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1、弹性力学数学基性力学数学基础2023/5/141第1页，本讲稿共43页第二章第二章 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2023/5/142第2页，本讲稿共43页第一第一节 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 A scalar is a quantity characterized by magnitudemagni
2、tude only,for example:only,for example:mass.mass.v矢量（矢量（vector）A vector is a quantity characterized by both A vector is a quantity characterized by both magnitudemagnitude and and directiondirection,such as,such as displacement,displacement,velocityvelocity.2023/5/143第3页，本讲稿共43页二、矢量的表示二、矢量的表示v大小和方向确
3、定分量大小和方向确定分量 A A is completely defined by its magnitude is completely defined by its magnitude A A and and by its three by its three direction anglesdirection angles 1 1,2 2 and and 3 3 矢量矢量A A在三个坐标轴上的投影（分量）在三个坐标轴上的投影（分量）Ax1x2x3123o2023/5/144第4页，本讲稿共43页v分量（投影）确定矢量分量（投影）确定矢量 已已知知分分量量，矢矢量量的的大大小小和和方方向向
4、可可由由几几何何关关系系得到得到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 take take on on the the values 1,2,and 3 unless oth
5、erwise stated.values 1,2,and 3 unless otherwise stated.2023/5/145第5页，本讲稿共43页三、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 之间的之间的夹角的余弦夹角的余弦x2x1x3x1x2x3The Cosine of The Angles Between xi and xj Axesx1x2x3x1a11a12
6、a13x2a21a22a23x3a31a32a33矢量在某轴上的投影矢量在某轴上的投影=分量在同一分量在同一轴投影的代数和轴投影的代数和2023/5/146第6页，本讲稿共43页 Using the above range convention,these equations may be written more compactly as所以应所以应所以应所以应有关系有关系有关系有关系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（类似于合力投影定理）2023/5/147第7页，本讲稿共43页记记坐标变换矩阵坐标变换矩阵则有则有2023/5/148
7、第8页，本讲稿共43页 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 It is important to notice that the r
8、epeated subscript j j in this equation is a so-called in this equation is a so-called dummy indexdummy index,which can equally well be replaced with another which can equally well be replaced with another subscript,say subscript,say k 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2023/5/149第9页，本讲稿共
9、43页四、正交关系四、正交关系（Orthogonality Relations）We We introduce introduce the the so-called so-called Kronecker Kronecker deltadelta symbol symbol ij ij defined as defined as Any Any set set of of vector vector components components A Ai i may may be be written written asas根据求和约定根据求和约定2023/5/1410第10页，本讲稿共43
10、页 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 form2023/5/1411第11页，本讲稿共43页五、矢量运算（五、矢量运算（Vector Operations）v矢量相加 The The result result of of addition addition or or subtraction subtraction of
11、 of two two vectors vectors A A and and B B is defined to be a third vector is defined to be a third vector C 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 vector A A is defined to be a second vector is defined to be a second vector C C2023/5/1412第12页
12、，本讲稿共43页v两个矢量的标量积（Scalar Product of two vectors）The The scalar scalar product product of of two two vectors vectors A A andand B B is expressible asis expressible asAB2023/5/1413第13页，本讲稿共43页v两个矢量的矢量积（Vector Product of Two Vectors）The The vector vector product product of of two two vectors vectors A
13、A andand B B is is to to be a third vector be a third vector C C perpendicular to perpendicular to A A andand B B where where e e denotes unit vector along the vector denotes unit vector along the vector C C,and iand i1 1,i i2 2,i i3 3 are unit vectors along are unit vectors along x x1 1,x x2 2 and
14、and x x3 3.ABC2023/5/1414第14页，本讲稿共43页 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 c
15、omponents of vector C C can 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2023/5/1415第15页，本讲稿共43页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.202
16、3/5/1416第16页，本讲稿共43页第二第二节 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 A Cartesian Cartesian tensor tensor of of order order one one is is defined defined as as a a quantity quantity having having three three components components T Ti i whose whose transformation transformation between between pri
17、med primed and and unprimed unprimed coordinate axes is governed by coordinate axes is governed by andandA first-order tensor is nothing more than a vector.A first-order tensor is nothing more than a vector.和和2023/5/1417第17页，本讲稿共43页v二阶笛卡尔张量二阶笛卡尔张量 Similarly,Similarly,a a Cartesian Cartesian tensor t
18、ensor of of order order two two is is defined defined as as a a quantity quantity having having nine nine componentscomponents T Tij ij whose whose transformation transformation between between primed primed and and unprimed unprimed coordinate coordinate axes axes is is governed governed by by the
19、the equationsequationsandoror2023/5/1418第18页，本讲稿共43页v高阶笛卡尔张量高阶笛卡尔张量Third-Third-and and higher-orderhigher-order Cartesian tensors are Cartesian tensors are defined analogously.defined analogously.v零阶笛卡尔张量零阶笛卡尔张量A Cartesian tensor of zeroth orderA Cartesian tensor of zeroth order is is defined to be
20、any quantity that is unchanged defined to be any quantity that is unchanged under coordinate transformation,that is,a under coordinate transformation,that is,a scalar.scalar.2023/5/1419第19页，本讲稿共43页 If If A Aij ij and and B Bij ij denote denote components components of of two two second-order second-
21、order tensors,tensors,the the addition addition or or subtraction subtraction of of these these tensors tensors is is defined defined to to be be a a third third tensor tensor of of second second order order having having components components C Cij ij given bygiven by二、笛卡尔张量的运算（二、笛卡尔张量的运算（Operation
22、 of Cartesian Tensors）vAddition of Cartesian Tensors The addition or subtraction of two Cartesian The addition or subtraction of two Cartesian tensors of the same order to be a third Cartesian tensors of the same order to be a third Cartesian tensor of the same order.tensor of the same order.2023/5/
23、1420第20页，本讲稿共43页vMultiplication of Cartesian Tensors The multiplication of Cartesian tensors can be 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,w
24、ith no repeated summation indices.An inner product of two Cartesian tensors is defined as an outer product 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.2023/5/1421第21页，本讲稿共43页v二阶张量的商规则（二阶张量的商规则（Quotient Rul
25、e for Second-Order Tensors）Suppose Suppose we we know know the the following following equation equation to to applyapply where where A Ai 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 quot
26、ient quotient rulerule states states the the components components T Tij ij are are indeed indeed the the components components of of a a second-order second-order Cartesian tensor.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2023/5/1422第22页，本讲稿共43页一、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
27、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 antisymmetric.二阶张量的九个分量可二阶张量的九个分量可以用以用3 3矩阵表示：矩阵表示：2023/5/1423第23页，本讲稿共43页例题例题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
28、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2023/5/1424第24页，本讲稿共43页证：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所有指标都是哑指标是哑指标2023/5/1425第25页，本讲稿共43页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 tha
29、t its operation on a vector 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 as2023/5/1426第26页，本讲稿共43页二二、对对称称张张量量的的特特征征值值和和特特征征
30、矢矢量量（Eigenvalues and Eigenvectors of Symmetric Tensors）Consider the equation where where T Tij ij 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
31、.Any Any nonzero nonzero vector vector n n satisfying satisfying this this equation equation is is known known as as unitunit eigenvector eigenvector of of the the tensor and tensor and is known as is known as eigenvalueeigenvalue .2023/5/1427第27页，本讲稿共43页Expand the equation and rearranging to get Th
32、e condition for a nontrivial solution of these homogeneous algebraic equations is that 2023/5/1428第28页，本讲稿共43页Equation yields the cubic equation are called first,second,and third invariant of the tensor T,respectively.where2023/5/1429第29页，本讲稿共43页 When the components Tij are those of a symmetric tens
33、or,it can easily be shown that cubic equation will have three real 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 ei
34、genvector associated with the eigen-value (1).In a similar way,we may also find two additional unit eigenvectors associated with the eigen-values (2)and (3).2023/5/1430第30页，本讲稿共43页 The above three unit eigenvectors are The above three unit eigenvectors are mutually perpendicular when mutually perpen
35、dicular when (1)(1),(2)(2),andand (3)(3)are all distinct.Consider two unit are all distinct.Consider two unit eigenvectors neigenvectors n(1)(1)and nand n(2)(2).These satisfy .These satisfy equationequation Multiplying Multiplying the the first first of of these these equations equations by by n n(2
36、)(2)i i and and the the second second by by n n(1)(1)i i and and subtracting,subtracting,we havewe have2023/5/1431第31页，本讲稿共43页That is On interchanging the dummy indices i and j in the first term on the left-hand side of this equation2023/5/1432第32页，本讲稿共43页Using Tij=Tji,we find that Hence,if (1)(2),t
37、hen 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.2023/5/1433第33页，本讲稿共43页三、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 Tenso
38、r）Choose Choose a a new new set set of of Cartesian Cartesian axes axes x x i i having having unit unit vectors vectors along along these these axes axes coincident coincident with with the the unit unit eigenvecotrs.eigenvecotrs.For For this this system system of of axes,axes,we havewe havex1x2x3x1
39、x2x3n(1)=i1n(2)=i2n(3)=i32023/5/1434第34页，本讲稿共43页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2023/5/1435第35页，本讲稿共43页 In In this this system system of of so-called so-called principal principal axesaxes defined defined by by the the unit unit eigenvectors eigenvectors n n(1)(1),n n(2)(2),n n(
40、3)(3),the the tensor components are therefore expressible astensor components are therefore expressible as The diagonal components are known as principal values of symmetric tensor T2023/5/1436第36页，本讲稿共43页 Consider Consider the the case case where where only only two two eigenvalues,eigenvalues,say
41、say (1)(1)and and (2)(2),are equal.We have,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.2023/5/1437第37页，本讲稿共43页 Consider Consider finally finally the the case case where where
42、 all all eigen-values eigen-values are equal,say,to are equal,say,to (1)(1).We have.We have so that the tensor components are expressible as Hence,the xi axes are already principal axes.axes are already principal axes.2023/5/1438第38页，本讲稿共43页第四第四节 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
43、包围的体积V，函数，函数P、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曲面积分和体积积分之间的关系即高斯积分公式公式2023/5/1439第39页，本讲稿共43页cos、cos、cos 为微分曲面为微分曲面dA的法线正方的法线正方向的方向余弦向的方向余弦令：令：则有：则有：2023/5/1440第40页，本讲稿共43页设设逗号逗号i “,i”表示前面表示前面的变量对坐标的变量对坐标 xi 求偏导求偏导数数则有高斯积分定理则有高斯积分定理的简写公式：的简写公式：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表示积分公式
44、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，德国数学家、物理学家和天文学家德国数学家、物理学家和天文学家2023/5/1441第41页，本讲稿共43页 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 components compon
45、ents 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 derivative respect respect to 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2023/5/1442第42页，本讲稿共43页 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2023/5/1443第43页，本讲稿共43页