高温高压环境下金属材料的本构关系研究
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摘要
高温高压条件下固体材料的本构关系是研究化学炸药爆轰、高能离子束或激光束以及高速撞击作用下,固体中状态量变化规律和运动特性必不可少的材料特性方程。为了精准描述和预测上述的状态量变化和运动学特征,本文根据陈大年等人[高压物理学报,19(2005)193]和彭建祥[博士论文,中国工程物理研究院研究生部,2006]的见解,着重分析了目前受到广泛应用的由Steinberg等人提出的[J. Appl. Phys.,51(1980)1498; J. Appl. Phys.,65(1989)1528]金属本构模型中存在的固有问题,并提出了一个新的金属本构函数表示式。获得的主要结果有：
(1)在全面分析了Burakovsky等人[Phys. Rev. B,71(2005)184118]、Nadal等人[J. Appl. Phys.,93(2003)2472]和俞宇颖等人[Chin. Phys. ett.,17(2008)64]针对Steinberg等人的金属材料本构模型提出的若干质疑及修改意见的基础上,提出了一个新的适用于密堆集晶体金属的本构模型,具体的函数表示式是式中,G和Y为剪切模量和屈服强度；β和n为硬化系数,εp为塑性应变；T0为参考温度,通常取零温或室温。式中的G(P,T0), ((?)G/(?)T)P,β,n和εp等值均可通过实验和理论计算方法求得。这个模型具有如下优点：
——与Steinberg等人、Nadal等人和俞宇颖等人的意见相比较,本模型中G随P的变化没有采用在一阶导数项截断的近似表示,而是采用了G(P)的全函数表示形式；
——与Steinberg等人、Nadal等人、Burakovsky等人的意见相比较,本模型中G随T的变化虽然仍采用了一阶导数项的近似表达,但抛弃了把(aG/(?)T)P用(aG/(?)T)0代替的做法,而是在向士凯等人[Phy.Rev. B70 (2004)174102]第一原理计算结果启示下采用了把((?)G/(?)T)P表示为以P为参数的函数式；
(2)以LY12铝合金为例,本文给出的本构函数式是
这组本构函数式的有效性得到了以下检验结果的支持：
——上列G(P,T)本构函数式是基于Sin'ko等人[J. Phys.:Condens. Matter, 4(2002)6989]和向士凯等人的第一原理计算结果导出的,因而具有坚实物理基础。此外,它的有效性还得到了Hiigoniot压缩线上声速测量结果和等熵压缩线上剪切模量计算结果(通过有限形变理论的Birch-Murnaghan方程)的支持；
——上列Y(P,T)本构模型的构建尽管在其物理构思上沿袭了Steinberg等人的做法,但由于本文G(P,T)本构函数式的不同,故在Y(P,T)本构函数式的具体形式上是与Steinberg等人的有所不同的。它的有效性得到了本文冲击加载实验结果(采用了Asay和Chhabildas基于各向同性硬化模型的数据处理方法)的支持。特别值得提出的是：实验结果再一次支持了G(P,T)/Y(P,T)常数近似关系在高压区的有效性(对于LY12铝合金,这个高压区的阀值是Pc≥30GPa)；但若关注到Pc的低压区,则G(P,T)/Y*(P,T)≈常数近似关系(式中Y*＝Y/[1+β(εP+εPi)]n)也是成立的。其实,这个结果也正是Steinberg等人构建他们本构模型的初衷。
——为了综合检验本文G(P,T)和Y(P,T)本构函数式在描述冲击加载下LY12铝合金状态量变化及其运动特性的有效性,文中利用冲击/卸载、冲击/冲击/卸载、冲击/再冲击以及冲击/卸载/冲击等四类复合加载过程的样品/窗口界面速度剖面实验结果进行了检验,发现模型预测值与实测值的符合程度是令人满意的,从而全面认定了本文本构模型的有效性。
(3)在发展实验技术方面,本文采用Asay等人[SAND-85-2009]建议的组合靶实验方案,解决了前人未能给出-20GPa以上压力区冲击/再冲击实验数据的难题(本文中给出了高达-70GPa预冲击压力点的实验数据),从而为满足今后精准测定更高压力区屈服强度的要求打通了一条新的技术途径。
最后应该说明,本文提出的高温高压区金属本构模型与前人提出的相比,尽管它具有更为坚实的物理基础和更少的近似假设,但仅能看着是初步的结果,因为它的普适性还需要通过更多类别金属的检验。即使对LY12铝合金来说,也还需要更多的实验数据和理论计算结果的检验,因为,只有这样才能对本构函数式中的有关材料参数值进行精确化修正。,为此目的,发展或优化现有的实验和理论计算方法也是值得重视的。在发展实验技术方面,看来进一步发展高温高压条件下的激光-超声(Laser-ultrasonic)和静态压缩下的屈服强度测量技术是必要的。
The constitutive relationship of solids in high-pressure and high-temperature conditions is a specific equation which is necessary for the research of chemical detonation, high-energy ion or laser beam, and high velocity impact. Some shortcomings in Steinberg and his colleagues'work [J. Appl. Phys.,51(1980) 1498; J. Appl. Phys.,65(1989) 1528] on high-pressure constitutive relationship were discussed, according to some viewpoints from Chen et al [Chin. J. High Pres. Phys, 19 (2005) 193] and Peng [Ph.D.Thesis(Mianyang, CAEP)(2006)], and then a new constitutive model was presented. The main conclusions are as following:
(1) A new constitutive model for closed packed crystal material was presented based on the analysis on researches of Burakovsky et al [Phys. Rev.,B71(2005)184118], Nadal et al [J.Appl.Phys.,93(2003)2472], and Yu et al [Chin.Phys. Lett.,17(2008) 64] on Steinberg's constitutive model. The new functions are Here G and Y are shear modulus and yield strength, respectively,βand n are work hardening coefficients,εP is plastic strain, To is the reference temperature, usually taken as room temperature. The values of G(P,T0), ((?)G/(?0T)p,β, n andεP can be obtained from experiments or calculations.
The merits of this model are:——Compared with the methods of Steinberg et al, Nadal et al, and Yu et al, the total function, rather than the first order approximate, was applied in describing the relationship between G and P.——Although the first order approximate function was still applied to describe the change of G with T in the new functions, ((?)G/(?)T)P was used instead of ((?)G/(?)T)0, in the light of Ab initio calculation results presented by Xiang et al [Phys. Rev. B70 (2004) 174102].
(2) Taken aluminum as an example, the constitutive functions are Following issues supports the validation of these functions:——The G(P,T) function which was deduced from the Ab initio calculation results presented by Sin'ko et al [J.Phys.:Condens. Matter,4(2002) 6989] and Xiang et al has an explicit physical foundation. It is also supported by the sound velocity measurement along Hugoniot and shear modulus calculations along isentropic (with Birch- Murnaghan equation which is based on finite strain theory).——Although the physical conception of the new Y(P,T) function is still followed Steinberg's work, the constitutive form is different because the G(P,T) has a different form. And the calculation with this model is in good agreement with shock experiment results (according to Asay and Chhabildas's isotropy hardening model). Specially, the hypothesis of G(P,T)/Y(P,T)≈cons. is validated by experimental results at high pressures (for LY12 alloy, the threshold is Pc>30GPa). While under low pressures, G(P,T)lY*P,T)≈cons. is suggested (here Y*=Y/[1+β(εP+εPi)]n). In fact, this hypothesis is also the original intention of Steinberg's work.——Experimental measured velocity profiles of sample-window interface in shock/release, shock/reshock/release, shock/reshock, and shock/ release/reshock experiments on LY12 alloy were analyzed to verify the G(P,T) and Y(P,T) functions. The experimental results are in good agreement with the constitutive model. Thus the validation of the new functions is thoroughly confirmed.
(3) In experiment technique field, the assembly sample method suggested by Asay et al [SAND-85-2009] was utilized, with which the puzzle that there are no reloading experiment data beyond 20GPa for aluminum was solved (～70GPa shock stress experiment data are reached in our experiments.). This is a new approach to measure yield strength accurately at higher pressures.
Finally, though the constitutive model presented in this paper is based on explicit physical foundation with less approximate, it is worth pointing out that more experiments on different metals are needed to confirm its universality. Even for LY12 alloy, it is needed more experimental data or calculation results to deduce precise parameters of the model. For this purpose, it is important to develop or optimize present experimental and calculation methods. So it is necessary to develop; the laser-ultrasonic technique and static high-pressure yield strength measurements under high-temperature and high-pressure.
(1)在全面分析了Burakovsky等人[Phys. Rev. B,71(2005)184118]、Nadal等人[J. Appl. Phys.,93(2003)2472]和俞宇颖等人[Chin. Phys. ett.,17(2008)64]针对Steinberg等人的金属材料本构模型提出的若干质疑及修改意见的基础上,提出了一个新的适用于密堆集晶体金属的本构模型,具体的函数表示式是式中,G和Y为剪切模量和屈服强度；β和n为硬化系数,εp为塑性应变；T0为参考温度,通常取零温或室温。式中的G(P,T0), ((?)G/(?)T)P,β,n和εp等值均可通过实验和理论计算方法求得。这个模型具有如下优点：
——与Steinberg等人、Nadal等人和俞宇颖等人的意见相比较,本模型中G随P的变化没有采用在一阶导数项截断的近似表示,而是采用了G(P)的全函数表示形式；
——与Steinberg等人、Nadal等人、Burakovsky等人的意见相比较,本模型中G随T的变化虽然仍采用了一阶导数项的近似表达,但抛弃了把(aG/(?)T)P用(aG/(?)T)0代替的做法,而是在向士凯等人[Phy.Rev. B70 (2004)174102]第一原理计算结果启示下采用了把((?)G/(?)T)P表示为以P为参数的函数式；
(2)以LY12铝合金为例,本文给出的本构函数式是
这组本构函数式的有效性得到了以下检验结果的支持：
——上列G(P,T)本构函数式是基于Sin'ko等人[J. Phys.:Condens. Matter, 4(2002)6989]和向士凯等人的第一原理计算结果导出的,因而具有坚实物理基础。此外,它的有效性还得到了Hiigoniot压缩线上声速测量结果和等熵压缩线上剪切模量计算结果(通过有限形变理论的Birch-Murnaghan方程)的支持；
——上列Y(P,T)本构模型的构建尽管在其物理构思上沿袭了Steinberg等人的做法,但由于本文G(P,T)本构函数式的不同,故在Y(P,T)本构函数式的具体形式上是与Steinberg等人的有所不同的。它的有效性得到了本文冲击加载实验结果(采用了Asay和Chhabildas基于各向同性硬化模型的数据处理方法)的支持。特别值得提出的是：实验结果再一次支持了G(P,T)/Y(P,T)常数近似关系在高压区的有效性(对于LY12铝合金,这个高压区的阀值是Pc≥30GPa)；但若关注到Pc的低压区,则G(P,T)/Y*(P,T)≈常数近似关系(式中Y*＝Y/[1+β(εP+εPi)]n)也是成立的。其实,这个结果也正是Steinberg等人构建他们本构模型的初衷。
——为了综合检验本文G(P,T)和Y(P,T)本构函数式在描述冲击加载下LY12铝合金状态量变化及其运动特性的有效性,文中利用冲击/卸载、冲击/冲击/卸载、冲击/再冲击以及冲击/卸载/冲击等四类复合加载过程的样品/窗口界面速度剖面实验结果进行了检验,发现模型预测值与实测值的符合程度是令人满意的,从而全面认定了本文本构模型的有效性。
(3)在发展实验技术方面,本文采用Asay等人[SAND-85-2009]建议的组合靶实验方案,解决了前人未能给出-20GPa以上压力区冲击/再冲击实验数据的难题(本文中给出了高达-70GPa预冲击压力点的实验数据),从而为满足今后精准测定更高压力区屈服强度的要求打通了一条新的技术途径。
最后应该说明,本文提出的高温高压区金属本构模型与前人提出的相比,尽管它具有更为坚实的物理基础和更少的近似假设,但仅能看着是初步的结果,因为它的普适性还需要通过更多类别金属的检验。即使对LY12铝合金来说,也还需要更多的实验数据和理论计算结果的检验,因为,只有这样才能对本构函数式中的有关材料参数值进行精确化修正。,为此目的,发展或优化现有的实验和理论计算方法也是值得重视的。在发展实验技术方面,看来进一步发展高温高压条件下的激光-超声(Laser-ultrasonic)和静态压缩下的屈服强度测量技术是必要的。
The constitutive relationship of solids in high-pressure and high-temperature conditions is a specific equation which is necessary for the research of chemical detonation, high-energy ion or laser beam, and high velocity impact. Some shortcomings in Steinberg and his colleagues'work [J. Appl. Phys.,51(1980) 1498; J. Appl. Phys.,65(1989) 1528] on high-pressure constitutive relationship were discussed, according to some viewpoints from Chen et al [Chin. J. High Pres. Phys, 19 (2005) 193] and Peng [Ph.D.Thesis(Mianyang, CAEP)(2006)], and then a new constitutive model was presented. The main conclusions are as following:
(1) A new constitutive model for closed packed crystal material was presented based on the analysis on researches of Burakovsky et al [Phys. Rev.,B71(2005)184118], Nadal et al [J.Appl.Phys.,93(2003)2472], and Yu et al [Chin.Phys. Lett.,17(2008) 64] on Steinberg's constitutive model. The new functions are Here G and Y are shear modulus and yield strength, respectively,βand n are work hardening coefficients,εP is plastic strain, To is the reference temperature, usually taken as room temperature. The values of G(P,T0), ((?)G/(?0T)p,β, n andεP can be obtained from experiments or calculations.
The merits of this model are:——Compared with the methods of Steinberg et al, Nadal et al, and Yu et al, the total function, rather than the first order approximate, was applied in describing the relationship between G and P.——Although the first order approximate function was still applied to describe the change of G with T in the new functions, ((?)G/(?)T)P was used instead of ((?)G/(?)T)0, in the light of Ab initio calculation results presented by Xiang et al [Phys. Rev. B70 (2004) 174102].
(2) Taken aluminum as an example, the constitutive functions are Following issues supports the validation of these functions:——The G(P,T) function which was deduced from the Ab initio calculation results presented by Sin'ko et al [J.Phys.:Condens. Matter,4(2002) 6989] and Xiang et al has an explicit physical foundation. It is also supported by the sound velocity measurement along Hugoniot and shear modulus calculations along isentropic (with Birch- Murnaghan equation which is based on finite strain theory).——Although the physical conception of the new Y(P,T) function is still followed Steinberg's work, the constitutive form is different because the G(P,T) has a different form. And the calculation with this model is in good agreement with shock experiment results (according to Asay and Chhabildas's isotropy hardening model). Specially, the hypothesis of G(P,T)/Y(P,T)≈cons. is validated by experimental results at high pressures (for LY12 alloy, the threshold is Pc>30GPa). While under low pressures, G(P,T)lY*P,T)≈cons. is suggested (here Y*=Y/[1+β(εP+εPi)]n). In fact, this hypothesis is also the original intention of Steinberg's work.——Experimental measured velocity profiles of sample-window interface in shock/release, shock/reshock/release, shock/reshock, and shock/ release/reshock experiments on LY12 alloy were analyzed to verify the G(P,T) and Y(P,T) functions. The experimental results are in good agreement with the constitutive model. Thus the validation of the new functions is thoroughly confirmed.
(3) In experiment technique field, the assembly sample method suggested by Asay et al [SAND-85-2009] was utilized, with which the puzzle that there are no reloading experiment data beyond 20GPa for aluminum was solved (～70GPa shock stress experiment data are reached in our experiments.). This is a new approach to measure yield strength accurately at higher pressures.
Finally, though the constitutive model presented in this paper is based on explicit physical foundation with less approximate, it is worth pointing out that more experiments on different metals are needed to confirm its universality. Even for LY12 alloy, it is needed more experimental data or calculation results to deduce precise parameters of the model. For this purpose, it is important to develop or optimize present experimental and calculation methods. So it is necessary to develop; the laser-ultrasonic technique and static high-pressure yield strength measurements under high-temperature and high-pressure.
引文
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