金属塑性加工原理 - Chapter 1A 数学与力学基础(1hr) - for printer

金属塑性加工原理（第1章补充）数学与力学基础李赛毅中南大学2013-091目的§1.1矢量、张量及其基本运算了解矢量与张量的基本性质及其运算法则：z矢量与张量的表征z矢量代数与张量代数（运算）§1.2变形与应变了解连续介质力学的基本概念z张量：变形梯度、位移梯度、速度梯度z变形z应变z。

23§1.1矢量、张量及其基本运算4矢量¾是从点x 指向点y 的线段，具有方向和长度¾是力、位移等物理量的合适表征方式●矢量：任意两点x 和y 之间的差别●点：对于三维Euclidean (欧几里德) 空间R 3中的元素，其位置可以用Cartesian (笛卡儿) 坐标表示：x {x 1, x 2, x 3} ∈R 3v r x yor v5Two vectors u and v compound according to the parallelogram law (平行四边形法则):w = u + vIf u and v are taken to be the adjacent sides of a parallelogram, then w is the diagonal （对角线）of the parallelogram:矢量的复合（和）：平行四边形法则wvu o 6The magnitude (or length) of a vector v is related to its components (v i ) through:222123i iv v v v v v ≡=++=v 矢量的大小Summation convention (or Einstein Summation):Summation is required over the repeated index (i= 1,2,3…full range) in the same term.Index notationEinstein 求和约定7矢量的标积Scalar product●亦称为：内积(Inner product ）、点积（Dot product)●定义：¾u, v :矢量u 和v 的长度¾Θ : 矢量u 和v 的夹角cos i i uv u v θ⋅==u v ●典型的物理意义:u ⋅v = 0Æ垂直（即正交）u ⋅v =uv Æ平行u ⋅v =−uv Æ反平行●标积的可交换性（commutative ）u ⋅v = v ⋅uv uθ8基矢量Base Vectors {e i }●定义：具有单位长度且彼此正交的一组矢量，亦称单位矢量.●考虑一个以O 为原点的包括3个基矢量{ei } 或{e 1, e 2, e 3}的正交坐标系，则:1e 3e 2e o 克罗内克符号Kronecker delta : δij其分量可以排列成一个3×3 的单位矩阵I （第一个指数为行，第二个指数为列:1 if = 0 if i j ij i ji jδ⎧⋅==⎨≠⎩e e 100010001⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦I9基矢量(2)●The coordinates of a point x can be defined by the scalar product●With the parallelogram law,any arbitrary vector can be expressed as a vector sum （矢量和）of the three base vectors(v 1, v 2, v 3 : components of v )31122331i i i ii v v v v v ==++==∑v e e e e e x1e 3e 2eo x-o()i i x x o =−⋅e vector vector i at point i x x =x e 位置矢量Position vector 10矢积Vector Product (1)●也称作：叉积Cross Product●The vector product of vectors u and v is denoted by u ×v .●It is a vector normal to the plane containing u and v , and oriented in the sense of a right-handed screw rotating from u to v .uvu ×v Its magnitude is equal to the area of a parallelogram two of whose sides are the vectors u and v :| u ×v | = uv sin θvu θ11矢积Vector Product (2)● A convenient expression for vector product in terms of components:ijk i j ku v ε×=u v e 1 if = 123, 312 or 231 (even permutations of 123)-1 if = 132, 213 or 321 (odd permutations of 123)0 if any two of are equalijk ijk ijk ijk ε⎧⎪=⎨⎪⎩No. vector product is NOT commutative !u ×v = −v ×u 12基矢量的手性Handedness of Base Vectors ● A set of orthonormal base vectors can be selected in two ways:(e 1×e 2) ⋅e 3= +1 Æright-handed coordinate basis .(e 1×e 2) ⋅e 3= −1 Æleft-handed coordinate basis .* In general, the triple scalar product in component form is given by()ijk i j ke u v w ×⋅=u v w 1e 3e 2e o(e 1×e 2)1e 3e 2e o (e 1×e 2)13坐标变换(1)●坐标选取的任意性：Many different choices are possible for the orthonormal base vectors and origin of the Cartesian coordinate system. Coordinate systems are usually chosen in different ways for convenience.●坐标变换的必要性：Change of coordinate system will be necessary in connecting quantities represented in different systems.● A vector is independent of the choice of coordinate system:¾Its direction and magnitude must not change. They are “invariants”.¾Its components will change with the choice.14●Consider a new orthonormal system consisting of right-handed base vectors (e’1, e’2, e’3)with thesame origin associated with (e 1, e 2, e 3).●A vector V can be expressed equally well in either coordinate system:V = v i e i = v ’i e ’i Note different values of the components in different coordinate system !What ’s the relationship between the two sets ofcomponents for the same vector?坐标变换(2)e ’1e 2e ’2e3e ’3e 1V15方向余弦Direction Cosines ●The two systems are related by nine direction cosines aij ,i.e. thecosine of the angle between the i -th primed and the j -th unprimed base vectors:In other word, a ij represent the components of e’i in e j 'ij i j a =⋅e e 'i ij ja =e e ●The set of direction cosines are not independent. There are six relationships (i, j = 1, 2, 3) between the nine direction cosines, and therefore only three of them are independent.''()() i j ik k jl l ik jl k lik jl kl ik jk ijaa a a a a a a δδ⋅=⋅=⋅===e e e e e eje 'ie θcos i iuv u v θ⋅==u v 16●The direction cosines can be arranged into a 3x 3 matrix R :●It is an orthogonal （正交）matrix, because each row (or column) is mutually orthogonal to the other two rows (or columns). Each row contains the direction cosines of the new axes in terms of the old axes. Each column contains the direction cosines of the old axes in terms of the new axes.I T =RR 1T −=R R det 1=±R 旋转矩阵Rotation Matrix●Properties of an orthogonal matrix:(:)ik jk ij ki kj ij a a or a a δδ==111213212223313233a a a a a a a a a ⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦R 行列式Determinant:17Inter-relationships between components of vectors in the two systems:j ji i v a v ′=These relations are called the laws of transformation for the components of vectors.矢量变换法则Transformation Law 111121312212223233132333'''v a a a v v a a a v v a a a v ⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥=⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦⎣⎦⎣⎦18小结●掌握矢量的定义、性质和基本运算¾Base vectors 基矢¾Scalar or dot product 标积/点积¾Vector or cross product 矢量积/叉积¾Transformation of axes 坐标变换¾Direction Cosines; Rotation matrix 方向余弦；旋转矩阵●熟悉符号¾Einstein Summation¾Kronecker delta: δij¾permutation （置换）tensor: εijk19张量Tensors●Tensors are mathematical representation of physical entities or phenomena (e.g. Stress, Strain, Elastic Stiffness).●They permit complicated behaviors to be described in a compact fashion that can be easily translated into numerical form. ¾Vectors are tensors of rank 1¾Scalars are tensors of rank 0●A tensor can be written as an organized collection of numbers. In physics, the numbers may be obtained as physical quantities that depend on a basis, and the collection is determined to be a tensor if the quantities transform appropriately under change of basis.20张量积Tensor Product (1)¾若u 和V 为矢量，则w 也称为外积（outer product ）¾所得张量的阶是各张量的阶之和¾对于给定的基矢{e i }:=⊗w u v●张量积, where ij i j ij i ji jw w u v =⊗=⊗=∑w u v e e unit dyads（单位并矢，构造高阶张量时需要）张量积 Tensor Product (2)● Examples of Matrix representation⎡ u1 ⎤ u ⊗ v = ⎢u2 ⎥ [ v1 v2 ⎢ ⎥ ⎢u3 ⎥ ⎣ ⎦ ⎡ u1v1 u1v2 v3 ] = ⎢u2 v1 u2 v2 ⎢ ⎢ u3v1 u3v2 ⎣ u1v3 ⎤ u2 v3 ⎥ ⎥ u3v3 ⎥ ⎦3-dimention, 1st-order 3-dimention, 1st-orderRank of tensor: 1 + 1 Number of components: 3×3u ⎤ ⎡v ⎡u u ⊗ v = ⎢ 11 12 ⎥ ⊗ ⎢ 11 ⎣u21 u22 ⎦ ⎣v212-dimention, 2nd-order 2-dimention, 2nd-order⎡ u11v11 u11v12 v12 ⎤ ⎢ u11v21 u11v22 =⎢ v22 ⎥ ⎢ u21v11 u21v12 ⎦ ⎢ ⎣u21v21 u21v22u12 v11 u12 v12 ⎤ u12 v21 u12 v22 ⎥ ⎥ u22 v11 u22 v12 ⎥ ⎥ u22 v21 u22 v22 ⎦21Rank of tensor: 2 + 2 Number of components: 4×4并矢积 Dyadic Product● Dyadic product is the special case of the tensor product between two vectors of the same dimensionu ⊗ v = (ui ei ) ⊗ (v j e j ) = ui v j ei ⊗ e j (α u) ⊗ v = u ⊗ (α v ) = α ( u ⊗ v ) u ⊗ (v + w) = u ⊗ v + u ⊗ w (u + v ) ⊗ w = u ⊗ w + v ⊗ w● Similar to vector product, outer product is NOT commutative, i.e.Proof:unit dyadsu ⊗ v ≠ v ⊗u22u ⊗ v = (ui ei ) ⊗ (v j e j ) = (ui v j )ei ⊗ e j where wij = ui v j v ⊗ u = (vi ei ) ⊗ (u j e j ) = (vi u j )ei ⊗ e j where wij = vi u jSaiyi Li (Central South University)11并矢举例: 位错滑移● We commonly form a dyad for the strain, m, produced on a slip system by combining unit vectors that represent slip direction b and slip plane normal n.m = b⊗n ( m ij = b i n j )εslip= Δγ m = Δγ b ⊗ nShear strainslip (ε ij = Δγ mij = Δγ bi n j )m － Schmid tensor ε － Strain tensor* 以后会详述23对称与反对称张量● A (2nd-order) tensor is said to be对称 symmetric (e.g. stress, strain tensors), ifT = TT (or Tij = Tji)反对称 anti-symmetric or skew-symmetric (e.g.rotation), if T = − TT (or Tij = − Tji) ● Any tensor can be decomposed （分解）into a symmetric and an anti-symmetric part, i.e. T=S+W where S = (T + TT)/2 W = (T − TT)/224Saiyi Li (Central South University)12张量变换 Tensor Transformations● Transformation of tensors follows the rules set up for vectors and unit vectors. Components change with transformation !′ T = Tij ei ⊗ e j = Trs e'r ⊗ e's = Tij (λri e'r ) ⊗ (λsj e's )Thuse 'i = aij e j or : ei = a ji e ' j= λri λsjTij e'r ⊗ e'sTrs = λ riλ sjTij ′25小结● 掌握张量的定义、性质和基本运算Tensor product, outer product, dyadic prodct 张量 积、外积、 并矢积 Unit dyads 单位并矢 Symmetric and anti-symmetric tensors Tensor transformation Scalar; Vector; Tensor26Saiyi Li (Central South University)13§1.2 变形与应变27目的§1.1 矢量、张量及其基本运算了解矢量与张量的基本性质及其运算法则：矢量与张量的表征 矢量代数与张量代数（运算）§1.2 变形的一般描述了解连续介质力学的基本概念张量：变形梯度、位移梯度、速度梯度 变形 应变。