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 材料工程  2019, Vol. 47 Issue (8): 59-81 PDF
http://dx.doi.org/10.11868/j.issn.1001-4381.2018.001412
0

#### 文章信息

LIU Pei-sheng, YANG Chun-yan, CHENG Wei

Study on property model for porous materials 3: mathematical deduction

Journal of Materials Engineering, 2019, 47(8): 59-81.
http://dx.doi.org/10.11868/j.issn.1001-4381.2018.001412

### 文章历史

Study on property model for porous materials 3: mathematical deduction
LIU Pei-sheng, YANG Chun-yan, CHENG Wei
Key Laboratory of Beam Technology of Ministry of Education, College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875, China
Abstract: Based on the "octahedral structure model" of three-dimensional reticulated porous materials, the mathematical deductions are introduced one by one for the mathematical relations of their basic physical and mechanical properties in this paper. The present review on these deductions covers the unidirectional tension and the multidirectional tension/compression of porous materials, as well as the conductivity and the fatigue property. Emphasis is placed on describing the equivalent circuit of the inner structure of porous materials, and the force action model both of quasi-rigid body and deformed body structure of porous materials under unidirectional tension. On this basis, the compressive strength is discussed, and the biaxial tension and triaxial tension/compression are mathematically deducted and analyzed. According to this octahedron model, the mathematical relations of mechanical properties can be also obtained from the deduction of unidirectional tension and compression, for porous materials under loading of non-direct tension and compression.
Key words: porous material    metal foam    ceramic foam    property model    mathematical deduction

1 多孔材料的电阻率

1.1 电阻率分析模型

 图 1 网状多孔材料电阻率分析模型 (a)立方格子导电单元；(b)导电单元等效电路 Fig. 1 Electrical resistivity analysis model of reticulated porous materials (a)cubic lattice of electrical conduction unit; (b)equivalent circuit of the conduction unit

1.2 数理关系推演

1.2.1 单元八面体有关尺寸计算 1.2.1.1 棱长(L)

 (1-1)

 (1-2)

1.2.1.2 棱径(r)

 (1-3)

 (1-4)
 (1-5)
1.2.1.3 中空孔径(r′)
 (1-6)
1.2.1.4 孔棱有效截面积(S)

 (1-7)
1.2.1.5 结点有效横截面积(S′)

 图 2 电流分析用结点纵向剖面图 Fig. 2 Sectional diagram of the node for electrical current analysis
 (1-8)

 (1-9)

 (1-10)
1.2.2 导电单元等效电路电阻率(ρ)

 (-11)

 (1-12)

1.2.2.1 半顶点电阻(R1)
 (1-13)
1.2.2.2 1/4侧点电阻(R2)
 (1-14)
1.2.2.3 棱电阻(R3)
 (1-15)

 (1-16)

 (1-17)

 (1-18)

 (1-19)

 (1-20)

2 多孔材料抗拉强度 2.1 抗拉强度分析模型

 图 3 网状多孔材料抗拉强度分析模型 (a)包容承载主单元八面体的立方格子单元；(b)承载单元中的孔棱受力分析图 Fig. 3 Tensile strength analysis model for reticulated porous materials (a)cubic lattice containing the unit octahedron that is loading; (b)force analysis diagram for pore strut in the loading unit

2.2 数理关系推演

 (2-1)

 (2-2)
2.2.1 准刚体结构受力模型

 (2-3)
 (2-4)

 (2-5)

 (2-6)

 (2-7)

 (2-8)

 (2-9)

σmax达到相应的许用应力[σ]时，σ即对应于多孔体整体的抗拉强度，故：

 (2-10)

 (2-11)

θ大于70%时，则为：

 (2-12)

K′和K的决定因素是多孔制品自身的结构状态。该结构状态不仅取决于制备工艺条件，而且相同制备工艺条件下也可随材质种类不同而不同。

2.2.2 变形体结构受力模型

 (2-13)
 (2-14)

 (2-15)

 (2-16)

 (2-17)

 (2-18)

σmax=[σ]时，式(2-18)中的σ即对应于多孔体整体的抗拉强度。同准刚体结构受力模型进行修正和处理，则当θ″=0或可视为足够小时可近似得到多孔体的σ为：

 (2-19)

 (2-20)

 (2-21)

 (2-22)
2.2.3 压缩强度问题

 (2-23)
2.2.4 补充说明

3 多孔材料的伸长率 3.1 伸长率分析模型

3.2 数理关系推演

 (3-1)

 (3-2)

 图 4 单元八面体所含孔隙(主孔)高度在多孔体拉伸前(a)和断裂后(b)的变化示意图 Fig. 4 Sketches of pore height in the unit octahedron before tension(a) and after rupture(b) of porous body
 (3-3)

 (3-4)

 (3-5)

 (3-6)

 (3-7)

 (3-8)

 (3-9)

 (3-10)

 (3-11)

 (3-12)

4 多孔材料弹性模量 4.1 弹性模量分析模型

 图 5 八面体孔隙单元拉伸变形分析图[33] (a)拉伸前; (b)拉伸后 Fig. 5 Tensile deformation analysis maps of octahedral pore unit[33] (a)before tension; (b)after tension
4.2 数理关系推演

Δα为：

 (4-1)

 (4-2)

 (4-3)

 (4-4)

 (4-5)

 (4-6)

 (4-7)
 (4-8)

 (4-9)

 (4-10)

 (4-11)

 (4-12)

 (4-13)

 (4-14)

 (4-15)
5 多孔材料双向拉伸 5.1 双向拉伸分析模型

 图 6 网状多孔材料双向拉伸分析模型 (a)八面体孔隙单元; (b)孔棱 Fig. 6 Analysis models for biaxial tension of reticulated porous materials (a)octahedron unit; (b)pore strut
5.2 数理关系推演 5.2.1 力分析

5.2.2 孔棱受力分析与关系推导

 (5-1)
 (5-2)
 (5-3)

 (5-4)
 (5-5)

A点作合力f方向的平行线与包含八面体的立方格子的棱边交于D点(见图 6)，则由式(5-4)和式(5-5)可得：

 (5-6)
 (5-7)

 (5-8)

 (5-9)

 (5-10)

 (5-11)

 (5-12)

 (5-13)
 (5-14)

 (5-15)

 (5-16)

 (5-17)

 (5-18)

 (5-19)

 (5-20)

 (5-21)

 (5-22)

 (5-23)
 (5-24)

 (5-25)

 (5-26)
5.3 双向拉压问题

6 三向拉压力学模型 6.1 三向拉压分析模型

 图 7 网状多孔材料三向拉伸分析模型[36-37] (a)八面体孔隙单元; (b)孔棱 Fig. 7 Analysis models for triaxial tension of reticulated porous materials[36-37] (a)octahedron unit; (b)pore strut
6.2 数理关系推演 6.2.1 孔棱受力总体分析

6.2.2 孔棱最大正应力推导

 (6-1)
 (6-2)
 (6-3)

f为:

 (6-4)

 (6-5)

 (6-6)
 (6-7)

 (6-8)

 (6-9)

 (6-10)
 (6-11)

 (6-12)
 (6-13)

 (6-14)

 (6-15)

 (6-16)
6.2.3 关系修正

 (6-17)

 (6-18)

 (6-19)

 (6-20)

6.2.4 简化表达

 (6-21)

 (6-22)

6.3 三向拉压问题

7 其他载荷形式力学模型 7.1 剪切载荷作用 7.1.1 物理模型

 图 8 剪切载荷作用下多孔体中孔隙单元的受力分析模型 (a)八面体单元; (b)孔棱 Fig. 8 Analysis model for pore unit of the porous body under shearing loads (a)octahedron unit; (b)pore strut
7.1.2 数理推演

 (7-1)

 (7-2)

 (7-3)

 (7-4)
 (7-5)

 (7-6)

 (7-7)

 (7-8)
7.1.3 多孔构件承载准则

 (7-9)

7.1.4 数理关系修正

 (7-10)

 (7-11)
7.2 扭转载荷作用(扭矩作用) 7.2.1 物理模型

7.2.2 数理推演 7.2.2.1 最大拉应力

 (7-12)

 (7-13)
7.2.2.2 简单情形分析举例

 图 9 多孔体圆轴扭转变形及其孔隙单元受力分析模型 (a)未变形的圆轴[41]; (b)变形后的圆轴[41]; (c)圆轴切应力分析单元[41]; (d)孔隙单元受力分析模型 Fig. 9 Torsional deformation of the porous shaft and the force analysis model for the pore unit (a)shaft before deformation[41]; (b)deformed shaft[41]; (c)analysis unit of shearing stress for the shaft[41]; (d)force analysis model for the pore unit

7.2.2.3 多孔构件承载准则

 (7-14)

7.2.2.4 数理关系修正

 (7-15)

 (7-16)

 (7-17)

 (7-18)

 (7-19)
7.3 弯曲载荷作用(弯矩作用) 7.3.1 物理模型

 图 10 弯曲载荷作用下多孔体中孔隙单元的受力分析模型 (a)弯矩M作用下的多孔元件[41]; (b)中性层左侧内部名义拉应力; (c)中性层右侧内部名义压应力 Fig. 10 Force analysis models of pore unit for the porous body under bending moment (a)porous component under bending moment M[41]; (b)internal nominal tensile stress on the left of the neutral layer; (c)internal nominal compressive stress on the right of the neutral layer
7.3.2 数理推演

 (7-20)

 (7-21)

 (7-22)

 (7-23)

 (7-24)
8 多孔材料疲劳性能 8.1 疲劳性能分析模型

8.2 数理关系推演 8.2.1 类应力疲劳

8.2.1.1 数理关系

 (8-1)

 (8-2)

 (8-3)

 (8-4)

8.2.1.2 对Fσ的分析

σmax达到弹性极限时，σMAX值若增加，则金属孔棱应变逐渐进入塑性区，式(8-4)不再适用。然而，对于同工艺同材质制备的多孔材料，在同一循环载荷条件下，若多孔体的Fσ越大，则应变超出弹性极限的距离也越远，故疲劳性能越差。因此，无论是在弹性区还是在塑性区内，Fσ都可作为衡量材料疲劳性能的指标。

8.2.2 类应变疲劳

 (8-5)

 图 11 等效孔径分析图 Fig. 11 Diagram for the equivalent size of pore diameter

b为：

 (8-6)

 (8-7)

 (8-8)
 (8-9)

 (8-10)

 (8-11)

 (8-12)

9 多孔材料比表面积 9.1 比表面积分析模型

9.2 数理关系推演

 (9-1)
 (9-2)
 (9-3)

 (9-4)

 (9-5)

 (9-6)

 (9-7)

 (9-8)

 (9-9)

，有：

 (9-10)

9.3 理论公式修正

 (9-11)

 (9-12)

10 结束语