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1、弹性力学数学基性力学数学基础2023/1/201第1页，共43页，编辑于2022年，星期六第二章第二章 数学基数学基础第一节第一节 标量和矢量标量和矢量第二节第二节 笛卡尔张量笛卡尔张量第三节第三节 二阶笛卡尔张量二阶笛卡尔张量第四节第四节 高斯积分定理高斯积分定理2023/1/202第2页，共43页，编辑于2022年，星期六第一第一节 标量和矢量量和矢量一、标量和矢量的定义（一、标量和矢量的定义（definition）v标量（标量（scalar）A scalar is a quantity characterized by magnitude only,for example:mass.v矢
2、量（矢量（vector）A vector is a quantity characterized by both magnitude and direction,such as displacement,velocity.2023/1/203第3页，共43页，编辑于2022年，星期六二、矢量的表示二、矢量的表示v大小和方向确定分量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在三个坐标轴上的投影（分量）A Ax1x2x3123o2023/1/2
3、04第4页，共43页，编辑于2022年，星期六v分量（投影）确定矢量分量（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 Ax1x2x3123o The The three three components components A A1 1,A A2 2,A A3 3 may may be be written written simply simply as as A Ai i with with the the range range conventionconvention,that that any any subscript subscript is is to to
4、 take take on on the the values 1,2,and 3 unless otherwise stated.values 1,2,and 3 unless otherwise stated.2023/1/205第5页，共43页，编辑于2022年，星期六三、坐标变换（三、坐标变换（Coordinate Transformation）考虑坐标原点重合的直角坐标系考虑坐标原点重合的直角坐标系 x 1,x 2,x 3 和和 x1,x2,x3 如图所示。如图所示。用用 aij 表示新旧坐标轴表示新旧坐标轴 x i 和和 xj 之间之间的夹角的余弦的夹角的余弦x2x1x3x1x2x
5、3The Cosine of The Angles Between xi and xj Axesx1x2x3x1a11a12a13x2a21a22a23x3a31a32a33矢量在某轴上的投影矢量在某轴上的投影=分量在同一轴分量在同一轴投影的代数和投影的代数和2023/1/206第6页，共43页，编辑于2022年，星期六 Using the above range convention,these equations may be written more compactly as所以应所以应有关系有关系x2x1x3x1x2x3A矢量矢量A向新坐标轴向新坐标轴x1投影（类似于合力投影定理）投影
6、（类似于合力投影定理）2023/1/207第7页，共43页，编辑于2022年，星期六记记坐标变换矩阵坐标变换矩阵则有则有2023/1/208第8页，共43页，编辑于2022年，星期六 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 impo
7、rtant 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.v同理，可得到由新坐标的分量表示旧坐标同理，可得到由新坐标的分量表示旧坐标系的分量系的分量2023/1/209第9页，共43页，编辑于2022年，星期六四、正交关系四、正交关系（Orthogonality Relations）We introduce the so-called Kronec
8、ker delta symbol ij defined as Any set of vector components Any set of vector components A Ai i may be written as may be written as根据求和约定根据求和约定2023/1/2010第10页，共43页，编辑于2022年，星期六 In a similar way,we may also obtain These equations are referred to as orthogonality relations.It thus follows that Above e
9、quation may be expressed in the form2023/1/2011第11页，共43页，编辑于2022年，星期六五、矢量运算（五、矢量运算（Vector Operations）v矢量相加 The result of addition or subtraction of two vectors A and B is defined to be a third vector Cv矢量与标量相乘 The The multiplication multiplication of of a a scalar scalar m m and and a a vector vecto
10、r A A is is defined to be a second vector defined to be a second vector C C2023/1/2012第12页，共43页，编辑于2022年，星期六v两个矢量的标量积（Scalar Product of two vectors）The scalar product of two vectors A and B is expressible asAB 2023/1/2013第13页，共43页，编辑于2022年，星期六v两个矢量的矢量积（Vector Product of Two Vectors）The vector produc
11、t of two vectors A and B is to be a third vector C perpendicular to A and B where where e e denotes unit vector along the vector denotes unit vector along the vector C C,and i,and i1 1,i i2 2,i i3 3 are unit vectors along are unit vectors along x x1 1,x x2 2 and and x x3 3.ABC 2023/1/2014第14页，共43页，编
12、辑于2022年，星期六 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.,e113)the components of vector the components of vector C C ca
13、n be written as can be written as利用符号利用符号eijk可以方便地可以方便地表示表示3阶行列式的值阶行列式的值2023/1/2015第15页，共43页，编辑于2022年，星期六v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.2023/1/2016第16页，共43
14、页，编辑于2022年，星期六第二第二节 笛卡笛卡尔张量量一、笛卡尔张量的定义一、笛卡尔张量的定义v一阶笛卡尔张量一阶笛卡尔张量 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 andandA first-order tensor is nothing more than a vector.A first-order t
15、ensor is nothing more than a vector.和和2023/1/2017第17页，共43页，编辑于2022年，星期六v二阶笛卡尔张量二阶笛卡尔张量 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 equationsandoror2023/1/2018第18页，共43
16、页，编辑于2022年，星期六v高阶笛卡尔张量高阶笛卡尔张量Third-and higher-order Cartesian tensors are defined analogously.v零阶笛卡尔张量零阶笛卡尔张量A Cartesian tensor of zeroth order is defined to be any quantity that is unchanged under coordinate transformation,that is,a scalar.2023/1/2019第19页，共43页，编辑于2022年，星期六 If Aij and Bij denote com
17、ponents 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（二、笛卡尔张量的运算（Operation of Cartesian Tensors）vAddition of Cartesian Tensors The addition or subtraction of two Cartesian tensors The
18、addition or subtraction of two Cartesian tensors of the same order to be a third Cartesian tensor of the of the same order to be a third Cartesian tensor of the same order.same order.2023/1/2020第20页，共43页，编辑于2022年，星期六vMultiplication of Cartesian Tensors The multiplication of Cartesian tensors can be
19、classified into two categories,outer 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 p
20、roduct followed by a contraction of the two;that is,by an equating of any index associated with one tensor to any index associated with the other.2023/1/2021第21页，共43页，编辑于2022年，星期六v二阶张量的商规则（二阶张量的商规则（Quotient Rule for Second-Order Tensors）Suppose we know the following equation to apply where where A A
21、i i denotes denotes components components of of an an arbitrary arbitrary vector,vector,B Bj j components components of of a a vector.vector.Then,Then,the the quotient quotient rulerule states states the the components components T Tij ij are are indeed indeed the the components of a second-order Ca
22、rtesian ponents of a second-order Cartesian tensor.书上有证明书上有证明下一章要利用这个法则下一章要利用这个法则2023/1/2022第22页，共43页，编辑于2022年，星期六一、对称张量和反对称张量的定义一、对称张量和反对称张量的定义v定义（定义（Definition）第三第三节 二二阶笛卡笛卡尔张量量 If Tij=Tji,then the tensor is said to be symmetric.On the other hand,if Tij=-Tji,then the tensor is said to be antisymme
23、tric.二阶张量的九个分量可以二阶张量的九个分量可以用用3 3矩阵表示：矩阵表示：2023/1/2023第23页，共43页，编辑于2022年，星期六例题例题2.1 试证明任意二阶张量可以表示为对称张量试证明任意二阶张量可以表示为对称张量 和反对称张量之和和反对称张量之和证：证：设设Tij 是任意二阶张量的分量，则有是任意二阶张量的分量，则有其中其中二阶对称张量二阶对称张量二阶反对称张量二阶反对称张量2023/1/2024第24页，共43页，编辑于2022年，星期六证：证：例题例题2.2 设设Aij 是二阶对称张量的分量，是二阶对称张量的分量，Bij 是二阶是二阶 反对称张量的分量，试证明关系
24、反对称张量的分量，试证明关系Aij Bij=0。因为因为所以所以所有指标都是所有指标都是哑指标哑指标2023/1/2025第25页，共43页，编辑于2022年，星期六v反对称张量的分量反对称张量的分量（Anti-symmetric Tensor Components）A special characteristic of an anti-symmetirc tensor is that its operation on a vector is equivalent to an appropriately defined vector-product operation.If Ai denote
25、s components of a vector and if Tij denotes components of a second-order anti-symmetric tensor,then where Wj denotes vector components defined as2023/1/2026第26页，共43页，编辑于2022年，星期六二二、对对称称张张量量的的特特征征值值和和特特征征矢矢量量（Eigenvalues and Eigenvectors of Symmetric Tensors）Consider the equation where where T Tij ij
26、 denotes denotes components components of of a a symmetric symmetric tensor,tensor,n ni i denotes denotes components components of of a a unit unit vector,vector,and and denotes denotes a a scalar.scalar.Any Any nonzero nonzero vector vector n n satisfying satisfying this this equation equation is i
27、s known known as as unitunit eigenvector eigenvector of of the the tensor tensor and and is is known known as as eigenvalueeigenvalue .2023/1/2027第27页，共43页，编辑于2022年，星期六Expand the equation and rearranging to get The condition for a nontrivial solution of these homogeneous algebraic equations is that
28、2023/1/2028第28页，共43页，编辑于2022年，星期六Equation yields the cubic equation are called first,second,and third invariant of the tensor T,respectively.where2023/1/2029第29页，共43页，编辑于2022年，星期六 When the components Tij are those of a symmetric tensor,it can easily be shown that cubic equation will have three real
29、roots.We denote these roots by (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 m
30、ay also find two additional unit eigenvectors associated with the eigen-values (2)and (3).2023/1/2030第30页，共43页，编辑于2022年，星期六 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 satisfy equation Multiplying M
31、ultiplying the the first first of of these these equations equations by by n n(2)(2)i i and the second by and the second by n n(1)(1)i i and subtracting,we have and subtracting,we have2023/1/2031第31页，共43页，编辑于2022年，星期六That is On interchanging the dummy indices i and j in the first term on the left-ha
32、nd side of this equation2023/1/2032第32页，共43页，编辑于2022年，星期六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),respectively.202
33、3/1/2033第33页，共43页，编辑于2022年，星期六三、对称张量的主轴和主值三、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 havex1x2x3x1x2x3n(1)=i1n(2)=i2n(3)=i32023/1/203
34、4第34页，共43页，编辑于2022年，星期六ij：i=j：非非对对角角线线元元素素为为零零非零元素在对角线上，就是特征值非零元素在对角线上，就是特征值2023/1/2035第35页，共43页，编辑于2022年，星期六 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 diagonal components are known as principal v
35、alues of symmetric tensor T2023/1/2036第36页，共43页，编辑于2022年，星期六 Consider the case where only two eigenvalues,say(1)and(2),are equal.We have provided only that the unit vectors i 1,i 2,i 3 be chosen such that i 3 lies along n(3)and i 1 and i 2 lie in any two mutually perpendicular directions.2023/1/2037
36、第37页，共43页，编辑于2022年，星期六 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 axes are already principal axes.2023/1/2038第38页，共43页，编辑于2022年，星期六第四第四节 高斯高斯积分定理分定理一、数学公式重写一、数学公式重写连续函数连续函数分片光滑曲面分片光滑曲面A包围的体积包围的体积V，函数，函数P、
37、Q、R在在A+V上连续上连续曲面积分和体积积分之间的关系即高斯积分公式曲面积分和体积积分之间的关系即高斯积分公式2023/1/2039第39页，共43页，编辑于2022年，星期六cos、cos、cos 为微分曲面为微分曲面dA的法线正方向的方的法线正方向的方向余弦向余弦令：令：则有：则有：2023/1/2040第40页，共43页，编辑于2022年，星期六设设逗号逗号i “,i”表示前面的表示前面的变量对坐标变量对坐标 xi 求偏导数求偏导数则有高斯积分定理的则有高斯积分定理的简写公式：简写公式：Bi 可以理解为矢量的分量可以理解为矢量的分量此乃高等数学中的高斯此乃高等数学中的高斯积分公式的简单
38、表示积分公式的简单表示卡尔卡尔 弗里德里希弗里德里希 高斯，高斯，1777-1855，德，德国数学家、物理学家和天文学家国数学家、物理学家和天文学家2023/1/2041第41页，共43页，编辑于2022年，星期六 If B denotes any scalar or scalar components of a vector or tensor,Greens theorem yields the following relation between surface and volume integrals:Where Where n ni i denotes denotes the the
39、components components of of unit unit normal normal vector vector of of the the surface surface area area dAdA,comma comma i i (i.e.,“,(i.e.,“,i i”)”)in in the the right-hand right-hand side side of of the the equation equation denotes denotes the the partial partial derivative respect to derivative respect to x xi i .二、广义的高斯二、广义的高斯 积分定理积分定理v标量函数标量函数2023/1/2042第42页，共43页，编辑于2022年，星期六 We note that,by addition,above equation also applies for vector components Bi such that and for second-order tensor components Bji such thatv矢量和二阶张量矢量和二阶张量2023/1/2043第43页，共43页，编辑于2022年，星期六