A Macroscopic Dynamic Constitutive Model for Ceramic Materials

TANG Ruitao XU Liuyun WEN Heming WANG Zihao

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Citation:

A Macroscopic Dynamic Constitutive Model for Ceramic Materials

    作者简介: TANG Ruitao (1989-), male, doctoral student, major in impact dynamics. E-mail: rttang@mail.ustc.edu.cn;
    通讯作者: WEN Heming, hmwen@ustc.edu.cn
  • 中图分类号: O347.3

A Macroscopic Dynamic Constitutive Model for Ceramic Materials

    Author Bio: TANG Ruitao (1989-), male, doctoral student, major in impact dynamics. E-mail: rttang@mail.ustc.edu.cn;
    Corresponding author: WEN Heming, hmwen@ustc.edu.cn ;
  • CLC number: O347.3

  • 摘要: 基于已有混凝土材料的相关研究,建立了陶瓷材料在动态载荷作用下的宏观本构模型。模型中状态方程采用多项式描述,强度面模型中考虑了压力相关性、Lode角效应、应变率效应、剪切损伤及拉伸软化等的影响。采用一个新的函数来描述陶瓷材料的强度面,其在较高压力下趋于一个平台值;并采用动态增强因子(DIF)考察剥除惯性效应后陶瓷材料的真实应变率效应。通过将模型预测的压力-体应变响应、准静态强度面以及应变率效应与相关实验数据进行对比,验证了该模型。单个单元测试模拟得到的结果与三轴实验数据以及侵彻实验数据高度吻合,进一步验证了此模型。为显示模型的优越性,还与JH-2模型的预测结果进行了比较。结果表明:所提出的本构模型能够很好地预测陶瓷材料在不同加载条件下的力学行为,且优于现有的模型。
  • Figure 1.  Comparison between the present model predictions (Eq.(9) and Eq.(10)) and the experimental data for ceramic materials

    Figure 2.  Comparisons between the present model predictions (Eq.(4) and Eq.(9)) with the experimental data

    Figure 3.  Comparison between the present model predictions (Eq.(6)) with available experimental data for different ceramic materials at different strain rates (Unit of strain rate: s–1)

    Figure 4.  Comparisons between the present model predictions (Eq.(4)) and the experimental data obtained from plate impact tests

    Figure 5.  Comparison between the experimental data[25] and the predictions by the present model of strength varies with pressure at different strain rates of AlN

    Figure 6.  Comparison of stress-strain curves for BeO under triaxial compression between the present model and experimental data[22]

    Figure 7.  Variation of pressure, effective stress with maximum principal strain for AlN under quasi-static uniaxial tension

    Figure 8.  Variation of pressure and effective stress with maximum principal strain for AlN under quasi-static biaxial tension

    Figure 9.  Numerically predicted relationship between pressure and maximum principal strain of AlN under both quasi-static triaxial tension

    Figure 10.  Schematic diagrams of the geometric dimensions of the projectile-target combination under the impact velocity of 1 500 m/s (Same materials used for all target configurations, and all configuration are axisymmetric.)

    Figure 11.  Comparison between the numerical results and the test data for the depth of penetration in AD99.7/RHA targets by flat-nosed tungsten alloy penetrators[35]

    Table 1.  Values of parameters for BeO in the present model

    Equation of state parametersConstitutive model parameters
    K1/GPaK2/GPaK3/GPaρc/(kg·m−3)FmWxWyS
    181.51 207.9−2 9913 03033.821.25
    Constitutive model parameters
    ${f_{{\rm{c}}}'}$/GPaft/GPaBG/GPaλmλslr
    1.50.151.21250.37.50.80.3
    下载: 导出CSV

    Table 2.  Values of parameters for AlN in the present model

    Equation of state parametersConstitutive model parameters
    ρc/(kg·m−3)K1/GPaK2/GPaK3/GPaK4/GPaK5/GPaK6/GPaFmWxWy
    3 229181.51 207.9−2 991181.9335.6−28333.82
    Constitutive model parameters
    ${f_{{\rm{c}}}'}$/GPaft/GPaBG/GPaSλmλslr
    30.31.71271.250.37.50.80.3
    下载: 导出CSV

    Table 3.  Values of various parameters for AlN ceramic (JH-2 model)

    Equation of state parametersConstitutive model parameters
    ρc/(kg·m−3)K1/GPaK2/GPaK3/GPaabCnm
    3 22920126001.361.00.0130.750.65
    Constitutive model parameters
    HEL/GPapHEL/GPaσHEL/GPaμHELT/GPaβd1d2
    9.05.06.00.024 20.321.00.021.85
    下载: 导出CSV

    Table 4.  Values of various parameters for AD99.7 ceramic[35] (The present model)

    Equation of state parametersConstitutive model parameters
    ρc/(kg·m−3)K1/GPaK1/GPaK3/GPaFmWxWyS
    3 809181.51 207.9−2 99133.821.25
    Constitutive model parameters
    ${f_{{\rm{c}}}'}$/GPaft/GPaBG/GPaλmλslr
    30.31.41350.37.50.80.3
    下载: 导出CSV

    Table 5.  Values of various parameters for tungsten alloy[35] (JC model)

    Constitutive model parameters
    ρc/(kg·m−3)G/GPaA1/GPaB1/GPaN1C1M1${\dot \varepsilon_{_0}}$/s−1
    17 6001221.5060.1770.120.0161.01.0
    Constitutive model parameters
    cp/(J·kg−1·K−1)Tm/KTr/KD1D2D3D4D5
    1341 7233002.00000
    Equation of state parameters
    Cs/(m·s−1)S1S2S3γ0A0
    4 0291.23001.540.4
    下载: 导出CSV

    Table 6.  Values of various parameters for RHA[35] (JC model)

    Constitutive model parameters
    ρc/(kg·m−3)G/GPaA1/GPaB1/GPaN1C1M1${\dot \varepsilon _{_{0}}}$/s−1
    7 800770.7920.510.260.0141.031.0
    Constitutive model parameters
    cp/(J·kg−1·K−1)Tm/KTr/KD1D2D3D4D5
    4771 7932940.053.44−2.120.0020.61
    Equation of state parameters
    Cs/(m·s−1)S1S2S3γ0A0
    4 5691.49002.170.460
    下载: 导出CSV
  • [1] FAHRENTHOLD E P. A continuum damage model for fracture of brittle solids under dynamic loading [J]. Journal of Applied Mechanics, 1991, 58(4): 904–909. doi: 10.1115/1.2897704
    [2] RAJENDRAN A M. Modeling the impact behavior of AD85 ceramic under multiaxial loading [J]. International Journal of Impact Engineering, 1994, 15(6): 749–768. doi: 10.1016/0734-743X(94)90033-H
    [3] JOHNSON G R, HOLMQUIST T J. An improved computational constitutive model for brittle materials [J]. High Pressure Science and Technology, 2008, 309(1): 981–984.
    [4] SIMHA C H, BLESS S, BEDFORD A, et al. Computational modeling of the penetration response of a high-purity ceramic [J]. International Journal of Impact Engineering, 2002, 27(1): 65–86. doi: 10.1016/S0734-743X(01)00036-7
    [5] RAVICHANDRAN G, SUBHASH G. A micromechanical model for high strain rate behavior of ceramics [J]. International Journal of Solids and Structures, 1995: 2627–2646.
    [6] ESPINOSA H D. On the dynamic shear resistance of ceramic composites and its dependence on applied multiaxial deformation [J]. International Journal of Solids and Structures, 1995, 32(21): 3105–3128. doi: 10.1016/0020-7683(94)00300-L
    [7] ESPINOSA H D, XU Y, BRAR N S. Micromechanics of failure waves in glass: Ⅱ, modeling [J]. Journal of the American Ceramic Society, 1997, 80(8): 2074–2085.
    [8] STEINBERG D J. Computer studies of the dynamic strength of ceramics [M]//Shock Waves. Berlin: Springer, 1992: 415–422.
    [9] XU H, WEN H M. A computational constitutive model for concrete subjected to dynamic loadings [J]. International Journal of Impact Engineering, 2016, 91: 116–125. doi: 10.1016/j.ijimpeng.2016.01.003
    [10] XU H, WEN H M. Semi-empirical equations for the dynamic strength enhancement of concrete-like materials [J]. International Journal of Impact Engineering, 2013, 60: 76–81. doi: 10.1016/j.ijimpeng.2013.04.005
    [11] ZHAO F Q, WEN H M. A comment on the maximum dynamic tensile strength of a concrete-like material [J]. International Journal of Impact Engineering, 2018, 115: 32–35. doi: 10.1016/j.ijimpeng.2018.01.009
    [12] ZHAO F Q, WEN H M. Effect of free water content on the penetration of concrete [J]. International Journal of Impact Engineering, 2018, 121: 180–190. doi: 10.1016/j.ijimpeng.2018.06.007
    [13] BRIDGMAN P W. Linear compressions to 30 000 kg/cm2, including relatively incompressible substances [J]. Proceedings of the American Academy of Arts and Sciences, 1949, 77(6): 189–234. doi: 10.2307/20023541
    [14] HART H V, DRICKAMER H G. Effect of high pressure on the lattice parameters of Al2O3 [J]. Journal of Chemical Physics, 1965, 43(7): 2265–2266. doi: 10.1063/1.1697121
    [15] SATO Y, AKIMOTO S. Hydrostatic compression of four corundum-type compounds: α-Al2O3, V2O3, Cr2O3, and α-Fe2O [J]. Journal of Applied Physics, 1979, 50(8): 5285–5291. doi: 10.1063/1.326625
    [16] BASSETT W A, WEATHERS M S, WU T C, et al. Compressibility of SiC up to 68.4 GPa [J]. Journal of Applied Physics, 1993, 74(6): 3824–3826. doi: 10.1063/1.354476
    [17] ROSENBERG Z, BRAR N S, BLESS S J. Dynamic high-pressure properties of AlN ceramic as determined by flyer plate impact [J]. Journal of Applied Physics, 1991, 70(1): 167–171. doi: 10.1063/1.350337
    [18] XIA Q, XIA H, RUOFF A L. Pressure induced rocksalt phase of aluminum nitride: a metastable structure at ambient condition [J]. Journal of Applied Physics, 1993, 73(12): 8198–8200. doi: 10.1063/1.353435
    [19] UENO M, ONODERA A, SHIMOMURA O, et al. X-ray observation of the structural phase transition of aluminum nitride under high pressure [J]. Physical Review B, 1992, 45(17): 10123. doi: 10.1103/PhysRevB.45.10123
    [20] ROSENBERG Z, YAZIV D, YESHURUN Y, et al. Shear strength of shock-loaded alumina as determined with longitudinal and transverse manganin gauges [J]. Journal of Applied Physics, 1987, 62(3): 1120–1122. doi: 10.1063/1.339721
    [21] BOURNE N K, MILLETT J, PICKUP I, et al. Delayed failure in shocked silicon carbide [J]. Journal of Applied Physics, 1997, 81(9): 6019–6023. doi: 10.1063/1.364450
    [22] FENG R, RAISER G F, GUPTA Y M, et al. Material strength and inelastic deformation of silicon carbide under shock wave compression [J]. Journal of Applied Physics, 1998, 83(1): 79–86. doi: 10.1063/1.366704
    [23] PICKUP I M, BARKER A K. Deviatoric strength of silicon carbide subject to shock [J]. AIP Conference Proceedings, 2000, 505(1): 573–576.
    [24] LEE M, BRANNON R M, BRONOWSKI D R. Uniaxial and triaxial compression tests of silicon carbide ceramics under quasi-static loading condition [R]. Albuquerque, New Mexico: Sandia National Laboratories, 2005.
    [25] CHEN W, RAVICHANDRAN G. Static and dynamic compressive behavior of aluminum nitride under moderate confinement [J]. Journal of the American Ceramic Society, 1996, 79(3): 579–584.
    [26] HEARD H C, CLINE C F. Mechanical behaviour of polycrystalline BeO, Al2O3 and AlN at high pressure [J]. Journal of Materials Science, 1980, 15(8): 1889–1897. doi: 10.1007/BF00550614
    [27] WILKINS M L, CLINE C F, HONODEL C A. Fourth progress report of light armor program [R]. Livermore: Lawrence Radiation Laboratories, 1969.
    [28] HOLMQUIST T J, TEMPLETON D W, BISHNOI K D. Constitutive modeling of aluminum nitride for large strain, high-strain rate, and high-pressure applications [J]. International Journal of Impact Engineering, 2001, 25(3): 211–231. doi: 10.1016/S0734-743X(00)00046-4
    [29] ZINSZNER J L, ERZAR B, FORQUIN P, et al. Dynamic fragmentation of an alumina ceramic subjected to shockless spalling: an experimental and numerical study [J]. Journal of the Mechanics and Physics of Solid, 2015, 85: 112–127. doi: 10.1016/j.jmps.2015.08.014
    [30] GALVEZ F, RODRIGUEZ J, SANCHEZ V. Tensile strength measurements of ceramic materials at high rates of strain [J]. Le Journal de Physique IV, 1997, 7(C3): 151.
    [31] GALVEZ F, RODRIGUEZ J, SANCHEZ V. The spalling of long bars as a reliable method of measuring the dynamic tensile strength of ceramics [J]. International Journal of Impact Engineering, 2002, 27(2): 161–177. doi: 10.1016/S0734-743X(01)00039-2
    [32] BOURNE N K. Shock-induced brittle failure of boron carbide [J]. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2002, 458: 1999–2006. doi: 10.1098/rspa.2002.0968
    [33] VOGLER T J, REINHART W D, CHHABILDAS L C. Dynamic behavior of boron carbide [J]. Journal of Applied Physics, 2004, 95: 4173–4183. doi: 10.1063/1.1686902
    [34] HAYUN S, PARIS V, DARIEL M P, et al. Static and dynamic mechanical properties of boron carbide processed by spark plasma sintering [J]. Journal of the European Ceramic Society, 2009, 29(16): 3395–3400. doi: 10.1016/j.jeurceramsoc.2009.07.007
    [35] LUNDBERG P, WESTERLING L, LUNDBERG B. Influence of scale on the penetration of tungsten rods into steel-backed alumina targets [J]. International Journal of Impact Engineering, 1996, 18(4): 403–416. doi: 10.1016/0734-743X(95)00049-G
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  • 收稿日期:  2019-12-06
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A Macroscopic Dynamic Constitutive Model for Ceramic Materials

    作者简介:TANG Ruitao (1989-), male, doctoral student, major in impact dynamics. E-mail: rttang@mail.ustc.edu.cn
    通讯作者: WEN Heming, hmwen@ustc.edu.cn
  • 1. CAS Key Laboratory for Mechanical Behavior and Design of Materials, University of Science and Technology of China, Hefei 230027, Anhui, China

摘要: 基于已有混凝土材料的相关研究,建立了陶瓷材料在动态载荷作用下的宏观本构模型。模型中状态方程采用多项式描述,强度面模型中考虑了压力相关性、Lode角效应、应变率效应、剪切损伤及拉伸软化等的影响。采用一个新的函数来描述陶瓷材料的强度面,其在较高压力下趋于一个平台值;并采用动态增强因子(DIF)考察剥除惯性效应后陶瓷材料的真实应变率效应。通过将模型预测的压力-体应变响应、准静态强度面以及应变率效应与相关实验数据进行对比,验证了该模型。单个单元测试模拟得到的结果与三轴实验数据以及侵彻实验数据高度吻合,进一步验证了此模型。为显示模型的优越性,还与JH-2模型的预测结果进行了比较。结果表明:所提出的本构模型能够很好地预测陶瓷材料在不同加载条件下的力学行为,且优于现有的模型。

English Abstract

  • Ceramic materials have been increasingly used in armors, personal armor system, aeronautic engineering and vehicle engineering due to its excellent ballistic performance. An understanding of the response and failure of ceramic materials under impact loading is of great significance for the design and assessment of military hardware and protective structures.

    As ballistic tests are expensive to perform and time consuming, numerical simulations have been widely employed in the design and optimization of ceramic armors. However, the accuracy of numerical simulations depends on the dynamic constitutive model for ceramic materials to a large extent. To this end, many models have been put forward[1-8]. Fahrenthold[1] proposed a continuum damage model for fracture of brittle solids under dynamic loading in which Weibull strength distribution was employed to account for the effects of flaw size distribution on the damage accumulation rate. The model was used to compute the depth of penetration in a steel plate impacted by a sphere of alumina without being applied to simulate the behavior of ceramic armors subjected to projectile impact. Rajendran[2] modeled the impact behavior of AD85 ceramic under multiaxial loading by assuming an existing distribution of micro-cracks and employing a criterion for crack growth based on theories of dynamic fracture mechanics. The strength of the intact material was rate dependent, and the inelastic deformation was modeled using an elastic-plastic cracking rule. The model predictions were found to be in good agreement with the test data obtained from plate impact tests. The most widely used constitutive model for ceramic materials in many commercial codes is the Johnson-Holmquist model (JH-2)[3]. The model employs two strength surfaces to describe the pressure-dependent compressive strength of the intact and failed materials, respectively. However, in the model strain rate effect is described by the same form of function adopted in the constitutive models for metals and concrete materials, which has been found to be inconsistent with the dynamic mechanical behavior in recent studies[9-12]. Furthermore, tensile softening cannot be accurately predicted, and no differentiation was made between the effects of strain rate and inertia (containment).

    The main objective of this paper is to establish a macroscopic dynamic constitutive model for ceramic materials by closely following the previous work on concrete[9] and on the basis of the experimental observations for some ceramic materials. Various equations of the constitutive model are given and compared with some available experimental data for ceramic materials in terms of pressure-volumetric response, quasi-static strength surface and strain rate effect. Furthermore, the model is also verified against the data for triaxial test by single element simulation approach.

    • A dynamic macroscopic constitutive model for ceramic materials is developed in the following sections based on the computational constitutive model for concrete subjected to dynamic loadings[9] with some modifications being made. There are two points which should be highlighted here, namely, (1) the pressure-volumetric response is described by polynomial equation and the phase transition (e.g. AlN ceramics) as the pressure increases to a certain value is considered; (2) a hyperbolic tangent function is employed to describe the pressure dependent shear strength surfaces of ceramic materials which level out at very high pressures.

    • Ceramics are complex granular materials, which contain a large number of micro cracks and voids just as concrete material. On the one hand, it has been observed experimentally that the volumetric strains of Al2O3 and SiC increase with increasing pressure[13-16] and the equation of state for this category of ceramic materials can be expressed as polynomial equation, viz.

      $\left\{\begin{aligned} & p = {K_1}\mu + {K_2}{\mu ^2} + {K_3}{\mu ^3} \\ & \mu = {V_0}/V - 1 = \rho/{\rho _0} - 1 \end{aligned} \right.$

      where p is pressure; $\mu $ is volumetric strain; K1, K2, K3 are bulk moduli for ceramic material; ${\rho _0} = 1/{V_0}$ and $\rho = 1/V$ indicate initial and current densities of ceramic material, respectively. On the other hand, the pressure-volumetric response of AlN has been observed experimentally to be quite different[17-19] and a phase transformation occurs from wurtzite structure to salt structure as pressure reaches a critical value. Thus, the equation of state for such ceramics as AlN can be written in the following form, namely

      $p = \begin{cases} {K_1}\mu + {K_2}{\mu ^2} + {K_3}{\mu ^3} & \quad\mu < {\mu _{{\rm{c}}1}} \\ {p_{\rm{c}}} & {\mu _{{\rm{c}}1}} \leqslant \mu \leqslant {\mu _{{\rm{c}}2}} \\ {K_4}(\mu - \mu ') + {K_5}{(\mu - \mu ')^2} + {K_6}{(\mu - \mu ')^3} & \quad\mu > {\mu _{{\rm{c}}2}} \end{cases}$

      where pc represents a constant pressure at which phase transformation occurs; K4, K5, K6 are bulk moduli for ceramic material after the phase transformation; $\mu '$ is an offset in volumetric strain due to phase transformation; ${\mu _{{\rm{c}}1}}$ and ${\mu _{{\rm{c}}2}}$ are volumetric strains at the beginning and end of phase transformation, respectively. For μ < 0, ceramic material is under tension condition. Thus, one obtains

      $p = {K_1}\mu $

    • On the basis of the previous studies on concrete[9-10] and the experimental observations for ceramic materials[17, 20-24], it is suggested that the strength surface of ceramic materials can be cast into the following form, i.e.

      $Y = \begin{cases} {3\left( {p + {f_{{\rm{tt}}}}} \right)r\left( {\theta,e} \right) }&\quad\;\; {p < 0} \\ {\left[ {{\rm{3}}{f_{{\rm{tt}}}}{\rm{ + }}\left( {{f_{{\rm{cc}}}} - 3{f_{{\rm{tt}}}}} \right) \times 3p/{f_{{\rm{cc}}}}} \right]r\left( {\theta,e} \right)}&{0 \leqslant p \leqslant {f_{\rm{c}}}/3} \\ {\left[ {{f_{{\rm{cc}}}} + Bf_{\rm{c}}'\tanh \left( {p/f_{\rm{c}}' - {f_{{\rm{cc}}}}/(3f_{\rm{c}}')} \right)} \right]r\left( {\theta,e} \right)}& \quad {p > {f_{\rm{c}}}/3} \end{cases}$

      in which Y is shear strength; ${f_{{\rm{cc}}}} = f_{\rm{c}}'{\psi _{\rm{c}}}{\eta _{\rm{c}}}$ and ${f_{{\rm{tt}}}} = {f_{\rm{t}}}{\psi _{\rm{t}}}{\eta _{\rm{t}}}$ are respective dynamic strengths of ceramic materials in uniaxial compression and tension, which take account of shear damage in the form of damage shape functions ($\eta $) and strain rate effects by dynamic increase factor (DIF); B is an empirical constant; $r\left( {\theta,e} \right)$ represents Lode effect. The dynamic increase factor in compression (ψc) can be obtained by the following equation[9-10], viz.

      ${\psi _{\rm{c}}} = \frac{{{f_{{\rm{cd}}}}}}{{f_{\rm{c}}'}} = \left( {{\psi _{\rm{t}}} - 1} \right)\frac{{{f_{\rm{t}}}}}{{f_{\rm{c}}'}} + 1$

      where fcd and ${f_{\rm{c}}'} $ are the dynamic and static uniaxial compressive strength, respectively, ft is the static uniaxial tensile strength, and ψt represents the dynamic increase factor in tension which is expressed as follows

      ${\psi _{\rm{t}}} = \frac{{{f_{{\rm{td}}}}}}{{{f_{\rm{t}}}}} = \left\{ {\left( {\frac{{{F_{\rm{m}}}}}{{{W_y}}} - 1} \right)\tanh \left[ {\left( {\lg \frac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}} - {W_x}} \right)S} \right] + 1} \right\}{W_y}$

      in which ftd is the dynamic uniaxial tensile strength, parameters Wx, Fm, Wy and S are constants to be determined experimentally, $\dot \varepsilon $ is the strain rate, ${\dot \varepsilon _0}$ is the reference strain rate usually taken to be ${\dot \varepsilon _0}$= 1.0 s−1.

      The multi-axial tests of ceramics show that the effective strain at maximum load (${\varepsilon _{\max }}$) increases almost linearly with increasing pressure. Furthermore, these tests also show ceramic compressive strength has no significant influence on this behavior. According to Eq.(18) in Ref.[9], the failure strain of ceramic is written as

      ${\varepsilon _{\rm f}} = \frac{{{\varepsilon _{\rm{m}}}}}{{{\lambda _{\rm{m}}}}}{\left( {\frac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}}} \right)^{0.02}} = \frac{1}{{{\lambda _{\rm{m}}}}}0.006\;5\max \left[ {1,1 + {\lambda _{\rm{s}}}\left( {\frac{p}{{{f_{\rm{c}}}}} - \frac{1}{3}} \right)} \right]{\left( {\frac{{\dot \varepsilon }}{{{{\dot \varepsilon }_0}}}} \right)^{0.02}}$

      where ${\varepsilon _{\rm{f}}}$is the failure strain of ceramic material, ${\varepsilon _{\rm{m}}}$is the strain corresponding to the maximum compressive strength, ${\lambda _{\rm{m}}}$ is the shear damage at which strength reaches its maximum value under compression, and ${\lambda _{\rm{s}}}$ is an empirical constant.

      The residual strength of the crushed ceramics is still high under the confining pressure. The residual strength surface for ceramic materials can be obtained from Eq.(4) by setting ftt = 0 and fcc = $f_{\rm{c}}'$ × r, namely

      $Y = \begin{cases} {3p \times r\left( {\theta,e} \right)}&{0 < p \leqslant {f_{\rm{c}}'} \times r/3} \\ {\left[ {{f_{\rm{c}}'} \times r + B{f_{\rm{c}}'}\tanh \left( {\dfrac{p}{{{f_{\rm{c}}'}}}-\dfrac{{{f_{\rm{c}}'}\times r}}{{3{f_{\rm{c}}'}}}} \right)} \right] \times r\left( {\theta,e} \right)}& \;\;\; {p > {f_{\rm{c}}'} \times r/3}\end{cases}$

      where r is a constant, $f_{\rm c}'$ × r represents the residual strength of concrete under quasi-static uniaxial compression. Meanwhile, the initial yield strength of concrete under quasi-static uniaxial compression is defined as $f_{\rm c}'$ × l.

    • The present model is verified against some available experimental data in terms of pressure-volumetric response, quasi-static strength surface and strain rate effect.

      Values of various parameters in equation of state can be determined by hydrostatic compression experiment and they are listed in Table 1. For Al2O3 and SiC ceramic materials, the relationship between pressure and volumetric strain (Eq.(1)) can be rewritten as

      Equation of state parametersConstitutive model parameters
      K1/GPaK2/GPaK3/GPaρc/(kg·m−3)FmWxWyS
      181.51 207.9−2 9913 03033.821.25
      Constitutive model parameters
      ${f_{{\rm{c}}}'}$/GPaft/GPaBG/GPaλmλslr
      1.50.151.21250.37.50.80.3

      Table 1.  Values of parameters for BeO in the present model

      $p = 1.815 \times {10^{11}}\mu + 1.208 \times {10^{12}}{\mu ^2} - 2.991 \times {10^{12}}{\mu ^3}$

      and, for AlN ceramic material, the relationship between pressure and volumetric strain (Eq.(2)) can be recast into the following form

      $p =\begin{cases} {1.815 \times {{10}^{11}}\mu + 1.208 \times {{10}^{12}}{\mu ^2} - 2.991 \times {{10}^{12}}{\mu ^3}}& \quad\;\;\;\; {\mu < 0.067}\\ {1.668 \times {{10}^{10}}}&{{\rm{0}}.{\rm{067}} \leqslant \mu \leqslant 0.330}\\ {1.819 \times {{10}^{11}}(\mu - 0.250) + 3.556 \times {{10}^{11}}{{(\mu - 0.250)}^2} - 2.830 \times {{10}^{11}}{{(\mu - 0.250)}^3}}& \quad\;\;\;\; {\mu > 0.330}\end{cases}$

      Fig.1 shows comparison of the present model predictions (Eq.(9) and Eq.(10)) with the experimental data for Al2O3 and SiC[13-16] and for AlN[17-19]. It can be seen from Fig.1 that good agreement is obtained.

      Figure 1.  Comparison between the present model predictions (Eq.(9) and Eq.(10)) and the experimental data for ceramic materials

      Fig.2(a) and Fig.2(b) show comparisons between the present model predictions (Eq.(4) with B = 1.4 for B4C and BeO and Eq.(9) with B = 1.7 for Al2O3 and AlN) and the test data[25-27] for the strength surfaces. The strength surfaces of Al2O3 and AlN obtained from the JH-2 model[28] are also shown in Fig.2(b). It is clear from Fig.2 that the present model predictions are in good agreement with available test results for ceramic materials. It is also clear from Fig.2(b) that the JH-2 model produces similar results to those of the present model for relatively low pressures whilst for higher pressure the present model levels out and the JH-2 model increases with increasing pressure.

      Figure 2.  Comparisons between the present model predictions (Eq.(4) and Eq.(9)) with the experimental data

      Values of parameters Fm, Wx and S in Eq.(6) for the dynamic increase factor in tension can be determined from tensile test data for ceramic materials at different strain rates. Fig.3 shows comparisons between the present model predictions (Eq.(6) with Fm = 3, Wx = 3.8 and S = 1.25) and available experimental data[29-31] for different ceramic materials at different strain rates. It is evident from Fig.3 that reasonable agreement is obtained.

      Figure 3.  Comparison between the present model predictions (Eq.(6)) with available experimental data for different ceramic materials at different strain rates (Unit of strain rate: s–1)

      Fig.4(a) and Fig.4(b) show comparisons between the theoretically predicted strength surfaces (Eq.(4)) and some experimental data obtained from plate impact tests on B4C[32-34], AlN[17], Al2O3[20] and SiC[21-23]. In the calculations, a strain rate of 105 s−1 is taken which is of a typical value in a plate impact test. Also shown in Fig.4(b) are the predictions from the JH-2 model. It is clear from Fig.4(a) that reasonable agreement is obtained between the present model predictions and the tests results for B4C which are somehow scattered whilst good agreement is achieved between Eq.(4) and the test data for AlN, Al2O3 and SiC as can be seen from Fig.4(b). It is also clear from Fig.4(b) that the JH-2 model has failed to predict the dynamic mechanical behavior of ceramic materials at higher confining pressure.

      Figure 4.  Comparisons between the present model predictions (Eq.(4)) and the experimental data obtained from plate impact tests

      Fig.5 shows comparison of the present model predictions and the experimental data obtained from SHPB (split Hopkinson pressure bar) tests[25]. Also shown in the figure are the predictions from the JH-2 model. In the tests, confining pressures of up to 230 MPa were applied, and a constant strain rate was employed (i.e. 500 s−1). The solid line indicates theoretically predicted quasi-static strength surface, and the broken line designates dynamic strength surface from the present model with a strain rate of 500 s−1. As can be seen from Fig.5 that a good agreement is obtained between the present model predictions and the experimental data whilst the JH-2 model underestimates the strength surface of AlN ceramic under the strain rate of 500 s−1. It should be stressed here that the present model predicts that the quasi-static and dynamic strength surfaces are parallel to each other, which has been confirmed/verified by the experimental data. It leads to further support for the accuracy and validity of the present constitutive model for ceramics.

      Figure 5.  Comparison between the experimental data[25] and the predictions by the present model of strength varies with pressure at different strain rates of AlN

      To demonstrate the quasi-static behavior of the present constitutive model, numerical tests are performed to evaluate stress-strain relationships for ceramics under various loading conditions, which includes: (a) triaxial compression with different confining pressures, (b) uniaxial tension, (c) biaxial tension and (d) triaxial tension. The numerical tests are carried out using single element simulation approach. Table 1 lists the values of various parameters in the present constitutive model for BeO ceramic. Table 2 and Table 3 list the values of various parameters for AlN ceramic employed in the present model and JH-2 model, respectively. Parameters in the present model are clearly defined in the previous paragraphs, and in JH-2 model. K1, K2, K3 are bulk moduli, and ρc is the density for ceramic material; a, b, C, m, n are strength surface parameters; pHEL, σHEL and μHEL are respectively the pressure, the equivalent stress and the volumetric strain at HEL (Hugoniot elastic limit); T is the maximum tensile hydrostatic pressure the material can withstand; β is strain rate parameter; d1, d2 are damage parameters.

      Equation of state parametersConstitutive model parameters
      ρc/(kg·m−3)K1/GPaK2/GPaK3/GPaK4/GPaK5/GPaK6/GPaFmWxWy
      3 229181.51 207.9−2 991181.9335.6−28333.82
      Constitutive model parameters
      ${f_{{\rm{c}}}'}$/GPaft/GPaBG/GPaSλmλslr
      30.31.71271.250.37.50.80.3

      Table 2.  Values of parameters for AlN in the present model

      Equation of state parametersConstitutive model parameters
      ρc/(kg·m−3)K1/GPaK2/GPaK3/GPaabCnm
      3 22920126001.361.00.0130.750.65
      Constitutive model parameters
      HEL/GPapHEL/GPaσHEL/GPaμHELT/GPaβd1d2
      9.05.06.00.024 20.321.00.021.85

      Table 3.  Values of various parameters for AlN ceramic (JH-2 model)

      Fig.6 shows the comparison of the numerically predicted stress-strain curves and the experimental data for BeO under triaxial compression with confinement pressures ranging from 0.1 GPa up to 1.0 GPa, as reported by Heard and Cline[26]. It is clear from Fig.6 that the numerical results are in reasonable agreement with the experimental observations.

      Figure 6.  Comparison of stress-strain curves for BeO under triaxial compression between the present model and experimental data[22]

      Fig.7 shows variations of pressure and effective stress with maximum principal strain of AlN under quasi-static uniaxial tension. It is evident from Fig.7 that the numerical results from the present model give an elastic-brittle softening response of AlN under uniaxial tension with a principal tensile strength of 0.3 GPa for AlN. It is also evident from the figure that the mechanic behavior of the ceramic under uniaxial tension predicted numerically by the JH-2 model is elastic-perfectly plastic with the minimum pressure of 0.31 GPa and the maximum effective stress of 0.93 GPa, which are obviously not in compliance with the actual tensile behavior of the ceramic.

      Figure 7.  Variation of pressure, effective stress with maximum principal strain for AlN under quasi-static uniaxial tension

      Fig.8 shows variations of pressure and effective stress with maximum principal strain of AlN under quasi-static biaxial tension. Elastic-brittle softening response of the ceramic is predicted numerically using the present model under biaxial tension with a principal tensile strength of 0.3 GPa. On the other hand, an elastic-perfectly plastic response of the ceramic under biaxial tension predicted numerically by the JH-2 model have the minimum pressure of 0.31 GPa and the maximum effective stress of 0.48 GPa.

      Figure 8.  Variation of pressure and effective stress with maximum principal strain for AlN under quasi-static biaxial tension

      Fig.9 shows numerically predicted relationship between pressure and the maximum principal strain of AlN under both quasi-static triaxial tension. It can be seen from the figure that the present model predicts the elastic-brittle softening response of AlN under triaxial tension with quasi-static principal tensile strength of 0.3 GPa. Whilst the material is no longer able to withstand any loads after the maximum pressure reached according to the JH-2 model predictions. Furthermore, no strain rate effect is found under hydrostatic tension in the JH-2 model since it hasn’t taken into account the strain rate effect in tension.

      Figure 9.  Numerically predicted relationship between pressure and maximum principal strain of AlN under both quasi-static triaxial tension

    • Numerical simulations are conducted in this section for the penetration of AD99.7/RHA target struck by flat-nosed projectiles as reported by Lundberg[35]. In the experiments, long tungsten projectiles with length-to-diameter ratio 15 were fired against the unconfined alumina with steel backing. The tests were carried out in three different scales with projectile lengths 30, 75 and 150 mm (corresponding diameters 2, 5 and 10 mm), respectively. The impact velocities were 1 500 m/s and 2 500 m/s. Fig.10 shows the schematic diagrams of the geometric dimensions of the projectile-target combination. Table 4, Table 5 and Table 6 list the values of various parameters for AD99.7 ceramic material in the present model, tungsten alloy and RHA materials in the JC model. Parameters A1 in the JC model is the initial yield strength under reference strain rate and reference temperature; B1, N1 are respectively the strain hardening modulus and index; C1 is strain rate hardening parameter; M1 is temperature softening index; ${\dot \varepsilon _0}$ is the reference strain rate; cp is heat capacity; Tm and Tr are the ambient temperature and the reference temperature (melting point), respectively; D1, D2, D3, D4 and D5 are failure criteria parameters; Cs is the intercept of shock wave velocity-particle velocity (us-up) curve; S1, S2, S3 are us-up curveslope coefficient; γ0 is Grüneisen parameter, A0 is the first order volume correction of γ0.

      Equation of state parametersConstitutive model parameters
      ρc/(kg·m−3)K1/GPaK1/GPaK3/GPaFmWxWyS
      3 809181.51 207.9−2 99133.821.25
      Constitutive model parameters
      ${f_{{\rm{c}}}'}$/GPaft/GPaBG/GPaλmλslr
      30.31.41350.37.50.80.3

      Table 4.  Values of various parameters for AD99.7 ceramic[35] (The present model)

      Constitutive model parameters
      ρc/(kg·m−3)G/GPaA1/GPaB1/GPaN1C1M1${\dot \varepsilon_{_0}}$/s−1
      17 6001221.5060.1770.120.0161.01.0
      Constitutive model parameters
      cp/(J·kg−1·K−1)Tm/KTr/KD1D2D3D4D5
      1341 7233002.00000
      Equation of state parameters
      Cs/(m·s−1)S1S2S3γ0A0
      4 0291.23001.540.4

      Table 5.  Values of various parameters for tungsten alloy[35] (JC model)

      Constitutive model parameters
      ρc/(kg·m−3)G/GPaA1/GPaB1/GPaN1C1M1${\dot \varepsilon _{_{0}}}$/s−1
      7 800770.7920.510.260.0141.031.0
      Constitutive model parameters
      cp/(J·kg−1·K−1)Tm/KTr/KD1D2D3D4D5
      4771 7932940.053.44−2.120.0020.61
      Equation of state parameters
      Cs/(m·s−1)S1S2S3γ0A0
      4 5691.49002.170.460

      Table 6.  Values of various parameters for RHA[35] (JC model)

      Figure 10.  Schematic diagrams of the geometric dimensions of the projectile-target combination under the impact velocity of 1 500 m/s (Same materials used for all target configurations, and all configuration are axisymmetric.)

      Fig.11 shows the comparison of total penetration depth between the numerical predictions and the experimental data[35]. PT represents the sum of thickness of the ceramic and penetration depth in the RHA back plate, and Lp is the length of the projectile. In the figure, the solid symbols indicate the experimental data and the hollow symbols represent the numerical results by the present model proposed in this paper. The data of different scales under the impact velocity of 1 500 m/s and 2 500 m/s are designated by square and triangle, respectively. For comparison, the numerical results obtained by Lundberg[35] with JH-2 model are also shown in Fig.11. The solid and broken lines represent the numerical results by JH-2 model at 1 500 m/s and 2 500 m/s respectively. It can be seen from Fig.11 that the numerical results from the present model are in good agreement with the experimental data. As can be seen from the figure that the numerical results for impact velocity of 1 500 m/s are in good agreement with the test data, whilst the results for impact velocity of 2 500 m/s are below the experimental observations[35]. Thus, the JH-2 model has failed to produce consistent results as compared to the experiments.

      Figure 11.  Comparison between the numerical results and the test data for the depth of penetration in AD99.7/RHA targets by flat-nosed tungsten alloy penetrators[35]

    • A dynamic macroscopic constitutive model for ceramic materials has been developed by closely following the previous work on concrete. The model captures the basic features of the mechanical response of ceramic materials including pressure hardening behavior, strain rate effects, strain softening behavior, path dependent behavior (Lode angle), and failure in both low and high confining pressures. In particular, a new function is used to characterize the pressure dependent shear strength surface of ceramic materials which levels out at very high pressures, and strain rate effect is taken into consideration by dynamic increase factor which excludes inertial effect.

      The present model has been compared with some available experimental data for ceramic materials. It transpires that the present model predictions are in good agreement with the experimental observations in terms of pressure-volumetric response, triaxial compression, quasi-static strength surface, dynamic strength surface, strain rate effect and depth of penetration. It also transpires that the present model is much more improved than the JH-2 model.

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