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1. 弹性模量矩阵元
对于足够小的变形,由胡克定律(Hooke's Law)可知,应力与应变成正比,即应力分量是应变分量的线性函数,用矩阵的形式可以表示为:
T1 | C11 | C12 | C13 | C14 | C15 | C16 | S1 | ||
T2 | C12 | C22 | C23 | C24 | C25 | C26 | S2 | ||
T3 | = | C13 | C23 | C33 | C34 | C35 | C36 | = | S3 |
T4 | C14 | C24 | C34 | C44 | C45 | C46 | S4 | ||
T5 | C15 | C25 | C35 | C45 | C55 | C56 | S5 | ||
T6 | C16 | C26 | C36 | C46 | C56 | C66 | S6 |
式中Cij就是我们通常所说的弹性模量,可以证明,上述刚度矩阵为对称阵,Cij=Cji,因此,弹性模量的独立张量元数目至多只有21个。晶系的对称性越高,独立的张量元数目就越少。需要指出的是,Cij的数目只与晶系有关,而与晶系中具体的对称类型无关。
下面分别讨论七种不同晶系的弹性模量矩阵元:
1.1 三斜晶系(Triclinic system)
三斜晶系是所有七大晶系中对称性最低的晶系,因此拥有最多的独立矩阵元,其形式为:
C11 | C12 | C13 | C14 | C15 | C16 |
C12 | C22 | C23 | C24 | C25 | C26 |
C13 | C23 | C33 | C34 | C35 | C36 |
C14 | C24 | C34 | C44 | C45 | C46 |
C15 | C25 | C35 | C45 | C55 | C56 |
C16 | C26 | C36 | C46 | C56 | C66 |
共有21个独立的刚度矩阵元,求解过程也因此较为复杂。
1.2 单斜晶系(Monoclinic system)
C11 | C12 | C13 | 0 | 0 | C16 |
C12 | C22 | C23 | 0 | 0 | C26 |
C13 | C23 | C33 | 0 | 0 | C36 |
0 | 0 | 0 | C44 | C45 | 0 |
0 | 0 | 0 | C45 | C55 | 0 |
C16 | C26 | C36 | 0 | 0 | C66 |
考虑对称性后,单斜晶系有11个独立的矩阵单元。
1.3 正交晶系(Orthorhombic system)
C11 | C12 | C13 | 0 | 0 | 0 |
C12 | C22 | C23 | 0 | 0 | 0 |
C13 | C23 | C33 | 0 | 0 | 0 |
0 | 0 | 0 | C44 | 0 | 0 |
0 | 0 | 0 | 0 | C55 | 0 |
0 | 0 | 0 | 0 | 0 | C66 |
从上式可以看出,正交晶系拥有相当高的对成性,其独立刚度矩阵元的数目为8个。
1.4 四方晶系(Tetragonal system)
1.4.1 四方晶系(4,-4,4/m)
对于具有4,-4,4/m对称操作的四方晶系,其弹性矩阵的形式为:
C11 | C12 | C13 | 0 | 0 | C16 |
C12 | C22 | C23 | 0 | 0 | -C16 |
C13 | C23 | C33 | 0 | 0 | 0 |
0 | 0 | 0 | C44 | 0 | 0 |
0 | 0 | 0 | 0 | C44 | 0 |
C16 | -C16 | 0 | 0 | 0 | C66 |
其独立刚度矩阵元的数目也为8个。
1.4.2 四方晶系(422,4mm,-42m,4/mmm) 对于具有422,4mm,-42m,4/mmm对称操作的四方晶系,其弹性矩阵的形式为: C11 C12 C13 0 0 0 C12 C11 C13 0 0 0 C13 C13 C33 0 0 0 0 0 0 C44 0 0 0 0 0 0 C44 0 0 0 0 0 0 C66 独立刚度矩阵元的数目仅为6个。
1.5 三角晶系(Trigonal system)
1.5.1三角晶系(3,3)
C11 | C12 | C13 | C14 | C15 | 0 |
C12 | C11 | C13 | -C14 | -C15 | 0 |
C13 | C13 | C33 | 0 | 0 | 0 |
C14 | -C14 | 0 | C44 | 0 | -C45 |
C15 | -C15 | 0 | 0 | C44 | C14 |
0 | 0 | 0 | -C45 | C14 | (C11-C12)/2 |
三角晶系(3,3)的独立刚度矩阵元的数目为8个。
1.5.2三角晶系(32,3m,32/m)
C11 | C12 | C13 | C14 | 0 | 0 |
C12 | C11 | C13 | -C14 | 0 | 0 |
C13 | C13 | C33 | 0 | 0 | 0 |
C14 | -C14 | 0 | C44 | 0 | 0 |
0 | 0 | 0 | 0 | C44 | C14 |
0 | 0 | 0 | 0 | C14 | (C11-C12)/2 |
三角晶系(32,3m,32/m)的独立刚度矩阵元的数目为6个。
1.6 六角晶系(Hexagonal system)
C11 | C12 | C13 | 0 | 0 | 0 |
C12 | C11 | C13 | 0 | 0 | 0 |
C13 | C13 | C33 | 0 | 0 | 0 |
0 | 0 | 0 | C44 | 0 | 0 |
0 | 0 | 0 | 0 | C44 | 0 |
0 | 0 | 0 | 0 | 0 | (C11-C12)/2 |
六角晶系共有5个独立的刚度矩阵元。
1.7 立方晶系(Cubic system)
C11 | C12 | C12 | 0 | 0 | 0 |
C12 | C11 | C12 | 0 | 0 | 0 |
C12 | C12 | C11 | 0 | 0 | 0 |
0 | 0 | 0 | C44 | 0 | 0 |
0 | 0 | 0 | 0 | C44 | 0 |
0 | 0 | 0 | 0 | 0 | C44 |
立方晶系是所有晶系中对称度最高的晶系,其独立的刚度矩阵元数目仅为3个。
至此,我们列出了所有七大晶系的刚度矩阵元,只要求出各晶系对应的所有独立矩阵元,即可得到晶体的刚度矩阵。
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