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Numerical simulation of tensile
deformation and failure of aluminium
alloys exposed to laser heating
Zewen Li, Mohsan Jelani, Zhonghua Shen, Xiaowu Ni,
Jianmin Zhang
Zewen Li, Mohsan Jelani, Zhonghua Shen, Xiaowu Ni, Jianmin Zhang,
"Numerical simulation of tensile deformation and failure of aluminium alloys
exposed to laser heating
," Proc. SPIE 11046, Fifth International Symposium on Laser Interaction with
Matter, 110461T (29 March 2019); doi: 10.1117/12.2524439
Event: Fifth International Symposium on Laser Interaction with Matter, 2018,
Changsha, China
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Numerical simulation of tensile deformation and failure of aluminium
alloys exposed to laser heating
Zewen Li[1], Mohsan Jelani[2]∗, Zhonghua Shen[1] [2], Xiaowu Ni[1] [2] and Zhang Jianmin[3]
[1]Advanced Launching Co-innovation Centre, Nanjing University of Science and Technology,
Nanjing, 210094, P. R. China
[2]School of Science, Nanjing University of Science and Technology, Nanjing, 210094, P. R. China
[3] State Key Laboratory of Laser Interaction with Matter, Northwest Institute of Nuclear
Technology, Xi’an 710024, China
Abstract
A 3D mathematical model was established for the investigation of the thermomechanical behavior of aluminum
alloys (Al-7075) under the combined action of tensile loading and laser irradiations. The transient temperature
fields and stress-strain field was obtained by using the finite element method. The Johnson–Cook’s constitutive
equation is implemented in the FE model to study the failure behaviour of alloy. The effects of various pre-loading
and laser power densities on the failure time, temperature distribution and the deformation behavior of aluminum
alloys are analyzed. The results indicate the significant reduction in failure time for higher laser power densities
and for high preloading values, which implies that preloading may contribute a significant role in the failure of the
material at elevated temperature. The numerical result agrees well with our previous experiment results, concluding
that the numerical model is reasonable.
Keyword: Aluminum alloys, fiber laser, tensile loading, thermomechanical effects, failure evolution
∗
Corresponding author: Tel/Fax:+86-25-84315687, Email: jelani373@gmail.com
Fifth International Symposium on Laser Interaction with Matter, edited by Yijun Zhao, Proc. of SPIE
Vol. 11046, 110461T · © 2019 SPIE · CCC code: 0277-786X/19/$18 · doi: 10.1117/12.2524439
Proc. of SPIE Vol. 11046 110461T-1
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1 INTRODUCTION
Aluminum alloys, because of their high strength to weight ratio, have been used as the primary structural material
for the automotive and aeronautic industries[1,2]. In aerospace engineering, various structural components may be
subjected to severe thermal loading which may be originated by aerodynamic heating, laser irradiation or by local
intense fire. The aluminum alloy's applications in preloaded environment demand for a more detailed investigation to its
thermomechanical behavior for determining the safety of materials.
High energy, continuous wave lasers are a proficient source of varying levels of machining or damaging to
structural materials. Studies about laser damage effect of aluminum were carried out frequently during past several
decades. However, the research about laser damage effect of aluminum under preloaded conditions is infrequent and
considerable. Medford et al.[3] had developed an analytical model to determine failure threshold and to predict the
thermal and structural response of aluminum alloys under combined effects of laser beam exposure and mechanical
loading. The specimens were irradiated by (CW) CO2 laser under tension or compression in the presence of tangential
subsonic flow. It was observed that combined laser exposure and mechanical loading significantly reduced the room
temperature tensile strength and threshold energy. Yang et al. [4] studied experimentally CW laser damage effect on steel
under preloaded invariable stretching stress. The 30CrMnSiA steel sample was preloaded with invariable stretching force
and then irradiated by YAG laser. An empirical formula was derived which relates the stretching stress and rupture
temperature. Moreover, a decrease in threshold rupture laser energy with the increase of stretching stress or laser power
density was reported. Long et al. [5] had presented the effects of laser power density, pre-loading and the thickness on the
failure time of the of the carbon fibre/epoxy composite laminates subjected to CW CO2 laser. Florando et al.
[6]
had
experimentally and numerically investigated the thermalmechanical behavior of 7075-T6 aluminum irradiated by CW
laser under a constant loading and found that as the samples are held under a constant load, the heating from the laser
profile causes non-uniform temperature and strain fields, and the strain-rate increases dramatically as the sample nears
failure.
In this paper, A 3D mathematical model was established for the investigation of the thermomechanical behavior of
aluminum alloys (Al-7075) under the combined action of tensile loadings and laser irradiations. The transient
temperature fields and stress-strain field was obtained by using the finite element method. The Johnson–Cook’s
constitutive equation is implemented in the FE model to study the failure behaviour of alloy. The effects of various
pre-loading and laser power densities on the failure time, temperature distribution and on the deformation behavior of
aluminum alloys are analyzed.
2 Mathematical model
2.1 The finite element model
The incident laser beam is aligned perpendicularly to the center area of 7075 alloy and assumed to be Gaussian
distribution, the tensile is loading on two edged sections of sheet. A three-dimensional finite element model is established
as shown in Fig. 1. The demension of the alloy sheet is 100×10×2(mm). The laser spot is located at the center of the
sheet, and radius is 3.5mm.
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Fig.1 3D schematic diagram for CW laser irradiated AL7075 under loading
2.2 Thermal model
The three-dimension thermal field of alloy sheet, T ( x, y, z, t ) , has been computed by solving the Heat Conduction
Differential Equation:
ρc
∂
∂
∂T
∂
∂T
∂
∂T
T=
(k
) + (k
)+ ( k
)
∂x ∂x
∂y ∂y ∂z ∂z
dt
(1)
where ρ , c and k are the density, heat capacity and thermal conductive coefficient.
The heat source from the laser was simulated as a surface heat flux boundary condition, where the heat flux is
proportional to the laser power denisity times the absorptivity:
−k
∂T
∂ns
= α I 0 f ( x, y )
(2)
Z =0
Meanwhile, the local convection and thermal radiation is gaven in:
−k
∂T
= h(T − T0 ) + σε (T 4 − T04 )
∂ns
(3)
The initial condition is:
∂T t = 0 = T0
(4)
Where T0 is the ambient temperature and it is taken as 298 K. Where is I0 the mean power density of incident laser at
the spot center, and α , h , ε represent the absorptivity, convetion coefficient and radiation emissivity of 7075 alloy,
respectively, σ is Stefan–Boltzmann constant, and f ( X , Y ) is the spatial distribution of the laser, for TEM00 mode
laser, it can be written as:
f ( X , Y ) = 2 exp( −2
I0 =
x2 + y 2
)
r02
Pi
π r02
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(5)
(6)
Where r0 is beam radius, its value is 3.5mm, Pi is the incident laser power.
2.3 Thermo-elastic constitutive theory
Materials expand or contract with temperature changes; therefore thermal strain depends on the present temperature
and the initial temperature, which is independent of stress. Generalized Hooke's law and strain-displacement relations:
1
⎡σ x −ν (σ y + σ z ) ⎤ + αΔT ⎫
⎪
⎣
⎦
E
⎪
1
ε y = ⎡⎣σ y −ν (σ z + σ x ) ⎤⎦ + αΔT ⎪
⎪⎪
E
⎬
1
ε z = ⎡⎣σ z −ν (σ x + σ y ) ⎤⎦ + αΔT ⎪
⎪
E
τ xy
τ yz
τ zx ⎪⎪
, γ yz =
, γ zx =
γ xy =
G
G
G ⎪⎭
εx =
(7)
Where ΔT =T − T0 indicates the change of temperature, T and T0 represent the present temperature and the initial
temperature. As the stress is a linear function of strain, the generalized stress (σ)–strain (ɛ) equation for linear elastic
materials is:
⎡σ xx ⎤ ⎡λ +2μ
⎢σ ⎥ ⎢
⎢ yy ⎥ ⎢ λ
⎢σ zz ⎥ ⎢ λ
⎢ ⎥=⎢
⎢σ yz ⎥ ⎢ 0
⎢σ zx ⎥ ⎢ 0
⎢ ⎥ ⎢
⎢⎣σ xy ⎥⎦ ⎢⎣ 0
λ
λ
λ
0
0
0
0
0
0
μ
λ +2μ
λ
λ +2μ
0
0
0
0
0
The shear modulus G and the Lame constants λ and μ
0 ⎤ ⎡ε xx − αT ΔT ⎤
⎢
⎥
0 ⎥⎥ ⎢ε yy − α T ΔT ⎥
0 ⎥ ⎢ε xx − α T ΔT ⎥
(8)
⎥
⎥⎢
0 ⎥ ⎢ 2ε yz
⎥
μ 0 ⎥ ⎢ 2ε zx ⎥
⎥
⎥⎢
0 μ ⎥⎦ ⎢⎣ 2ε xy
⎥⎦
are related to the Young's modulus E and Poisson's
0
0
0
0
ratio ν as:
λ=
Eν
E
；G=
=μ
(1 + ν )(1 − 2ν )
2(1 + ν )
(9)
2.4 The Johnson–Cook plasticity and damage model
The stress–strain behavior of the AL-7075 is described using the empirical Johnson–Cook flow strength model[6] for
this study which is fit for common aluminum alloys, where the flow stress σ flow can be described in the following
manner:
σ flow =(A+Bε n )[1+Cln(
ε&
)](1-T *m )+C p p
ε&0
(10)
Where
T* =
T − T0
Tmelt − T0
(11)
The parameter A is nominally the yield strength at the reference strain rate ε&0 , B and n are the work hardening
parameters, C represents the strain rate dependence, m is the temperature dependence, Cp is the pressure dependence,
Tmelt is the melt temperature, and T0 is the reference temperature, taken as the ambient room temperature. The main
assumptions with the model are that the strain rate dependence will follow linearly with the log of the strain rate, and the
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temperature dependence will scale in a power law manner with T*.
A Johnson–Cook failure model is used to describe the failure of the material, where the strain to failure can be
described by
ε f =(D0 +D D exp(D3σ * ))(1+D 4 ln(
ε&
))(1+D5T * )
&
ε0
(12)
Where σ * is the stress triaxiality and is defined as
σ* =
σ mean
σ Von _ Mises
(13)
The mean stress is the average of the three principal stresses
σ mean =
σ1 + σ 2 + σ 3
3
(14)
and the Von Mises stress is
(σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 1 − σ 3 ) 2
(15)
2
While empirical in nature, the physics behind the Johnson–Cook damage model is that the failure strain will depend on
σ Von _ Mises =
the stress triaxiality σ * in an exponential manner, modified by the constants DD and D3. The term D0 represents the
void nucleation term, and the model also has terms D4 and D5 that account for the strain rate and temperature
dependence of the failure strain, respectively. Damage in a material element is governed by the element damage
parameter that follows the equation:
Delement = ∑
Δε
(16)
εf
Where Δε is the incremental accumulation of strain in the element, and is εf the failure strain. The material element is
determined to have failed when the damage parameter in the element reaches unity. Once the failure criterion is reached,
the code erodes the element, which simulates the generation and coalescences of voids. And the material is determined to
have totally failed when the minimum of the damage parameter in the cross section along the Z direction reaches unity.
Dtotal = Min(∑
Δε
εf
)
(17)
cross − sec tion
3 Results and Discussion
In present study, we calculate the temperature field and stress-strain field of AL-7075 in these situations, laser
power density is 1500, 1900, 2300 and 2700W/cm2 combine a constant preloading under 242, 333, 423 and 514MPa.
3.1 Temperature
The results of the COMSOL Multiphysics can be displayed in many demensions. The 3D temperature distribution
and contour line of a quarter of the model is shown for 1500W/cm2 laser power density and 242MPa preloading at 15.8s
in Fig. 2. This results indicate that the temperature distribution decreases not very fast in the y and z direction, the
difference temperature between front and rear surface and between middle and edge on front surafce in the y direction is
about 20K. The temperature distribution decreases very rapidly in the x direction. It can be seen the maximum
temperature on the front surface of the workpiece is 642K at 15.8 s when it failures totally, which is lower than the
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melting point temperature (750K) of AL-7075 alloy. It means that the AL-7075 is not melt when the fracture failure
occurs. The temperature evolution of center spot for different laser power densities under 242MPa preloading is shown
in Fig.3. As the laser strarts to irradiate the temperature of the work increase gradually, and the maximum temperatrue
(failure temperature) is different for difference laser power density under the same prelaoding.
Fig.2 The 3D temperature distribution a quarter of the model for the case of 1500W/cm2 and 242MPa at 15.8s
Fig.3 The temperature evolution of center spot for different laser power density under 242MPa preloading
3.2 Stress-Strain behavior
The yield stress and Strain evolution of the middle of the workpiece for different laser power densities under
242MPa preloading in shown in Fig.4. For 1500W/cm2, as the laser starts to irradiate, the temperature rises, and the yield
stress decreases gradually, when the yield stress reduce to 242MPa, the decreasing speed of the yield stress reduces and
the strain begin to increase. And the yield stress decreases to 214MPa at 9s and keeps on that level until the workpiece is
damaged finally. Meanwhile, the strain increasing dramatically. This is because the yield stress is lower than the
preloading and the plastic deformation of the workpiece occur in the laser irradiated zone, so the strain start to increase
dramatically. And the increasing of strain induces the hardness effect to the workpieces. While the increase of yield stress
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induce by hardness effect equals to the decrease of yield stress induce by temperature effect, the yield stress keep a
dynamic equilibrium. As the laser continue radiate, the strain continues increasing until the damge parameter reach unity,
the workpiece is damaged finally. Specially, the duration for the yield stress keep balance is much shorter than the
duration for yield stress deceasing under 242MPa preloading. For higher laser power density situation, the drop speed of
the yield stress is inceasing due to the faster speed of the temperature rise, and the dynamic equilibrium value of the
yield stress decreases and the duration of the dynamic equilibrium is shorter at the same time.
Fig.4 The yield stress and Strain evolution of the middle of the workpiece for different laser power densities under 242MPa preloading
(solid line:Yield stress, dash line: Strain)
Fig. 5. Failure Strain variation for Al-7075 subjected to various laser power densities and preloading.
(a) Strain versus preload for various laser power densities (b) Strain versus laser power density for various preloads.
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Failure Strain(the strain when the damage parameter of the element reach unity) variation for Al-7075 subjected to
various laser power densities and preloading is shown in Fig. 5. As shown in the Fig.5(a), the failure strain decreases as
the preloading increases at the same laser power density. Specially, the effect is much more remarkable for 1500W/cm2.
From Fig.5(b), the failure strain decreases as the laser power density increases for 242MPa prelaoding, but for higher
preloading, the failure strain increases as the laser power density increasing firstly and reach maximum at 1900W/cm2,
and then start to decreases.
3.3 Failure Temperature
Figure 6 expresses the failure temperature (the temperature of the center point of the front surface of the specimen
when it gets fractured) of Al-7075 as a function of preload and laser power density, respectively. As shown in the
Fig.6(a), the failure temperature decreases as the preloading increases at the same laser power density. The failure
temperature is almost same for 1900W/cm2 and 2300W/cm2, and for 1500W/cm2 and 2700W/cm2 under higher
preloading. From Fig.6(b), the failure temperature increases as the laser power density increasing firstly and reach to a
platform between 1900W/cm2 and 2300W/cm2, and then start to decreases. It means that the failure temperature has a
maximum in the range of 1500-2700W/cm2.
Fig. 6. Failure temperatrue variation for Al-7075 subjected to various laser power densities and preloading.(a) Failure Temperatrue
versus preload for various laser power densities (b) Failure Temperatrue versus laser power density for various preloads.
3.4 Failure Time
The Dcross-section evolution of the middle of the workpiece is shown in Fig. 7. From Fig.7(a), Dcross-section evolution
behavior is similar to strain which is shown in the Fig.4. For 242MPa preloading, the speed of the decease of failure time
reduce along with the increasing of laser power density, but it not affect by inceasing the preloading value for
1500W/cm2 that is shown in Fig.7(b). By compare Fig.7(a) and Fig(b), it is found that the mechanical preloading has a
greater impact to the failure time than the laser heating for AL-7075.
The Fig. 8(a and b) represents the variation of failure time under fixed laser power and fixed tensile preloading
respectively for Al-7075 type specimens. The Fig. 8(a) shows the decrease of failure time with the increase of laser
power densities, this behavior is more obvious in the case of lower laser power densities. The almost similar trend as of
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Al-6061 specimens in the failure time reduction with increasing preload and laser power is noticed [7-9]. In the Fig. 8(b)
shows that the failure time decrease linearly along with the increase of the preloadings. And the slope of failure time
decrease as the laser power density increasing.
Fig.7 The Dcross-section evolution of the middle of the workpiece
(a) for different power density under 242MPa (b) for different preloading at 1500W/cm2
Fig. 8. Failure time variation for Al-7075 subjected to various laser power densities and preloading.
(a) Failure time versus preload for various laser power densities (b) Failure time versus laser power density for various preloads.
4 CONCLUSION
In this paper, A 3D mathematical model was established for the investigation of the thermomechanical behavior of
aluminum alloys (Al-7075) under the combined action of tensile loadings and laser irradiations. The damage process is
analysed according to the temperature and the stress-strain evolution. The effects of various pre-loading and laser power
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densities on the failure strain, failure temperature and failure time of aluminum alloys are analyzed. The results indicate
the significant reduction in failure time for higher laser power densities and for high preloading values, which implies
that preloading may contribute a significant role in the failure of the material at elevated temperature. The numerical
result agrees well with our previous experiment results, concluding that the numerical model is reasonable.
5 ACKNOWLEDGMENTS
This work is supported by the National Natural Science Foundation of China (Grant No. 61605079) and the
Research Foundation of State Key Laboratory of Laser Interaction with Matter (SKLLIM1604).
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