弹性力学数学基础

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,*,土木工程专业：弹性力学,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,11/25/2024,1,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,11/25/2024,2,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 example:,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,11/25/2024,3,二、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 angles,1,2,and,3,矢量A在三个坐标轴上的投影（分量）,A,x,1,x,2,x,3,1,2,3,o,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,11/25/2024,4,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,A,x,1,x,2,x,3,1,2,3,o,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愈伞箕弹性力学数学基础弹性力学数学基础,11/25/2024,5,三、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,x,2,x,1,x,3,x,1,x,2,x,3,The Cosine of The Angles Between xi and xj Axes,x,1,x,2,x,3,x,1,a,11,a,12,a,13,x,2,a,21,a,22,a,23,x,3,a,31,a,32,a,33,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,11/25/2024,6,Using the above range convention,these equations may be written more compactly as,所以应有关系,x,2,x,1,x,3,x,1,x,2,x,3,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,11/25/2024,7,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,11/25/2024,8,We may achieve a further simplification by adopting the summation convention requiring 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,11/25/2024,9,四、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,11/25/2024,10,In a similar way,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,11/25/2024,11,五、矢量运算（Vector Operations,）,矢量相加,The result of addition or subtraction of two vectors,A,and,B,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,11/25/2024,12,两个矢量的标量积（,Scalar Product of two vectors,）,The scalar product of two vectors,A,and,B,is expressible as,A,B,嗅倔犬瘴痉娜石类想饮妹滚隧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,11/25/2024,13,两个矢量的矢量积（,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.,A,B,C,二糙氦铝藏谆厌淌煎贴筛牺壬骤壶纸脆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,11/25/2024,14,If the symbol,e,ijk,is defined as follows:,e,ijk,=+1,for,i,=1,j,=2,k,=3 or any even number of permutations of this arrangement(e.g.,e,312,),e,ijk,=-1,for odd permutations of,i,=1,j,=2,k,=3 (e.g.,e,132,),e,ijk,=0,for two or more indices equal(e.g.,e,113,),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,11/25/2024,15,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 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,11/25/2024,16,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,T,i,whose transformation between primed and unprimed coordinate axes is governed by,and,A first-order tensor is 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,11/25/2024,17,二阶笛卡尔张量,Similarly,a Cartesian tensor of order two is defined as a quantity having nine components,T,ij,whose transformation between primed and unprimed coordinate axes is governed by the equations,and,or,or,颗救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,11/25/2024,18,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,11/25/2024,19,If,A,ij,and,B,ij,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,C,ij,given by,二、笛卡尔张量的运算（Operation of Cartesian Te