彈性力學數(shù)學基礎

1、彈性力學數(shù)學基礎2022/9/181第1頁，共43頁，2022年，5月20日，9點37分，星期三第二章 數(shù)學基礎第一節(jié) 標量和矢量第二節(jié) 笛卡爾張量第三節(jié) 二階笛卡爾張量第四節(jié) 高斯積分定理2022/9/182第2頁，共43頁，2022年，5月20日，9點37分，星期三第一節(jié) 標量和矢量一、標量和矢量的定義（definition）標量（scalar） A scalar is a quantity characterized by magnitude only, for example: mass.矢量（vector） A vector is a quantity characterized b

2、y both magnitude and direction, such as displacement, velocity.2022/9/183第3頁，共43頁，2022年，5月20日，9點37分，星期三二、矢量的表示大小和方向確定分量 A is completely defined by its magnitude A and by its three direction angles1 , 2 and 3 矢量A在三個坐標軸上的投影（分量）Ax1x2x3123o2022/9/184第4頁，共43頁，2022年，5月20日，9點37分，星期三分量（投影）確定矢量 已知分量，矢量的大小和方向

3、可由幾何關系得到Ax1x2x3123o 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. 2022/9/185第5頁，共43頁，2022年，5月20日，9點37分，星期三三、坐標變換（Coordinate Transformation） 考慮坐標原點重合的直角坐標系 x1, x2, x3 和 x1, x2, x3

4、 如圖所示。 用 aij 表示新舊坐標軸 xi 和 xj 之間的夾角的余弦x2x1x3x1x2x3The Cosine of The Angles Between xi and xj Axesx1x2x3x1a11a12a13x2a21a22a23x3a31a32a33矢量在某軸上的投影=分量在同一軸投影的代數(shù)和2022/9/186第6頁，共43頁，2022年，5月20日，9點37分，星期三 Using the above range convention, these equations may be written more compactly as所以應有關系x2x1x3x1x2x3A矢

5、量A向新坐標軸x1投影（類似于合力投影定理）2022/9/187第7頁，共43頁，2022年，5月20日，9點37分，星期三記坐標變換矩陣則有2022/9/188第8頁，共43頁，2022年，5月20日，9點37分，星期三 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 ca

6、se, 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，可得到由新坐標的分量表示舊坐標系的分量2022/9/189第9頁，共43頁，2022年，5月20日，9點37分，星期三四、正交關系 （Orthogonality Relations） We introduce the so-called

7、Kronecker delta symbol ij defined as Any set of vector components Ai may be written as根據(jù)求和約定2022/9/1810第10頁，共43頁，2022年，5月20日，9點37分，星期三 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 form2022

8、/9/1811第11頁，共43頁，2022年，5月20日，9點37分，星期三五、矢量運算（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矢量與標量相乘 The multiplication of a scalar m and a vector A is defined to be a second vector C2022/9/1812第12頁，共43頁，2022年，5月20日，9點37分，星期三兩個矢量的標

9、量積（Scalar Product of two vectors） The scalar product of two vectors A and B is expressible asAB2022/9/1813第13頁，共43頁，2022年，5月20日，9點37分，星期三兩個矢量的矢量積（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 al

10、ong the vector C, and i1, i2, i3 are unit vectors along x1, x2 and x3 .ABC2022/9/1814第14頁，共43頁，2022年，5月20日，9點37分，星期三 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 =

11、1, j = 2, k = 3 (e.g., e132 ) eijk = 0 for two or more indices equal (e.g., e113 ) the components of vector C can be written as利用符號eijk可以方便地表示3階行列式的值2022/9/1815第15頁，共43頁，2022年，5月20日，9點37分，星期三標量三重積（Scalar Triple Product） The scalar triple product or box product A B C is a scalar product of two vector

12、s, in which any vector is a vector product of other two vectors, i.e.2022/9/1816第16頁，共43頁，2022年，5月20日，9點37分，星期三第二節(jié) 笛卡爾張量一、笛卡爾張量的定義一階笛卡爾張量 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 b

13、y andA first-order tensor is nothing more than a vector.和2022/9/1817第17頁，共43頁，2022年，5月20日，9點37分，星期三二階笛卡爾張量 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 equationsandor

14、or2022/9/1818第18頁，共43頁，2022年，5月20日，9點37分，星期三高階笛卡爾張量Third- and higher- order Cartesian tensors are defined analogously. 零階笛卡爾張量A Cartesian tensor of zeroth order is defined to be any quantity that is unchanged under coordinate transformation, that is, a scalar.2022/9/1819第19頁，共43頁，2022年，5月20日，9點37分，星

15、期三 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、笛卡爾張量的運算（Operation of Cartesian Tensors）Addition of Cartesian Tensors The addition or subtraction of two Carte

16、sian tensors of the same order to be a third Cartesian tensor of the same order.2022/9/1820第20頁，共43頁，2022年，5月20日，9點37分，星期三Multiplication of Cartesian Tensors The multiplication of Cartesian tensors can be classified into two categories, outer products and inner products. The outer products of two te

17、nsors 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 of the two; that is, by an equating of any index associate

18、d with one tensor to any index associated with the other. 2022/9/1821第21頁，共43頁，2022年，5月20日，9點37分，星期三二階張量的商規(guī)則（Quotient Rule for Second-Order Tensors） Suppose we know the following equation to apply where Ai denotes components of an arbitrary vector, Bj components of a vector. Then, the quotient rule

19、states the components Tij are indeed the components of a second-order Cartesian tensor. 書上有證明下一章要利用這個法則2022/9/1822第22頁，共43頁，2022年，5月20日，9點37分，星期三一、對稱張量和反對稱張量的定義定義（Definition）第三節(jié) 二階笛卡爾張量 If Tij = Tji , then the tensor is said to be symmetric. On the other hand, if Tij = -Tji , then the tensor is said

20、 to be antisymmetric. 二階張量的九個分量可以用33矩陣表示：2022/9/1823第23頁，共43頁，2022年，5月20日，9點37分，星期三例題2.1 試證明任意二階張量可以表示為對稱張量 和反對稱張量之和證：設Tij 是任意二階張量的分量，則有其中二階對稱張量二階反對稱張量2022/9/1824第24頁，共43頁，2022年，5月20日，9點37分，星期三證：例題2.2 設Aij 是二階對稱張量的分量， Bij 是二階 反對稱張量的分量，試證明關系Aij Bij =0。因為所以所有指標都是啞指標2022/9/1825第25頁，共43頁，2022年，5月20日，9點3

21、7分，星期三反對稱張量的分量（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 denotes components of a vector and if Tij denotes components of a second-order anti-symmet

22、ric tensor, then where Wj denotes vector components defined as2022/9/1826第26頁，共43頁，2022年，5月20日，9點37分，星期三二、對稱張量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 s

23、calar. Any nonzero vector n satisfying this equation is known as unit eigenvector of the tensor and is known as eigenvalue .2022/9/1827第27頁，共43頁，2022年，5月20日，9點37分，星期三Expand the equation and rearranging to get The condition for a nontrivial solution of these homogeneous algebraic equations is that 20

24、22/9/1828第28頁，共43頁，2022年，5月20日，9點37分，星期三Equation yields the cubic equation are called first, second, and third invariant of the tensor T, respectively.where2022/9/1829第29頁，共43頁，2022年，5月20日，9點37分，星期三 When the components Tij are those of a symmetric tensor, it can easily be shown that cubic equation w

25、ill 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 eigenvector associated with the ei

26、gen-value (1) . In a similar way, we may also find two additional unit eigenvectors associated with the eigen-values (2) and (3) .2022/9/1830第30頁，共43頁，2022年，5月20日，9點37分，星期三 The above three unit eigenvectors are mutually perpendicular when (1) , (2) , and (3) are all distinct . Consider two unit eige

27、nvectors n(1) and n(2) . These satisfy equation Multiplying the first of these equations by n(2)i and the second by n(1)i and subtracting, we have2022/9/1831第31頁，共43頁，2022年，5月20日，9點37分，星期三That is On interchanging the dummy indices i and j in the first term on the left-hand side of this equation2022/

28、9/1832第32頁，共43頁，2022年，5月20日，9點37分，星期三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.20

29、22/9/1833第33頁，共43頁，2022年，5月20日，9點37分，星期三三、對稱張量的主軸和主值（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)=i32022/9/18

30、34第34頁，共43頁，2022年，5月20日，9點37分，星期三ij：i=j：非對角線元素為零非零元素在對角線上，就是特征值2022/9/1835第35頁，共43頁，2022年，5月20日，9點37分，星期三 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

31、 values of symmetric tensor T2022/9/1836第36頁，共43頁，2022年，5月20日，9點37分，星期三 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. 2022/9/1837第37頁，共43頁，2022年，5月20日，9點37分，星期三 Consider finally the case where all eigen-values are equal, say, to (1) . We have so that the tensor compo