Structure and Elasticity of Garnet under High Pressure by First-Principles Simulation

YANG Longxing; LIU Lei; LIU Hong; YI Li; GU Xiaoyu

doi:10.11858/gywlxb.20190785

Volume 33 Issue 6

Nov 2019

Turn off MathJax

Article Contents

Abstract

References

Chinese Journal of High Pressure Physics > 2019 > 33(6): 060104.

LI Zhengpeng, QU Yandong. Dynamic Response Analysis of Buried X70 Steel Pipe near Weld Zone under Blast Loads[J]. Chinese Journal of High Pressure Physics, 2020, 34(3): 034204. doi: 10.11858/gywlxb.20190831

Citation:

YANG Longxing, LIU Lei, LIU Hong, YI Li, GU Xiaoyu. Structure and Elasticity of Garnet under High Pressure by First-Principles Simulation[J]. Chinese Journal of High Pressure Physics, 2019, 33(6): 060104. doi: 10.11858/gywlxb.20190785

Citation:

PDF( 1685 KB)

Structure and Elasticity of Garnet under High Pressure by First-Principles Simulation

doi: 10.11858/gywlxb.20190785

Key Laboratory of Earthquake Forecasting, United Laboratory of High-Pressure Physics andEarthquake Science, Institute of Earthquake Forecasting, CEA, Beijing 100036, China

Received Date: 28 May 2019
Rev Recd Date: 12 Jun 2019
Issue Publish Date: 25 Sep 2019

Abstract

Abstract

Garnet is an important component of the upper mantle and mantle transition zone, and its properties under high temperature and pressure are of great significance to understand the composition, structure and dynamic process of mantle. Therefore, the crystal structure and elastic properties of pyrope, almandine, spessartite, uvarovite, grossular and andradite under 0–16 GPa, the six most common garnet in the Earth, were calculated by first principle method. The results show the unit cell volume of pyralaspite (pyrope, almandine, spessartite) is smaller than that of ugrandite (uvarovite, grossular and andradite), and the density of pyralaspite is higher than that of ugrandite except for pyrope. During structural compression, the volume change of polyhedron is from large to small as [XO₈] dodecahedron, [YO₆] octahedron and [SiO₄] tetrahedron, and their ratio is close to 3∶2∶1, indicating that the compression mechanism of garnet is mainly controlled by the dodecahedron. The variation of bond angle shows that tetrahedron and octahedron of the ugrandite would be more regular under high pressure; while the tetrahedron of pyralaspite becomes more irregular under high pressure. The bulk modulus of garnet increases with the increase of almandine, and decreases with the increase of uvarovite and grossular; while the shear modulus of garnet increases with the increase of grossular, and decreases with the increase of almandine and uvarovite. The wave velocity of pyralaspite is smaller than that of ugrandite except for pyrope. Calculation results show that the wave velocities of garnet intersect with the typical wave velocity model of the Earth near 410 km, proving that garnet is an important component of the mantle, and the existence of garnet and its solid solution with different compositions may have an important influence on the wave velocity structure of the Earth’s mantle.
- garnet,
- crystal structure,
- elasticity,
- high pressure,
- first principle simulation

FullText(HTML)

作为液体和气体长距离运输的一种重要方式，管道运输在国家经济发展和国民生活中发挥着重要作用。然而，随着城镇化进程的加速，城市管网系统密集分布，爆炸作用引起的管道安全问题受到国内外广泛关注^[1-3]。都的箭等^[4]通过实验研究发现，正对爆心管段背面受到很大的轴向拉应力作用，且管道受爆炸载荷的影响主要与爆心距有关。Ji等^[5]研究了X70钢管在局部爆炸载荷下的动力响应，发现管道的挠度和损伤程度随炸药量和接触面积的增大而增大，且壁厚对管道损伤和失效后的运动有重要作用。数值模拟是研究爆炸问题的一种重要方法，只要方法得当，模拟效果可与实际情况相吻合^[6-7]。为此，梁政等^[8]利用数值模拟方法研究了管道埋深、药量和管道壁厚因素对爆炸载荷下的埋地管道动力响应的影响。房冲^[9]通过模拟研究发现，在爆炸载荷下充水管道的变形量、位移和峰值压强都比内空管道小。余洋等^[10]采用野外实验与数值计算相结合的方法研究了初始条件对钢质方管在侧向局部爆炸载荷作用下损伤破坏效应的影响。

迄今为止，对爆炸载荷作用下焊缝区附近埋地钢管的动力响应的相关研究鲜有报道。基于此，以两种含Y型焊缝（坡口有2 mm余高焊缝和坡口无余高焊缝）的埋地X70钢管为例，采用有限元软件ANSYS/LS-DYNA，数值模拟研究爆炸载荷作用下焊缝区附近埋地X70钢管的动力响应规律，以期为埋地管线附近的爆破施工设计和埋地管线的安全防护提供一定的理论参考。

1. 有限元模型

1.1 计算模型

采用cm-g-μs单位制，建立由TNT炸药、黄土和焊接管道组成的计算模型，如图1所示。模型纵向长38.4 cm，管道中心到模型侧面的宽度为130.0 cm，模型整体高271.6 cm，其中：TNT炸药为边长14.0 cm的正方体，采用中心起爆方式；焊接管道为外径1 016.0 mm、壁厚14.6 mm的X70钢管。焊缝选取两种尺寸^[11]，分别为Y型坡口有余高(H = 2.0 mm)焊缝和Y型坡口无余高(H = 0)焊缝，如图2所示。为了提高计算收敛速度，将焊缝尺寸设计图进行适当的简化，简化模型如图3所示。两种焊缝均不考虑分层焊接工艺的影响，焊缝与管道采用共节点方式连接。

图 1 计算模型

Figure 1. Calculation model

下载: 全尺寸图片幻灯片

图 2 焊缝的设计尺寸

Figure 2. Design of weld size

下载: 全尺寸图片幻灯片

图 3 焊缝的简化模型

Figure 3. Simplified diagram of weld model

下载: 全尺寸图片幻灯片

考虑到计算模型的对称性，取1/2模型建模。炸药、黄土、管道及焊缝选用SOLID164六面体实体单元，用扫掠方式划分网格，并对焊缝位置进行网格细化处理。炸药和黄土采用欧拉网格，焊接管道和焊缝采用拉格朗日网格，运用任意拉格朗日-欧拉算法及管土间流固耦合算法模拟爆炸载荷作用下埋地焊接管道的动力响应。在土体外侧和底面设置透射边界条件，模型对称面施加对称约束。

1.2 计算工况

为了初步揭示爆炸载荷作用下两种焊缝形式的埋地焊接管道的动力响应规律，选取药包尺寸为14.0 cm × 14.0 cm × 14.0 cm的TNT炸药，对埋深为1.5 m的焊缝有余高（H = 2.0 mm）管道（管道A）和焊缝无余高（H = 0）管道（管道B），在炸高分别为60.0、85.0和110.0 cm的3种条件下的6种工况进行模拟计算，如表1所示，其中，h_B为炸高。

表 1 计算工况

Table 1. Calculation conditions

Weld type	Buried depth of pipeline/m	Size of TNT/(cm × cm × cm)	h_B/cm
No weld reinforcement (H = 0)	1.5	14.0 × 14.0 × 14.0	60.0, 85.0, 110.0
Weld reinforcement (H = 2.0 mm)	1.5	14.0 × 14.0 × 14.0	60.0, 85.0, 110.0

下载: 导出CSV

| 显示表格

1.3 材料参数

TNT炸药选用高能炸药模型（Mat_High_Explosive_Burn）和JWL状态方程定义。JWL状态方程表达式为

${p_{\rm{z}}}= A\left( {1 - \frac{\omega }{{{R_1}\nu }}} \right){{\rm{e}}^{ - {R_1}\nu }} + B\left( {1 - \frac{\omega }{{{R_2}\nu }}} \right){{\rm{e}}^{ - {R_2}\nu }} + \frac{{\omega E}}{\nu }$

(1)

式中：p_z为爆炸产物的压力，A、B、R₁、R₂、ω为TNT材料常数，v为爆炸产物的相对比容，E为炸药初始内能。炸药密度ρ_z、爆速D以及JWL状态方程参数见表2^[12]。

表 2 炸药材料参数^[12]

Table 2. Material parameters of explosive^[12]

ρ_z/(g·cm^–3)	D/(m·s^–1)	p/GPa	A/GPa	B/GPa	R₁	R₂	ω	E/(J·cm^–3)
1.58	6 930	21	373.77	3.75	4.15	0.90	0.35	6 000

下载: 导出CSV

| 显示表格

黄土选用泡沫模型（Mat_Soil_and_Foam）描述。该材料模型的应力屈服常数f为

$f = {S\!_{ij}}{\delta _{ij}}/2 - \left( {{a_0} + {a_1}{p_{\rm{t}}} + {a_2}{{p_{\rm{t}}}^2}} \right)$

(2)

式中：S_ij为土体材料的Cauchy偏应力张量，δ_ij为土体材料的Kronecker系数，a₀、a₁、a₂分别为土体摩擦角、土体黏聚力和土体爆炸动载效应的影响系数，p_t为土体压力。a₀、a₁、a₂由土工实验测得的内摩擦角和土壤黏聚力参数确定，土体密度ρ_t、剪切模量G、体积模量K等参数见表3^[13-14]。

表 3 土体材料参数^[13–14]

Table 3. Material parameters of soil^[13–14]

ρ_t/(g·cm^–3)	G/MPa	K/MPa	a₀/Pa²	a₁/Pa	a₂
1.8	41.14	87.87	2.12 × 10⁸	5.23 × 10³	3.22 × 10^–2

下载: 导出CSV

| 显示表格

X70钢管道和焊缝均采用双线性随动材料模型（Mat_Plastic_Kinematic）描述，遵循von Mises屈服准则，其表达式为

$\sigma =\begin{cases} {{E_{\rm{s}}}\varepsilon }&{\varepsilon \leqslant {\varepsilon _{\rm{e}}}}\\ {{\sigma _{\rm{y}}} + {E_{\rm{t}}}(\varepsilon - {\varepsilon _{\rm{e}}})}&{\varepsilon > {\varepsilon _{\rm{e}}}} \end{cases}$

(3)

式中：σ为应力；σ_y为屈服应力；E_s为弹性模量；E_t为切线模量，0 < E_t < E_s；ε为应变；ε_e为弹性极限应变。管道和焊缝的具体材料参数见表4^[15-18]，其中，μ为泊松比。

表 4 管道及焊缝材料参数^[15–18]

Table 4. Material parameters of pipe and weld bead^[15–18]

Material	ρ/(g·cm^–3)	μ	E_s/GPa	E_t/GPa	σ_y/GPa
X70-pipeline^[15–16]	7.90	0.3	210	13.5	0.48
Weld bead^[17–18]	7.25	0.3	220	15.3	0.55

下载: 导出CSV

| 显示表格

2. 结果与讨论

2.1 管道应力分析

图4为边长14.0 cm的正方体TNT炸药爆炸时，炸高h_B为60.0 cm，埋深为1.5 m的两种X70管道焊缝附近的von-Mises应力云图。由图4可以看出：当传播时间为1 440 μs时，爆炸应力波阵面已经接触管道；当传播时间为1 600 μs时，焊缝有2.0 mm余高的管道A和焊缝无余高的管道B的最大应力增幅分别为81.4 MPa和43.0 MPa；当传播时间为1 920 μs时，管道A和管道B的最大应力均大于焊缝与管道的材料屈服应力，且应力沿迎爆面正对爆心位置向外扩展，其中管道A的应力呈“十”字形扩展，而管道B的应力以椭圆形向四周扩展；在3 520 μs时，管道应力集中主要沿裂缝位置发展，管道A和管道B的应力最大值分别为601.2 MPa和591.0 MPa；在6 080 μs时，管道A和管道B继续变形但应力减小，应力最大值分别减小到581.8 MPa和565.8 MPa；在9 120 μs时，管道A和管道B的应力集中基本消失。

图 4 不同时刻X70钢管道的von Mises应力

Figure 4. von Mises stress of X70 steel pipe at different moments

下载: 全尺寸图片幻灯片

图5和图6分别为两种管道外表面上正对爆心位置的焊缝与管道分界面处焊缝单元与管道单元的应力时程曲线。在管道受爆炸载荷作用阶段，两种管道的应力在大约480 μs内呈跳跃式上升。其主要原因是管道为瞬时受力，一部分爆炸能量使管道变形并向管道四周传递，导致焊缝与管道分界面处两个典型单元的应力呈降低趋势，此现象与图4的应力云图吻合。根据应力集中系数和余高关系的经验公式^[19]可得：管道A和管道B的应力集中系数分别为1.016和1.008，即随着余高增大，应力集中系数逐渐增大。对比图5和图6可知，管道A的焊缝单元应力峰值较高，应力下降趋势相对较陡。这也说明焊缝余高的存在使得焊缝与管道分界面的截面尺寸突变增大，从而导致焊缝有余高的焊接管道受应力集中的影响较大。在1 912 μs时，图5（管道A）和图6（管道B）的焊缝单元应力最大，分别约为560.0 MPa（焊缝的屈服强度为550 MPa）和545.6 MPa。同时，管道A的焊缝处首先达到管道屈服强度（480 MPa），按照von Mises屈服准则，管道A开始进入局部塑性变形阶段，此时管道B的应力尚未达到材料的屈服强度。

图 5 管道A(H = 2.0 mm)典型单元的应力时程曲线

Figure 5. Stress-time curves of classic element of A pipe (H = 2.0 mm)

下载: 全尺寸图片幻灯片

图 6 管道B(H = 0)典型单元的应力时程曲线

Figure 6. Stress-time curves of classic element of B pipe (H = 0)

下载: 全尺寸图片幻灯片

2.2 管道位移分析

埋地X70管道的迎爆面和背爆面的最大位移如表5所示。从表5可知，由于爆炸冲击波的一部分能量在土中被耗散，且随着爆炸冲击波在土中传播距离的增大，两种焊缝形式管道的迎爆面和背爆面的最大位移均呈现减小的趋势。当炸高h_B从60.0 cm增加到85.0 cm以及从85.0 cm增加到110.0 cm时，管道A和管道B迎爆面的最大位移减小量分别为2.303 cm、0.715 cm和2.300 cm、0.572 cm，而管道A和管道B背爆面的最大位移减小量分别为0.391 cm、0.235 cm和0.373 cm、0.280 cm。两种焊缝形式管道迎爆面的最大位移减小量大于背爆面，这是由于爆炸冲击载荷在土中传播后直接作用于管道迎爆面，对管道迎爆面产生的影响较大，土体对管道背爆面具有一定的支撑作用，从而减小了管道背爆面位移。在相同炸高下管道A比管道B的最大位移大，且在炸高为60.0、85.0和110.0 cm时，两种焊缝形式管道迎爆面的最大位移差值分别为0.270、0.267和0.124 cm，即随着炸高的增大，两种焊缝形式的埋地焊接管道最大位移的差值逐渐减小，也说明当炸高较小时，管道A整体抵抗变形的能力弱于管道B。然而，随着炸高的增大，作用于管道的能量减小^[20]，管道塑性变形较小，使得这种现象逐渐模糊。

表 5 埋地X70管道的迎爆面和背爆面的最大位移

Table 5. Maximum displacement of explosion-front and explosion-back surfaces of buried X70 pipeline

Types of weld	h_B/cm	Maximum displacement/cm
Types of weld	h_B/cm	Explosion-front surface	Explosion-back surface
Weld reinforcement (H = 2.0 mm)	60.0	5.482	0.846
	85.0	3.179	0.455
	110.0	2.464	0.220
No weld reinforcement (H = 0)	60.0	5.212	0.943
	85.0	2.912	0.570
	110.0	2.340	0.290

下载: 导出CSV

| 显示表格

2.3 管道等效应变分析

表6为两种不同类型焊缝的埋地焊接管道在不同炸高下的最大等效应变统计。从表6可知，管道A和管道B的最大等效应变均随炸高的增大而减小。当炸高从60.0 cm增大到85.0 cm时，管道A和管道B的最大等效应变分别减小约58.12%和61.13%；当炸高从85.0 cm增大到110.0 cm时，管道A和管道B的最大等效应变分别减小约45.92%和38.05%，在炸高相同时，管道A的最大等效应变大于管道B，且管道A的最大等效应变位于焊缝余高表面，而管道B的最大等效应变在焊缝与管道处一定范围内沿纵向分布。这在一定程度上说明管道B能更好地协调焊缝与管道分界处的应变，有利于保障焊缝与管道的局部协同变形性能。

表 6 不同炸高时埋地X70管道的最大等效应变

Table 6. Maximum effective strain of buried X70 pipeline with different blasting heights

Types of weld	h_B/cm	Peak effective strain/10^–3
Weld reinforcement (H = 2.0 mm)	60.0	9.937
	85.0	4.162
	110.0	2.251
No weld reinforcement (H = 0)	60.0	6.877
	85.0	2.673
	110.0	1.656

下载: 导出CSV

| 显示表格

2.4 管道振速分析

表7为不同炸高下两种焊缝形式管道的迎爆面和背爆面处焊缝位置中心单元X方向的最大振动速度。从表7可以看出，两种焊缝形式管道的迎爆面和背爆面的最大振动速度均随着炸高增大而减小，且迎爆面的最大振速均大于背爆面。这说明迎爆面受爆炸地震波的影响较大。当炸高h_B为60.0、85.0和110.0 cm时，管道B的迎爆面的最大振动速度较管道A大，迎爆面差值分别为1.600、0.539和0.329 m/s，而背爆面差值在0.200 m/s以内。管道峰值速度随着管壁厚度的增大而减小^[12]，由于管道A增加了管道在焊缝位置的径向厚度，可将其视为管道焊缝位置的壁厚增大导致管道A的峰值振速减小。这说明管道A抵抗爆炸振动的性能优于管道B，且在炸高为60.0 cm时，管道A抵抗振动性能的优势较为明显。

表 7 埋地X70管道的迎爆面和背爆面最大振速

Table 7. Maximum vibration velocity of explosion-front and explosion-back surfaces of buried X70 pipeline

Types of weld	h_B/cm	Maximum vibration velocity/(m·s^–1)
Types of weld	h_B/cm	Explosion-front surface	Explosion-back surface
Weld reinforcement (H = 2.0 mm)	60.0	22.748	4.431
	85.0	9.316	2.817
	110.0	4.503	1.693
No weld reinforcement (H = 0)	60.0	24.348	4.294
	85.0	9.855	2.867
	110.0	4.832	1.746

下载: 导出CSV

| 显示表格

图7为不同炸高时两种焊缝形式的管道典型单元的速度时程曲线。当炸高h_B分别为60.0、85.0和110.0 cm时，管道A和管道B达到最大振速的时间分别为2 560 μs和2 560 μs、4 500 μs和4 600 μs、7 200 μs和7 200 μs，两种焊缝形式的管道达到最大振速的时间差值均在100 μs以内。这说明两种焊缝形式的管道达到最大振速的时间主要受炸高的影响，受焊缝形式的影响较小。

图 7 管道典型单元的速度时程曲线

Figure 7. Velocity-time curve of typical pipeline elements

下载: 全尺寸图片幻灯片

3. 结　论

（1）当炸高为60.0 cm时，两种焊缝形式的埋地X70焊接管道在爆炸载荷作用下焊缝位置均出现应力集中，但焊缝有余高的管道受应力集中影响较大，且会先于焊缝无余高管道进入屈服阶段。

（2）当炸高为60.0～110.0 cm时，由于爆炸载荷直接作用于迎爆面，且管土间的相互作用对管道背爆面具有一定的支撑作用，两种焊缝形式管道迎爆面的最大位移均大于背爆面的最大位移。当炸高为60.0、85.0 cm时，焊缝有余高的管道整体抵抗变形的能力明显弱于焊缝无余高的管道。

（3）焊缝无余高管道较焊缝有余高管道在焊缝与管道分界处的应变更为协调，能更好地保障焊缝与管道的局部协同变形性能。

（4）在相同的爆炸载荷下，焊缝有余高管道抵抗振动的性能优于焊缝无余高管道。药量相同条件下，相对于焊缝形式，炸高对含焊缝区管道的最大振速起主要作用。

References(44)

References

[1]	PALKE A C, STEBBINS J F, GEIGER C A, et al. Cation order-disorder in Fe-bearing pyrope and grossular garnets: a ²⁷Al and ²⁹Si MAS NMR and ⁵⁷Fe Mossbauer spectroscopy study [J]. American Mineralogist, 2015, 100(2/3): 536–547.
[2]	范大伟, 李博, 陈伟, 等. 石榴子石族矿物状态方程研究进展 [J]. 高压物理学报, 2018, 32(1): 010101. doi: 10.11858/gywlxb.20170597 FAN D W, LI B, CHEN W, et al. Research progress of the equation of state for garnet minerals [J]. Chinese Journal of High Pressure Physics, 2018, 32(1): 010101. doi: 10.11858/gywlxb.20170597
[3]	FEI Y, BERTKA C M. Phase transitions in the Earth’s mantle and mantle mineralogy [J]. Mantle Petrology: Field Observations and High Pressure Experimentation, 1999, 6: 189–207.
[4]	RINGWOOD A E. Phase transformations and their bearing on the constitution and dynamics of the mantle [J]. Geochimica et Cosmochimica Acta, 1991, 55(8): 2083–2110. doi: 10.1016/0016-7037(91)90090-R
[5]	IRIFUNE T, RINGWOOD A E. Phase transformations in subducted oceanic crust and buoyancy relationships at depths of 600–800 km in the mantle [J]. Earth and Planetary Science Letters, 1993, 117(1/2): 101–110.
[6]	WANG Z, JI S. Elasticity of six polycrystalline silicate garnets at pressure up to 3.0 GPa [J]. American Mineralogist, 2001, 86(10): 1209–1218. doi: 10.2138/am-2001-1009
[7]	CONRAD P G, ZHA C S, MAO H K, et al. The high-pressure, single-crystal elasticity of pyrope, grossular, and andradite [J]. American Mineralogist, 1999, 84(3): 374–383. doi: 10.2138/am-1999-0321
[8]	BABUŠKA V, FIALA J, KUMAZAWA M, et al. Elastic properties of garnet solid-solution series [J]. Physics of the Earth and Planetary Interiors, 1978, 16(2): 157–176. doi: 10.1016/0031-9201(78)90086-9
[9]	ANDERSON O L, NAFE J E. The bulk modulus-volume relationship for oxide compounds and related geophysical problems [J]. Journal of Geophysical Research, 1965, 70(16): 3951–3963. doi: 10.1029/JZ070i016p03951
[10]	ANDERSON D L, ANDERSON O L. The bulk modulus-volume relationship for oxides [J]. Journal of Geophysical Research, 1970, 75(26): 3494–3500.
[11]	HAZEN R M. Crystal structures and compressibilities of pyrope and grossular to 60 kbar [J]. American Mineralogist, 1978, 63(3/4): 297–303.
[12]	LEGER J M, REDON A M, CHATEAU C. Compressions of synthetic pyrope, spessartine and uvarovite garnets up to 25 GPa [J]. Physics and Chemistry of Minerals, 1990, 17(2): 161–167.
[13]	ZHANG L, AHSBAHS H, KUTOGLU A, et al. Single-crystal hydrostatic compression of synthetic pyrope, almandine, spessartine, grossular and andradite garnets at high pressures [J]. Physics and Chemistry of Minerals, 1999, 27(1): 52–58. doi: 10.1007/s002690050240
[14]	TAKAHASHI T, LIU L G. Compression of ferromagnesian garnets and the effect of solid solutions on the bulk modulus [J]. Journal of Geophysical Research, 1970, 75(29): 5757–5766. doi: 10.1029/JB075i029p05757
[15]	HUANG S, CHEN J. Equation of state of pyrope–almandine solid solution measured using a diamond anvil cell and in situ synchrotron X-ray diffraction [J]. Physics of the Earth and Planetary Interiors, 2014, 228: 88–91. doi: 10.1016/j.pepi.2014.01.014
[16]	MILANI S, NESTOLA F, ALVARO M, et al. Diamond-garnet geobarometry: the role of garnet compressibility and expansivity [J]. Lithos, 2015, 227: 140–147. doi: 10.1016/j.lithos.2015.03.017
[17]	FAN D, XU J, MA M, et al. P-V-T equation of state of spessartine-almandine solid solution measured using a diamond anvil cell and in situ synchrotron X-ray diffraction [J]. Physics and Chemistry of Minerals, 2015, 42(1): 63–72. doi: 10.1007/s00269-014-0700-2
[18]	MURAKAMI M, SINOGEIKIN S V, LITASOV K, et al. Single-crystal elasticity of iron-bearing majorite to 26 GPa: implications for seismic velocity structure of the mantle transition zone [J]. Earth and Planetary Science Letters, 2008, 274(3/4): 339–345.
[19]	FAN D W, WEI S Y, LIU J, et al. High pressure X-ray diffraction study of a grossular-andradite solid solution and the bulk modulus variation along this solid solution [J]. Chinese Physics Letters, 2011, 28(7): 076101. doi: 10.1088/0256-307X/28/7/076101
[20]	FAN D W, KUANG Y, XU J, et al. Thermoelastic properties of grossular-andradite solid solution at high pressures and temperatures [J]. Physics and Chemistry of Minerals, 2017, 44(2): 137–147. doi: 10.1007/s00269-016-0843-4
[21]	DU W, CLARK S M, WALKER D. Thermo-compression of pyrope-grossular garnet solid solutions: non-linear compositional dependence [J]. American Mineralogist, 2014, 100(1): 215–222.
[22]	GILLAN M J, ALFÈD, BRODHOLT J, et al. First-principles modelling of Earth and planetary materials at high pressures and temperatures [J]. Reports on Progress in Physics, 2006, 69(8): 2365–2441. doi: 10.1088/0034-4885/69/8/R03
[23]	PERDEW J P, ZUNGER A. Self-interaction correction to density-functional approximations for many-body systems [J]. Physical Review B, 1981, 23(10): 5048–5079. doi: 10.1103/PhysRevB.23.5048
[24]	WENTZCOVITCH R M, MARTINS J L, PRICE G D. Ab initio molecular dynamics with variable cell shape: application to MgSiO₃ [J]. Physical Review Letters, 1993, 70(25): 3947–3950. doi: 10.1103/PhysRevLett.70.3947
[25]	DA SILVA C, STIXRUDE L, WENTZCOVITCH R M. Elastic constants and anisotropy of forsterite at high pressure [J]. Geophysical Research Letters, 1997, 24(15): 1963–1966. doi: 10.1029/97GL01756
[26]	KARKI B B, STIXRUDE L, WENTZCOVITCH R M. High-pressure elastic properties of major materials of Earth’s mantle from first principles [J]. Reviews of Geophysics, 2015, 39(4): 507–534.
[27]	LIU L, DU J G, ZHAO J, et al. Elastic properties of hydrous forsterites under high pressure: first-principle calculations [J]. Physics of the Earth and Planetary Interiors, 2009, 176(1): 89–97.
[28]	LIU L, DU J, LIU W, et al. Elastic behavior of (Mg_xFe_{1- x})₂SiO₄ olivine at high pressure from first-principles simulations [J]. Journal of Physics and Chemistry of Solids, 2010, 71(8): 1094–1097. doi: 10.1016/j.jpcs.2010.03.013
[29]	LIU L, DU J G, LIU H, et al. Differential stress effect on the structural and elastic properties of forsterite by first-principles simulation [J]. Physics of the Earth and Planetary Interiors, 2014, 233: 95–102. doi: 10.1016/j.pepi.2014.06.010
[30]	LIU L, LV C J, ZHUANG C Q, et al. Effects of differential stress on the structure and Raman spectra of calcite from first-principles calculations [J]. American Mineralogist, 2016, 101(8): 1892–1897. doi: 10.2138/am-2016-5558
[31]	CEPERLEY D M, ALDER B J. Ground state of the electron gas by a stochastic method [J]. Physical Review Letters, 1980, 45(7): 566–569. doi: 10.1103/PhysRevLett.45.566
[32]	NIELSEN O H, MARTIN, RICHARD M. First-principles calculation of stress [J]. Physical Review Letters, 1983, 50(9): 697–700. doi: 10.1103/PhysRevLett.50.697
[33]	VANDERBILT D. Soft self-consistent pseudopotentials in a generalized eigenvalue formalism [J]. Physical Review B, 1990, 41(11): 7892–7895. doi: 10.1103/PhysRevB.41.7892
[34]	NIELSEN O H, MARTIN R M. Quantum-mechanical theory of stress and force [J]. Physical Review B, 1985, 32(6): 3780–3791. doi: 10.1103/PhysRevB.32.3780
[35]	马艳梅, 彭刚, 李敏. 镁铝石榴子石的高压X射线衍射研究 [J]. 高压物理学报, 2008, 22(3): 305–308. doi: 10.3969/j.issn.1000-5773.2008.03.014 MA Y M, PENG G, LI M. X-ray diffraction investigation of pyrope under pressure [J]. Chinese Journal of High Pressure Physics, 2008, 22(3): 305–308. doi: 10.3969/j.issn.1000-5773.2008.03.014
[36]	WEBB S L. The elasticity of the upper mantle orthosilicates olivine and garnet to 3 GPa [J]. Physics and Chemistry of Minerals, 1989, 16(7): 684–692.
[37]	ZOU Y, GRÉAUX S, IRIFUNE T, et al. Thermal equation of state of Mg₃Al₂Si₃O₁₂ pyrope garnet up to 19 GPa and 1700 K [J]. Physics and Chemistry of Minerals, 2012, 39(7): 589–598. doi: 10.1007/s00269-012-0514-z
[38]	HAZEN R M, DOWNS R T, CONRAD P G, et al. Comparative compressibilities of majorite-type garnets [J]. Physics and Chemistry of Minerals, 1994, 21(5): 344–349.
[39]	BASS J D. Elasticity of uvarovite and andradite garnets [J]. Journal of Geophysical Research: Solid Earth, 1986, 91(B7): 7505–7516. doi: 10.1029/JB091iB07p07505
[40]	BASS J D. Elasticity of grossular and spessartite garnets by Brillouin spectroscopy [J]. Journal of Geophysical Research, 1989, 94(B6): 7621–7628. doi: 10.1029/JB094iB06p07621
[41]	O’NEILL B, BASS J, R. SMYTH J, et al Elasticity of a grossular-pyrope-almandine garnet [J]. Journal of Geophysical Research Solid Earth, 1989, 94(B12): 17819–17824. doi: 10.1029/JB094iB12p17819
[42]	SATO Y, AKAOGI M, AKIMOTO S I. Hydrostatic compression of the synthetic garnets pyrope and almandine [J]. Journal of Geophysical Research, 1978, 83(B1): 335–338. doi: 10.1029/JB083iB01p00335
[43]	徐光宪, 王祥云. 物质结构[M]. 2版. 北京: 高等教育出版社, 1987: 621–622. XU G X, WANG X Y. Material structure [M]. 2nd ed. Beijing: Higher Education Press, 1987: 621–622.
[44]	LI L, WEIDNER D J, BRODHOLT J, et al. Ab initio molecular dynamic simulation on the elasticity of Mg₃Al₂Si₃O₁₂ pyrope [J]. Journal of Earth Science, 2011, 22(2): 169–175. doi: 10.1007/s12583-011-0169-6

Relative Articles

[1]	ZHAO Yong-Nian, ZHANG Zhi-Lin, CUI Qi-Liang, LIU Zhen-Xian, ZOU Guang-Tian. The Compression Behavior of Raman Active Mode E2g of Graphite and the Pressure Effect of Force Constant of CC Bonding[J]. Chinese Journal of High Pressure Physics, 1992, 6(1): 48-53 . doi: 10.11858/gywlxb.1992.01.007
[2]	ZHOU Jian-Shi. The High Oxygen Pressure Synthesis of Metallic LaCuO3 and the XPS Study on Cu3+ State[J]. Chinese Journal of High Pressure Physics, 1992, 6(1): 7-14 . doi: 10.11858/gywlxb.1992.01.002
[3]	LIU Hong-Jian, LI Li-Ping, WANG Yi-Feng, SU Wen-Hui. Structural Characters and the Conductivity of Nd2-xCexCuO4 Samples prepared under Atmosphere and High Pressure[J]. Chinese Journal of High Pressure Physics, 1991, 5(3): 161-168 . doi: 10.11858/gywlxb.1991.03.001
[4]	ZHAO Yong-Nian, LIU Zhen-Xian, CUI Qi-Liang, XU Yi-Qiu, FAN Yu-Guo, ZOU Guang-Tian. Studies of Raman Spectra for MnF2 at High-Pressures[J]. Chinese Journal of High Pressure Physics, 1990, 4(1): 1-6 . doi: 10.11858/gywlxb.1990.01.001
[5]	LIU Zhen-Xian, CUI Qi-Liang, ZHAO Yong-Nian, ZOU Guang-Tian. Influence of Pressure-Transmitting Media on the Lattice Vibration and Phase Transition Pressure-High Pressure Raman Spectra Studies of -Bi2O3[J]. Chinese Journal of High Pressure Physics, 1990, 4(2): 81-86 . doi: 10.11858/gywlxb.1990.02.001
[6]	SUN Bao-Quan, WANG Yi-Feng, LIU Hong-Jian, LI Li-Ping, SU Wen-Hui. The Influences of Cool- and Hot-Pressure Treatment on the Phase Transition and Superconductivity of YBa2Cu3O7-[J]. Chinese Journal of High Pressure Physics, 1990, 4(4): 246-253 . doi: 10.11858/gywlxb.1990.04.002
[7]	WANG Ru-Ju, LI Feng-Ying, YAO Yu-Shu, WANG Ji-Fang, LIN Xue-Shi. Studies of Phase Transition in LADTGS Crystal with Ultrasonic Method[J]. Chinese Journal of High Pressure Physics, 1989, 3(2): 156-158 . doi: 10.11858/gywlxb.1989.02.009
[8]	SUN Feng-Guo, LIN Hua-Ling. A New Calculation of Energy and Polarizability of Hydrogen-Like Atom under High Pressure[J]. Chinese Journal of High Pressure Physics, 1989, 3(3): 246-249 . doi: 10.11858/gywlxb.1989.03.011
[9]	CHEN Xu, JIN Xiao-Gang, YANG Mu-Song. Experimental Studies of Hugoniot for Powder Mixture of BaCO3 and TiO2 and Shock Wave Synthesis of BaTiO3[J]. Chinese Journal of High Pressure Physics, 1989, 3(1): 67-77 . doi: 10.11858/gywlxb.1989.01.009
[10]	ZHANG Shu-Hua. Second and Third Order Elastic Constants of Titanium Alloy TC-6[J]. Chinese Journal of High Pressure Physics, 1989, 3(2): 172-175 . doi: 10.11858/gywlxb.1989.02.012
[11]	Lü Zhi, DAI Yu-Zhi. Influence of Diamond Particle-Size on Wearability of High Pressure Synthetic Polycrystal Diamond[J]. Chinese Journal of High Pressure Physics, 1988, 2(1): 85-88 . doi: 10.11858/gywlxb.1988.01.012
[12]	GU Yuan, WANG Yong-Gang, MAO Chu-Sheng, NI Yuan-Long, WU Feng-Chun, MA Min-Xun. Preliminary Experiments on the Equation of State of Materials at High-Pressure Produced by Laser Driven Shock Waves[J]. Chinese Journal of High Pressure Physics, 1988, 2(2): 165-170 . doi: 10.11858/gywlxb.1988.02.011
[13]	YIN Xiu-Jun, SHEN Zhong-Yi, ZHANG Yun, HONG Jing-Xin. A Measurement of the Curie Temperature in Ferromagnetic Materials[J]. Chinese Journal of High Pressure Physics, 1988, 2(3): 259-263 . doi: 10.11858/gywlxb.1988.03.009
[14]	WANG De-Xin, LIU Pei-Luan, JIAO Qing-Yu, XUE Yong-Jin, YANG Guo-Qing. Experimental Investigation on Densification of Sintered Diamond Polycrystals with Inclusions[J]. Chinese Journal of High Pressure Physics, 1988, 2(1): 89-91 . doi: 10.11858/gywlxb.1988.01.013
[15]	WANG Yi-Feng, SU Wen-Hui, QIAN Zheng-Nan, MA Xian-Feng, YAN Xue-Wei. A Study on the High Temperature-High Pressure Stability and the X-Ray Phase Analysis for Compounds R2Fe4/3W2/3O7 with Pyrochlore Structure[J]. Chinese Journal of High Pressure Physics, 1988, 2(4): 296-304 . doi: 10.11858/gywlxb.1988.04.002
[16]	WANG Gui-Chao. An Empirical Expression of Elastic Sound Speed of Materials at High Pressures[J]. Chinese Journal of High Pressure Physics, 1988, 2(1): 92-95 . doi: 10.11858/gywlxb.1988.01.014
[17]	WU Shao-Zeng, ZHANG Cheng-Xiang, WU Li-Xin. Calculation of the Gruneisen Parameter and Its Pressure Dependence for NaCl[J]. Chinese Journal of High Pressure Physics, 1987, 1(2): 155-164 . doi: 10.11858/gywlxb.1987.02.009
[18]	YAN Xue-Wei, MA Xian-Feng, NI Jia-Zan, QIAN Zheng-Nan, WANG Yi-Feng, SU Wen-Hui. Synthesis of Jewelry Jadeite at High Pressure and High Temperature[J]. Chinese Journal of High Pressure Physics, 1987, 1(1): 76-80 . doi: 10.11858/gywlxb.1987.01.010
[19]	CUI Shuo-Jing, XU Da-Peng, SU Wen-Hui, WANG Fu-Song, WANG Sheng-Long. An Investigation on Electric Properties of the Polyaniline Complexes under High Pressure[J]. Chinese Journal of High Pressure Physics, 1987, 1(1): 71-75 . doi: 10.11858/gywlxb.1987.01.009
[20]	WANG Ru-Ju, WANG Ji-Fang. Elastic Behaviour of the Gadolinium Gallium Garnet Crystal under High Pressure[J]. Chinese Journal of High Pressure Physics, 1987, 1(1): 32-38 . doi: 10.11858/gywlxb.1987.01.005

Supplements(0)

Cited By

Proportional views

Proportional views

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Figures(7) / Tables(2)

Get Citation

PDF

XML

Article Metrics

Article views(9754) PDF downloads(59)