1、Engineering Solid Mechanics,Chapter 2 Mathematical Preliminaries （數(shù)學(xué)基礎(chǔ)）,Xiaojun Yan 閆曉軍 School of Jet Propulsion動(dòng)力學(xué)院,Content（內(nèi)容）,2,Page 1,Introduction Theory of elasticity Mathematical Preliminaries Analysis of Stress Analysis of Strain Elastic Stress-strain Relations Solution of the Elastic Problem

2、 Continuum plasticity and Microplasticity Characteristics and modeling of uniaxial behavior Deformation Yield criteria Plastic stress-strain relations Microplasticity Kinematics of large deformations and continuum mechanics,2.1Scalar （標(biāo)量） 2.2Vector （矢量） 2.3Indicial Notation and Summation Convention

3、（指標(biāo)記法與求和約定） 2.4Coordinate Transformations （坐標(biāo)變換） 2.5Tensors（張量） 2.6Kronecker Delta Symbol（克羅內(nèi)克符號(hào)） 2.7Alternating Symbol（交錯(cuò)張量） 2.8Salar Product,Vector Product an Triple Product （標(biāo)量積矢量積和三重積） 2.9Scalar Vector Fields（標(biāo)量場(chǎng)和矢量場(chǎng)） 2.10 Divergence Theorem (散度定理),2,Chapter 2 Mathematical Preliminaries,Page 1,2

4、.1Scalar （標(biāo)量）,2,material density （材料的密度） Youngs modulus E （楊氏模量 ） Poissons ratio （泊松比 ） shear modulus G. （剪切彈性模量 ）,Examples of scalar（標(biāo)量）：,Page 2,2,2.2Vector（矢量）,displacement of material points （位移） rotation of material points （轉(zhuǎn)角）,Examples of vector（矢量）：,Page 3,2.2Vector（矢量）,？,2,Examples of vector（

5、矢量）：,Page 4,2.3 Indicial Notation and Summation Convention（指標(biāo)記法與求和約定）,2,Indicial Notation and Summation Convention,Page 5,2,Indicial Notation and Summation Convention,2.3 Indicial Notation and Summation Convention（指標(biāo)記法與求和約定）,Page 6,2,Indicial Notation and Summation Convention,2.3 Indicial Notation a

6、nd Summation Convention（指標(biāo)記法與求和約定）,Page 7,Differentiation Notation(微分的記法),Use a comma to indicate differentiation（用逗號(hào)表示微分）,2,2.3 Indicial Notation and Summation Convention（指標(biāo)記法與求和約定）,Page 8,Hamilton算子 讀音：納普拉(Nabla) 代爾（del）,矢量。,2.4 Coordinate Transformation（坐標(biāo)變換）,It is convenient and in fact necessar

7、y to express elasticity variables and field equations in several different coordinate systems. The situation requires the development of particular transformation rules for scalar, vector, and high-order variables.,2,Page 9,2,2.4 Coordinate Transformation（坐標(biāo)變換）,Page 10,2,An arbitrary vector ：,2.4 Co

8、ordinate Transformation（坐標(biāo)變換）,Page 11,2.4 Coordinate Transformation（坐標(biāo)變換）,Example:,2,Page 12,2.4 Coordinate Transformation（坐標(biāo)變換）,2,Page 13,Example:,2.4 Coordinate Transformation（坐標(biāo)變換）,Note:,Above relations constitute the transformation laws for the Cartesian components of a vector under a change of

9、coordinate frame The vector is unaltered, and only its components are changed.,2,Page 14,2.5 Tensors （張量）,2,Definition of Tenors,Page 15,2.5 Tensors （張量）,Distinction between the components and the tensor,2,Page 16,2,2.5 Tensors （張量）,Tesnor Equations (張量相等):,Addition （張量相加)：,Properties of tensors.,Pa

10、ge 17,2,Multiplication. (張量相乘)：,2.5 Tensors （張量）,Tensor Equation. (張量方程)：,Properties of tensors.,Page 18,Constraction. (張量縮并)：,2.5 Tensors （張量）,2,Properties of tensors.,Page 19,27 quantities,3 quantities,Third order,First order,2.5 Tensors （張量）,2,Example:,Properties of tensors.,Page 20,Symmetry. (對(duì)稱

11、張量)：,Skew Symmetry. (斜對(duì)稱張量)：,2.5 Tensors （張量）,2,Properties of tensors.,Page 21,Isotropic Tensors. (各向同性張量)： Components have the same value in all coordinate systems.,2.5 Tensors （張量）,Quotient Rule. (商法則)：,2,Properties of tensors.,Page 22,For any tensor b, c is a scalar, then a is a tensor,2.5 Tensor

12、s （張量）,Quotient Rule. (商法則)：,Alternative Theorems (交錯(cuò)法則)： If,Page 23,is a vector for any vector u; OR,is a tensor for any tensor b;OR,is a scalar for any vectors u,Then a are tensors,2,2.6 Kronecker Delta Symbol（克羅內(nèi)克符號(hào)）,Page 24,2,2.7 Alternating Symbol（交錯(cuò)張量）,Page 25,2,2.8 Scalar Product, Vector Prod

13、uct an Triple Product (標(biāo)量積矢量積和三重積),The scalar product of vectors a and b is defined as :(矢量a和b的標(biāo)量積定義為),The scalar product. (標(biāo)量積)：,Page 26,The vector product of vectors a and b is defined as :(矢量積定義為),2,2.8 Scalar Product, Vector Product an Triple Product (標(biāo)量積矢量積和三重積),The vector product. (矢量積)：,Page

14、27,2,The form of Triple product (三重積),2.8 Scalar Product, Vector Product an Triple Product (標(biāo)量積矢量積和三重積),The triple product. (三重積)：,Page 28,2,2.9 Scalar field and vector field （標(biāo)量場(chǎng)和矢量場(chǎng)）,Gradient of a scalar field .(標(biāo)量場(chǎng)的梯度), is treated as an operator .(作為一個(gè)算子)：,Page 29,Vector: 在該點(diǎn)變化率最大的方向，及變化率值,2,2.9S

15、calar field and vector field （標(biāo)量場(chǎng)和矢量場(chǎng)）,Divergence of a vector .(矢量場(chǎng)的散度),The dot product of the operator with a vector defines the divergence of that vector .(算子與一個(gè)矢量的點(diǎn)積定義為這個(gè)矢量的散度)。,Page 30,Scalar: 矢量場(chǎng)中一點(diǎn)通量對(duì)體積的變化率，源的強(qiáng)度,2,2.9 Scalar field and vector field （標(biāo)量場(chǎng)和矢量場(chǎng)）,Curl of a vector .(矢量場(chǎng)的旋度),The corss product of operator and a vector is defined as curl.(算子與一個(gè)矢量的叉積定義為這個(gè) 矢量的散度),Page 31,Vector: 最大的環(huán)量面密度,Divergence T