Int J Mech Mater Des (2014) 10:93–107 DOI 10.1007/s10999-013-9233-y
Numerical implementation of strain rate dependent thermo viscoelastic constitutive relation to simulate the mechanical behavior of PMMA Uzair Ahmed Dar • Wei Hong Zhang Ying Jie Xu
•
Received: 23 July 2013 / Accepted: 17 October 2013 / Published online: 30 October 2013 Ó Springer Science+Business Media Dordrecht 2013
Abstract In this work, a nonlinear viscoelastic constitutive relation was implemented to describe the mechanical behavior of a transparent thermoplastic polymer polymethyl methacrylate (PMMA). The quasi-static and dynamic response of the polymer was studied under different temperatures and strain rates. The effect of temperature was incorporated in elastic and relaxation constants of the constitutive equation. The incremental form of constitutive model was developed by using Poila–Kirchhoff stress and Green strain tensors theory. The model was implemented numerically by establishing a user defined material subroutine in explicit finite element (FE) solver LSDYNA. Finite element models for uniaxial quasistatic compressive test and high strain rate split Hopkinson pressure bar compression test were built to verify the accuracy of material subroutine. Numerical results were validated with experimental stress strain curves and the results showed that the model successfully predicted the mechanical behavior of
U. A. Dar (&) W. H. Zhang (&) Y. J. Xu Engineering Simulation and Aerospace Computing (ESAC), Key Laboratory of Contemporary Design and Integrated Manufacturing Technology, Northwestern Polytechnical University, Xi’an 710072, Shaanxi, China e-mail:
[email protected] W. H. Zhang e-mail:
[email protected] Y. J. Xu e-mail:
[email protected]
PMMA at different temperatures for low and high strain rates. The material model was further engaged to ascertain the dynamic behavior of PMMA based aircraft windshield structure against bird impact. A good agreement between experimental and FE results showed that the suggested model can successfully be employed to assess the mechanical response of polymeric structures at different temperature and loading rates. Keywords Constitutive model Dynamic behavior Numerical simulation Strain rate Viscoelastic material
1 Introduction Polymethyl methacrylate (PMMA) is a transparent thermoplastic polymer widely used in various commercial and defense applications. In aeronautical industry it is particularly used to manufacture air craft canopies, bubble windows and windshields owing to its exceptional optical clarity, process shape adaptability, low density and high strength. During service, this material is subjected to temperature and impact loadings and it can perform quite differently under unique loading conditions. Therefore, it is necessary to completely understand the dynamic behavior of PMMA under complex loading conditions which will help to optimize the design and safety of air craft transparencies. The mechanical properties of polymers
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are highly dependent on temperature and rate of deformation or distinctively called strain rate. PMMA has been extensively investigated in past on its tensile and compressive strengths at different temperatures and strain rates. At higher strain rates PMMA exhibits an increase in elastic modulus and yield strength while decrease in failure strain comparing to quasi-static or very low strain rate. Conversely, there is a decrease in elastic modulus and yield strength with increase in temperature (Huang and Guo 2007). PMMA is a strain rate sensitive material which behaves differently from quasi static to dynamic loading showing different stress strain response and failure modes. At higher temperature it shows decreasing flow stress and strain hardens at much higher strains than low temperature conditions (Moy et al. 2011). The fracture toughness of this polymer is also the function of loading rate and is significantly higher at higher strain rates (Weerasooriya et al. 2006). Some researcher such as Arruda et al. (1995), Chen et al. (2002), Lee and Kim (2003) and Li and Lambros (2001) investigated the effects of both the strain rate and temperature and reported a substantial change in mechanical response, thermal softening, strain hardening and deformation behavior of PMMA. Other researchers extended their experimental and analytical outcomes to develop material constitutive relations to model the stress–strain response of these polymeric materials. Gao et al. (2011) studied the tensile and compressive behavior of MYDB-3 PMMA at different loading rates and temperatures and found out that the stress–strain response of this material differs significantly under tension and compression loadings. On the basis of their experimental results they proposed a tensile constitutive model which effectively predicted the dynamic mechanical behavior of the material. Frank and Brockman (2001) combined nonlinear viscoelasticity with viscoplasticity in their model and included the effect of hydrostatic pressure, strain hardening and strain rate to capture the nonlinear response of glassy polymers. They implemented the model in finite element code and successfully predicted the impact response of polycarbonate. Drozdov (1999) derived constitutive relations to model the nonlinear viscoelastic response of glassy polymer at glass transition temperature at small strains. Richeton et al. (2006) presented a temperature and strain rate dependent three dimensional constitutive model for amorphous polymers. The model was validated in compression for
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various temperatures and strain rates for PMMA, polycarbonate and polyamideimide. Zhu Zhaoxiang, Wang Lili and Tang Zhiping investigated the mechanical behavior of epoxide resin at lower and higher strain rates. On the basis of their experimental results they developed a nonlinear viscoelastic constitutive relationship called Zhu–Wang–Tang (ZWT) viscoelastic material model (Tang 1981). This model was further augmented by Zhou and Wang (Zhou et al. 1992; Wang et al. 2010) by introducing damage variable with nonlinear elastic terms and proposed damage modified ZWT model. The model was successfully employed to predict the damage and failure of PMMA at high strain rates (Wang and Yue 2010). Although, ZWT model has been previously implemented in finite element analysis programs to assess the dynamic response of PMMA based structures (Wang and Yue 2010; Zhu et al. 2009). However, the temperature effect was not incorporated in those models. As described earlier that temperature variations along with high strain rate can have significant effect on the mechanical response of PMMA. It is therefore necessary to investigate the combined effect of temperature and strain rate to exercise the most stringent design criteria for PMMA structures. Hence, ZWT rate dependent nonlinear viscoelastic constitutive model coupled with temperature effects was developed in this study. The model was implemented by establishing a user defined material subroutine in FE solver LS-DYNA. Uniaxial compressive test and split Hopkinson pressure bar (SHPB) test are the most established techniques to measure stress strain response of material at low and high strain rates, respectively. These two techniques were numerically employed to check the validity of material subroutine at different loading rates. The material model was then further applied to evaluate the dynamic response of a full scale PMMA windshield against bird impact.
2 Strain rate dependent thermo-viscoelastic ZWT model Zhu–Wang–Tang model in its rheological form consists of five elements including one nonlinear spring and two Maxwell elements connected in parallel as shown in Fig. 1. Equation 1 represents the differential form and Eq. 2 represents the integral form of ZWT nonlinear viscoelastic constitutive relation.
Mechanical behavior of PMMA
95
r ¼ rE ðeÞ þ E1
e_ e
ts h1
ds þ E2
0
e_ e
ts h2
r ¼ rE ðe; TÞ þ E1 ðTÞ ð1Þ þ E2 ðTÞ
e_ e
e_ e 0 ts h2 ðTÞ
ts h1 ðTÞ
ds
ð4Þ
ds
where
0
ð2Þ The term rE ðeÞ describes strain rate independent nonlinear elastic response of material and is represented by Eq. 3. Where E0, a and b are the corresponding elastic constants. rE ðeÞ ¼ E0 e þ ae2 þ be3 :
Zt
Zt
0
ds:
ð3Þ
ð5Þ
For quasi static and low strain rate testing conditions, the time of loading varies from 1 to 102 s. The relaxation time (h2) for high strain rate Maxwell element varies from 1 to 102 ls. Under these conditions, this high frequency Maxwell element will relax at the beginning of loading and will not contribute to material response, therefore it can be neglected. The ZWT equation then reduces to Eq. 6:
(a) 200
25°C 60°C 100°C
Strain rate = 0.1 s-1
160
120
Increase in temperature
80
40
0 0
0.025
0.05
0.075
0.1
0.125
0.15
Strain
(b) 200
0.001 s-1 0.1 s-1
Temperature= 25° C
150
Stress (MPa)
The second and third integral term in Eq. 2 describe the low and high strain rate viscoelastic response of two Maxwell elements. E1, E2 and h1, h2 are the elastic constants and stress relaxation time for corresponding Maxwell elements, respectively. g is viscous coefficient which describe the viscous characteristic of polymer and h = g/E. s is the time variable and t is the loading time. Suo et al. (2006) performed a number of experiments and recorded the stress strain response of PMMA at different temperatures and strain rates. With the increase of temperature, the flow stress decreases whereas it increases with the increase in strain rate as shown in Fig. 2. The yield stress of specimen as a function of temperature and strain rates is shown in Fig. 3. The experimental results at various temperatures show almost similar viscoelastic stress strain response for a certain strain rate. Therefore, it is assumed that stress relaxation time (h) and elastic constants (E, a and b) are function of temperature T. If Eq. 2 can be written with temperature dependent relaxation time and elastic constants, then Eq. 4 represents the thermo-viscoelastic form of ZWT model.
rE ðe; TÞ ¼ E0 ðTÞe þ aðTÞe2 þ bðTÞe3 :
Stress (MPa)
or drE ðeÞ oe or1 or2 ¼ þ þ ot ot ot de ot drE ðeÞ oe r1 r2 þ E1 þ E2 ¼ de ot h1 h2 Zt Zt
100 Increase in strain rate
50
0 0
0.025
0.05
0.075
0.1
0.125
0.15
Strain
Fig. 1 Rheological ZWT model
Fig. 2 Stress-strain response of PMMA with the increase of a temperature, b strain rate
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(a)
U. A. Dar et al.
300
deduced on basis of updated lagrangian approach and by using second Poila Kirchhoff stress tensor and Green strain tensor (Yapeng et al. 1987). The Kirchhoff stress tensor Sij of ‘r’ in Eq. 4 can be represented as;
200
M2 1 Sij ¼ SEij þ SM ij þ Sij
500
0.001 s-1 0.1 s-1 550 s-1 Y=141.6-1.2T Y=191.7-1.4T Y=418.5-2.8T
Yield Stress (MPa)
400
100
0 0
20
40
60
80
100
120
Temperature (°C)
(b)
500
Yield Stress (MPa)
M2 1 SEij , SM ij and Sij are nonlinear elastic stress tensor, viscoelastic stress tensor for low and high strain rate Maxwell elements respectively and can be represented by the following equations:
SEij ¼ E0 ðTÞAEkl þ aðTÞEij2 þ bðTÞEij3 Zt ts oE kl 1 e h1 ðTÞ ds A SM ij ¼ E1 ðTÞ os
25°C 60°C 100°C Y=232+19.6ln Y=135+11.8ln Y=86.6+10ln
400
ð8Þ
ð9aÞ ð9bÞ
0 300 2 SM ij ¼ E2 ðTÞ
200
Zt
oEkl e A os
ts h2 ðTÞ
ð9cÞ
ds
0
Eij is the Green strain tensor and A is threedimensional isotropic elasticity matrix and represented as Eq. 10.
100
0 0.0001
0.001
0.01
0.1
1
10
100
1000
Strain rate (s -1)
A¼
Fig. 3 Yield stress as function of a temperature b strain rate
r ¼ rE ðe; TÞ þ E1 ðTÞ
Zt
e_ e
ts h1 ðTÞ
ds
ð6Þ
0
In contrast, under dynamic loading and high strain rate testing conditions, the time of loading varies from 1 to 102 ls and relaxation time (h1) varies from 0.1 to 1 s. There is not enough time for low frequency Maxwell element to relax till the end of loading. It behaves like an elastic solid of stiffness E1 and reduces to a single spring element. Eq. 4 then conforms into Eq. 7: Zt hts r ¼ rE ðe; TÞ þ E1 ðTÞe þ E2 ðTÞ e_ e 2 ðTÞ ds ð7Þ 0
3 Incremental form of ZWT model
1 ð1 þ mÞð1 2mÞ 2 1m m m 6 6 m 1m m 6 6 m 1m 6 m 6 6 0 0 0 6 6 6 0 0 0 4 0 0 0
0
0
0
0
0
0
12m 2
0
0
12m 2
0
0
7 0 7 7 7 0 7 7 0 7 7 7 0 7 5 12m 2
ð10Þ The incremental form of nonlinear elastic stress tensor SEij can be obtained as DSEij ¼ E0 ðTÞADEkl þ 2aðTÞEij ADEkl þ 3bðTÞEij2 ADEkl
ð11Þ
The incremental form of low strain rate stress tensor can be obtained as;
1 SM ij
M1 M1 1 DSM ij ¼ Sij;tþDt Sij;t
For three dimensional finite element analysis, the incremental form of ZWT constitutive equation was
123
3
0
and for very small interval Dt,
ð12Þ t oEkl os
¼
tþDt oEkl os
kl ¼ DE Dt
Mechanical behavior of PMMA
1 SM ij;t
¼ E1 ðTÞ
Zt
oEt A kl e os
ts h1 ðTÞ
97
ds
0
DEkl t Ah1 ðTÞ 1 e h1 ðTÞ Dt Zt oEkltþDt tþDts e h1 ðTÞ ds ¼ E1 ðTÞ A os 0 ! htþDt DEkl ¼ E1 ðTÞ Ah1 ðTÞ 1 e 1 ðTÞ Dt
¼ E1 ðTÞ 1 SM ij;tþDt
1 DSM ij ¼ E1 ðTÞ
DEkl Dt t Ah1 ðTÞ 1 e h1 ðTÞ e h1 ðTÞ Dt
ð13Þ
ð14Þ ð15Þ
Similarly, the incremental form of high strain rate 2 stress tensor SM ij can be obtained as; DEkl Dt t 2 Ah2 ðTÞ 1 e h2 ðTÞ e h2 ðTÞ ð16Þ DSM ij ¼ E2 ðTÞ Dt Combining Eqs. 11, 15 and 16, the incremental form of the Poila–Kirchhoff stress tensor, Sij, of total stress ‘r’ in time t can be defined as; DSij ¼ E0 ðTÞADEkl þ 2aðTÞEij ADEkl þ 3bðTÞEij2 ADEkl |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} nonlinear elastic stress tensor
DEkl Dt t Ah1 ðTÞ 1 e h1 ðTÞ e h1 ðTÞ þ E1 ðTÞ Dt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} low strain rate stress tensor
DEkl Dt t Ah2 ðTÞ 1 e h2 ðTÞ e h2 ðTÞ þ E2 ðTÞ Dt |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} high strain rate stress tensor
ð17Þ StþDt ij
¼
Stij
þ DSij
ð18Þ
a user defined material subroutine (*MAT_USER_ DEFINED_MATERIAL_MODELS). The material subroutine was developed with in multiple source routine file (DYN21.F) provided by LS DYNA. The subroutine consists of material density, shear and bulk modulus along with seven material parameters. In subroutine, the stress and strain tensors were stored in defined set of arrays and history variables. The martial parameters were identified from experimental stress strain curves and are presented in the following section. In order to validate the accuracy of material subroutine, finite element simulations of quasi-static compressive test and split Hopkinson pressure bar compression test of a PMMA specimen were performed. The results from numerical simulations were compared with experimental data and were found in close agreement. 4.1 Determination of material parameters The experimental results show that all the stress stain curves of PMMA display similar trend and strain rate sensitivity with the increase in temperature. Therefore, all the parameters of ZWT model that is, elastic constants (E0, a, b, E1 and E2) and relaxation time (h1, h2) are considered to be temperature dependent. From the experimental stress strain data the material constant for PMMA at different temperatures were determined through curve fitting and the parameters are listed in Table 1. The comparison between experimentally and theoretically determined stress strain curves of PMMA is shown in Fig. 4. The results showed that ZWT model predicted the material response at different temperature and strain rates with fair accuracy.
4 Implementation and validation of material model
4.2 Uniaxial compressive testing
The incremental form of ZWT model described in Eq. 17 was implemented in LS-DYNA by developing
To validate the numerical model for quasi-static and lower strain rates, FE simulations of uniaxial
Table 1 Material constants for ZWT model T (°C)
E0 (GPa)
a (GPa)
b (GPa)
E1 (GPa)
h1 (s)
E2 (GPa)
h2 (s)
25
4.16
-47.00
153.43
1.107
0.74
2.918
9.5E-04
40
3.98
-53.80
237.90
0.987
0.59
2.458
8.3E-04
60 80
2.79 2.46
-41.11 -42.56
194.20 239.00
1.274 1.326
0.30 0.26
2.099 1.217
9.2E-05 1.8E-04
100
1.65
-35.40
230.90
1.604
0.20
1.194
5.2E-05
123
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U. A. Dar et al.
compression tests were performed in accordance with the experiments. In uniaxial compression test, the specimen was sandwiched between two rigid plates and compressed under the application of constant vertical load F(t). The load is applied according to desired rate of strain. The configuration for uniaxial compression test with a cylindrical specimen between two rigid steel plates is shown in Fig. 5. The length (l) and diameter (d) of specimen was 6.5 mm, giving a length to diameter ratio (l/d) of 1. The dimensions of the specimen were taken similar as used in experiments. The specimen was meshed with eight node SOLID164 element type. The fixed boundary condition was employed at the bottom plate while a constant load depending upon required strain rate was applied at the upper plate. The nominal longitudinal strain (en) can be calculated by dividing the change in length (Dl) between top and bottom interface of specimen to its original length (lo).
500
Experimental Theoretical
550 s-1 25°C
400
Stress (MPa)
40°C
en ¼
Dl l lo ¼ lo lo
ð19Þ
And the true longitudinal strain (et) is given by; et ¼ lnð1 þ en Þ
ð20Þ
The results from simulations were compared with experimental results (Suo et al. 2006). The FE predicted stress strain response at various temperatures agrees well with experimental results and are shown in Fig. 6. 4.3 Split Hopkinson pressure bare testing The split Hopkinson pressure bar testing technique is used to characterize the mechanical behavior of material at high strain rates ranging from 102 to 104 s-1. In its simplest form, SHPB test apparatus consist of a striker bar, an incident bar, a transmission bar and a strain acquisition system. All the pressure bars have same diameter and are made up of similar material. Typically, a gas gun is used to launch the striker bar to impact the incident bar at some velocity depending upon the required strain rate to be induced in specimen. The specimen is held in between incident and transmitted bar. The schematic of conventional SHPB test setup is shown in Fig. 7.
300
4.3.1 SHPB test method
0.1 s-1
200 25°C 40°C 0.001
100
s-1 60°C 80°C
0 0.00
0.02
0.04
0.06
0.08
Strain
Fig. 4 Comparison of experimentally and theoretically predicted stress–strain response
During testing, the striker bar with initial velocity impacts the incident bar and a rectangular compression wave is produced in it. The amplitude of the wave depends on velocity, material and dimensions of the striker bar. The wave travels through the length of the incident bar and when it reaches at the end of the incident bar, a portion of it is transmitted into the specimen as a compressive pulse while rest is reflected
Fig. 5 Simulation model of quasi-static compressive test of PMMA specimen
F (t)
Specimen 6.5 mm
123
Mechanical behavior of PMMA
99
160 Experiment Simulation 25°C
Stress (MPa)
120
40°C
1D compressive test Strain rate=0.001 s-1
80
60°C 80°C 40 100°C
0 0
0.02
0.04
0.06
0.08
Strain
measured from strain gages A and B located on the incident and transmitted bars (Fig. 7). In SHPB testing, it is assumed to have one dimensional wave propagation and dynamic stress equilibrium in the specimen (eT = eI ? eR) which implies that the incident pulse must be proportional to the stress in the specimen to achieve constant strain rate. By employing classical SHPB theory (Meyers 1994), the stress (rs), strain (es) and strain rate ðe_s Þ in the specimen can be computed from the following equations; 2 DB rs ðtÞ ¼ EB eT ðtÞ ð21Þ Ds
240
2CB es ðtÞ ¼ Ls
Experiment Simulation
200
Stress (MPa)
eR ðtÞ
ð22Þ
0
25°C 160 1D compressive test Strain rate=0.1 s-1
Zt
d 2CB es ¼ eR ðtÞ dt Ls
40°C
e_s ðtÞ ¼
60°C
Ds and Ls are diameter and length of the specimen, DB is the diameter, CB is wave propagation speed pffiffiffiffiffiffiffiffiffiffiffiffiffi CB ¼ EB =qB , EB is elastic modulus and qB is
120
80
80°C
40
100°C
ð23Þ
density of pressure bars. 0 0
0.02
0.04
0.06
0.08
Strain
4.3.2 SHPB simulation model
Fig. 6 Comparison of FE and experimental results
back as an elastic tensile pulse. The reflection and transmission is due to the difference in cross sectional area and acoustic impedance mismatch between the specimen and pressure bars. The wave is then transmitted from specimen to transmit bar, the amplitude of transmitted wave depends upon the material properties of the specimen. The incident strain (eI), reflected strain (eR) and transmitted strain (eT) are
A three dimensional finite element model of SHPB setup was developed in LS-DYNA as shown in Fig. 8. The specimen, striker, incident and transmitter bars were meshed with SOLID 164 eight node solid elements. A mesh sensitivity analysis was carried out separately and the final selected mesh consists of 20 elements along circumferential direction and 3 mm element size along the length of pressure bars. The simulation model of SHPB setup consists of 94,080 Ls
Striker bar V
Strain gauge A
Incident bar Incident wave
Ds
Strain gauge B
Transmitter bar
Transmitted wave
DB
Reflected wave
εR εI time
εT
strain
strain
Specimen
time
Fig. 7 Schematic of split Hopkinson pressure bar testing setup
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U. A. Dar et al.
Fig. 8 Finite element model of SHPB Table 2 Material properties and dimensions of SHPB Pressure bar
Length (mm)
Diameter (mm)
Material
Density (Kg m-3)
Elastic modulus (GPa)
Wave velocity (m/s)
Compressive strength (GPa)
Poisson’s ratio
Striker
250
12.7
Maraging steel
8,060
198
4,956
2.6
0.3
Incident
1,200
12.7
Transmitter
1,200
12.7
480
200
(a)
400
50 0 -50
320 40°C
240
100°C 160
-100
80 -150
Incident stress pulse
0
-200 0
100
200
300
400
500
0
0.02
0.04 Strain
Time (µs) 100
Transmitter Bar stress (MPa)
25°C
SHPB test Strain rate=550 s-1
100
Stress (MPa)
Incident Bar stress (MPa)
150
Experiment Simulation
Reflected stress pulse
90
Fig. 10 Comparison (e9 = 550 s-1)
(b)
70 60 50 40 30 20 10 400
450
500
Time (µs)
Fig. 9 Simulation results of stress pulses generated in a incident pressure bar, b transmitter bar
123
FE
and
experimental
0.08
results
Transmitted stress pulse (inverted)
80
0 350
of
0.06
solid elements in total. The pressure bars are modeled with isotropic elastic material model. The material of striker and pressure bars is Maraging steel and the characteristic dimensions and material properties of bars are listed in Table 2. Automatic surface to surface contact was defined between pressure bars and specimen. In simulations, striker bar impacts the incident bar with initial velocity of 7.5 m/s which in turn produces an average strain rate of 550 s-1. The amplitude and duration of incident, reflected and transmitted stress waves were recorded at central element on incident and transmitter bars and the results are shown in Fig. 9. On
Mechanical behavior of PMMA
101
structural damage to critical aircraft components. Over the years, the virtual experiment or commonly regarded as FE simulation, is prove to be an effective tool to predict the dynamic response of air craft components against bird impact. These simulations have effectively reduced the experimentation cost. After successful implementation of ZWT constitutive model in the preceding section, the model was further applied to assess the impact response of PMMA based windshield of a military aircraft against bird strike. A three dimensional finite element model for bird windshield impact problem was built for this purpose. Fig. 11 Simulation model for bird–windshield impact
5.1 FE Modeling of windshield structure
impacting the striker bar to incident bar, a compressive stress wave is produced which travel along the bar and reaches at the center of the bar in 127 ls. The amplitude of the compressive wave reaches an average value of 153 MPa in incident bar. The stress wave reaches at specimen interface in 374 ls and a portion of it is transmitted to transmitter bar through specimen while rest is reflected back to the incident bar. An average stress value of 44 MPa is transmitted to transmitter bar while an average of 94 MPa was reflected back to the incident bar. The dynamic response of specimen was calculated from reflected and transmitted strains pulses (eT, eR) measured from incident and transmitter bars, respectively. The results from simulations were compared with experimental results and were found in good agreement as shown in Fig. 10. The results prove the validity of material model at higher strain rate loading conditions. The verified numerical model was further used to investigate the dynamic behavior of a full scale aircraft windshield against bird impact and is discussed in the ensuing section.
The windshield structure consists of windshield, gasket and surrounding frame. The simulation model of windshield structure is shown in Fig. 11. The windshield considered in the simulations is manufactured from monolithic, uniform cross section aeronautical standard PMMA. The corresponding material properties have been discussed in detail in Sect. 4. ZWT material model was employed by using LSDYNA user defined material subroutine to ascertain the dynamic behavior of windshield under bird impact. The finite element mesh of windshield consists of 40,000 SOLID 164 elements. The material of surrounding frame is wrought aluminum alloy and its material properties are provided in Table 3. The frame was modeled as plastic kinematic material with solid element type. The rubber gasket between frame and windshield was modeled with Moony Rivlin hyperplastic material model with hour glassing control. Material parameters for rubber gasket are listed in Table 4. 5.2 FE Modeling of bird
5 Bird–windshield impact analysis Bird impact is considered to be one of the most significant threats to aircrafts and it causes serious
The real bird is a complex make up of flesh, blood and pneumatic bones and it is difficult to implement the actual bird model in any finite element program. Substitute bird modeling approach has been proposed
Table 3 Material parameters of surrounding frame Density (Kg m-3)
Elastic modulus (GPa)
Tangent modulus (MPa)
Poisson’s ratio
Yield strength (MPa)
Failure strain
2,780
72
690
0.3
313
0.28
123
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U. A. Dar et al.
distortion. SPH approach is based on interpolation theory and smoothing kernel functions. In this formulation, the bird is characterized as set of discrete interacting particles which represent the interpolation points at which the material properties are determined. The field variable of any discrete particle can be computed through interpolation of neighboring particles. The 1.8 kg bird was modeled as right cylinder of 120 mm diameter and 180 mm length as shown in Fig. 12. The SPH model of bird consists of 32,220 nodes each weighing 0.0557 g lumped mass and average distance of 4 mm between two nodes. Before applying the proposed bird model to bird– windshield impact simulations, the model was first validated through flexible flat plate impact test experiment. The bird with initial velocity of 146 m/s was impacted on 6.35 mm thick aluminum plate and corresponding plastic deformation of the plate was measured as shown in Fig. 13. The resultant plastic deformation after impact was determined and compared with experimental results (Welsh et al. 1986). The displacement time plot at the center of plate is shown in Fig. 14. The FE predicted maximum central deformation d of 44.6 mm was in close agreement with experimentally determined value of 41.275 mm. The validated material model for bird was then used in full scale bird–windshield impact problem.
Table 4 Moony Rivlin material parameter for rubber gasket Density (Kg m-3)
Elastic modulus (MPa)
Poisson’s ratio
A (MPa)
B (MPa)
1,100
310
0.49
0.378
0.977
in literature in which the bird is either modeled as elasto plastic material with certain failure criterion or through equation of state (EOS) approach with pressure volume relationship of water. Various EOS such as polynomial, Murnaghan and Gruneisen form of equations can be used to model the behavior of bird (Meguid et al. 2008; Jenq et al. 2007; Johnson and Holzapfel 2006). In present simulations, the bird was modeled as soft body which acts like fluid on the windshield structure. An elastic–plastic–hydrodynamic material model with polynomial EOS was employed to predict the behavior bird. In polynomial form of equation of state, the pressure P is calculated by the following equation (LS-DYNA Keyword User Manual, Ver. 971 2006): P ¼ Co þ C1 l þ C2 l2 þ C3 l3
ð24Þ
where; l¼
q 1 qo
ð25Þ
C0, C1, C2 and C3 are material constants and l is the ratio of current density q to initial density qo . The physical properties for substitute bird and corresponding equation of state parameters are provided in Table 5. During impact, the elements in bird mesh distort severely due to high rate of deformation. This severe mesh distortion can seriously impair the progress of calculations due to significant decrease in explicit time step. Instead of using classical finite element mesh, a mesh less smoothed particle hydrodynamics (SPH) technique was used to model the bird in present case. SPH approach has advantages over other approaches such as lagrangian or Arbitrary Lagrangian–Eulerian (ALE) formulation due its numerical stability, constant time step and no mesh
5.3 Simulations and results FE simulations were carried out to predict the impact response of windshield structure against bird impact. In LS-DYNA, automatic node to surface contact was defined between bird and windshield while surface to surface tied contact between windshield, gasket and surrounding frame was used to model the interaction between bird and windshield structure. Numerical simulations were performed for a range impact velocities in which the bird was impacted at cental point of windshield. With the increase in bird impact velocity, the deformation in the windshield and its corresponding components increases due to increased
Table 5 Material properties and EOS parameters for bird Density (Kg m-3) 900
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Shear modulus (GPa) 2
Yield stress (MPa) 0.02
EOS parameters C0 0
C1
C2 9
2.1 9 10
6.2 9 10
C3 9
10.1 9 109
Mechanical behavior of PMMA
103
120 mm
180 mm
Fig. 12 SPH model for bird
Fig. 13 Bird impact on flexible aluminum plate
50
Displacement (mm)
40
30
20
10
0 0
0.5
1
1.5
2
Time (ms)
Fig. 14 Central displacement of aluminum plate
2.5
3
kinetic energy of bird and higher impact force. The effect of elevated temperature was also simulated to observe the dynamic response of windshield at different temperatures. Figure 15 shows the sequence of impact events at various time intervals when bird was impacted at initial velocity of 65 m/s. Figure also shows the equivalent stress fields in different windshield components during impact. The upper and lower end of windshield surrounding frame remains in the state of higher stress during the impact. The bird body crumbles and slides along the surface of windshield and the whole impact event lasts for 14 ms. The response of windshield against different impact velocities was studied and it was observed that resultant normal displacement at impact point
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Surrounding Frame
Gasket
Windshield
Cross sectional view
Windshield structure
104
t = 2 ms
4 ms
6 ms
8 ms
Fig. 15 Bird impact on windshield structure
increases with the increase of velocity as shown in Fig. 16. Also the time to reach maximum displacement decreases with the increase in impact velocity. The simulation results for maximum normal displacement at 65 m/s bird velocity was compared with experimental results of Bai and Sun (2005) and were found in good agreement. The numerically determined maximum normal displacement of 13.9 mm at 4.9 ms was in close comparison with experimentally measured normal displacement of 12 mm at 4.5 ms. The magnitude of equivalent stress increases with the increase of velocity. At 65 m/s impact velocity the value of maximum stress was 38.7 MPa which
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increased to 69.5 MPa with the increase in velocity to 125 m/s as shown in Fig. 17. The effect of temperature rise was also considered and its effect on maximum displacement and stress was studied. Figure 18 shows the displacement time plots at central point of windshield at bird impact velocity of 65 m/s. The results show that the maximum normal displacement increases with the increase in temperature. The maximum displacement increases up to two times when the temperature increases from 25 to 100 °C. The equivalent stress decreases with the increase of temperature as shown in Fig. 19. A 32 % decrease in maximum stress was recorded with the increase in
Mechanical behavior of PMMA 40
105
(37.4)
(29.7)
25
30
(21.8)
20 (13.9)
15
40°C 60°C 80°C 100°C
35
Stress (MPa)
25 20 15
10
10
5
5 0
0 0
2
4
6
0
8
2
4
8
10
12
Fig. 19 Effect of temperature on equivalent stress
Fig. 16 Normal displacement at central point of windshield
16
80
65 m/s 85 m/s 105 m/s 125 m/s
70
Kinetic Energy Hourglass Energy Internal Energy Energy ratio
14
System Energy (KJ)
60
Stress (MPa)
6
Time (ms)
Time (ms)
50 40 30 20 10
2
12 1.5
10 8
1 6 4
Energy ratio
Displacement (mm)
35 30
40
65 m/s 85 m/s 105 m/s 125 m/s
0.5
2
0 0
2
4
6
8
0
0 0
1
2
Time (ms)
3
4
5
6
Time (ms)
Fig. 17 Equivalent stress history
Fig. 20 Energy balance of bird impact simulation
30 25
Displacement (mm)
of any PMMA structure with acceptable accuracy. However, the trivial difference in experimental and FE results can be attributed to material parameters identification of both bird and windshield.
40°C 60°C 80°C 100°C
20 15
5.4 Energy balance
10 5 0 0
2
4
6
8
10
12
Time (ms)
Fig. 18 Effect of temperature on central normal displacement
temperature from 25 to 100 °C. From the results it can be realized that the temperature incorporated ZWT model can successfully predict the dynamic behavior
In explicit FE impact analysis, the energy balance is not necessarily kept due to addition of hourglass energy. Using reduced integration elements formulations in LS-DYNA simulations, hourglass energy may arise due to artificial forces applied to node to prevent specious zero-energy modes (hour glassing). It is therefore necessary to monitor the time-history of energy components in order to identify that the total energy of the system is conserved. At any time t during the analysis the total energy T of the system can be expressed as;
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T ¼ Ek þ E p
ð26Þ
Ek is the initial kinetic energy and Ep is the potential energy. Ep can be further divided into its components, initial internal strain energy (Ei) and external work done (We) as follow: Ep ¼ Ei þ W e
ð27Þ
At any time t, the total energy of the system can be divided as (LS-DYNA Keyword user manual, Ver. 971 2006; LS-DYNA Theory Manual v1.0 2006): 0 T 0 ¼ Ek0 þ Ei0 þ Es0 þ Er0 þ Ed0 þ Ehg
ð28Þ
where Ek0 is kinetic energy, Ei0 internal energy, Es0 sliding interface energy, Er0 rigid-wall energy, Ed0 is 0 is hourglass energy at any damping energy and Ehg time during analysis. Normalizing total energy with respect to initial energy of the system the energy ratio (E.R) can be defined as: E:R ¼
T0 ¼1 Ek þ E i þ W e
ð29Þ
In the current bird strike analysis the contributing components of energy are kinetic, internal and hourglass energy while other components having negligible effect are therefore neglected. As a bench mark, the hourglass energy should be less than 10 % of peak internal energy (LS-DYNA Aerospace Working Group, Ver 12.1. 2012). The time history of system energies during simulation is shown in Fig. 20. The ratio of hourglass energy to peak internal energy remains less than 0.1 during simulation which justifies the robustness of analysis.
6 Conclusions Polymethyl methacrylate is a strain rate and temperature sensitive polymeric material. The flow stress in this material decreases with the increase in temperature while increases with the increase of strain rate. A temperature and strain rate dependent nonlinear viscoelastic material model was implemented in explicit finite element hydro-code to determine the mechanical response of PMMA. The model successfully predicted the mechanical behavior of PMMA for temperature ranging from 25 to 100 °C and strain rate ranging from 10-3 to 102 s-1 up to 8 % of strain. The efficacy of material subroutine was verified through finite
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element simulations of uniaxial compressive testing under low and high strain rate conditions. The numerically predicted mechanical response was in close agreement with experimental data. The validated numerical model was used to assess the dynamic response of full scale windshield structure against bird impact. The model successfully predicted the impact response of windshield for a range of temperature and impact velocities. Acknowledgments This work is supported by 973 Program (2012CB025904), 111 Project (B07050) and National Natural Science Foundation of China (51221001).
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