM micrograph of the sample deformed at temperature of 300°C with the strain rate of 10 s−1 with dislocation walls.By raising the deformation temperature, dislocation slip, climb, and annihilation correspondingly increase, together with vacancy diffusion rate, resulting in easier activation of dynamic recovery and even dynamic recrystallization to overcome the hardening. Figure4 shows, for the sample deformed at 0.01 s−1/450°C, subgrains formed at grain boundaries, evidence of dynamic recrystallization, which can greatly contribute to the decrease of the flow stress. However, with the increase of strain rate, the time for energy accumulation, dynamic recovery, and recrystallization is shortened, and the true stress would increase.Figure 4
TEM micrograph of the sample deformed at temperature of 450°C with the strain rate of 0.01 s−1.


### 3.2. Deformation Constitutive Equation

Usually, the relationship regarding flow stress, deformation temperature, and strain rate can be expressed by Arrhenius equation. Zener-Hollomon parameterZ in an exponent-type equation represents the effects of strain rate and temperature on the deformation behaviors. Consequently, Arrhenius equation can be used to connect Z parameter with stress [18, 19]. Hence,(1)
Z
=
ε
˙
e
x
p
Q
R
T
,
(2)
ε
˙
=
A
F
σ
e
x
p
-
Q
R
T
,where(3)
F
σ
=
σ
n
′
α
σ
<
0.8
e
x
p
β
σ
α
σ
>
1.2
s
i
n
h
α
σ
n
∀
σ
,in which ε
˙ is the stain rate (s−1),Q is the activation energy of hot deformation (kJmol−1),R is the universal gas constant (8.31 Jmol−1K−1),T is the absolute temperature (K), and σ is the flow stress (MPa) for a given strain, whileA, α, and n, n
′ are the material constants, α
=
β
/
n
′.Equations (4) and (4) are the substitution of the power law and exponential law of F
(
σ
) into (1) for the low-stress (α
σ
<
0.8) and high-stress level (α
σ
>
1.2), respectively:(4)
ε
˙
=
B
σ
n
′
,
(5)
ε
˙
=
B
′
e
x
p
β
σ
1
2
,where B and B
′ are the material constants. To find out the material constants in the above equations, deformation strain of peak stress is used here as an example.Equations (6) and (6) are the logarithm form of (4) and (4), respectively: (6)
l
n
σ
=
1
n
′
l
n
ε
˙
-
1
n
′
l
n
B
,
(7)
σ
=
1
β
l
n
ε
˙
-
1
β
l
n
B
′
.Substituting the values of the peak stress and corresponding strain rate at different deformation temperatures into (6) and (6) can obtain the relationship regarding the stress and strain rate (in logarithm form) shown in Figure 5.Figure 5
Relationships between strain rate and peak stresses: (a)l
n
ε
˙ and l
n
σ; (b) l
n
ε
˙ and σ.
(a)
(b)As can be seen in Figure5, a group of straight lines approximated is close to parallel. The slopes of these lines have similar values and values of n
′ and β at different deformation temperatures can be derived by linear fitting method. As α
=
β
/
n
′, each corresponding value of α can be calculated. By average, the mean value of α is 8.0128 × 10−3.Equation (1) can be expressed as follows, fit for all the stress level σ:(8)
ε
˙
=
A
s
i
n
h
α
σ
n
e
x
p
-
Q
R
T
.Taking the logarithm of (8) for both sides gives(9)
l
n
s
i
n
h
α
σ
=
l
n
ε
˙
n
+
Q
n
R
T
-
l
n
A
n
.Substituting the peak stress and relevant strain rate for all the tested temperatures into (9) gives the relationship shown in Figure 6. It is not hard to obtain the value ofn, 9.343.Figure 6
Relationship between ln[sinh(ασ)] and l
n
ε
˙.For a fixed strain rate, derivative of (8) gives(10)
Q
=
R
n
d
l
n
s
i
n
h
α
σ
d
1
/
T
.According to (10), l
n
s
i
n
h
α
σ can be regarded as a function of 1
/
T, shown in Figure 7, and the activation energy can be computed as 229.546 kJ/mol by averaging the values of Q at four different strain rates.Figure 7
Relationship between ln[sinh(ασ)] and temperature.On the basis of the value ofn and Q known, it is easy to get the value ofA, 5.17 × 1022 s−1, 9.53 × 1022 s−1, 8.3 × 1022 s−1, and 3.63 × 1022 s−1 for 300°C, 350°C, 400°C, and 450°C, respectively.Then, (8) can be expressed below by substituting the values obtained above:(11)
ε
˙
=
A
s
i
n
h
8.0128
×
1
0
-
3
σ
9.343
e
x
p
-
2.29546
×
1
0
5
R
T
.Finally, Zener-Hollomon parameterZ in (1) can be represented as follows:(12)
Z
=
ε
˙
e
x
p
2.29546
×
1
0
5
R
T
=
A
s
i
n
h
8.0128
×
1
0
-
3
σ
9.343
.According to (8) and (1), relationships between the flow stress σ and Zener-Hollomon parameter Z can be described below:(13)
σ
=
1
α
l
n
Z
A
1
/
n
+
Z
A
2
/
n
+
1
1
/
2
.Or(14)
σ
=
124.8
l
n
Z
A
1
/
9.343
+
Z
A
2
/
9.343
+
1
1
/
2
.Depending on the deformation temperature, the constantA in (11) to (14) is 5.17 × 1022 s−1, 9.53 × 1022 s−1, 8.3 × 1022 s−1, and 3.63 × 1022 s−1 for 300°C, 350°C, 400°C, and 450°C, respectively.


### 3.3. Modification of Zener-Hollomon Parameter

The above-developed constitutive equations for 2026 Al alloy at peak stress state were verified by comparison of the values from experiments and constitutive equation; differences between them are shown in Table1. In the table, error is defined as(15)
E
r
r
o
r
=
σ
c
-
σ
m
σ
m
×
100
%
.
σ
c is the stress calculated by the constitutive equation, and σ
m is the peak stress measured by experiments.Table 1
Comparisons of the predicated (σ
P) and measured (σ
E) peak stress (unit: MPa) before the modification of Z parameter.
Strain rate (s−1)
Temperature (°C)
σ
E
σ
P
Error analysis
Error (%)
Mean (%)
Standard deviation (%)
0.01
300
160
239.18
49.49
29.20
19.42
350
115
126.34
9.86
400
80
67.53
15.58
450
72
41.85
41.87
0.1
300
191
277.79
45.44
27.49
15.24
350
132
150.90
14.32
400
100
84.64
15.36
450
83
54.09
34.83
1
300
228
318.32
39.61
25.35
13.74
350
160
177.83
11.14
400
125
104.57
16.34
450
105
68.99
34.30
10
300
240
360.24
50.10
26.83
17.67
350
186
206.67
11.11
400
150
127.03
15.31
450
125
86.51
30.79Apparently, errors between the experiment and predicated results range from 9.86% to 50.10%, indicating the need to modify the constitutive equation.Usually, four different stages, namely, work hardening stage (stage 1), stable stage (stage 2), softening stage (stage 3), and steady stage (stage 4), show up on the true strain-true stress curves of typical steels, which can be explained well from the viewpoint of dynamic recovery and dynamic recrystallization under hot deformation [36, 37]. However, for almost all of the Al alloys, precipitation of the second phases inevitably occurs in the hot deformation procedure; 2026 Al alloy is no exception. From Figure 8, the SEM and TEM microstructures of the sample deformed at 350°C with the strain rate of 1 s−1 and the final strain of 0.6; the matrix was occupied by numerous tiny white particles (most of them are Al2Cu), a combining effect of temperature and strain whose influence on flow stress cannot be neglected.Figure 8
Microstructure after hot deformation at true strain of 0.6 with the deformation temperature of 350°C and strain rate of 1 s−1: (a) backscattered electron image and (b) TEM micrograph.
(a)
(b)Theoretically, by hindering recrystallization or being coherent withα-Al, precipitation can enhance the material strength to some degree, reflected by the rise of the true stress level on the true stress-true strain curve [33, 34]. Typically speaking, precipitation itself is a complicated factor whose impact on the true stress is relevant to temperature. Therefore, the influence of temperature on the flow stresses should be enforced here for the nature of 2026 Al alloy.Through numerous repeated calculations and validation, it is found that, to make theZ parameter better conform to the real, (1) should multiply with e
x
p
(
22.93975
X
-
135.508
). So, the revised Z
′ parameter can be expressed as (16)
Z
′
=
e
x
p
22.93975
X
-
135.508
ε
˙
e
x
p
Q
R
T
,where factorX represents the deformation temperature in °C; it represents the influences of the temperature on the precipitation behavior of the alloy.Therefore, the flow stress constitutive equation expressed as (13) can be renewed as(17)
σ
=
1
α
l
n
Z
′
A
1
/
n
+
Z
′
A
2
/
n
+
1
1
/
2
.Parameters ofα,n, and A in the equation are not changed as before, while Z is the modified one from (16).


### 3.4. Verification of the Revised Constitutive Equation

Table2 shows the comparisons between the peak stresses calculated from the revised constitutive equation and the experiment data.Table 2
Comparisons of the predicated (σ
P) and measured (σ
E) peak stress (unit: MPa) after the modification of Z parameter.
Strain rate (s−1)
Temperature (°C)
σ
E
σ
P
Error analysis
Error (%)
Mean (%)
Standard deviation (%)
0.01
300
160
169.37
5.86
3.03
2.4
350
115
115.28
0.24
400
80
81.72
2.15
450
72
69.22
3.87
0.1
300
191
202.22
5.87
4.14
2.03
350
132
138.54
4.96
400
100
101.21
1.21
450
83
86.76
4.53
1
300
228
238.21
4.48
2.53
1.44
350
160
164.37
2.73
400
125
123.29
1.37
450
105
106.60
1.52
10
300
240
246.77
2.82
2.62
0.72
350
186
192.33
3.40
400
150
147.50
1.66
450
125
128.22
2.58As observed from the values, agreement between the calculated and measured values is good, and the maximum relative error is only 5.87%. The results indicated that the modifiedZ parameter can give a good estimate of the peak stresses for 2026 Al alloy under different hot compression temperature and strain rate and can be used in guidance of the hot processing of 2026 Al alloy.


## 3.1. True Stress and Strain

The true stress-strain curves obtained from hot compression tests for 2026 Al alloy are shown in Figure2. For aluminum alloys of high stacking fault energy, dynamic recovery is the main reason for the softening phenomenon [35], which is confirmed by the occurrence of dynamic softening on the curves in this work, for the flow stresses increase monotonically up to an almost steady state (e.g., samples deformed at strain rate of 0.01 s−1). As can be seen from the figures, the strain rate and temperature significantly affect the flow stresses. The true stress level will rise once deformation temperature decreases and strain rate increases.Figure 2
Typical true stress-strain curves for 2026 Al alloy under different deformation temperatures with strain rates of (a) 0.01 s−1; (b) 0.1 s−1; (c) 1 s−1; and (d) 10 s−1.
(a)
(b)
(c)
(d)Under low temperatures, restraint of the number of slip systems reduces the mobility at grain boundaries, thus increasing the flow stress. This was approved further by the dislocation walls in Figure3, for the TEM observation of the sample at 300°C/10 s−1, which shows the highest flow stress during compression.Figure 3
TEM micrograph of the sample deformed at temperature of 300°C with the strain rate of 10 s−1 with dislocation walls.By raising the deformation temperature, dislocation slip, climb, and annihilation correspondingly increase, together with vacancy diffusion rate, resulting in easier activation of dynamic recovery and even dynamic recrystallization to overcome the hardening. Figure4 shows, for the sample deformed at 0.01 s−1/450°C, subgrains formed at grain boundaries, evidence of dynamic recrystallization, which can greatly contribute to the decrease of the flow stress. However, with the increase of strain rate, the time for energy accumulation, dynamic recovery, and recrystallization is shortened, and the true stress would increase.Figure 4
TEM micrograph of the sample deformed at temperature of 450°C with the strain rate of 0.01 s−1.


## 3.2. Deformation Constitutive Equation

Usually, the relationship regarding flow stress, deformation temperature, and strain rate can be expressed by Arrhenius equation. Zener-Hollomon parameterZ in an exponent-type equation represents the effects of strain rate and temperature on the deformation behaviors. Consequently, Arrhenius equation can be used to connect Z parameter with stress [18, 19]. Hence,(1)
Z
=
ε
˙
e
x
p
Q
R
T
,
(2)
ε
˙
=
A
F
σ
e
x
p
-
Q
R
T
,where(3)
F
σ
=
σ
n
′
α
σ
<
0.8
e
x
p
β
σ
α
σ
>
1.2
s
i
n
h
α
σ
n
∀
σ
,in which ε
˙ is the stain rate (s−1),Q is the activation energy of hot deformation (kJmol−1),R is the universal gas constant (8.31 Jmol−1K−1),T is the absolute temperature (K), and σ is the flow stress (MPa) for a given strain, whileA, α, and n, n
′ are the material constants, α
=
β
/
n
′.Equations (4) and (4) are the substitution of the power law and exponential law of F
(
σ
) into (1) for the low-stress (α
σ
<
0.8) and high-stress level (α
σ
>
1.2), respectively:(4)
ε
˙
=
B
σ
n
′
,
(5)
ε
˙
=
B
′
e
x
p
β
σ
1
2
,where B and B
′ are the material constants. To find out the material constants in the above equations, deformation strain of peak stress is used here as an example.Equations (6) and (6) are the logarithm form of (4) and (4), respectively: (6)
l
n
σ
=
1
n
′
l
n
ε
˙
-
1
n
′
l
n
B
,
(7)
σ
=
1
β
l
n
ε
˙
-
1
β
l
n
B
′
.Substituting the values of the peak stress and corresponding strain rate at different deformation temperatures into (6) and (6) can obtain the relationship regarding the stress and strain rate (in logarithm form) shown in Figure 5.Figure 5
Relationships between strain rate and peak stresses: (a)l
n
ε
˙ and l
n
σ; (b) l
n
ε
˙ and σ.
(a)
(b)As can be seen in Figure5, a group of straight lines approximated is close to parallel. The slopes of these lines have similar values and values of n
′ and β at different deformation temperatures can be derived by linear fitting method. As α
=
β
/
n
′, each corresponding value of α can be calculated. By average, the mean value of α is 8.0128 × 10−3.Equation (1) can be expressed as follows, fit for all the stress level σ:(8)
ε
˙
=
A
s
i
n
h
α
σ
n
e
x
p
-
Q
R
T
.Taking the logarithm of (8) for both sides gives(9)
l
n
s
i
n
h
α
σ
=
l
n
ε
˙
n
+
Q
n
R
T
-
l
n
A
n
.Substituting the peak stress and relevant strain rate for all the tested temperatures into (9) gives the relationship shown in Figure 6. It is not hard to obtain the value ofn, 9.343.Figure 6
Relationship between ln[sinh(ασ)] and l
n
ε
˙.For a fixed strain rate, derivative of (8) gives(10)
Q
=
R
n
d
l
n
s
i
n
h
α
σ
d
1
/
T
.According to (10), l
n
s
i
n
h
α
σ can be regarded as a function of 1
/
T, shown in Figure 7, and the activation energy can be computed as 229.546 kJ/mol by averaging the values of Q at four different strain rates.Figure 7
Relationship between ln[sinh(ασ)] and temperature.On the basis of the value ofn and Q known, it is easy to get the value ofA, 5.17 × 1022 s−1, 9.53 × 1022 s−1, 8.3 × 1022 s−1, and 3.63 × 1022 s−1 for 300°C, 350°C, 400°C, and 450°C, respectively.Then, (8) can be expressed below by substituting the values obtained above:(11)
ε
˙
=
A
s
i
n
h
8.0128
×
1
0
-
3
σ
9.343
e
x
p
-
2.29546
×
1
0
5
R
T
.Finally, Zener-Hollomon parameterZ in (1) can be represented as follows:(12)
Z
=
ε
˙
e
x
p
2.29546
×
1
0
5
R
T
=
A
s
i
n
h
8.0128
×
1
0
-
3
σ
9.343
.According to (8) and (1), relationships between the flow stress σ and Zener-Hollomon parameter Z can be described below:(13)
σ
=
1
α
l
n
Z
A
1
/
n
+
Z
A
2
/
n
+
1
1
/
2
.Or(14)
σ
=
124.8
l
n
Z
A
1
/
9.343
+
Z
A
2
/
9.343
+
1
1
/
2
.Depending on the deformation temperature, the constantA in (11) to (14) is 5.17 × 1022 s−1, 9.53 × 1022 s−1, 8.3 × 1022 s−1, and 3.63 × 1022 s−1 for 300°C, 350°C, 400°C, and 450°C, respectively.


## 3.3. Modification of Zener-Hollomon Parameter

The above-developed constitutive equations for 2026 Al alloy at peak stress state were verified by comparison of the values from experiments and constitutive equation; differences between them are shown in Table1. In the table, error is defined as(15)
E
r
r
o
r
=
σ
c
-
σ
m
σ
m
×
100
%
.
σ
c is the stress calculated by the constitutive equation, and σ
m is the peak stress measured by experiments.Table 1
Comparisons of the predicated (σ
P) and measured (σ
E) peak stress (unit: MPa) before the modification of Z parameter.
Strain rate (s−1)
Temperature (°C)
σ
E
σ
P
Error analysis
Error (%)
Mean (%)
Standard deviation (%)
0.01
300
160
239.18
49.49
29.20
19.42
350
115
126.34
9.86
400
80
67.53
15.58
450
72
41.85
41.87
0.1
300
191
277.79
45.44
27.49
15.24
350
132
150.90
14.32
400
100
84.64
15.36
450
83
54.09
34.83
1
300
228
318.32
39.61
25.35
13.74
350
160
177.83
11.14
400
125
104.57
16.34
450
105
68.99
34.30
10
300
240
360.24
50.10
26.83
17.67
350
186
206.67
11.11
400
150
127.03
15.31
450
125
86.51
30.79Apparently, errors between the experiment and predicated results range from 9.86% to 50.10%, indicating the need to modify the constitutive equation.Usually, four different stages, namely, work hardening stage (stage 1), stable stage (stage 2), softening stage (stage 3), and steady stage (stage 4), show up on the true strain-true stress curves of typical steels, which can be explained well from the viewpoint of dynamic recovery and dynamic recrystallization under hot deformation [36, 37]. However, for almost all of the Al alloys, precipitation of the second phases inevitably occurs in the hot deformation procedure; 2026 Al alloy is no exception. From Figure 8, the SEM and TEM microstructures of the sample deformed at 350°C with the strain rate of 1 s−1 and the final strain of 0.6; the matrix was occupied by numerous tiny white particles (most of them are Al2Cu), a combining effect of temperature and strain whose influence on flow stress cannot be neglected.Figure 8
Microstructure after hot deformation at true strain of 0.6 with the deformation temperature of 350°C and strain rate of 1 s−1: (a) backscattered electron image and (b) TEM micrograph.
(a)
(b)Theoretically, by hindering recrystallization or being coherent withα-Al, precipitation can enhance the material strength to some degree, reflected by the rise of the true stress level on the true stress-true strain curve [33, 34]. Typically speaking, precipitation itself is a complicated factor whose impact on the true stress is relevant to temperature. Therefore, the influence of temperature on the flow stresses should be enforced here for the nature of 2026 Al alloy.Through numerous repeated calculations and validation, it is found that, to make theZ parameter better conform to the real, (1) should multiply with e
x
p
(
22.93975
X
-
135.508
). So, the revised Z
′ parameter can be expressed as (16)
Z
′
=
e
x
p
22.93975
X
-
135.508
ε
˙
e
x
p
Q
R
T
,where factorX represents the deformation temperature in °C; it represents the influences of the temperature on the precipitation behavior of the alloy.Therefore, the flow stress constitutive equation expressed as (13) can be renewed as(17)
σ
=
1
α
l
n
Z
′
A
1
/
n
+
Z
′
A
2
/
n
+
1
1
/
2
.Parameters ofα,n, and A in the equation are not changed as before, while Z is the modified one from (16).


## 3.4. Verification of the Revised Constitutive Equation

Table2 shows the comparisons between the peak stresses calculated from the revised constitutive equation and the experiment data.Table 2
Comparisons of the predicated (σ
P) and measured (σ
E) peak stress (unit: MPa) after the modification of Z parameter.
Strain rate (s−1)
Temperature (°C)
σ
E
σ
P
Error analysis
Error (%)
Mean (%)
Standard deviation (%)
0.01
300
160
169.37
5.86
3.03
2.4
350
115
115.28
0.24
400
80
81.72
2.15
450
72
69.22
3.87
0.1
300
191
202.22
5.87
4.14
2.03
350
132
138.54
4.96
400
100
101.21
1.21
450
83
86.76
4.53
1
300
228
238.21
4.48
2.53
1.44
350
160
164.37
2.73
400
125
123.29
1.37
450
105
106.60
1.52
10
300
240
246.77
2.82
2.62
0.72
350
186
192.33
3.40
400
150
147.50
1.66
450
125
128.22
2.58As observed from the values, agreement between the calculated and measured values is good, and the maximum relative error is only 5.87%. The results indicated that the modifiedZ parameter can give a good estimate of the peak stresses for 2026 Al alloy under different hot compression temperature and strain rate and can be used in guidance of the hot processing of 2026 Al alloy.


## 4. Conclusions

In this study, the effects of deformation parameters on the true stress-strain behavior of 2026 Al alloy were investigated. According to the experiment data, a constitutive equation incorporating the effects of strain rates and temperatures is obtained, while the errors between the experiment and predicated results range from 9.86% to 50.10%. So a revised constitutive equation is derived by compensation of temperature considering the impact of the second phase on the true stress; a good agreement between the predicated and the experiment results indicated that the revised constitutive equation can give an accurate prediction of the peak stress for 2026 Al alloy and can be used in the hot processing of the 2026 Al alloy. In addition, pictures about the microstructures were obtained using scanning electron microscopy (SEM) and transmission electron microscopy (TEM) for verification and assistance.



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*Source: 3803472-2016-08-11.xml*