张量计算简介

补充介绍：塑性力学课程涉及的张量n1kkkn1jjjn1iii2211xaxaxaxaxaxaSnn显然，指标i,j,k与求和无关，可用任意字母代替。为简化表达式，引入Einstein求和约定：每逢某个指标在一项中重复一次，就表示对该指标求和，指标取遍正数1，2，…，n。重复的指标称为哑标。于是kkjjiixaxaxaS以求和公式为例iiixba是违约的，求和时要保留求和号n1iiiixba双重求和31i31jjiijxxaS简写成jiijxxaS展开式（9项）333323321331322322221221311321121111xxaxxaxxaxxaxxaxxaxxaxxaxxaS三重求和（27项）kkxxxaxxxaSjiijk31i31j31kjiijk例外的情形：111ECR222ECR333ECRiiiiiECECR出现双重指标但不求和时，在指标下方加划线以示区别，或用文字说明（如i不求和）。规定：这里i相当于一个自由指标，而i只是在数值上等于i，并不与i求和。坐标系的变换关系321332313322212312111321eeeeeeqqqqqqqqqjjiiiiieeeqq关于张量的定义旧坐标系：321xxxO单位基向量：},,{321eee新坐标系：单位基向量：321xxxO},,{321eee也可写为或iieQe'jijiji),cos(eeeeq设a为任意向量，其在新、旧坐标系下的分量分别为},,{},,,{321321aaaaaaiiiiaqaiiiiaqa该向量的分量满足坐标变换关系321332313322212312111321aaaqqqqqqqqqaaa321332313322212312111321aaaqqqqqqqqqaaaT或aQa''aQaT向量或一阶张量二阶张量对于直角坐标系321xxxO则该物理量定义一个二阶张量。jijjiijiTqqTjiT中的物理量当进行坐标变换时，若满足TQTQT'j'i'jjiijiTqqTQTQT'T二阶张量力学分析中用张量表述的例子－各向同性材料的广义胡克定律xxzzyyxxxxGE2211yyzzyyxxyyGE2211zzzzyyxxzzGE2211xyxyG2yzyzG2zxzxG2klklijjlikijE211如果用张量形式描述klijkllkklijklijLL3131一般的经典弹性本构关系或应力应变关系也可按矩阵形式写出本构张量ijklL323232233231321332213212323332223211233223232331231323212312233323222311313231233131311331213112313331223111133213231331131313211312133313221311213221232131211321212112213321222111123212231231121312211212123312221211333233233331331333213312333333223311223222232231221322212212223322222211113211231131111311211112113311221111LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLijkl322331132112332211323232233231321332213212323332223211233223232331231323212312233323222311313231233131311331213112313331223111133213231331131313211312133313221311213221232131211321212112213321222111123212231231121312211212123312221211333233233331331333213312333333223311223222232231221322212212223322222211113211231131111311211112113311221111322331132112332211LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL并按矩阵形式写出本构关系对于广义胡克定律，其本构矩阵可以写为1000000000100000000010000000001000000000100000000010000000002112121000000212112100000021212111ELijkl3223311321123322113223311321123322111000000000100000000010000000001000000000100000000010000000002112121000000212112100000021212111E而广义胡克定律定义的弹性本构关系可以写为考虑到应力张量和应变张量的对称性，可以简写成231312332211232323132312233323222311132313131312133313221311122312131212123312221211332333133312333333223311222322132212223322222211112311131112113311221111231312332211222LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLL而各向同性材料广义胡克定法律的本构矩阵元素可以写为2111333322221111ELLL211332222333311113322112211ELLLLLL)1(2232313131212ELLL关于克罗内克尔符号的有关计算3ii3111332211iiijkjikjijijikjik332211三项中最多能保留一项，k=i或k=j那一项。),(jiforsumnoijjjijijiiijkjikjijijikjik332211iiijijiiijij332211333322221111333332323131232322222121131312121111ilkljkijiljlijjlkljkiljlijkljkijkljkij三项最多能保留k=j那一项。),(jiforsumnoaaijjjijijkjikaajijijikjikaaaa332211jijiaajjjijiaaaa332211三项只能保留k=j那一项。),(jiforsumnoaajjjj张量的点积张量的双点积