Optimized Design and Energy Absorption of TPU Material Based on Hierarchical Structure
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摘要: 选择柔性热塑性聚氨酯（TPU）为原料制备实验试样。基于面心立方层级结构，改变结构的直梁样式、间距，并将蜂窝结构层引入结构中，通过实验和有限元分析，研究层级结构在准静态加载下的变形模式和吸能特性。实验结果与有限元分析得到的载荷-位移曲线吻合较好。结果表明：与初始面心立方层级结构相比，增大直梁的振幅，调整直梁间距为2 mm，即增加塑性铰个数时，又或将蜂窝结构层引入层级结构，可以大幅提高层级结构的能量吸收性能；减小间距、增多塑性铰数目对改善层级结构吸能能力的效果最优；D-1构型与M-1构型相比，比吸能提高了46%；调整塑性铰位置时，层级结构会发生屈曲，不利于层级结构的能量吸收；与M-1构型相比，D-2和D-3构型的比吸能分别降低了27%和34%。Abstract: In this paper, the deformation behavior and energy absorption performance of the hierarchical structure were investigated based on the face-centered cubic hierarchical structure to change the amplitude and distance of the hierarchical structure’s straight beams and introduce the honeycomb structure layer. The quasi-static compression behavior of the proposed hierarchical structure was carried out through finite element methods (FEM) and experimental tests. Thermoplastic polyurethanes (TPU) as the raw material was used to prepare the experimental samples. The force-displacement curves obtained from the FEM and experimental tests are consistent with each other. Compared with the original face-centered cubic layered structure, the conclusion can be obtained: the new hierarchical structure has better energy absorption performance than the face-centered cubic hierarchical structure. The introduction of sinusoidal beams, the increase of plastic hinges, the adjustment of the angle between layers, and the introduction of honeycomb structure layers can enhance the energy absorption ability.
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The corrugated interface between different fluids grow when accelerated from a low-density fluid to a high-density fluid, which is called Rayleigh-Taylor (RT) instability[1-2].This phenomenon may also occur in metals, but usually under a high pressure and at a high strain-rate, but differs most distinctly from the RT instability in fluids in its strength effect of the metal, which can stabilize the perturbation growth[3-4] and make the metallic RT instability more complex and difficult.Here, it is also affected by the loading state and the properties of the metallic materials.The metallic RT instability at high pressure and strain-rate can be observed in inertial confinement fusion[5], supernova explosion[6], asteroid collision[7], the motion of earth's inner core and plate tectonics[8], and so on.Therefore, the metallic RT instability is currently a major concern for researchers and receives a great deal of academic attention.
In theoretical studies about the metallic interface instability, dispersion relations of the perturbation growth are derived mainly based on the energy[9-11] or force equilibrium[12-13].However, the previous linear analysis cannot predict the perturbation growth accurately just by applying the perfect plastic constitutive relation and constant pressure loading state.Based on the energy balance, a perturbation growth equation using Steinberg-Guinan (SG) and the Johnson-Cook constitutive models, as well as a variable pressure loading process consistent with experiments, has been derived that precisely predicts the growth of metallic RT instabilities driven by detonation and laser plasma.However, the linear analysis still has its limitations and does not take full account of the loading procedure.
Experimental studies of the metallic RT instability started in the 1970s.The pioneering experimental research[14] was the perturbation growth of a flat aluminum plate accelerated by the expansion of detonation products, which was observed using a high-energy X-ray facility.What was achieved then inspired researchers, and the similar equipment was utilized in later research[15-17].In the USA and Russia particularly, numerous numerical simulations and experimental investigations for the metallic RT instability have been carried out, but have mainly concentrated on the perturbation growth and such influencing factors as the initial amplitude, the wavelength and material properties.Igonin and Ignatova et al.[18-19] experimentally and numerically studied the dynamic behaviors of copper (Cu) and tantalum (Ta) subjected to both shock and shockless loading by employing a perturbation growth method.They observed that the formation of the bi-periodic twin structures resulted in an initial loss of the shear strength of Cu, but failed to observe localization in Ta.Olson et al.[16] experimentally studied the effects of the grain size and material processing on the RT perturbation growth of Cu.They found that both the single-crystal orientation and the strain hardening due to the material processing can affect the perturbation growth, but the polycrystalline grain size cannot.For the plane detonation, the loading pressure is generally about 30 GPa.To enhance the loading pressure, Henry de Frahan et al.[17] studied the beryllium RT instability using an iron flyer plate to impact the second high explosive (HE) to raise the pressure to 50 GPa in their experiments, and combined numerical simulations to calibrate the feasibility of different constitutive models.When the sample is driven by electromagnetism[20-21] or laser[4, 22], the loading pressure can be further increased.Very extreme conditions of pressures over 1 000 GPa and strain rates of 108 s-1 have been achieved at the National Ignition Facility, USA, where the RT instability experiment in vanadium was carried out, and constitutive models in solid phase were tested by comparing simulations with experiments measuring the perturbation growth[23] under the extreme conditions mentioned.
In the metallic interface instability, the perturbation growth is related to and arrested by the material strength.Moreover, some investigations have demonstrated that the material strength increases under these extreme conditions.Results from the metallic RT experiments and computations by Barnes et al.[14] show that the yield strength of 1100-0 aluminum is over 10 times larger than the standard parameter, and the yield strength of 304 stainless steel also increases by more than three times.Using the SG constitutive strength model, calculations of plasma-driven quasi-isentropic RT experiments of Al-6061-T6 using the Omega laser at a peak drive pressure of 20 GPa indicate that its yield strength is a factor of about 3.6 times over the ambient value[22].In Park et al.'s[4] plasma-driven quasi-isentropic polycrystalline vanadium RT experiments using the Omega laser with a peak drive pressure of 100 GPa, the measured RT growth was substantially lower than predictions using the existing constitutive models (SG and Preston-Tonks-Wallace) that work well at low pressures and long timescales.Using the SG model, the simulations agree with the RT experimental data when the initial strength is raised by a factor of 2.3.Therefore, the SG and Preston-Tonks-Wallace models underestimate the strength of vanadium under very high pressures and strain rates.Belof et al.[24] first measured the dynamic strength of iron undergoing solid-solid phase transition by using RT instability.In conjunction with detailed hydrodynamic simulations, the analysis results revealed significant strength enhancement of the dynamically generated ε-Fe and reverted α′-Fe, comparable in magnitude to the strength of austenitic stainless steels.Therefore, the metallic RT instability was suggested and used as a tool for evaluating the material strength of solids at high pressures and high strain rates[3, 25], and then for modifying or developing new constitutive models for these conditions[26-27].
In view of the dominant role of the material strength in metallic interface instabilities, and the limitations of existing constitutive models at high pressures and high strain rates, we aimed to investigate the material strength and its effects on metallic interface instabilities.In this paper, we also conducted an RT instability experiment in explosion-driven aluminum, and measured the perturbation growth using X-ray radiography.In combination with elastic-plastic hydrodynamic simulations, we investigated the dynamic behavior of metallic RT instabilities and the role of the material strength in these.
1. Experimental Setup
Following that of Barnes et al., [14]our experiment used the setup as shown in Fig. 1(a), where we have a sketch of the experimental setup consisting of a detonator, a booster, plane wave lens, JO-9159 HE (100 mm in diameter and 50 mm in thickness), an aluminum sample, and a vacuum.Fig. 1(b) shows the experimental sample of aluminum with a diameter of 65 mm and a thickness of 1.5 mm in the central region.An initial sinusoidal perturbation was machined on the side of the aluminum sample facing the HE.The perturbation amplitude and wavelength were 0.25 mm and 6 mm, respectively.The HE products crossed the void of 3.5 mm between the sample and HE and accumulated on the perturbation interface of the sample, providing a smooth rise to peak pressure and a quasi-isentropic drive.Moreover, the void between the sample and HE can ensure that the temperature of the sample at high pressures remain below the melting point[22].
Figure 1. Sketch of the experimental setup and sampleIn the experiment, X-ray radiography was used to record the evolution of the perturbation interface from the JO-9159 explosive detonation at zero time.A Doppler pin system was used to measure the history of the free surface velocity, which can be integrated to obtain the corresponding free surface displacement.Fig. 1(c) shows the distribution of measurement points of the free surface velocity, where both points 1 and 2 correspond to the wave trough positions with one wavelength interval, and point 3 corresponds to the wave crest position with 1.5 wavelength intervals from point 2.
2. Numerical Methods
Based on our indoor hydrodynamic code of compressible multi-viscous flow and turbulence (MVFT)[28-30], we developed a detonation and shock dynamics code with high fidelity by considering the explosive detonation and the elastic-plastic behavior of the material.This code can be used to study the physical problem of multi-materials, large deformation, and strong shock.The governing equations in conserved form are as follows
{∂∂t∫VρdV=−∮SρuinidS∂∂t∫VρujdV=−∮SPnjdS−∮SρuiujnidS+∮SsijnidS∂∂t∫VρEdV=−∮SuiPnidS−∮SρuiEnidS+∮SsijujnidS (1) where i and j represent the x, y, and z directions; V is the control volume, S the surface of control volume, n the unit vector of the external normal direction, ρ, uk (where k=i, j), p, and E are the density, velocity, pressure, and total energy of per unit mass; and sij the deviation stress tensor.
The physical problem as described by Eq.(1) was decomposed into three one-dimensional problems.For each of them, the physical quantities in a cell were interpolated and reconstructed using a piecewise parabolic method (PPM).The one-dimensional problem was then resolved using a two-step Euler algorithm:First the physical quantities were solved by the Lagrange matching, and then remapped back to the stationary Euler meshes.The effect of material strength, explosive detonation, and artificial viscosity were implemented in the Lagrange step.The multi-material interface was captured by applying a volume-of-fluid (VOF) method.
2.1 Equation of State
In our numerical simulations, the equation of state (EOS) for the explosive and aluminum are the Jones-Wilkins-Lee (JWL) and Mie-Grüneisen equations of state, respectively.The Jones-Wilkins-Lee equation of state is
p(ρ,T)=A(1−ωR1v)e−R1v+B(1−ωR2v)e−R2v+ωˉEv (2) where v=ρ0/ρ is specific volume; A, B, R1, R2, and ω are constants; and E is the internal energy per unit volume.Table 1 lists the JWL EOS parameters of the JO-9159 explosive.The Mie-Grüneisen equation of state is
Table 1. Equation of state parameters of JO-9159 explosiveρ/(g·cm-3) pCJ/GPa DCJ/(km·s-1) A/GPa B/GPa R1 R2 ω 1.86 36 8.862 934.8 12.7 4.6 1.1 0.37 | Show Table
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p=ρ0c2μ[1+(1−γ0/2)μ−aμ2/2][1−(S1−1)μ−S2μ2μ+1−S3μ3(μ+1)2]2+(γ0+aμ)ˉE (3) where μ=ρ/ρ0-1 is the relative compression, ρ0 the initial density, c the sound velocity at zero pressure, γ0 the Grüneisen coefficient, and a, S1, S2, and S3 are constants (in Table 2).
Table 2. Mie-Grüneisen equation of state parameters of aluminumρ/(g·cm-3) c/(km·s-1) γ0 a S1 S2 S3 2.703 5.22 1.97 0.47 1.37 0 0 | Show TableDownLoad:
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2.2 Constitutive Model
In our simulations, the elastic-plastic behavior of aluminum at high pressures and high strain rates was described using the SG constitutive model.The SG model introduces pressure, temperature, and strain-rate terms into the elastic-plastic constitutive equation, while the coupling effect of pressure and strain rate on flow stress was characterized by the separating variables.Additionally, as the flow stress in the SG model relies on pressure, there is a coupling relationship between the material constitutive equation and the equation of state, which indicates the feature of pressure hardening of metal under high pressure.The dynamic yield strength YSG and the shear modulus G determined by the SG model are expressed as
YSG=Y0[1+β(εp+εi)]n[1+Apη−1/3−B(T−300)] (4) G=G0[1+Apη−1/3−B(T−300)] (5) where Y0 and G0 are the initial yield strength and the shear modulus, respectively; β and n are the material strain hardening coefficient and the hardening index, respectively; A is the pressure hardening coefficient; η=ρ/ρ0 is the material compression ratio; and B is the temperature softening coefficient (in Table 3).
Table 3. Steinberg-Guinan constitutive model parameters of aluminumY0/GPa Ymax/GPa G0/GPa β n A/GPa-1 B/(10-3K-1) 0.29 0.68 27.6 125 0.1 0.0652 0.616 | Show TableDownLoad:
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3. Results and Discussions
In our experiment, X-ray radiography recorded an image of the perturbed interface at 7.78 μs, as shown in Fig. 2(a), from which we obtained the amplitude of 0.77 mm simultaneously by image processing.In the simulations, the mesh size was 15.6 μm, and Tables 2 and 3 list the parameters of the Mie-Grüneisen EOS and the SG constitutive model of aluminum, respectively.
Figure 2. Comparisons of the perturbed interface between experiment and numerical simulations ((a) Experimental image, (b) Simulated image at normal strengths Y0 and G0, (c) Simulated image at 10 times the normal strengths Y0 and G0)Fig. 3 shows the pressure histories of the crest (solid line) and the trough (dashed line) on the loading surface, which increase continuously and smoothly in a short time and form an approximate quasi-isentropic drive.Afterwards, the expansion of the detonation products decelerates gradually, and the loading pressure on the interface rises slowly.However, the pressure at the trough ascends faster than that at the crest, and the peak pressure at the trough is also relatively larger, because the detonation products converges at the trough and diverges at the crest.The average peak pressure is about 25 GPa, and the strain rate reaches 106 s-1.The loading pressure then reduces gradually, which is attributed to the decrease of the expansion pressure of the detonation products and the unloading effect of the rarefaction wave reflecting from the free surface.
Figure 3. Pressure histories of crest and trough at the loading surfaceFig. 4 shows several contours of local pressure (a), density (b), and temperature (c) at 6 different times after the arrival of the detonation products at the loading surface, from left to right and top to bottom, at 6.36, 6.5, 6.7, 6.9, 7.1 and 7.3 μs.They exhibit an evolution process of the perturbed interface and the interaction of the wave and the interface.The blue part of the temperature contour is the aluminum sample, and the sample temperature remains below 500 K and far below the melting point of 1 200 K, which indicates that the sample is in the elastic-plastic state all the time.
Figure 4. Contours of local pressure (a), density (b), and temperature (c) at 6.36, 6.5, 6.7, 6.9, 7.1, and 7.3 μs from left to right and top to bottom after the arrival of detonation products at the loading surfaceFig. 2(b) shows an image of the sample at 7.78 μs when the initial yield strength Y0 and the shear modulus G0 are normal values.Fig. 5 shows a comparison of the perturbation amplitudes between the experiment and numerical simulations, where the square symbol corresponds to the experimental result, and the solid black line corresponds to the numerical results when Y0 and G0 are normal.The simulated amplitude is much larger than that in the experiment when using normal values of Y0 and G0.This is because the aluminum strengthens under such conditions, and the SG constitutive model underestimates its strength, which can suppress the perturbation growth.
Figure 5. Growth histories of the perturbation amplitudeFig. 6 shows the time histories of the free surface velocity (a) and displacement (b) at 3 measurement points (dot-dot-dashed lines:experiment; solid, dashed, and dotted lines:simulations), which agree well with each other.Therefore, the calculations of detonation of the explosive and the thermodynamic state of the sample are exact.
Figure 6. Time histories of the free surface velocity (a) and displacement (b)Fig. 7 shows the calculated time histories of the strain at the crest (solid line) and trough (dashed line) of the loading surface.The deformation at the trough is much larger than that at the crest because the trough of the sample is in a stronger tensile stress state, which is the main mechanism for the deformation of the perturbation interface.Fig. 8 shows the time histories of the dynamic yield strength at the crest (solid line) and trough (dashed line) of the loading surface, calculated using the SG constitutive model, similar to the profile of the loading pressure, which demonstrate that the material strength increased as did the loading pressure under a certain condition.
Figure 7. Time histories of strain at the crest and trough of the loading surface
Figure 8. Time histories of yield strength at the crest and trough of the loading surfaceMoreover, when the loading pressure reaches a peak, the dynamic yield strength was up to 3 times that of the initial value.In fact, the normal SG model is generally calibrated by conventional Hopkinson and Taylor impacts experiment with a lower strain rate.Under the current loading conditions (loading pressure of 25 GPa and strain rate of 106 s-1), the strength is not great enough to suppress the perturbation growth.However, when the initial yield strength Y0 and the shear modulus G0 increase to 10 times that of the normal values, good agreement between the experiment and simulation is achieved, as shown in Fig. 2(c) for the perturbation interface and in Fig. 5 for the perturbation amplitude with the dashed line.Therefore, the material strength intensively stabilizes the perturbation growth.The dotted lines in Fig. 5 are fitted lines from the simulated results, indicating that the perturbation amplitude grows exponentially over time.
We studied the effect of the initial yield strength and the initial shear modulus of the material on the evolution and growth of the perturbed interface.Figs. 9(a), 10(a), and 11(a) show the calculated growth histories of perturbation amplitude, strain histories, and dynamic yield strength histories at the trough of the loading surface, respectively, when the initial yield strength is fixed at the normal value and as the initial shear modulus increases gradually.The growth of the perturbation amplitude exhibits no change even when the initial shear modulus increases to 10 times that of the normal value, which means that the initial shear modulus has no influence on the material deformation and does not affect the dynamic yield strength.
Figure 9. Growth histories of the perturbation amplitude for different values of initial shear modulus (a) and yield strength (b)
Figure 10. Time histories of strain at the trough of the loading surface for different values of initial shear modulus (a) and yield strength (b)
Figure 11. Time histories of yield strength at the trough of the loading surface for different values of initial shear modulus (a) and yield strength (b)Figs. 9(b), 10(b), and 11(b) show the numerical results when the initial shear modulus is fixed and the initial yield strength gradually increases to 10 times that of the normal value.These indicate that, with the increase of the initial yield strength, the dynamic yield strength also increases, the material deformation is retarded, and the perturbation growth is suppressed markedly.Therefore, the initial shear modulus of the material exerts no effect on the growth of the metallic RT instability within a certain range, while the initial yield strength has an obvious effect on it.
4. Conclusions
We have established an experimental setup and developed a numerical simulation method to investigate the RT instability in metallic materials driven by explosion.We also studied the RT instability in aluminum, and drew the following conclusions:
(1) The perturbation amplitude grows following an exponential law over time.
(2) When using the normal physical property parameters of aluminum, simulated evolution of the perturbed interface agrees with experiment only qualitatively, and there is a big quantitative difference between them because the aluminum strengthens under high pressures and at high strain rates, and the SG constitutive model underestimates its strength as being not great enough to suppress the perturbation growth.
(3) When the initial yield strength and the initial shear modulus increase to 10 times their normal values, the numerical and experimental results are in good agreement both qualitatively and quantitatively.The underlying physical mechanism is the stabilization effect of material strength on the perturbation growth.Moreover, the initial shear modulus has no influence on the perturbation growth within a certain range whereas the initial yield strength does influence it strongly.Therefore, the material strength dominates the evolution of the metallic RT instability.
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图 1 面心立方层级结构(a)和不同振幅下的正弦曲线梁(b)
Figure 1. Face-centered cubic hierarchical structure (a) and sinusoidal beams of different amplitudes (b)
图 3 蜂窝结构层：(a)正三角形，(b)内凹形
Figure 3. Honeycomb structure layer: (a) regular triangle，(b) re-entrant
图 5 不同网格尺寸下M-1构型的载荷-位移曲线(a)和能量时程曲线(b)
Figure 5. Force-displacement curves of M-1 with different mesh sizes (a) and time history of energy (b)
图 6 实验试样 (a) 和CMT5305万能试验机 (b)
Figure 6. Experimental samples (a) and CMT5305 universal testing machine (b)
图 7 (a) 拉伸试样及尺寸和 (b) TPU材料的应力-应变曲线
Figure 7. (a) Tensile specimens and its dimensions，(b) stress-strain curve of TPU
图 8 调整幅值(a)、间距(b)、角度(c)以及插入蜂窝结构层(d)时层级结构试样的载荷-位移曲线
Figure 8. Force-displacement curves of hierarchical structure sample: adjustment of (a) amplitude, (b) distance, (c) angle and (d) introducing honeycomb structural layers
图 9 M-1试样不同阶段变形模式的实验与有限元模拟结果对比
Figure 9. Experimental and finite element simulation comparison of deformation modes of M-1 specimens at different stages
图 10 M-1试样的实验与有限元模拟载荷-位移曲线对比
Figure 10. Comparison of experimental and finite element simulation of force-displacement curves for M-1 specimens
图 11 不同振幅层级结构在不同应变下的变形
Figure 11. Deformation of hierarchical structures of different amplitude under various strains
图 12 不同振幅层级结构的EA和SEA
Figure 12. EA and SEA of hierarchical structure ofdifferent amplitudes
图 13 不同间距层级结构在不同应变下的变形
Figure 13. Deformation of hierarchical structures of different distance under various strains
图 14 不同间距层级结构的EA和SEA
Figure 14. EA and SEA of hierarchical structure ofdifferent distance
图 15 不同角度层级结构在不同应变下的变形
Figure 15. Deformation of hierarchical structures of different angle under various strains
图 16 不同角度层级结构的EA和SEA
Figure 16. EA and SEA of hierarchical structure ofdifferent straight beam angles
图 17 不同应变下蜂窝层级结构的变形情况
Figure 17. Deformation of honeycomb hierarchical structures under various strains
图 18 蜂窝层级结构的EA和SEA
Figure 18. EA and SEA of hierarchical structure with honeycomb structure layer
表 1 4种设计方法获得的几何模型的具体结构特征
Table 1. Specific structural features of geometric models obtained by four design methods
Model Specific structural features Distance/mm Amplitude/mm Angle/(º) M-1 3 0 0, 90 AM-1 3 0.25 0, 90 AM-2 3 0.50 0, 90 AM-3 3 1.00 0, 90 D-1 2 0 0, 90 D-2 2-4-2 0 0, 90 D-3 4-2-4 0 0, 90 AN-1 0 0, 45 AN-2 0 0, 45, 135 AN-3 0 0, 90, 45, 135 AN-4 0 0, 30, 60, 90 H-1 Honeycomb structure layer: equilateral triangle H-2 Honeycomb structure layer: re-entrant 
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表 2 打印试样的质量
Table 2. Mass of the 3D printed sample
Model Mass/g Model Mass/g M-1 6.92 AN-1 9.01 AM-1 7.63 AN-2 8.72 AM-2 7.74 AN-3 8.38 AM-3 9.39 AN-4 7.73 D-1 9.26 H-1 10.05 D-2 6.66 H-2 9.15 D-3 6.71
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