2014-04-17 21:21:03 感谢硕士生伊国辉整理此文
2015-03-05 09:31:31 修正公式错误
2016-06-30 09:14:23 增加不同晶系代码. 感谢 吴俊彦 校对格公式.
背景原理简介
各向异性广泛存在各种材料之中. 什么是各向异性呢? 简单来说, 就是晶体在不同方向有不同的性能. 我们很容易把各向异性和不均匀性混淆, 其实这是两个完全不同的概念. 各向异性就是说材料的性能与方向有关, 而不均匀性是指材料的性能与部位有关. 拿单晶来说, 内部任一点, 结构与性能都是相同的, 但不同方向却有不同的性能$.$
晶体是各向异性的, 不同方向性能不一样, 而且有着严格的对称性. 这里要说一下, 对称性是一个很了不起的性质, 在数学, 物理学中都有着广泛的应用, 而且在材料性质的分析中也常用到. 多晶材料存在择优取向, 也有一定的各向异性.
各种材料都有弹性, 大多数材料的弹性性质也具有各向异性. 例如, 在立方晶体中[111]方向通常比[100]方向更难压缩(stiff). 当我们对材料施加载荷, 材料会发生相应的形变, 在弹性范围内, 形变遵循胡克(Hook)定律, 即应力与应变是线性关系, 可以表示为 σ = C ε σ=Cε σ σ 是应力, ε ε 是应变, C C 为杨氏模量(或称弹性模量), 也常用 E E 或 Y Y 来表示.
材料不同方向上弹性模量不同, 我们怎么描述这种不同呢? 最好用数学方法, 建立数学框架, 准确直观地将弹性各项异性描述出来. 下面我们就进行这种数学的描述. 本人数学水平有限, 不能一步一步推导, 但我们可以简要理解一下推导过程. 弹性各项异性的推导就是利用张量和群论推广胡克定律. 我们先只考虑低阶弹性常数. 考虑二阶的 C i j k l Cijkl , 原来的胡克定律 σ = C ε σ=Cε 可推广为为矩阵形式
<span class="MathJax" id="MathJax-Element-9-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">σ=(<mtable rowspacing="4pt" columnspacing="1em">σ1σ2σ3σ4σ5σ6)=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13C<mrow class="MJX-TeXAtom-ORD">14C<mrow class="MJX-TeXAtom-ORD">15C<mrow class="MJX-TeXAtom-ORD">16C<mrow class="MJX-TeXAtom-ORD">21C<mrow class="MJX-TeXAtom-ORD">22C<mrow class="MJX-TeXAtom-ORD">23C<mrow class="MJX-TeXAtom-ORD">24C<mrow class="MJX-TeXAtom-ORD">25C<mrow class="MJX-TeXAtom-ORD">26C<mrow class="MJX-TeXAtom-ORD">31C<mrow class="MJX-TeXAtom-ORD">32C<mrow class="MJX-TeXAtom-ORD">33C<mrow class="MJX-TeXAtom-ORD">34C<mrow class="MJX-TeXAtom-ORD">35C<mrow class="MJX-TeXAtom-ORD">36C<mrow class="MJX-TeXAtom-ORD">41C<mrow class="MJX-TeXAtom-ORD">42C<mrow class="MJX-TeXAtom-ORD">43C<mrow class="MJX-TeXAtom-ORD">44C<mrow class="MJX-TeXAtom-ORD">45C<mrow class="MJX-TeXAtom-ORD">46C<mrow class="MJX-TeXAtom-ORD">51C<mrow class="MJX-TeXAtom-ORD">52C<mrow class="MJX-TeXAtom-ORD">53C<mrow class="MJX-TeXAtom-ORD">54C<mrow class="MJX-TeXAtom-ORD">55C<mrow class="MJX-TeXAtom-ORD">56C<mrow class="MJX-TeXAtom-ORD">61C<mrow class="MJX-TeXAtom-ORD">62C<mrow class="MJX-TeXAtom-ORD">63C<mrow class="MJX-TeXAtom-ORD">64C<mrow class="MJX-TeXAtom-ORD">65C<mrow class="MJX-TeXAtom-ORD">66](<mtable rowspacing="4pt" columnspacing="1em">ε1ε2ε3ε4ε5ε6)=<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C<mi mathvariant="bold-italic">ε" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">σ = ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 21 C 31 C 41 C 51 C 61 C 12 C 22 C 32 C 42 C 52 C 62 C 13 C 23 C 33 C 43 C 53 C 63 C 14 C 24 C 34 C 44 C 54 C 64 C 15 C 25 C 35 C 45 C 55 C 65 C 16 C 26 C 36 C 46 C 56 C 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ = C ε σ=(σ1σ2σ3σ4σ5σ6)=[C11C12C13C14C15C16C21C22C23C24C25C26C31C32C33C34C35C36C41C42C43C44C45C46C51C52C53C54C55C56C61C62C63C64C65C66](ε1ε2ε3ε4ε5ε6)=Cε
可以证明, 刚度矩阵 C C 为对称阵, C i j = C j i Cij=Cji . 因此, 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, C i j Cij 的数目只与晶系有关, 而与晶系中具体的对称类型无关.
C C 的逆矩阵 S S 称为柔顺矩阵. 利用 S S 可得到杨氏弹性模量的一般表达式. 我们用与[100], [010], [001]三个晶向的方向余弦来表示任意方向的杨氏模量. 设 l 1 , l 2 , l 3 l1,l2,l3 为空间某一方向与晶体主轴的方向余弦, 空间任一方向的杨氏模量 E E 的大小只与方向有关, 具体表达式如下
<span class="MathJax" id="MathJax-Element-18-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11l14+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">(l1l2<mo stretchy="false">)2+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">(l1l3<mo stretchy="false">)2+2S<mrow class="MJX-TeXAtom-ORD">14<mo stretchy="false">(l2l3l12<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">15<mo stretchy="false">(l3l13<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">16<mo stretchy="false">(l2l13<mo stretchy="false">)<mtd />+S<mrow class="MJX-TeXAtom-ORD">22l24+2S<mrow class="MJX-TeXAtom-ORD">23<mo stretchy="false">(l2l3<mo stretchy="false">)2+2S<mrow class="MJX-TeXAtom-ORD">24<mo stretchy="false">(l3l23<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">25<mo stretchy="false">(l1l3l22<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">26<mo stretchy="false">(l1l23<mo stretchy="false">)<mtd /><mtd />+S<mrow class="MJX-TeXAtom-ORD">33l34+2S<mrow class="MJX-TeXAtom-ORD">34<mo stretchy="false">(l2l33<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">35<mo stretchy="false">(l1l33<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">36<mo stretchy="false">(l1l2l32<mo stretchy="false">)<mtd /><mtd /><mtd />+S<mrow class="MJX-TeXAtom-ORD">44<mo stretchy="false">(l2l3<mo stretchy="false">)2+2S<mrow class="MJX-TeXAtom-ORD">45<mo stretchy="false">(l1l2l32<mo stretchy="false">)+2S<mrow class="MJX-TeXAtom-ORD">46<mo stretchy="false">(l1l3l22<mo stretchy="false">)<mtd /><mtd /><mtd /><mtd />+S<mrow class="MJX-TeXAtom-ORD">55<mo stretchy="false">(l1l3<mo stretchy="false">)2+2S<mrow class="MJX-TeXAtom-ORD">56<mo stretchy="false">(l2l3l12<mo stretchy="false">)<mtd /><mtd /><mtd /><mtd /><mtd /><mtd />+S<mrow class="MJX-TeXAtom-ORD">66<mo stretchy="false">(l1l2<mo stretchy="false">)2" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 l 4 1 + 2 S 12 ( l 1 l 2 ) 2 + S 22 l 4 2 + 2 S 13 ( l 1 l 3 ) 2 + 2 S 23 ( l 2 l 3 ) 2 + S 33 l 4 3 + 2 S 14 ( l 2 l 3 l 2 1 ) + 2 S 24 ( l 3 l 3 2 ) + 2 S 34 ( l 2 l 3 3 ) + S 44 ( l 2 l 3 ) 2 + 2 + 2 + 2 + 2 + S 15 ( l 3 l 3 1 ) S 25 ( l 1 l 3 l 2 2 ) S 35 ( l 1 l 3 3 ) S 45 ( l 1 l 2 l 2 3 ) S 55 ( l 1 l 3 ) 2 + 2 + 2 + 2 + 2 + 2 + S 16 ( l 2 l 3 1 ) S 26 ( l 1 l 3 2 ) S 36 ( l 1 l 2 l 2 3 ) S 46 ( l 1 l 3 l 2 2 ) S 56 ( l 2 l 3 l 2 1 ) S 66 ( l 1 l 2 ) 2 1/E=S11l14+2S12(l1l2)2+2S13(l1l3)2+2S14(l2l3l12)+2S15(l3l13)+2S16(l2l13)+S22l24+2S23(l2l3)2+2S24(l3l23)+2S25(l1l3l22)+2S26(l1l23)+S33l34+2S34(l2l33)+2S35(l1l33)+2S36(l1l2l32)+S44(l2l3)2+2S45(l1l2l32)+2S46(l1l3l22)+S55(l1l3)2+2S56(l2l3l12)+S66(l1l2)2
写为矩阵元素加和的形式为
<span class="MathJax" id="MathJax-Element-19-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=<mo movablelimits="false">∑<mrow class="MJX-TeXAtom-ORD">m=13<mo movablelimits="false">∑<mrow class="MJX-TeXAtom-ORD">n=13<mo movablelimits="false">∑<mrow class="MJX-TeXAtom-ORD">p=13<mo movablelimits="false">∑<mrow class="MJX-TeXAtom-ORD">q=13S<mrow class="MJX-TeXAtom-ORD">mnpqlmlnlplq<mtd />=<mo movablelimits="false">∑<mo stretchy="false">(<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">LT<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">L<mo stretchy="false">)<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">L=<mo stretchy="false">(l12,l22,l33,l2l3,l1l3,l1l2<mo stretchy="false">)" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E L = ∑ m = 1 3 ∑ n = 1 3 ∑ p = 1 3 ∑ q = 1 3 S m n p q l m l n l p l q = ∑ ( S L T L ) = ( l 2 1 , l 2 2 , l 3 3 , l 2 l 3 , l 1 l 3 , l 1 l 2 ) 1/E=∑m=13∑n=13∑p=13∑q=13Smnpqlmlnlplq=∑(SLTL)L=(l12,l22,l33,l2l3,l1l3,l1l2)
一种较对称, 方便推导的形式为
<span class="MathJax" id="MathJax-Element-20-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11l14+S<mrow class="MJX-TeXAtom-ORD">22l24+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">23<mo stretchy="false">)<mo stretchy="false">(l2l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">55+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l1l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">66+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l1l2<mo stretchy="false">)2<mtd />+2[<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">14+S<mrow class="MJX-TeXAtom-ORD">56<mo stretchy="false">)l12+S<mrow class="MJX-TeXAtom-ORD">24l22+S<mrow class="MJX-TeXAtom-ORD">34l32]l2l3<mtd />+2[S<mrow class="MJX-TeXAtom-ORD">15l12+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">25+S<mrow class="MJX-TeXAtom-ORD">46<mo stretchy="false">)l22+S<mrow class="MJX-TeXAtom-ORD">35l32]l1l3<mtd />+2[S<mrow class="MJX-TeXAtom-ORD">16l12+S<mrow class="MJX-TeXAtom-ORD">26l22+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">36+S<mrow class="MJX-TeXAtom-ORD">45<mo stretchy="false">)l32]l1l2" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 l 4 1 + S 22 l 4 2 + S 33 l 4 3 + ( S 44 + 2 S 23 ) ( l 2 l 3 ) 2 + ( S 55 + 2 S 13 ) ( l 1 l 3 ) 2 + ( S 66 + 2 S 12 ) ( l 1 l 2 ) 2 + 2 [ ( S 14 + S 56 ) l 2 1 + S 24 l 2 2 + S 34 l 2 3 ] l 2 l 3 + 2 [ S 15 l 2 1 + ( S 25 + S 46 ) l 2 2 + S 35 l 2 3 ] l 1 l 3 + 2 [ S 16 l 2 1 + S 26 l 2 2 + ( S 36 + S 45 ) l 2 3 ] l 1 l 2 1/E=S11l14+S22l24+S33l34+(S44+2S23)(l2l3)2+(S55+2S13)(l1l3)2+(S66+2S12)(l1l2)2+2[(S14+S56)l12+S24l22+S34l32]l2l3+2[S15l12+(S25+S46)l22+S35l32]l1l3+2[S16l12+S26l22+(S36+S45)l32]l1l2
这三种不同的表达形式, 可根据需要选择使用.
不同晶系的杨氏弹性模量
上面杨氏弹性模量的公式有些复杂, 好在除三斜晶系外, 大多数晶体都具有对称性. 考虑到晶体的对称性, 某些弹性常数必定为零, 而某些则相等, 所以对具有对称性的晶体, 相应的的杨氏模量公式简单些. 下面两张图总结了不同晶系刚度矩阵和柔顺矩阵的特点, 以及不同晶系杨氏弹性模量的公式, 后者为许多文献所引用.
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:
<span class="MathJax" id="MathJax-Element-21-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13C<mrow class="MJX-TeXAtom-ORD">14C<mrow class="MJX-TeXAtom-ORD">15C<mrow class="MJX-TeXAtom-ORD">16<mtd />C<mrow class="MJX-TeXAtom-ORD">22C<mrow class="MJX-TeXAtom-ORD">23C<mrow class="MJX-TeXAtom-ORD">24C<mrow class="MJX-TeXAtom-ORD">25C<mrow class="MJX-TeXAtom-ORD">26<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33C<mrow class="MJX-TeXAtom-ORD">34C<mrow class="MJX-TeXAtom-ORD">35C<mrow class="MJX-TeXAtom-ORD">36<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">44C<mrow class="MJX-TeXAtom-ORD">45C<mrow class="MJX-TeXAtom-ORD">46<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">55C<mrow class="MJX-TeXAtom-ORD">56<mtd /><mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 22 C 13 C 23 C 33 C 14 C 24 C 34 C 44 C 15 C 25 C 35 C 45 C 55 C 16 C 26 C 36 C 46 C 56 C 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C13C14C15C16C22C23C24C25C26C33C34C35C36C44C45C46C55C56C66]
<span class="MathJax" id="MathJax-Element-22-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">13S<mrow class="MJX-TeXAtom-ORD">14S<mrow class="MJX-TeXAtom-ORD">15S<mrow class="MJX-TeXAtom-ORD">16<mtd />S<mrow class="MJX-TeXAtom-ORD">22S<mrow class="MJX-TeXAtom-ORD">23S<mrow class="MJX-TeXAtom-ORD">24S<mrow class="MJX-TeXAtom-ORD">25S<mrow class="MJX-TeXAtom-ORD">26<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33S<mrow class="MJX-TeXAtom-ORD">34S<mrow class="MJX-TeXAtom-ORD">35S<mrow class="MJX-TeXAtom-ORD">36<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">44S<mrow class="MJX-TeXAtom-ORD">45S<mrow class="MJX-TeXAtom-ORD">46<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">55S<mrow class="MJX-TeXAtom-ORD">56<mtd /><mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 22 S 13 S 23 S 33 S 14 S 24 S 34 S 44 S 15 S 25 S 35 S 45 S 55 S 16 S 26 S 36 S 46 S 56 S 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S13S14S15S16S22S23S24S25S26S33S34S35S36S44S45S46S55S56S66]
共有21个独立的矩阵元, 杨氏模量
<span class="MathJax" id="MathJax-Element-23-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11l14+S<mrow class="MJX-TeXAtom-ORD">22l24+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">23<mo stretchy="false">)<mo stretchy="false">(l2l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">55+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l1l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">66+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l1l2<mo stretchy="false">)2<mtd />+2[<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">14+S<mrow class="MJX-TeXAtom-ORD">56<mo stretchy="false">)l12+S<mrow class="MJX-TeXAtom-ORD">24l22+S<mrow class="MJX-TeXAtom-ORD">34l32]l2l3<mtd />+2[S<mrow class="MJX-TeXAtom-ORD">15l12+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">25+S<mrow class="MJX-TeXAtom-ORD">46<mo stretchy="false">)l22+S<mrow class="MJX-TeXAtom-ORD">35l32]l1l3<mtd />+2[S<mrow class="MJX-TeXAtom-ORD">16l12+S<mrow class="MJX-TeXAtom-ORD">26l22+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">36+S<mrow class="MJX-TeXAtom-ORD">45<mo stretchy="false">)l32]l1l2" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 l 4 1 + S 22 l 4 2 + S 33 l 4 3 + ( S 44 + 2 S 23 ) ( l 2 l 3 ) 2 + ( S 55 + 2 S 13 ) ( l 1 l 3 ) 2 + ( S 66 + 2 S 12 ) ( l 1 l 2 ) 2 + 2 [ ( S 14 + S 56 ) l 2 1 + S 24 l 2 2 + S 34 l 2 3 ] l 2 l 3 + 2 [ S 15 l 2 1 + ( S 25 + S 46 ) l 2 2 + S 35 l 2 3 ] l 1 l 3 + 2 [ S 16 l 2 1 + S 26 l 2 2 + ( S 36 + S 45 ) l 2 3 ] l 1 l 2 1/E=S11l14+S22l24+S33l34+(S44+2S23)(l2l3)2+(S55+2S13)(l1l3)2+(S66+2S12)(l1l2)2+2[(S14+S56)l12+S24l22+S34l32]l2l3+2[S15l12+(S25+S46)l22+S35l32]l1l3+2[S16l12+S26l22+(S36+S45)l32]l1l2
2 单斜晶系(Monoclinic system)
考虑对称性后, 单斜晶系有13个独立的矩阵单元:
<span class="MathJax" id="MathJax-Element-24-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">130C<mrow class="MJX-TeXAtom-ORD">150<mtd />C<mrow class="MJX-TeXAtom-ORD">22C<mrow class="MJX-TeXAtom-ORD">230C<mrow class="MJX-TeXAtom-ORD">250<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">330C<mrow class="MJX-TeXAtom-ORD">350<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">440C<mrow class="MJX-TeXAtom-ORD">46<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">550<mtd /><mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 22 C 13 C 23 C 33 0 0 0 C 44 C 15 C 25 C 35 0 C 55 0 0 0 C 46 0 C 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C130C150C22C230C250C330C350C440C46C550C66]
<span class="MathJax" id="MathJax-Element-25-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">130S<mrow class="MJX-TeXAtom-ORD">150<mtd />S<mrow class="MJX-TeXAtom-ORD">22S<mrow class="MJX-TeXAtom-ORD">230S<mrow class="MJX-TeXAtom-ORD">250<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">330S<mrow class="MJX-TeXAtom-ORD">350<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">440S<mrow class="MJX-TeXAtom-ORD">46<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">550<mtd /><mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 22 S 13 S 23 S 33 0 0 0 S 44 S 15 S 25 S 35 0 S 55 0 0 0 S 46 0 S 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S130S150S22S230S250S330S350S440S46S550S66]
<span class="MathJax" id="MathJax-Element-26-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11l14+S<mrow class="MJX-TeXAtom-ORD">22l24+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">23<mo stretchy="false">)<mo stretchy="false">(l2l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">55+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l1l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">66+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l1l2<mo stretchy="false">)2<mtd />+2[S<mrow class="MJX-TeXAtom-ORD">15l12+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">25+S<mrow class="MJX-TeXAtom-ORD">46<mo stretchy="false">)l22+S<mrow class="MJX-TeXAtom-ORD">35l32]l1l3" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 l 4 1 + S 22 l 4 2 + S 33 l 4 3 + ( S 44 + 2 S 23 ) ( l 2 l 3 ) 2 + ( S 55 + 2 S 13 ) ( l 1 l 3 ) 2 + ( S 66 + 2 S 12 ) ( l 1 l 2 ) 2 + 2 [ S 15 l 2 1 + ( S 25 + S 46 ) l 2 2 + S 35 l 2 3 ] l 1 l 3 1/E=S11l14+S22l24+S33l34+(S44+2S23)(l2l3)2+(S55+2S13)(l1l3)2+(S66+2S12)(l1l2)2+2[S15l12+(S25+S46)l22+S35l32]l1l3
3 正交晶系(Orthorhombic system)
正交晶系拥有相当高的对称性, 其独立矩阵元的数目为9个.
<span class="MathJax" id="MathJax-Element-27-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13000<mtd />C<mrow class="MJX-TeXAtom-ORD">22C<mrow class="MJX-TeXAtom-ORD">23000<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">550<mtd /><mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 22 C 13 C 23 C 33 0 0 0 C 44 0 0 0 0 C 55 0 0 0 0 0 C 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C13000C22C23000C33000C4400C550C66]
<span class="MathJax" id="MathJax-Element-28-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">13000<mtd />S<mrow class="MJX-TeXAtom-ORD">22S<mrow class="MJX-TeXAtom-ORD">23000<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">550<mtd /><mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 22 S 13 S 23 S 33 0 0 0 S 44 0 0 0 0 S 55 0 0 0 0 0 S 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S13000S22S23000S33000S4400S550S66]
<span class="MathJax" id="MathJax-Element-29-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11l14+S<mrow class="MJX-TeXAtom-ORD">22l24+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">23<mo stretchy="false">)<mo stretchy="false">(l2l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">55+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l1l3<mo stretchy="false">)2+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">66+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l1l2<mo stretchy="false">)2" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 l 4 1 + S 22 l 4 2 + S 33 l 4 3 + ( S 44 + 2 S 23 ) ( l 2 l 3 ) 2 + ( S 55 + 2 S 13 ) ( l 1 l 3 ) 2 + ( S 66 + 2 S 12 ) ( l 1 l 2 ) 2 1/E=S11l14+S22l24+S33l34+(S44+2S23)(l2l3)2+(S55+2S13)(l1l3)2+(S66+2S12)(l1l2)2
4 四方晶系(Tetragonal system)
4.1 四方晶系 4 , 4 ˉ , 4 / m 4,4ˉ,4/m
对于具有 4 , 4 ˉ , 4 / m 4,4ˉ,4/m 对称操作的四方晶系, 其独立矩阵元的数目为7个:
<span class="MathJax" id="MathJax-Element-32-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">1300C<mrow class="MJX-TeXAtom-ORD">16<mtd />C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">1300−C<mrow class="MJX-TeXAtom-ORD">16<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 11 C 13 C 13 C 33 0 0 0 C 44 0 0 0 0 C 44 C 16 − C 16 0 0 0 C 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C1300C16C11C1300−C16C33000C4400C440C66]
<span class="MathJax" id="MathJax-Element-33-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">1300S<mrow class="MJX-TeXAtom-ORD">16<mtd />S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">1300−S<mrow class="MJX-TeXAtom-ORD">16<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 11 S 13 S 13 S 33 0 0 0 S 44 0 0 0 0 S 44 S 16 − S 16 0 0 0 S 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S1300S16S11S1300−S16S33000S4400S440S66]
<span class="MathJax" id="MathJax-Element-34-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(l14+l24<mo stretchy="false">)+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l12+l22<mo stretchy="false">)l32+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">66+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l1l2<mo stretchy="false">)2<mtd />+2S<mrow class="MJX-TeXAtom-ORD">16<mo stretchy="false">(l12−l22<mo stretchy="false">)l1l2" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 ( l 4 1 + l 4 2 ) + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( l 2 1 + l 2 2 ) l 2 3 + ( S 66 + 2 S 12 ) ( l 1 l 2 ) 2 + 2 S 16 ( l 2 1 − l 2 2 ) l 1 l 2 1/E=S11(l14+l24)+S33l34+(S44+2S13)(l12+l22)l32+(S66+2S12)(l1l2)2+2S16(l12−l22)l1l2
4.2 四方晶系 422 , 4 m m , 4 ˉ 2 m , 4 / m m m 422,4mm,4ˉ2m,4/mmm
对于具有 422 , 4 m m , 4 ˉ 2 m , 4 / m m m 422,4mm,4ˉ2m,4/mmm 对称操作的四方晶系, 独立矩阵元的数目仅为6个:
<span class="MathJax" id="MathJax-Element-37-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13000<mtd />C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">13000<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 11 C 13 C 13 C 33 0 0 0 C 44 0 0 0 0 C 44 0 0 0 0 0 C 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C13000C11C13000C33000C4400C440C66]
<span class="MathJax" id="MathJax-Element-38-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">13000<mtd />S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">13000<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">66]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 11 S 13 S 13 S 33 0 0 0 S 44 0 0 0 0 S 44 0 0 0 0 0 S 66 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S13000S11S13000S33000S4400S440S66]
<span class="MathJax" id="MathJax-Element-39-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(l14+l24<mo stretchy="false">)+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l12+l22<mo stretchy="false">)l32+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">66+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l1l2<mo stretchy="false">)2" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 ( l 4 1 + l 4 2 ) + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( l 2 1 + l 2 2 ) l 2 3 + ( S 66 + 2 S 12 ) ( l 1 l 2 ) 2 1/E=S11(l14+l24)+S33l34+(S44+2S13)(l12+l22)l32+(S66+2S12)(l1l2)2
5 三方晶系(Trigonal system)
5.1 三方晶系 3 , 3 ˉ 3,3ˉ
三方晶系 3 , 3 ˉ 3,3ˉ 的独立矩阵元的数目为7个.
<span class="MathJax" id="MathJax-Element-42-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13C<mrow class="MJX-TeXAtom-ORD">14C<mrow class="MJX-TeXAtom-ORD">150<mtd />C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">13−C<mrow class="MJX-TeXAtom-ORD">14−C<mrow class="MJX-TeXAtom-ORD">150<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">440−C<mrow class="MJX-TeXAtom-ORD">15<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">44C<mrow class="MJX-TeXAtom-ORD">14<mtd /><mtd /><mtd /><mtd /><mtd /><mrow class="MJX-TeXAtom-ORD">C<mrow class="MJX-TeXAtom-ORD">11−C<mrow class="MJX-TeXAtom-ORD">122]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 11 C 13 C 13 C 33 C 14 − C 14 0 C 44 C 15 − C 15 0 0 C 44 0 0 0 − C 15 C 14 C 11 − C 12 2 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C13C14C150C11C13−C14−C150C33000C440−C15C44C14C11−C122]
<span class="MathJax" id="MathJax-Element-43-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">13S<mrow class="MJX-TeXAtom-ORD">14S<mrow class="MJX-TeXAtom-ORD">150<mtd />S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">13−S<mrow class="MJX-TeXAtom-ORD">14−S<mrow class="MJX-TeXAtom-ORD">150<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">440−S<mrow class="MJX-TeXAtom-ORD">15<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">44S<mrow class="MJX-TeXAtom-ORD">14<mtd /><mtd /><mtd /><mtd /><mtd />2<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">11−S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 11 S 13 S 13 S 33 S 14 − S 14 0 S 44 S 15 − S 15 0 0 S 44 0 0 0 − S 15 S 14 2 ( S 11 − S 12 ) ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S13S14S150S11S13−S14−S150S33000S440−S15S44S142(S11−S12)]
<span class="MathJax" id="MathJax-Element-44-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(l14+l24+2l12l22<mo stretchy="false">)+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l12+l22<mo stretchy="false">)l32+2S<mrow class="MJX-TeXAtom-ORD">14<mo stretchy="false">(3l12−l22<mo stretchy="false">)l2l3+2S<mrow class="MJX-TeXAtom-ORD">15<mo stretchy="false">(l12−3l22<mo stretchy="false">)l1l3<mtd />=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(1−l32<mo stretchy="false">)2+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(1−l32<mo stretchy="false">)l32+2S<mrow class="MJX-TeXAtom-ORD">14<mo stretchy="false">(3l12−l22<mo stretchy="false">)l2l3+2S<mrow class="MJX-TeXAtom-ORD">15<mo stretchy="false">(l12−3l22<mo stretchy="false">)l1l3" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 ( l 4 1 + l 4 2 + 2 l 2 1 l 2 2 ) + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( l 2 1 + l 2 2 ) l 2 3 + 2 S 14 ( 3 l 2 1 − l 2 2 ) l 2 l 3 + 2 S 15 ( l 2 1 − 3 l 2 2 ) l 1 l 3 = S 11 ( 1 − l 2 3 ) 2 + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( 1 − l 2 3 ) l 2 3 + 2 S 14 ( 3 l 2 1 − l 2 2 ) l 2 l 3 + 2 S 15 ( l 2 1 − 3 l 2 2 ) l 1 l 3 1/E=S11(l14+l24+2l12l22)+S33l34+(S44+2S13)(l12+l22)l32+2S14(3l12−l22)l2l3+2S15(l12−3l22)l1l3=S11(1−l32)2+S33l34+(S44+2S13)(1−l32)l32+2S14(3l12−l22)l2l3+2S15(l12−3l22)l1l3
5.2 三方晶系 32 , 3 m , 3 ˉ m 32,3m,3ˉm
三方晶系 32 , 3 m , 3 ˉ m 32,3m,3ˉm 独立矩阵元的数目为6个.
<span class="MathJax" id="MathJax-Element-47-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13C<mrow class="MJX-TeXAtom-ORD">1400<mtd />C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">13−C<mrow class="MJX-TeXAtom-ORD">1400<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">44C<mrow class="MJX-TeXAtom-ORD">14<mtd /><mtd /><mtd /><mtd /><mtd /><mrow class="MJX-TeXAtom-ORD">C<mrow class="MJX-TeXAtom-ORD">11−C<mrow class="MJX-TeXAtom-ORD">122]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 11 C 13 C 13 C 33 C 14 − C 14 0 C 44 0 0 0 0 C 44 0 0 0 0 C 14 C 11 − C 12 2 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C13C1400C11C13−C1400C33000C4400C44C14C11−C122]
<span class="MathJax" id="MathJax-Element-48-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">13S<mrow class="MJX-TeXAtom-ORD">1400<mtd />S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">13−S<mrow class="MJX-TeXAtom-ORD">1400<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">44S<mrow class="MJX-TeXAtom-ORD">14<mtd /><mtd /><mtd /><mtd /><mtd />2<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">11−S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 11 S 13 S 13 S 33 S 14 − S 14 0 S 44 0 0 0 0 S 44 0 0 0 0 S 14 2 ( S 11 − S 12 ) ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S13S1400S11S13−S1400S33000S4400S44S142(S11−S12)]
<span class="MathJax" id="MathJax-Element-49-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(l14+l24+2l12l22<mo stretchy="false">)+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l12+l22<mo stretchy="false">)l32+2S<mrow class="MJX-TeXAtom-ORD">14<mo stretchy="false">(3l12−l22<mo stretchy="false">)l2l3<mtd />=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(1−l32<mo stretchy="false">)2+S<mrow class="MJX-TeXAtom-ORD">33l34<mtd />+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(1−l32<mo stretchy="false">)l32+2S<mrow class="MJX-TeXAtom-ORD">14<mo stretchy="false">(3l12−l22<mo stretchy="false">)l2l3" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 ( l 4 1 + l 4 2 + 2 l 2 1 l 2 2 ) + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( l 2 1 + l 2 2 ) l 2 3 + 2 S 14 ( 3 l 2 1 − l 2 2 ) l 2 l 3 = S 11 ( 1 − l 2 3 ) 2 + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( 1 − l 2 3 ) l 2 3 + 2 S 14 ( 3 l 2 1 − l 2 2 ) l 2 l 3 1/E=S11(l14+l24+2l12l22)+S33l34+(S44+2S13)(l12+l22)l32+2S14(3l12−l22)l2l3=S11(1−l32)2+S33l34+(S44+2S13)(1−l32)l32+2S14(3l12−l22)l2l3
6 六方晶系(Hexagonal system)
六方晶系共有5个独立的矩阵元.
<span class="MathJax" id="MathJax-Element-50-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">13000<mtd />C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">13000<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd /><mrow class="MJX-TeXAtom-ORD">C<mrow class="MJX-TeXAtom-ORD">11−C<mrow class="MJX-TeXAtom-ORD">122]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 11 C 13 C 13 C 33 0 0 0 C 44 0 0 0 0 C 44 0 0 0 0 0 C 11 − C 12 2 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C13000C11C13000C33000C4400C440C11−C122]
<span class="MathJax" id="MathJax-Element-51-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">13000<mtd />S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">13000<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">33000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />2<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">11−S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 11 S 13 S 13 S 33 0 0 0 S 44 0 0 0 0 S 44 0 0 0 0 0 2 ( S 11 − S 12 ) ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S13000S11S13000S33000S4400S4402(S11−S12)]
<span class="MathJax" id="MathJax-Element-52-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(l14+l24+2l12l22<mo stretchy="false">)+S<mrow class="MJX-TeXAtom-ORD">33l34+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(l12+l22<mo stretchy="false">)l32<mtd />=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(1−l32<mo stretchy="false">)2+S<mrow class="MJX-TeXAtom-ORD">33l34+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">)<mo stretchy="false">(1−l32<mo stretchy="false">)l32" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 ( l 4 1 + l 4 2 + 2 l 2 1 l 2 2 ) + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( l 2 1 + l 2 2 ) l 2 3 = S 11 ( 1 − l 2 3 ) 2 + S 33 l 4 3 + ( S 44 + 2 S 13 ) ( 1 − l 2 3 ) l 2 3 1/E=S11(l14+l24+2l12l22)+S33l34+(S44+2S13)(l12+l22)l32=S11(1−l32)2+S33l34+(S44+2S13)(1−l32)l32
7 立方晶系(Cubic system)
立方晶系是所有晶系中对称度最高的晶系, 其独立矩阵元数目仅为3个, C 11 , C 12 , C 44 C11,C12,C44
<span class="MathJax" id="MathJax-Element-54-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C=[<mtable rowspacing="4pt" columnspacing="1em">C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12C<mrow class="MJX-TeXAtom-ORD">12000<mtd />C<mrow class="MJX-TeXAtom-ORD">11C<mrow class="MJX-TeXAtom-ORD">12000<mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">11000<mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />C<mrow class="MJX-TeXAtom-ORD">44]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">C = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ C 11 C 12 C 11 C 12 C 12 C 11 0 0 0 C 44 0 0 0 0 C 44 0 0 0 0 0 C 44 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ C=[C11C12C12000C11C12000C11000C4400C440C44]
<span class="MathJax" id="MathJax-Element-55-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=[<mtable rowspacing="4pt" columnspacing="1em">S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">12000<mtd />S<mrow class="MJX-TeXAtom-ORD">11S<mrow class="MJX-TeXAtom-ORD">12000<mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">11000<mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">4400<mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">440<mtd /><mtd /><mtd /><mtd /><mtd />S<mrow class="MJX-TeXAtom-ORD">44]" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">S = ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S 11 S 12 S 11 S 12 S 12 S 11 0 0 0 S 44 0 0 0 0 S 44 0 0 0 0 0 S 44 ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ S=[S11S12S12000S11S12000S11000S4400S440S44]
<span class="MathJax" id="MathJax-Element-56-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">1<mrow class="MJX-TeXAtom-ORD">/E=S<mrow class="MJX-TeXAtom-ORD">11<mo stretchy="false">(l14+l24+l34<mo stretchy="false">)+<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">44+2S<mrow class="MJX-TeXAtom-ORD">12<mo stretchy="false">)<mo stretchy="false">(l12l22+l12l32+l22l32<mo stretchy="false">)<mtd />=S<mrow class="MJX-TeXAtom-ORD">11−2<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">11−S<mrow class="MJX-TeXAtom-ORD">12−<mrow class="MJX-TeXAtom-ORD">S<mrow class="MJX-TeXAtom-ORD">442<mo stretchy="false">)<mo stretchy="false">(l12l22+l22l32+l12l32<mo stretchy="false">)" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 / E = S 11 ( l 4 1 + l 4 2 + l 4 3 ) + ( S 44 + 2 S 12 ) ( l 2 1 l 2 2 + l 2 1 l 2 3 + l 2 2 l 2 3 ) = S 11 − 2 ( S 11 − S 12 − S 44 2 ) ( l 2 1 l 2 2 + l 2 2 l 2 3 + l 2 1 l 2 3 ) 1/E=S11(l14+l24+l34)+(S44+2S12)(l12l22+l12l32+l22l32)=S11−2(S11−S12−S442)(l12l22+l22l32+l12l32)
杨氏模量的极值
对于立方晶体, 我们可以用与[100], [010], [001]三个晶向的方向余弦来表示任意方向的杨氏模量, 结果如下
<span class="MathJax" id="MathJax-Element-57-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"><mrow class="MJX-TeXAtom-ORD">1E=S<mrow class="MJX-TeXAtom-ORD">11−2<mo stretchy="false">(S<mrow class="MJX-TeXAtom-ORD">11−S<mrow class="MJX-TeXAtom-ORD">12−<mrow class="MJX-TeXAtom-ORD">S<mrow class="MJX-TeXAtom-ORD">442<mo stretchy="false">)<mo stretchy="false">(l12l22+l22l32+l32l12<mo stretchy="false">)<mtd />=S<mrow class="MJX-TeXAtom-ORD">11+<mo stretchy="false">(1−A<mo stretchy="false">)S<mrow class="MJX-TeXAtom-ORD">44<mo stretchy="false">(l12l22+l22l32+l32l12<mo stretchy="false">)A=2<mrow class="MJX-TeXAtom-ORD">S<mrow class="MJX-TeXAtom-ORD">11−S<mrow class="MJX-TeXAtom-ORD">12S<mrow class="MJX-TeXAtom-ORD">44<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">S=<mrow class="MJX-TeXAtom-ORD"><mi mathvariant="bold">C<mrow class="MJX-TeXAtom-ORD">−1" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">1 E A S = S 11 − 2 ( S 11 − S 12 − S 44 2 ) ( l 2 1 l 2 2 + l 2 2 l 2 3 + l 2 3 l 2 1 ) = S 11 + ( 1 − A ) S 44 ( l 2 1 l 2 2 + l 2 2 l 2 3 + l 2 3 l 2 1 ) = 2 S 11 − S 12 S 44 = C − 1 1E=S11−2(S11−S12−S442)(l12l22+l22l32+l32l12)=S11+(1−A)S44(l12l22+l22l32+l32l12)A=2S11−S12S44S=C−1
其中 S 11 , S 12 , S 44 S11,S12,S44 分别为立方晶体的三个独立的弹性柔顺系数, 柔顺矩阵 S S 与弹性矩阵 C C 的矩阵互为逆矩阵. l 1 , l 2 , l 3 l1,l2,l3 为空间某一方向与晶体主轴的方向余弦. A A 为各向异性值. 因此, 知道了三个柔顺弹性常数的值, 即可求得空间任一方向的杨氏模量 E E , E E 的大小只与方向有关.
由于 l 2 1 + l 2 2 + l 2 3 = 1 , l 1 , l 2 , l 3 ∈ [ 0 , 1 ] l12+l22+l32=1,l1,l2,l3∈[0,1] , 可以知道杨氏模量的两个极值为
<span class="MathJax" id="MathJax-Element-66-Frame" tabindex="0" data-mathml="<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">E1=<mrow class="MJX-TeXAtom-ORD">1S<mrow class="MJX-TeXAtom-ORD">11E2=<mrow class="MJX-TeXAtom-ORD">1S<mrow class="MJX-TeXAtom-ORD">11+<mrow class="MJX-TeXAtom-ORD">13<mo stretchy="false">(1−A<mo stretchy="false">)S<mrow class="MJX-TeXAtom-ORD">44" role="presentation" style="margin:0px;padding:0px;display:inline;line-height:normal;text-align:left;word-spacing:normal;word-wrap:normal;float:none;direction:ltr;max-width:none;max-height:none;min-width:0px;min-height:0px;border:0px;position:relative;">E 1 E 2 = 1 S 11 = 1 S 11 + 1 3 ( 1 − A ) S 44 E1=1S11E2=1S11+13(1−A)S44
前者对应于坐标轴方向, 后者对应于体对角线方法. 根据各向异性值 A A 与1的大小不同, 相应于极小或极大值.
对于其他晶系杨氏模量的极值, 不易得到解析公式, 直接使用数值方法搜索即可.
杨氏弹性模量各向异性的图示
为了直观地表达弹性模量的各向异性, 人们常常将其用三维图来表示. 这种各向异性的直观图示方法具有一般性, 在科学数据可视化中经常遇到. 量子化学中常用的原子轨道的角度分布图就是一例. 具体原理是, 在球坐标系 ( r , θ , ϕ ) (r,θ,ϕ) 中, 对仅依赖于方向的函数 F ( θ , ϕ ) F(θ,ϕ) 中, 做曲面 r = F ( θ , ϕ ) r=F(θ,ϕ) . 显然, 当 F F 为常数时, 此曲面为球面, 各个方向函数值相同, 不存在各向异性; 当 F F 随 ( θ , ϕ ) (θ,ϕ) 变化时, 曲面便可表示出函数值的变化.
mathematica中可使用球坐标绘图函数SphericalPlot3D来做出这种图, 很多文献中的图就是利用这个函数做的, 请参考这个函数的说明 和相应的弹性模量示例 .
考虑到Matlab使用更广泛些, 下面给出基于Matlab的绘图方法.
利用Matlab绘制各向异性图时, 有两种实现方法. 一种是利用球坐标绘图, 像mathematica那样. 虽然Matlab没有球坐标绘图函数, 但可以先将球坐标转换为直角坐标然后再绘图, 也不是很麻烦. 另一种方法是直接使用直角坐标, 利用等值面函数, 绘制函数 r − E = 0 r−E=0 的等值面.
下面的代码绘制几种金属的杨氏模量三维各向异性曲面, 弹性常数来源于这里 .
matlab 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
function Aniso
clc;clear;close all;
%% 处理数据, 计算矩阵以及弹性模量的极值 % 单斜晶系测试 % C=zeros(6); % C(1,1)=125;C(1,2)=87; C(1,3)=90; C(1,4)=0; C(1,5)=-9;C(1,6)=0; % C(2,2)=169;C(2,3)=105;C(2,4)=0; C(2,5)=-7;C(2,6)=0; % C(3,3)=128;C(3,4)=0; C(3,5)=11;C(3,6)=0; % C(4,4)=53;C(4,5)=0; C(4,6)=-0.6; % C(5,5)=36;C(5,6)=0; % C(6,6)=48; % for i=2:6; for j=1:i-1; C(i,j)=C(j,i); end; end % 立方晶系
C11=240.20 ; C12= 125.60 ; C44= 28.20 ; % Nb % C11=522.40; C12= 160.80; C44=204.40; % W % C11=107.30; C12= 60.90; C44= 28.30; % Al % C11=346.70; C12= 250.70; C44= 76.50; % Pt % C11=231.40; C12= 134.70; C44=116.40; % Fe % C11=124.00; C12= 93.40; C44= 46.10; % Ag % C11= 49.50; C12= 42.30; C44= 14.90; % Pb % C11= 13.50; C12= 11.44; C44= 8.78; % Li
C=zeros (6 );
C(1 :3 ,1 :3 )=C12;
for i =1 :3 ; C(i ,i )=C11; end for i =4 :6 ; C(i ,i )=C44; end
S=inv(C);
S11=S(1 ,1 ); S12=S(1 ,2 ); S13=S(1 ,3 ); S14=S(1 ,4 ); S15=S(1 ,5 ); S16=S(1 ,6 );
S22=S(2 ,2 ); S23=S(2 ,3 ); S24=S(2 ,4 ); S25=S(2 ,5 ); S26=S(2 ,6 );
S33=S(3 ,3 ); S34=S(3 ,4 ); S35=S(3 ,5 ); S36=S(3 ,6 );
S44=S(4 ,4 ); S45=S(4 ,5 ); S46=S(4 ,6 );
S55=S(5 ,5 ); S56=S(5 ,6 );
S66=S(6 ,6 );
% 立方晶系极值公式
A=2* (S11- S12)/ S44;
Emax=1/ S11; Emin=1/ (S11+ (1- A)* S44/3 );
if (A>1 ); Emin=1/ S11; Emax=1/ (S11+ (1- A)* S44/3 ); end
fprintf(' A=%9.4f Emin=%9.4f Emax=%9.4fn', A, Emin, Emax); %% 使用球坐标作图
[theta, phi]=meshgrid ( linspace (0 ,pi ), linspace (0 ,2* pi ) );
L1=sin (theta).* cos (phi);
L2=sin (theta).* sin (phi);
L3=cos (theta);
% 三斜晶系杨氏模量公式, 可用于任意晶系
E=S11 * L1.^4 + S22 * L2.^4 + S33 * L3.^4 ... + (S44+2* S23) * (L2.* L3).^2 + (S55+2* S13) * (L1.* L3).^2 + (S66+2* S12) * (L1.* L2).^2 ... + 2* ((S14+ S56) * L1.^2 + S24 * L2.^2 + S34 * L3.^2 ) .* L2.* L3 ... + 2* ( S15 * L1.^2 + (S25+ S46) * L2.^2 + S35 * L3.^2 ) .* L1.* L3 ... + 2* ( S16 * L1.^2 + S26 * L2.^2 + (S36+ S45) * L3.^2 ) .* L1.* L2;
% 立方晶系 % E=S11+(1-A)*S44*( (L1.*L2).^2+(L2.*L3).^2+(L3.*L1).^2 );
E=1./ E;
x=E.* L1; y=E.* L2; z=E.* L3;
% 或使用函数转为直角坐标 % [x,y,z] = sph2cart(v, pi/2-u,E);
surf(x,y,z, E, ' FaceColor' ,' interp' , ' EdgeColor' ,' none' );
% 作模量的某一切面图 % [X,Y,Z]=meshgrid(linspace(-Emax,Emax)); % contourslice(X,Y,Z,X,x,y,z,[0 0]) %% 或使用直角坐标等值面方法作图 % [x,y,z]=meshgrid(linspace(-Emax,Emax)); % r=sqrt(x.^2+y.^2+z.^2); % L1=x./r; L2=y./r; L3=z./r; % % % 立方晶系 % E=S11+(1-A)*S44*( (L1.*L2).^2+(L2.*L3).^2+(L3.*L1).^2 ); % E=1./E; % v=r-E; % % p=patch(isosurface(x,y,z,v,0)); % isocolors(x,y,z,E,p); % isonormals(x,y,z,v,p); % set(p,'FaceColor','interp','EdgeColor','none'); %% 设置图片格式, 输出图片
axis tight; title ' Nb A=0.5' ;
view(45 ,30 ); daspect([1 1 1 ]);
camlight; lighting phong;
colormap jet; % 低版本Matlab默认的填色模式
cbar=colorbar; title(cbar, ' GPa' );
set(gca,' position' ,[0.12 ,0.05 , 0.6 ,0.85 ]);
set(gcf,' position' ,[20 ,20 , 1000 ,900 ]);
set(gcf, ' PaperPositionMode' , ' auto' );
print(gcf,' - dpng' ,' - r300' ,' Nb.png' )
end
注意
matlab默认的渲染颜色取决于Z轴大小, 这不符合我们的要求, 因为我们需要用颜色表示E的大小, 这样图形更直观.
matlab球坐标转换函数使用的球坐标采用数学约定, 与物理上常用的不同, 使用仰角El, 而非俯视角 θ θ , E l + θ = π / 2 El+θ=π/2
不同晶系杨氏模量的表达式不同, 只要把代码里E的表达式修改成相应的方程即可.
杨氏模量在某一平面内的截面图形可利用极坐标或直角坐标绘制, 原理类似.
为了让大家有一个更直观的了解, 我们把具有不同各向异性值的立方金属选取具有代表性几个, 列于下表
几种金属的弹性数据(单位: GPa) 金属 C11 C12 C44 S11 S12 S44 A Emin Emax 铌Nb 240.20 125.60 28.20 0.0065 -0.0022 0.0355 0.49 80.01 153.95 钨W 522.40 160.80 204.40 0.0025 -0.0007 0.0062 1.13 446.71 493.65 铝Al 107.30 60.90 28.30 0.0158 -0.0057 0.0353 1.22 63.20 75.57 铂Pt 346.70 250.70 76.50 0.0073 -0.0031 0.0131 1.59 136.29 210.51 铁Fe 231.40 134.70 116.40 0.0076 -0.0028 0.0086 2.41 132.28 283.34 银Ag 124.00 93.40 46.10 0.0229 -0.0098 0.0217 3.01 43.75 120.44 铅Pb 49.50 42.30 14.90 0.0951 -0.0438 0.0671 4.14 10.52 40.23 锂Li 13.50 11.44 8.78 0.3328 -0.1526 0.1139 8.52 3.00 21.23
相应的三维杨氏模量图如下
参考资料
T. C. T. Ting; On Anisotropic Elastic Materials for which Young’s Modulus E(n) is Independent of n or the Shear Modulus G(n,m) is Independent of n and m; J Elasticity 81(3):271-292, 2006; 10.1007/s10659-005-9016-2
Kevin M. Knowles, Philip R. Howie; The Directional Dependence of Elastic Stiffness and Compliance Shear Coefficients and Shear Moduli in Cubic Materials; J Elast 120(1):87-108, 2014; 10.1007/s10659-014-9506-1
Matlab绘图高级部分
科学计算可视化: 三维流场绘图
Applied Mechanics of Solids: Constitutive Models Relations between Stress and Strain
球坐标
转载本文请联系原作者获取授权，同时请注明本文来自李继存科学网博客。 https://wap.sciencenet.cn/blog-548663-786478.html
上一篇：
正则表达式基础知识 下一篇：
希腊字母与拉丁字母的对应
