metals Article
A New Experimental and Numerical Framework for Determining of Revised J–C Failure Parameters Cunxian Wang 1 , Tao Suo 1,2, *, Yulong Li 1,2 , Pu Xue 1,2 and Zhongbin Tang 1,2 1
2
*
School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China;
[email protected] (C.W.);
[email protected] (Y.L.);
[email protected] (P.X.);
[email protected] (Z.T.) Shaanxi Key Laboratory of Impact Dynamics and Engineering Application (IDEA), Northwestern Polytechnical University, Xi’an 710072, China Correspondence:
[email protected]; Tel.: +86-159-0920-8902
Received: 23 April 2018; Accepted: 27 May 2018; Published: 30 May 2018
Abstract: Since damage evolutions of materials play important roles in simulations, such as ballistic impacts and collisions, a new experimental and numerical method is established to determine the revised Johnson–Cook (JC) failure parameters of a 2618 aluminum alloy and a Ti-6Al-4V titanium alloy. Not only the strain distributions, but also the stress triaxialities of designed specimens with different notches, are analyzed and revised using the finite element (FE) model. Results show that the largest strain concentrated on the surface of the circumferential area where the initial damage happened, which coincided with the practical damage evolution in the FE model. The complete damage strain, which denoted the largest strain before fracture calculated by the picture, is put forward to replace the traditional failure strain. Consequently, the digital image correlation (DIC) method and the micro speckle are carried out to measure the complete strain from the circumferential area. In addition, the relationships between the complete damage strain, the revised stress triaxiality, the strain rate and the temperature are established by conducting the quasi-static and dynamic experiments under different temperatures. Finally, the simulations for the ballistic impact tests are conducted to validate the accuracy of the parameters of the revised JC damage model. Keywords: damage evolutions; stress triaxiality; complete damage strain; the digital image correlation; revised Johnson–Cook (JC) damage model
1. Introduction Designing a containment structure for a fan blade is important for aero-engines to improve safety and reliability, even when it is really an expensive and time-consuming process. As the full-scale conventional containment test costs millions of dollars, it always follows relatively inexpensive simulations. These simulations can be used to predict effects of many factors on ballistic performances and failure modes of advanced aero-engine alloys, and also contribute to design and optimization of containment systems. The dynamic mechanical properties of materials under high strain rates can be described through a relationship between a flow stress and strain, a strain rate, a temperature, and a deformation history, known as a constitutive relationship or a constitutive model. The constitutive models have become the core factors, which can significantly influence efficiency and accuracy of simulations [1,2]. In addition, recent reports indicated that ductile fracture was a complicated process, which was influenced by strain rate, temperature, deformation history, stress triaxiality and microstructure of a material [3–5]. As the same as the constitutive model, the damage evolutions of materials also play important roles in some simulations, such as ballistic impacts and collisions.
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Theoretically, deformation of metal materials during an impact process is not merely the compression, tension or torsion, but composed of at least two kinds of deformation modes [6]. Consequently, stress states at different locations of a material always tend to be different. Actually, damage modes of a material with different loading histories are complex, thus making the investigation on the effect of stress state or stress triaxiality on damage evolution becomes necessary and valuable [7,8]. The efforts to investigate the performance of ductile fracture were started in the 1970s. It caps more than 40 years of widespread concerns with the influence of stress triaxiality on ductile fracture [9–11]. Bridgman reported hydrostatic stress could influence failure of materials, and stress triaxiality was represented as a result of the effect of hydrostatic stress [12]. Fouad et al. carried out some tests on metals and put forward that the failure strain changed with the variation of stress triaxiality [13–15]. Khan et al. presented a three-dimensional progressive failure analysis (PFA) of laminated composite structures, and failure of an isotopic-type material under hydrostatic failure was modeled from an analytical and finite element stand [16]. In addition, the relation between the failure strain and the stress triaxiality is not linear or monotonic. Researches also indicated that the stress triaxiality can be used to distinguish the fracture mechanism of metals. Since the development of constitutive models, many researchers have put forward that the failure strain can be influenced not only by the stress triaxiality but also by other factors, such as strain rate and temperature [17,18]. Johnson and Cook developed a special failure criterion to characterize the damage in the JC constitutive model, which was already widely used in research and engineering, and the damage was derived from the following cumulative damage law [19]: ∆ε D=∑ (1) εf where ∆ε is an increment of an effective plastic strain during an increment in loading, and D is a damage parameter. D = 0 is assumed to be the initial status, and failure is then allowed to occur when D = 1. A failure strain considers stress triaxiality, strain rate and temperature, which can be defined as: ε f = [ D1 + D2 exp( D3 σ∗ )][1 + D4 Inε∗ ][1 + D5 T ∗ ]
(2)
where σ∗ denote the mean stress normalized by the effective stress which is often referred to as stress triaxiality, and ε∗ is the normalized effective plastic strain rate. The parameters (i.e., D1 , D2 , D3 , D4 , and D5 ) are fracture constants of a material. Both the failure strain and the accumulation of damage are functions of stress triaxiality, strain rate, and temperature. Failed elements are then removed from the FE model with an element erosion algorithm. Research by Kim and Tvergaard also indicated that a microscopic mechanism of ductile failure was influenced by stress triaxiality [20,21]. As is known, nucleation, growth and coalescence of micro-voids can be considered as the mechanism of the ductile fracture of metals from the micro mechanical point of view. Researches by Anderson et al. pointed out that the growth of voids might be directly affected by the stress triaxiality and put forward that the occurrence of ductile fracture could be determined by the stress triaxiality, which was common in compressive and tensile tests for metals with good plasticity [22]. Researches on ductile fracture and failure models have been conducted for a long time; however, few investigations take an accurate method into account to obtain failure parameters. Specifically, there were few reports on measurement of failure strains. In addition, there is no close connection with the measurement of the failure strain when the stress triaxialities are revised. In the present work, a new experimental and numerical method was established to determine the revised JC failure parameters of a 2618 aluminum alloy and a Ti-6Al-4V titanium alloy. A series of specimens were designed, and strain distributions of the designed specimens were analyzed by using the FE model. Loading speeds and stress triaxialities of the samples with different notches were also revised by using the FE model to ensure the same strain rate. The digital image correlation (DIC) method and the micro speckle were conducted to measure a strain field since the simulation results
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correlation (DIC) method and the micro speckle were conducted to measure a strain field since the simulation results clearly indicated that the largest strain concentrated on surface of the clearly indicatedarea. that the on surface of thefracture circumferential area. circumferential Thelargest largeststrain strainconcentrated from the strain field before calculated byThe the largest picture strain from the strain field before fracture calculated by the picture was considered as the complete was considered as the complete strain, which was used to replace the failure strain. In addition, the strain, which was used to replace the failure strain. In addition, thestress relationships between the complete relationships between the complete damage strain, the revised triaxiality, the strain rate and damage strain, the revised stress triaxiality, theout strain and the temperature established by the temperature were established by carrying therate quasi-static and dynamicwere experiments under carrying out the quasi-static and dynamic experiments under different temperatures. Finally, by using different temperatures. Finally, by using the parameters of the revised JC damage model, the the parameters revised JC damage model, the simulations forthe theaccuracy ballisticof impact tests were simulations for of thethe ballistic impact tests were conducted to validate parameters. conducted to validate the accuracy of parameters. 2. Materials and Testing Specimens 2. Materials and Testing Specimens 2.1. Materials 2.1. Materials The 2618 aluminum alloy was provided by AECC Commercial Aircraft Engine Co., LTD The 2618 aluminum alloy was provided by AECC Commercial Aircraft Engine Co., LTD., (Shanghai, China). Ti-6Al-4V titanium projectiles, which were used for the ballistic impact tests, (Shanghai, China). Ti-6Al-4V titanium projectiles, which were used for the ballistic impact tests, were purchased from Western Superconducting Technologies Co., Ltd. (Xi’an, China). A series of were purchased from Western Superconducting Technologies Co., Ltd., (Xi’an, China). A series of mechanical tests for the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy were conducted to mechanical tests for the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy were conducted to estimate the uniformity and the stability of the materials, and results showed the flow stresses and estimate the uniformity and the stability of the materials, and results showed the flow stresses and the the fracture strain of the specimens taken from three directions were almost identical, thus proving fracture strain of the specimens taken from three directions were almost identical, thus proving the the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy were isotropic. 2618 aluminum alloy and the Ti-6Al-4V titanium alloy were isotropic. 2.2. Specimens Specimens for for the the JC JC Constitutive Constitutiveand andDamage DamageModel Model 2.2. Smooth cylinder cylinder specimens specimens with with dimensions dimensions of of Φ5 Φ5 ×× 55 mm mm were were used used to to characterize characterize both both Smooth quasi-static and dynamic compressive elastic–plastic behaviors of the 2618 aluminum alloy and the quasi-static and dynamic compressive elastic–plastic behaviors of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. Furthermore, the quasi-static specimens with a diameter of 5 mm and the Ti-6Al-4V titanium alloy. Furthermore, the quasi-static specimens with a diameter of 5 mm and the smallerdynamic dynamicspecimens specimenswith witha adiameter diameter 4 mm shown in Figure were to obtain smaller ofof 4 mm (as(as shown in Figure 1a) 1a) were usedused to obtain not not only the tensile elastic–plastic behavior but also the damage characteristics. In addition, the only the tensile elastic–plastic behavior but also the damage characteristics. In addition, the cylinder cylinder specimens with different notch radii and pure torsion specimens were employed to specimens with different notch radii and pure torsion specimens were employed to determine the determine the fracture behaviors stress triaxialities. Figure 1b shows with the specimens fracture behaviors under differentunder stress different triaxialities. Figure 1b shows the specimens different with different notch radii for quasi-static tensile experiments, where r is the radius of minimum notch radii for quasi-static tensile experiments, where r is the radius of the minimum the cross section, cross section, and R is the radius of the circumferential notch. The pure torsion specimens were and R is the radius of the circumferential notch. The pure torsion specimens were designed based on designed based on the Chinese standard GB/T228-2002. the Chinese standard GB/T228-2002.
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Figure 1. (a) Specimens for dynamic tensile experiments; and (b) specimens with different notches Figure 1. (a) Specimens for dynamic tensile experiments; and (b) specimens with different notches radii for quasi-static tensile experiments. radii for quasi-static tensile experiments.
3. Determination of JC Failure Parameters 3. Determination of JC Failure Parameters Quasi-static compressive experiments and tensile experiments were performed using the Quasi-static compressive experiments and tensile experiments were performed using the DNS-100 DNS-100 electronic universal testing machine, which had a maximum load capacity of 10 kN, and electronic universal testing machine, which had a maximum load capacity of 10 kN, and the quasi-static the quasi-static torsion experiments were carried out by using an electronic torsion testing machine. torsion experiments were carried out by using an electronic torsion testing machine. In addition, In addition, the high strain rate experiments were conducted under different strain rates by the high strain rate experiments were conducted under different strain rates by employing the split employing the split Hopkinson bar. The strain, the stress and the strain rate of the specimen can be calculated based on the one-dimensional elastic-stress wave theory [23], which can be shown as:
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Hopkinson bar. The strain, the stress and the strain rate of the specimen can be calculated based on the A E εcan σ s =which (3) one-dimensional elastic-stress wave theory [23], be shown as: t A
s
A (3) σs = E 2C t ε t εs = 0 A sε R dτ (4) 0 ls Z 2C0 t εs = ε R dτ (4) l 2C εs =s 00 ε R (5) ls 0 . 2C εs = εR (5) ls where ε R and ε T are the reflected and transmitted strains which can be measured by the strain where ε R and ε T are the reflected and transmitted strains which can be measured by the strain gages gages stuck on the input and output bars, respectively; C0 is the velocity of a longitudinal elastic stuck on the input and output bars, respectively; C0 is the velocity of a longitudinal elastic wave; E is wave; E is the Young’s modulus and A is the cross-sectional area of the loading bars; ls and As are the Young’s modulus and A is the cross-sectional area of the loading bars; ls and As are the length and the length and the cross-sectional area of the specimen, respectively; and σ is the engineering the cross-sectional area of the specimen, respectively; and σs is the engineerings stress and ε s is the stress and εstrain. engineering s is the engineering strain. Theoretically, Theoretically, the the variations variations of strain strain at the the circumferential circumferential area were not the the same same when when we applied the same loading speed, as shown in Figure Figure 2a. 2a. Thus, a corresponding FE model model combining combining the determine thethe loading speed for for eacheach kindkind of specimen, which was the DIC DICmethod methodwas wasperformed performedtoto determine loading speed of specimen, which supported to avoid the the strain rate effect. modelwas wasestablished establishedbyby using was supported to avoid strain rate effect.InIndetail, detail, the FE model using thethe JC JC constitutive parameters, which were obtained from the experiments, and the strain rates were constitutive parameters, which were obtained from the experiments, and the strain were calculated the DIC method, thethe maximum strain in the calculated by byusing usingthe themaximum maximumstrain-to-time strain-to-timeratio. ratio.For For the DIC method, maximum strain in strain-field-to-time ratioratio was was regarded as theasstrain rate ofrate specimens with notches. The strain the strain-field-to-time regarded the strain of specimens with notches. The levels strain were same make calculation easier, as shown the Figure Moreover, 72 images levelskept werethe kept thetosame tothe make the calculation easier, as in shown in the 2c. Figure 2c. Moreover, 72 were captured per second a high-speed camera, and the calculation chose one image every images were captured perbysecond by a high-speed camera, and the process calculation process chose one −3/s, 24 images. strain rate the quasi-static experimentexperiment was set as was 10−3set /s,as so10 the loading speeds image everyThe 24 images. Theofstrain rate of the quasi-static so the loading of specimens with cross sections of 1.5 mm, 2.0mm, mm2.0 andmm 2.5 and mm2.5 notch setwere as 0.00135 mm/s, speeds of specimens with cross sections of 1.5 mmwere notch set as 0.00135 0.0015 0.0017 mm/s,mm/s, respectively. mm/s, mm/s 0.0015 and mm/s and 0.0017 respectively.
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Figure 2. (a) Variations of strain at the circumferential area when the same loading speed was Figure 2. (a) Variations of strain at the circumferential area when the same loading speed was applied to applied to the 2618 aluminum alloy; (b) strain distribution of specimens with 1.5 mm notch for the the 2618 aluminum alloy; (b) strain distribution of specimens with 1.5 mm notch for the 2618 aluminum 2618 aluminum alloy simulated by the FE model; and (c) determination of the strain rates by the DIC alloy simulated by the FE model; and (c) determination of the strain rates by the DIC method. method.
In In general, general, the the equivalent equivalent failure failure strain strain is is calculated calculated by by [12]: [12]: d0 ε fε = 2 ln 0 f = 2ln d
(6) (6)
where where dd00 and d denote the initial and final diameters of specimens. However, the deformation of the notch complicated. Thus, the accuracy of the of equivalent failure strain wasstrain always influenced notch area areawas was complicated. Thus, the accuracy the equivalent failure was always by many factors, suchfactors, as necking uncertainty the measure. Themeasure. strain distribution specimens influenced by many suchand as necking andin uncertainty in the The strainofdistribution of specimens with 1.5 mm notch in the tensile tests was illustrated by using the FE model, as shown in Figure 2b. The simulation results clearly indicated that the largest strain concentrated on the
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with 1.5 mm notch in the tensile tests was illustrated by using the FE model, as shown in Figure 2b. surface of the circumferential area. Therefore, the equivalent failure strain was revised by The simulation results clearly indicated that the largest strain concentrated on the surface of the introducing the complete damage strain, which was defined as the maximum strain on the surface circumferential area. Therefore, the equivalent failure strain was revised by introducing the complete before the fracture. damage strain, which was defined as the maximum strain on the surface before the fracture. Furthermore, the Bridgman method to determine the stress triaxialities was also revised, Furthermore, the Bridgman method to determine the stress triaxialities was also revised, caused by caused by the fact that the stress triaxiality was not constant during the tensile deformation. the fact that the stress triaxiality was not constant during the tensile deformation. Consequently, the FE Consequently, the FE models were also established to revise the stress triaxialities. Figure 3a models were also established to revise the stress triaxialities. Figure 3a sketches the relationships sketches the relationships between the stress triaxiality and the strain for three kinds of specimen. between the stress triaxiality and the strain for three kinds of specimen. As is shown, the initial As is shown, the initial stress triaxialities in the FE model were the same as the stress triaxialities stress triaxialities in the FE model were the same as the stress triaxialities calculated by the Bridgman calculated by the Bridgman equation, and each of the stress triaxialities increased to a stable value equation, and each of the stress triaxialities increased to a stable value accompanied with the plastic accompanied with the plastic deformation. The stress triaxialities of the tensile specimens with deformation. The stress triaxialities of the tensile specimens with notches can be revised by replacing notches can be revised by replacing the Bridgman’s results with the stable stress triaxialities the Bridgman’s results with the stable stress triaxialities calculated by FE model. In addition, the stress calculated by FE model. In addition, the stress triaxiality of the smooth specimen was revised by triaxiality of the smooth specimen was revised by considering the stress triaxiality before fracture, considering the stress triaxiality before fracture, as shown in Figure 3b. Results from the tensile as shown in Figure 3b. Results from the tensile experiments showed the increase of stress triaxiality was experiments showed the increase of stress triaxiality was almost uniform after the necking almost uniform after the necking happened. Thus, the revised stress triaxiality can be calculated by: happened. Thus, the revised stress triaxiality can be calculated by: a t t−2t− t1 1 1 1 1 ∗ a * · 2 1 σ = + σ 3= + +2 lnln 11+ (7)(7) ⋅ 3 2 22R R t2 t2 where tt11 is when thethe necking starts, and tand fracture 2 is the where is the thetime timespent spent when necking starts, t2 istime thespent time when spentthe when the happens. fracture By calculating the pixels, Equation (7) can be replaced with: happens. By calculating the pixels, Equation (7) can be replaced with: 1 11 1 N00 t −t2t − t1 N ∗ * 2 1 σ = σ 3=+ 2+ lnln 11++ 4N ⋅ · (8)(8) 3 2 4 N11 t2 t2 Finally, the model waswas revised by establishing the relationships betweenbetween the complete Finally, theJC JCdamage damage model revised by establishing the relationships the damage strain, the revised stress triaxiality, the strain rate and the temperature. complete damage strain, the revised stress triaxiality, the strain rate and the temperature.
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Figure 3. (a) Relationships between the stress triaxiality and the strain for the specimens with three Figure 3. (a) Relationships between the stress triaxiality and the strain for the specimens with three kinds of notches; and (b) schematic diagram for the revised strain triaxiality of the smooth kinds of notches; and (b) schematic diagram for the revised strain triaxiality of the smooth quasi-static quasi-static tensile specimens. tensile specimens.
As mentioned above, the maximum strain concentrated on the surface of the circumferential As mentioned above, the maximum strain concentrated on the surface of the circumferential area. Therefore, the DIC method was applied to measure the strain field during experiments, and area. Therefore, the DIC method was applied to measure the strain field during experiments, and the the maximum strain in the strain field before fracture calculated by the picture was considered as maximum strain in the strain field before fracture calculated by the picture was considered as the the complete damage strain, as shown in Figure 4a. It is worth mentioning that the measurement of complete damage strain, as shown in Figure 4a. It is worth mentioning that the measurement of the complete damage strain faced the challenge caused by the minor area of the notch. The micro the complete damage strain the challenge the minor of the notch.byThe micro speckle was performed usingfaced an airbrushing toolcaused and anbyenamel paintarea manufactured GUNZE Japan, as shown in Figure 4b.
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speckle was performed using an airbrushing tool and an enamel paint manufactured by GUNZE Japan, as shown 4b.REVIEW Metals 2018, in 8, xFigure FOR PEER 6 of 12
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Figure 4. (a) Strain fields calculated by using the DIC method and micro speckles; and (b) heating Figure 4. (a) Strain fields calculated by using the DIC method and micro speckles; and (b) heating furnace with high-temperature resistant glass, an airbrushing tool and an enamel paint. furnace with high-temperature resistant glass, an airbrushing tool and an enamel paint.
Dynamic tensile tests were performed to get the parameter D4 of the revised JC damage model. Dynamic tensile testsinduced were performed to get the of the revised JC damage The multiple-stress-wave fracture needed to parameter be avoidedDto4 ensure the accuracy of themodel. strain The multiple-stress-wave induced be avoided ensure the accuracy of the strainthe at at the fracture. Thus, the pulsefracture widthneeded of thetostress wavetowas improved to guarantee the fracture. Thus, the pulse width of the stress wave was improved to guarantee the single-stress-wave single-stress-wave induced fracture, which was validated by checking the high-speed video. The induced fracture, which was validatedthe byinitial checking the high-speed video. DIC method DIC method was also used to calculate damage strain instead of toolThe measuring, such was as a also used to calculate the initial damage strain instead of tool measuring, such as a vernier caliper. vernier caliper. Afterwards, the parameter D5 of the revised JC damage model was determined Afterwards, thequasi-static parameter Dtensile revised JC damage based the quasi-static 5 of thetests based on the at 373 K, 473model K andwas 573determined K. For both theon quasi-static and tensile tests at 373 K, 473 K and 573 K. For both the quasi-static and dynamic tests at high temperatures, dynamic tests at high temperatures, two heating furnaces with the high-temperature resistant glass two heating furnaces the the high-temperature were not only to heattothe specimen, were used not only with to heat specimen, butresistant also to glass provide anused observation port capture the but also to provide an observation port to capture the pictures. Furthermore, a blue light was added to pictures. Furthermore, a blue light was added to increase the image definition of the micro speckle increase the image definition of the speckle in the high-temperature It is in worthy mentioning in the high-temperature tests. It ismicro worthy mentioning that the failuretests. strains the torsion tests that the failure strains in the torsion tests were calculated by using the shear fracture strain as follows: were calculated by using the shear fracture strain as follows: √ ε f = γf / 3 (9) εf =γf / 3 (9) where γ f is the shear fracture strain which can be calculated by a torsion angle, and this method is the shear fracture which can be calculated by a torsion along angle,the and thisdirection method where γ the based on that thestrain deformation of specimens is homogeneous axial f isassumption until fracture. is based on the assumption that the deformation of specimens is homogeneous along the axial direction until fracture. 4. Results and Discussion 4. Results Discussion Figureand 5 shows the true stress versus the true strain curves of the 2618 aluminum alloy and the Ti-6Al-4V titanium tested at different strain rates. In curves addition, the 2618 fittedaluminum curves were alsoand added Figure 5 showsalloy the true stress versus the true strain of the alloy the for comparison. The parameters of the JC constitutive model [3] are listed in Table 1 based on an Ti-6Al-4V titanium alloy tested at different strain rates. In addition, the fitted curves were also equation as follows: added for comparison. The parameters of the JC constitutive model m [3] are listed in Table 1 based on .∗ σ = [ A + Bεn ][1 + CInε ][1 − T ∗ ] (10) an equation as follows: .∗
m where σ and ε are the effective stress plastic (10) σ =and [ A +effective Bε n ][1 + CIn ε* ][1strain, − T * ] respectively. ε is the normalized effective plastic strain rate; n denotes the work hardening exponent; A, B, C and m are the constants of where the effective stressas:and effective plastic strain, respectively. ∗ is the σ and the material; andεtheare quantity T* is defined normalized effective plastic strain rate; n denotes the work hardening exponent; A, B, C and m are T∗ = (T − /( Tm − T (11) the constants of the material; and the quantity T*Tis as: r )defined r)
T * = (T − Tr ) / (Tm − Tr )
(11)
where Tm is the melting temperature, and Tr is the reference temperature which is typically taken as 293 K, and the reference strain rates for both the two materials were set as 0.001. The FE models for the revision of strain rates and stress triaxialities were conducted by using these parameters.
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where Tm is the melting temperature, and Tr is the reference temperature which is typically taken as 293 K, and the reference strain rates for both the two materials were set as 0.001. The FE models for the Metals 8, x FOR PEER REVIEW revision of2018, strain rates and stress triaxialities were conducted by using these parameters. 7 of 12
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Figure 5. (a) True stress versus true strain curves of the 2618 aluminum alloy; (b) fitting of
Figure 5. (a) True stress versus true strain curves of the 2618 aluminum alloy; (b) fitting of temperature temperature sensitivity for the 2618 aluminum alloy; (c) true stress versus true strain curves of the sensitivity for the 2618 aluminum alloy; (c) true stress versus true strain curves of the Ti-6Al-4V Ti-6Al-4V titanium alloy; and (d) fitting of temperature sensitivity for the Ti-6Al-4V titanium alloy. titanium alloy; and (d) fitting of temperature sensitivity for the Ti-6Al-4V titanium alloy. Table 1. JC constitutive parameters of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy.
Table 1. JC constitutive parameters of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. Parameters A (MPa) 2618 Aluminum alloy 360 Parameters A (MPa) Ti-6Al-4V titanium alloy 1019.5
B 315 B 674.1
C m n 0.003 2.4 0.44 C m 0.03 0.457 0.92
n
2618 Aluminum alloy 360 315 0.003 2.4 0.44 Ti-6Al-4V alloy between 1019.5the complete 674.1 0.03 Figure 6 shows thetitanium relationships damage0.457 strain, 0.92 the revised stress triaxiality, the strain rate and the temperature of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. As be seen, the between complete the damage strains of both strain, two materials decreased the Figure 6 shows thecan relationships complete damage the revised stressas triaxiality, stress triaxialities increased. However, the equivalent failure strains were relatively larger than the the strain rate and the temperature of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. complete damage strains, which were calculated by the DIC method, and this tendency was As can be seen, the complete damage strains of both two materials decreased as the stress triaxialities weakened with the increase of stress triaxiality. It was considered that the necking effect influenced increased. However, thefailure equivalent strains relativelyequations. larger than thethe complete the accuracy of the strainfailure calculated by were the Bridgman Take case of damage the strains, which were calculated the DICThe method, andmorphologies this tendency was weakened with the increase Ti-6Al-4V titanium alloy forbyexample. fracture indicated that the failure mode of stress It was considered that necking effectwith influenced the accuracy of the failure was triaxiality. changed from the ductile fracture to the the brittle fracture the increase of stress triaxiality. obvious by dimples were found in the ductile extension of the specimen notch, strainThe calculated the Bridgman equations. Take the case ofregion the Ti-6Al-4V titaniumwithout alloy for example. which were hardly seenindicated in the specimen the 1.5 mmwas notch, as shown Figure 7a,b.fracture In The fracture morphologies that thewith failure mode changed frominthe ductile a transient crack existed dimples became shallowdimples and less were obvious in thein the to theaddition, brittle fracture with thezone increase ofwhere stressthe triaxiality. The obvious found center locationregion of the fracture, which proved that the complete damage from the ductile extension of the specimen without notch, which were strains hardlycalculated seen in the specimen surfaces of the specimens avoided the error caused by the change of fracture mode. In addition, the with the 1.5 mm notch, as shown in Figure 7a,b. In addition, a transient crack zone existed where complete damage strains increased with the strain rate, and the toughness of material decreased the dimples became shallow and less obvious in the center location of the fracture, which proved under high strain rates. Figure 7c shows the fracture morphology of smooth tensile specimen under that the damage strains surfaces of the specimens avoided the error 2000complete s−1. The secondary cracks calculated were found from in thethe fracture, which indicated that the specimens
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caused by the change of fracture mode. In addition, the complete damage strains increased with the strain rate, and the toughness of material decreased under high strain rates. Figure 7c shows the fracture morphology of smooth tensile specimen under 2000 s−1 . The secondary cracks were found Metals 2018, 8, x FORindicated PEER REVIEW 8 of 12 in the fracture, which that the specimens suffered from the tension–shear coupling in the high strain rate tests. Consequently, the complete damage strain increased since the tension–shear suffered from the tension–shear coupling in the high strain rate tests. Consequently, the complete couplingdamage induced the decreasing stress triaxiality. Finally, the revised JC failure parameters of the strain increased since the tension–shear coupling induced the decreasing stress triaxiality. 2618 aluminum alloy and the Ti-6Al-4V titanium alloy obtained the results, as listed in Finally, the revised JC failure parameters of the 2618were aluminum alloy by andfitting the Ti-6Al-4V titanium Table 2. alloy were obtained by fitting the results, as listed in Table 2.
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Figure 6. (a) Relationship between the complete damage strain and the revised stress triaxiality of the
Figure 6.2618 (a) Relationship the complete damage strain andstrain the revised stress stress triaxiality of aluminum alloy;between (b) relationship between the complete damage and the revised the 2618 aluminum alloy; (b) relationship between the complete damage strain and the revised triaxiality of the Ti-6Al-4V titanium alloy; (c) relationship between the complete damage strain and stress rate of the titanium 2618 aluminum (d) relationship between the the complete strain and and the triaxialitythe of strain the Ti-6Al-4V alloy;alloy; (c) relationship between completedamage damage strain the strain rate of the Ti-6Al-4V titanium alloy; (e) relationship between the complete damage strain strain rate of the 2618 aluminum alloy; (d) relationship between the complete damage strain and the strain rate of the Ti-6Al-4V titanium alloy; (e) relationship between the complete damage strain and the temperature of the 2618 aluminum alloy; and (f) relationship between the complete damage strain and the temperature of the Ti-6Al-4V titanium alloy.
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temperature of the 2618 aluminum alloy; and (f) relationship between the complete damage 9 of 12 Metalsand 2018,the 8, 396 strain and the temperature of the Ti-6Al-4V titanium alloy.
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Figure 7. (a) Fracture morphology in the ductile extension region of the smooth tensile specimen; Figure 7. (a) Fracture morphology in the ductile extension region of the smooth tensile specimen; (b) (b) fracture morphology in the ductile extension region of the specimen with the 1.5 mm notch; and (c) fracture morphology in the ductile extension region of the specimen with the 1.5 mm notch; and (c) fracture morphology of smooth tensile specimen under 2000 s−1 . fracture morphology of smooth tensile specimen under 2000 s−1. Table 2. Revised JC failure parameters of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. Table 2. Revised JC failure parameters of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. Parameters Parameters
D1D1
2618 aluminum alloy 0.032 0.032 2618 aluminum alloy Ti-6Al-4V titanium alloy Ti-6Al-4V titanium alloy 0.021 0.021
D D22
0.662 0.662 0.132 0.132
D33 D DD4 4 D 5 D5 −1.771 −1.771 0.0131 0.0131 0.867 0.867 −1.1 0.0238 0.0238 3.451 3.451 −1.1
5. Validation 5. Validation In previous study, study,ballistic ballisticimpact impacttests testson on2222mm mm 2618 aluminum plates were carried with In previous 2618 aluminum plates were carried outout with two two kinds of Ti-6Al-4V titanium projectiles to investigate the impact response at a nominal kinds of Ti-6Al-4V titanium alloy alloy projectiles to investigate the impact response at a nominal velocity velocity of 210m/s. Considering the excellent ability of nonlinear analysis, we developed twoelement finite of 210 m/s. Considering the excellent ability of nonlinear analysis, we developed two finite element models by using the commercial software ABAQUS to validate the accuracy of the revised models by using the commercial software ABAQUS to validate the accuracy of the revised parameters. parameters. The mesh pattern and size dependency were at first for the an purpose of The mesh pattern and size dependency were examined at first examined for the purpose of finding optimized finding an optimized mesh for stability, accuracy, and efficiency of the impact analysis. Both the mesh for stability, accuracy, and efficiency of the impact analysis. Both the Ti-6Al-4V titanium alloy Ti-6Al-4V and titanium alloy projectilealloy and plate the 2618 alloy the plate were modeled with the projectile the 2618 aluminum werealuminum modeled with eight-node underintegrated eight-nodesolid underintegrated hexagonal solid elements (C3D8R). in-plane mesh pattern wasand kept hexagonal elements (C3D8R). The in-plane mesh pattern wasThe kept the same for all models, the the same for all models, and the central part of the target was divided into three different regions, central part of the target was divided into three different regions, as shown in Figure 8. In addition, as shown in Figure 8. In addition, the mesh density gradually coarsening from the inner region the mesh density was gradually coarsening from thewas inner region (the potential impact region) to the (the potential impact region) to the outer. Mesh transition between regions was good enough outer. Mesh transition between regions was good enough to prevent stress wave reflections from to the prevent stress wave reflections from the regions. Simply refining mesh does boundary of regions. Simply refining the boundary mesh doesofnot necessarily improve thethe accuracy sincenot the necessarily improve the accuracy the material-model parameters were calibrated for a specific material-model parameters were since calibrated for a specific mesh size and a mode of failure changes mesh size and a mode of failure changes during the simulation. Furthermore, there was no clear during the simulation. Furthermore, there was no clear theoretical guideline on the required mesh theoretical guideline on the required mesh density for the range of impacts covered in this study. density for the range of impacts covered in this study. On account of softening effects and mesh On account of softening effects and mesh dependent failure algorithms, the only reasonable dependent failure algorithms, the only reasonable methodology to find the most appropriate mesh size methodology to find the most appropriate mesh size was trial and error, while the results against a was trial and error, while the results against a controlled test data were compared. Subsequently, it was controlled test data were compared. Subsequently, it was possible to arrive at an optimum after possible to arrive at an optimum after meshing were used, and some guidelines for that particular meshing were used, and some guidelines for that particular case were drawn. Thus, the thickness of case were drawn. Thus, the thickness of each target was modeled with one mesh density in this study, each target was modeled with one mesh density in this study, while more similar experiments were while more similar experiments were performed during the calibration. For mesh patterns in the performed during the calibration. For mesh patterns in the thickness direction, the number of thickness direction, the number of through thickness elements was chosen to be 11 for the 22 mm through thickness elements was chosen to be 11 for the 22 mm target as a baseline, since target as a baseline, since implementation of reduced integration solid elements required at least implementation of reduced integration solid elements required at least three elements through the three elements through the thickness to be able to capture the bending deformation modes accurately. thickness to be able to capture the bending deformation modes accurately. A viscosity based A viscosity based stabilization method was used during the simulations to prevent hourglass modes stabilization method was used during the simulations to prevent hourglass modes of the reduced of the reduced integration elements. A contact behavior between the projectile and the target was integration elements. A contact behavior between the projectile and the target was attained by attained by using a penalty based single-surface-type contact algorithm that used a nodal constraint using a penalty based single-surface-type contact algorithm that used a nodal constraint formulation with an element erosion scheme. formulation with an element erosion scheme.
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(a) (a)
(b)
Figure Ti-6Al-4V solid solid plate plate Figure8.8.(a) (a)Exact Exactmesh mesh used used for for the the 2618 2618 aluminum aluminum alloy alloy plates and the Ti-6Al-4V Figure 8. (a) Exact mesh used for the 2618 aluminum alloy plates and the Ti-6Al-4V solid plate projectile projectileand andaahollow hollowblade bladeprojectile; projectile;and and(b) (b) exact exact mesh mesh used for the 2618 aluminum projectile aluminum alloy alloy plates plates and a hollow blade projectile; and (b) exact mesh used for the 2618 aluminum alloy plates and the andthe theTi-6Al-4V Ti-6Al-4Vhollow hollowblade bladeprojectile. projectile. and Ti-6Al-4V hollow blade projectile.
Byusing usingthe theparameters parametersof ofthe the revised revised JC JC failure failure criteria criteria in ABAQUS, the By the impact impact responses responsesof of By using the parameters of the revised JC2618 failure criteriaalloy in ABAQUS, the impact responses the Ti-6Al-4V titanium alloy projectile and the aluminum plate were examined based on the Ti-6Al-4V titanium alloy projectile and the 2618 aluminum were examined based on of the Ti-6Al-4V titanium alloy projectile andimpacted the 2618 aluminum alloy plate were examinedblade based explicit analysis scheme. For the plates plates ananexplicit analysis scheme. For the impacted by solid plate projectiles projectiles and and hollow hollow blade on an explicit analysis scheme. For the plates impacted by solid plate projectiles and hollow blade projectiles,simulations simulationsillustrated illustratedthat that there there was was no no perforation perforation on the targets, projectiles, targets, which which was was the the same same projectiles, simulations As illustrated there no perforation on morphology the targets, which was the same as the experiments. experiments. shown that in Figure Figurewas 9a,b, the damage damage asas the As shown in 9a,b, the morphology and and the the size size were were the experiments. As shown in Figure 9a,b, the damage morphology and the size were comparatively comparatively coincident coincident with with the the experiments. experiments. Moreover, Moreover, the the cracks comparatively cracks on on the the back back of of the the 2618 2618 coincident with theplates experiments. Moreover, the cracks on the the backexperiments of the 2618 aluminum alloy plates aluminum alloy were predicted as well, and both and the simulations aluminum alloy plates were predicted as well, and both the experiments and the simulations were predicted as well, andthat boththe thepossible experiments and the simulations indicated thedirection same result the that indicated thesame same result initiating origin and the indicated the result that the possible initiating origin and the propagation propagation direction of of the the possible initiating origin and the propagation direction of the cracks for the 2618 aluminum alloy cracksfor forthe the2618 2618aluminum aluminum alloy alloy plates plates impacted impacted by by solid solid plate plate projectiles cracks projectiles and and hollow hollow blade blade plates impacted by solid plate projectilesthe andfailure hollow bladethe projectiles were different. In conclusion, projectiles were different. In conclusion, mode, dent formation, the perforation shape projectiles were different. In conclusion, the failure mode, the dent formation, the perforation shape the failure mode, the dent formation, perforation shape thedamage 2618 aluminum alloy plates in the the2618 2618 aluminum alloy plates in inthe the simulations and of the ofofthe aluminum alloy plates the simulations and the damage of of the the Ti-6Al-4V Ti-6Al-4V titanium titanium simulations and the damage of the well Ti-6Al-4V titanium alloywhich projectiles all corresponded well to the alloy projectiles all corresponded to the test results, was attributed to alloy projectiles all corresponded well to the test results, which was attributed to the the revised revised JC JC test results, which was attributed to the revised JC failure parameters. failure parameters. failure parameters.
(a) (a)
(b) (b)
Figure 9. (a) Damage morphologies of the 2618 aluminum alloy plates impacted by the Ti-6Al-4V Figure 9. 9. (a) (a) Damage aluminum alloy plates impacted by the solid Figure Damage morphologies morphologiesofofthe the2618 2618 aluminum alloy plates impacted by Ti-6Al-4V the Ti-6Al-4V solid plate projectiles and hollow blade projectiles; and (b) damage morphologies of the Ti-6Al-4V plate plate projectiles and hollow bladeblade projectiles; and (b) damage morphologies of the Ti-6Al-4V solid solid andthe hollow projectiles; solid plateprojectiles projectile and hollow blade projectile.and (b) damage morphologies of the Ti-6Al-4V plate projectile and the hollow blade projectile. solid plate projectile and the hollow blade projectile.
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6. Summary and Conclusions A new experimental and numerical method was carried out to determine the revised JC failure parameters of the 2618 aluminum alloy and the Ti-6Al-4V titanium alloy. The specimens with different notches were designed, and then the loading speeds, the strain distributions and the stress triaxialities were all analyzed and revised by using the FE model. Results clearly indicated that the largest strain concentrated on the surface of the circumferential area. Thus, the traditional failure strain in the JC failure model was replaced by the complete damage strain, which was defined as the largest strain before fracture. Therefore, the DIC method and the micro speckle were conducted to measure the strain field. A series of quasi-static and dynamic experiments under different temperatures were conducted to support the establishment of the relationships between the complete damage strain, the revised stress triaxiality, the strain rate and the temperature. The results showed the Bridgman failure strains were relatively larger than the complete damage strains, which were calculated by the DIC method, and this tendency was weakened with the increase of stress triaxiality. In addition, the complete damage strains increased with the strain rate, and the toughness of material decreased under high strain rates, since the tension–shear coupling induced the decreasing stress triaxiality. Finally, the comparisons between the ballistic impact tests and simulations were conducted. The failure mode, the dent formation, the perforation shape of the 2618 aluminum alloy plates and the damage of the Ti-6Al-4V titanium alloy projectiles in the simulations all corresponded well to the experiments, thus proving the accuracy of the revised JC failure parameters. Author Contributions: Conceptualization, Y.L. and Z.T.; data curation, C.W.; formal analysis, C.W. and Y.L.; investigation, C.W.; methodology, T.S. and P.X.; project administration, T.S. and Y.L.; resources, T.S. and P.X.; validation, C.W.; writing of the original draft, C.W. Funding: This research was funded by National Natural Science Foundation of China: Nos. 11772268, 11522220 and 11527803. Acknowledgments: The authors would like to acknowledge Liang Hou from AECC Commercial Aircraft Engine Co., Ltd. and Xiang Wang from Jiangsu Tie Mao Glass Co., Ltd. in China. Conflicts of Interest: The authors declare that there are no conflicts of interest regarding the publication of this article.
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