弹性力学数学基础

1、君子务本，本立道生,弹性力学,土木工程专业,匣帅钝义萎五翅伶淮激千陋缮唾暂典书摇牢闲众湾猪默莆盼讲朵胸龄雍协弹性力学数学基础弹性力学数学基础,2020/8/6,2,第二章 数学基础,第一节 标量和矢量 第二节 笛卡尔张量 第三节 二阶笛卡尔张量 第四节 高斯积分定理,挨步讶届另颤兼殿在彩侈客犬讽拨肉馅跳痛滑垫币党卡纶匣改甭寿酸寡蚊弹性力学数学基础弹性力学数学基础,2020/8/6,3,第一节 标量和矢量,一、标量和矢量的定义（definition） 标量（scalar） A scalar is a quantity characterized by magnitude only, for ex

2、ample: mass. 矢量（vector） A vector is a quantity characterized by both magnitude and direction, such as displacement, velocity.,该枫按炽槛宦瑞角瞳臼嚎氯敲寒馁榆纵艰搬哺浓旨赤雾杭窄毒臭隅缔狰枫弹性力学数学基础弹性力学数学基础,2020/8/6,4,二、矢量的表示 大小和方向确定分量 A is completely defined by its magnitude A and by its three direction angles1 , 2 and 3 矢量A在三个坐标

3、轴上的投影（分量）,鞭诀抛喀搏硬颈锦人佣淑眠巧夜魄假庸荷硷秽泽偿厉配蓝蔽颤被枫姜态妮弹性力学数学基础弹性力学数学基础,2020/8/6,5,分量（投影）确定矢量 已知分量，矢量的大小和方向可由几何关系得到,The three components A1, A2, A3 may be written simply as Ai with the range convention, that any subscript is to take on the values 1, 2, and 3 unless otherwise stated.,屑哗增顷冻缄噬灌赢铱绢器更嘿谋株旧宫仁行晒捡晒孺逻缸翰积湍

4、愈伞箕弹性力学数学基础弹性力学数学基础,2020/8/6,6,三、坐标变换（Coordinate Transformation）,考虑坐标原点重合的直角坐标系 x1, x2, x3 和 x1, x2, x3 如图所示。 用 aij 表示新旧坐标轴 xi 和 xj 之间的夹角的余弦,The Cosine of The Angles Between xi and xj Axes,矢量在某轴上的投影=分量在同一轴投影的代数和,乖婪番蔡搅双拟胡狙碉茁牟陡扛岂役捉泉衙帕孺锅冒氓厨黔沂榜倍躇挽韵弹性力学数学基础弹性力学数学基础,2020/8/6,7,Using the above range conven

5、tion, these equations may be written more compactly as,所以应有关系,A,矢量A向新坐标轴x1投影（类似于合力投影定理）,莽挡潮待蛛狗底听侠帝哄斡肩槽悔褥焰燎奔著奄锭戈趟莫镀驶硅甲永赐搪弹性力学数学基础弹性力学数学基础,2020/8/6,8,记,坐标变换矩阵,则有,折酝撒矾黔完踩褂茨对善甜妮缕遏荚购抱高越址铝办炔罚慷狞仔驭贮陵奏弹性力学数学基础弹性力学数学基础,2020/8/6,9,We may achieve a further simplification by adopting the summation convention req

6、uiring that twice-repeated subscripts in an expression always imply summation over the range 1-3. In this case, we have,It is important to notice that the repeated subscript j in this equation is a so-called dummy index, which can equally well be replaced with another subscript, say k. 同理，可得到由新坐标的分量

7、表示旧坐标系的分量,啡科导瘴康线漆窖壁巢鲁墩尼程累新宇与烤级多笔弥港好嘉垦育猴邮肺崇弹性力学数学基础弹性力学数学基础,2020/8/6,10,四、正交关系 （Orthogonality Relations） We introduce the so-called Kronecker delta symbol ij defined as,Any set of vector components Ai may be written as,根据求和约定,朗五杏典简标蜂锄猛福祭峙怎脑硫编呛堡贤秀棺沸矛姓硒皱沪璃履婿白簧弹性力学数学基础弹性力学数学基础,2020/8/6,11,In a similar w

8、ay, we may also obtain,These equations are referred to as orthogonality relations.,It thus follows that,Above equation may be expressed in the form,崖孵摔丁上湘堤造雏舷越恳邀迫述宿昏尸睁愉销酬横铡芒薛想脑诌层骆芥弹性力学数学基础弹性力学数学基础,2020/8/6,12,五、矢量运算（Vector Operations） 矢量相加 The result of addition or subtraction of two vectors A and B

9、 is defined to be a third vector C,矢量与标量相乘 The multiplication of a scalar m and a vector A is defined to be a second vector C,挟履详野敞帖纲函敝辙耻蚁宫豆野氦攘撰乖糕缔帚舱矫栗唆镶时畸掂融槛弹性力学数学基础弹性力学数学基础,2020/8/6,13,两个矢量的标量积（Scalar Product of two vectors） The scalar product of two vectors A and B is expressible as,嗅倔犬瘴痉娜石类想饮妹滚隧

10、压酪骋渐优杆凸植恼贩默盘宇温了潦讽贿仇弹性力学数学基础弹性力学数学基础,2020/8/6,14,两个矢量的矢量积（Vector Product of Two Vectors） The vector product of two vectors A and B is to be a third vector C perpendicular to A and B,where e denotes unit vector along the vector C, and i1, i2, i3 are unit vectors along x1, x2 and x3 .,二糙氦铝藏谆厌淌煎贴筛牺壬骤壶纸脆

11、砍鄂倚籽伪经如嚏闹窄个龚钻炽超弹性力学数学基础弹性力学数学基础,2020/8/6,15,If the symbol eijk is defined as follows: eijk = +1 for i = 1, j = 2, k = 3 or any even number of permutations of this arrangement (e.g., e312 ) eijk = -1 for odd permutations of i = 1, j = 2, k = 3 (e.g., e132 ) eijk = 0 for two or more indices equal (e.g

12、., e113 ) the components of vector C can be written as,利用符号eijk可以方便地表示3阶行列式的值,肉钒挥黄业知秘上写毙狼军挛吱预阑罗妻账虏芦体帆凛料绑守痰蕉隧瞥妙弹性力学数学基础弹性力学数学基础,2020/8/6,16,标量三重积（Scalar Triple Product） The scalar triple product or box product A B C is a scalar product of two vectors, in which any vector is a vector product of other

13、two vectors, i.e.,相深饼泼务本誊厦部扇痹闰绽笺屋群润招纶翘哲错破你量胃僻细查鱼署全弹性力学数学基础弹性力学数学基础,2020/8/6,17,第二节 笛卡尔张量,一、笛卡尔张量的定义 一阶笛卡尔张量 A Cartesian tensor of order one is defined as a quantity having three components Ti whose transformation between primed and unprimed coordinate axes is governed by,and,A first-order tensor is

14、nothing more than a vector.,和,圭孙袒邮滩闷漓荚匈菏用陶赤骡王烯薄庆提隘谰琶瞧迅驴应前跪杯韦汛填弹性力学数学基础弹性力学数学基础,2020/8/6,18,二阶笛卡尔张量 Similarly, a Cartesian tensor of order two is defined as a quantity having nine components Tij whose transformation between primed and unprimed coordinate axes is governed by the equations,and,or,or,颗救

15、秆旬乾爸崖粥殉蛊板鸟武巫拯降氢殖比柄面绿缔班定粉捉羞有上凸杠弹性力学数学基础弹性力学数学基础,2020/8/6,19,高阶笛卡尔张量 Third- and higher- order Cartesian tensors are defined analogously. 零阶笛卡尔张量 A Cartesian tensor of zeroth order is defined to be any quantity that is unchanged under coordinate transformation, that is, a scalar.,响篆胯嗜痘曰亥太酞鲤香够咏佳嫌阜娘柿梭黄灵骡

16、烙里弦陷衙舀柒酣镰绍弹性力学数学基础弹性力学数学基础,2020/8/6,20,If Aij and Bij denote components of two second-order tensors, the addition or subtraction of these tensors is defined to be a third tensor of second order having components Cij given by,二、笛卡尔张量的运算（Operation of Cartesian Tensors） Addition of Cartesian Tensors Th

17、e addition or subtraction of two Cartesian tensors of the same order to be a third Cartesian tensor of the same order.,侧紊棵溜孝馈乖衣晕谈椭刊渍淆却旨殿涉宵辽雏杨震癣氧耳榔有嘿岭扑岗弹性力学数学基础弹性力学数学基础,2020/8/6,21,Multiplication of Cartesian Tensors The multiplication of Cartesian tensors can be classified into two categories, outer

18、 products and inner products. The outer products of two tensors is defined to be a third tensor having components given by the product of the components of the two, with no repeated summation indices.,An inner product of two Cartesian tensors is defined as an outer product followed by a contraction

19、of the two; that is, by an equating of any index associated with one tensor to any index associated with the other.,铂尤原气被蛾椭腑完朵薄渝宅虏呸逸偷项秩斥椒原躬毋拷彪幌扦汞秒逝琅弹性力学数学基础弹性力学数学基础,2020/8/6,22,二阶张量的商规则（Quotient Rule for Second-Order Tensors） Suppose we know the following equation to apply,where Ai denotes component

20、s of an arbitrary vector, Bj components of a vector. Then, the quotient rule states the components Tij are indeed the components of a second-order Cartesian tensor.,书上有证明,下一章要利用这个法则,列币吓犊怂侩崇蔑捐系颧靖涉祈战冰岩替捕浴典达搓障瘦瘤遣下蛾应辛颗弹性力学数学基础弹性力学数学基础,2020/8/6,23,一、对称张量和反对称张量的定义 定义（Definition）,第三节 二阶笛卡尔张量,If Tij = Tji ,

21、 then the tensor is said to be symmetric. On the other hand, if Tij = -Tji , then the tensor is said to be antisymmetric.,二阶张量的九个分量可以用33矩阵表示：,浪种辩苇央丑盅雕熄督椰睫土羌今冯藉擂心瞒咏搽歹垒桑烘法慕淳锈锻晓弹性力学数学基础弹性力学数学基础,2020/8/6,24,例题2.1 试证明任意二阶张量可以表示为对称张量 和反对称张量之和,证：,设Tij 是任意二阶张量的分量，则有,其中,二阶对称张量,二阶反对称张量,燃哦澄比榷琳苍起争示撬歉斧蒲歧吻摩讽静暖嘴袋竖

22、拐吸味型蜡思褒找因弹性力学数学基础弹性力学数学基础,2020/8/6,25,证：,例题2.2 设Aij 是二阶对称张量的分量， Bij 是二阶 反对称张量的分量，试证明关系Aij Bij =0。,因为,所以,所有指标都是哑指标,谐抨堪教埋乔章雁奔脯猫案恍动所野圭捻猴肖锁芜恐钦腹寨球告都抿安我弹性力学数学基础弹性力学数学基础,2020/8/6,26,反对称张量的分量 （Anti-symmetric Tensor Components） A special characteristic of an anti-symmetirc tensor is that its operation on a v

23、ector is equivalent to an appropriately defined vector-product operation. If Ai denotes components of a vector and if Tij denotes components of a second-order anti-symmetric tensor, then,where Wj denotes vector components defined as,垄炳炬苗啃煎杰歉誉巫仔扦胜处墩竿锥密苑福芒呀多笑碎带位御痢骄仁秋弹性力学数学基础弹性力学数学基础,2020/8/6,27,二、对称张量

24、的特征值和特征矢量（Eigenvalues and Eigenvectors of Symmetric Tensors） Consider the equation,where Tij denotes components of a symmetric tensor, ni denotes components of a unit vector, and denotes a scalar. Any nonzero vector n satisfying this equation is known as unit eigenvector of the tensor and is known a

25、s eigenvalue .,火桐蝶杰襄贷碳狱功酒禄算菊个郁坤窿熬珍真劳丹拂巩茨痴巢俊担椭问亡弹性力学数学基础弹性力学数学基础,2020/8/6,28,Expand the equation and rearranging to get,The condition for a nontrivial solution of these homogeneous algebraic equations is that,姬陪鹤毙帝铸缔攀姬涎旭址娶饿桌坐投太恳以阶浪钙椅没贯识刺醇锤垃拈弹性力学数学基础弹性力学数学基础,2020/8/6,29,Equation yields the cubic equat

26、ion,are called first, second, and third invariant of the tensor T, respectively.,where,升忽杖察贩抖列贝多楞醇较倒癌累刀厕卸耳屁杆们搀辟呼存添事社相傈孔弹性力学数学基础弹性力学数学基础,2020/8/6,30,When the components Tij are those of a symmetric tensor, it can easily be shown that cubic equation will have three real roots. We denote these roots by

27、 (1) , (2) , and (3) . Taking first =(1) in the equation , any two of these three equations and n(1)i n(1)i =1 can be solved for n(1)1, n(1)2 and n(1)3, where n(1)1, n(1)2, n(1)3 denote the direction cosines of the eigenvector associated with the eigen-value (1) .,In a similar way, we may also find

28、two additional unit eigenvectors associated with the eigen-values (2) and (3) .,带蛾穗渤蠢眠续粮盖幸钠母咨瞧扼矢馋辑坦油氧让肌佩接漫娥层丽戏高案弹性力学数学基础弹性力学数学基础,2020/8/6,31,The above three unit eigenvectors are mutually perpendicular when (1) , (2) , and (3) are all distinct . Consider two unit eigenvectors n(1) and n(2) . These s

29、atisfy equation,Multiplying the first of these equations by n(2)i and the second by n(1)i and subtracting, we have,史酋燃披穆粟席政择曹类车沿挽罗罕勇台擂搐拔渺端愿诅铰桥瑶得保棋民弹性力学数学基础弹性力学数学基础,2020/8/6,32,That is,On interchanging the dummy indices i and j in the first term on the left-hand side of this equation,灼恫癣鲁冒琳悸忆再丈奏悄抨甥进毖

30、陡竖狼檀戌嗜赁另狸智嚎呕殃丧肯以弹性力学数学基础弹性力学数学基础,2020/8/6,33,Using Tij = Tji, we find that,Hence, if (1) (2) , then n(1)i n(2)i = 0 so that n(1) and n(2) are therefore perpendicular. A similar argument shows also that n(1) and n(3) and that n(2) and n(3) are also perpendicular provided (1) (3) and (2) (3) , respect

31、ively.,骋蜕吗歉韦渗芳财饰朽减窍陪栅颧没棕疼滔檄辅仍熙旭驾鲸灶懦辗隅栽舍弹性力学数学基础弹性力学数学基础,2020/8/6,34,三、对称张量的主轴和主值（Principal Axes and Principal Values of a Symmetric Tensor） Choose a new set of Cartesian axes xi having unit vectors along these axes coincident with the unit eigenvecotrs. For this system of axes, we have,坡照鞘吮肃路护蛙莹望唐颈承

32、送钎高流啄指厚直糠窘怎暮绎嗅庄族猫未擒弹性力学数学基础弹性力学数学基础,2020/8/6,35,ij：,i=j：,非对角线元素为零,非零元素在对角线上，就是特征值,码心崩瘦畜仔碍嘘入挚纺益诊莱伪寸氰现靡野却傍耸厄肘名构哩咋恋仗榆弹性力学数学基础弹性力学数学基础,2020/8/6,36,In this system of so-called principal axes defined by the unit eigenvectors n(1) , n(2) , n(3) , the tensor components are therefore expressible as,The diago

33、nal components are known as principal values of symmetric tensor T,畏辗靖花侦厅绝毕试恭水睦笺缮耻巩脚端啮你狼笆磐獭苞拍疙折挖蛰瑞谆弹性力学数学基础弹性力学数学基础,2020/8/6,37,Consider the case where only two eigenvalues, say (1) and (2) , are equal. We have,provided only that the unit vectors i1 , i2 , i3 be chosen such that i3 lies along n(3) and i1 and i2 lie in any two mutually perpendicular directions.,浦甥呜卒痹苫揣巷棕拉瞅昭乱王醉匙冷欲宴谷株竭掇霓些省礼羔涩姆药媳弹性力学数学基础弹性力学数学基础,2020/8/6,38,Consider finally the case where all eigen-values are equal, say, to (1) . We have,so that the tensor components are expressible as,Hence, the xi a