Math Review: Vectors and Tensors for Rheological Applications Presented by Randy H. Ewoldt University of Illinois at Urbana-Champaign U. of Minnesota Rheological Measurements Short Course June 2016 MathReview-1
The 4 Key Phenomena of Rheology * 1. Viscoelasticity 2. Shear-Thinning 3. Extensional-Thickening 4. Rod-Climbing * Macosko (1994) Photos & Videos from the Ewoldt Research Group (YouTube.com)
Key Question: What happens when I push on it? (to keep track, we will need vectors & tensors) Some examples you will see this week velocity field rate of strain v v v v x y z γ v v T stress σ xx xy xz Force j ij yx yy yz Area i zx zy zz Image: wikipedia.org MathReview-3
Why we're here Objective: help you survive the mathematical parts of the course involving vectors, tensors, and their operations. Approach: Big idea: keep info, use in equations Review details: notation, gradient, products. MathReview-4
Keep info, use in equations. MathReview-5
Three Basic Concepts Scalar: has magnitude but no direction Examples: Density (r), temperature (T), pressure (p), and speed (magnitude of velocity). Scalars are represented by plain-font letters. Vector: has magnitude and one direction. Examples: Velocity (v), displacement (u), force (f), and temperature gradient (T). Vectors are represented by lower-case boldface letters. TENSOR: has magnitude and two directions. Examples: Stress (T, t), strain (B, C), velocity gradient (v), and displacement gradient (F = u). Tensors are represented by upper-case boldface letters. MathReview-6
Scalars Although they may vary with position, they are defined by a single number. T T( x, y) For convenience, consider temperature. In the plot: red = high blue = low We can all look at the picture and tell where hot and cold are. earth.nullschool.net (June 16, 2016, 7:00pm) at start of math review MathReview-7
Vectors Scalar plots do not completely describe vector fields Note that other possible fields would have the same speed at each point. Speed Velocity earth.nullschool.net (June 16, 2016, 7:00pm) at start of math review MathReview-8
Representing Vectors In three-space, a vector can be decomposed into three unit cardinal components (e i ) If you are traveling 2 mph east, 3 mph north, and 0 mph up, your velocity is exactly determined. It is often convenient to express vectors as ordered triples. Likewise, we will often refer to components by subscript. x 3 e 1 e 3 e 2 v x 2 v 2mph v1 3mph v 2 0 v 3 mph mph 2 e 3 e 0e 1 2 3 x 1 MathReview-9
Magnitude of a Vector an invariant independent of direction The magnitude of a vector is often a useful quantity (speed = magnitude of velocity) This is easily attained by the Pythagorean theorem. v v 3 v 2 v 1 v v 1 2 v 2 2 v 3 2 MathReview-10
The Gradient: a special vector Suppose that I have a multi-dimensional (e.g., x,y) scalar field (such as temperature). I would like to be able to define how it changes. T T( x, y) represents change MathReview-11
The Gradient The gradient of the scalar is a vector field that captures the change of the scalar with respect to position. The gradient's direction is that in which the field changes fastest. The gradient's magnitude is equal to the derivative of the scalar field in the direction of maximum change. MathReview-12
A Field and Its Gradient Note that the arrows point towards black (high) regions. At the maximum (or minimum), the gradient is zero. The gradient is also zero at the saddle point. MathReview-13
A Field and Its Gradient Maxima are marked by stars Saddles are marked by circles (on the 3D plot only) Consider the point with the triangle... MathReview-14
The Gradient Vector T T T T ex e y ez grad( T ) x y z T x T y T z nabla, del, or directional derivative : i x i e i e e e x y z x y z MathReview-15
The Gradient Vector T T T T ex e y ez x y z T x T y T z grad( T ) Gradient = vector of slopes in each direction Gradient of scalar Field: must consider slopes in 3 directions nabla, del, or directional derivative : i x i e i e e e x y z x y z MathReview-16
Exercise If p = 2x 2 y-z, what is p? MathReview-17
Adding Vectors Two vectors may be added by placing them "tail to head" or by adding their respective components. v+w w v w v i i v w i v1 w1 v1 w1 v v ; w w ; v w v w 2 2 2 2 v3 w3 v3 w3 MathReview-18
Exercise If v = 1 2 and w = 3 3-2, what is v + w? 0 MathReview-19
Vector Multiplication Scalar and Vector: just what you might guess (next page) Vector and Vector: 3 different options Dot product (reduce complexity): vector vector = scalar Cross product (maintain complexity): vector vector = vector Dyadic product (increase complexity): vector vector = TENSOR MathReview-20
Scalar times Vector A vector is multiplied by a scalar by multiplying ( scaling ) each component of the vector. v1 rv1 rv r v rv 2 2 v r 3 v3 MathReview-21
Dot Product of Two Vectors The dot (or scalar, or inner) product of two vectors is a scalar Interpretation: measure of the directional overlap between two vectors. Calculation: summation of the products of corresponding components of each vector v w v w v w v w v w i v v v v v i 2 2 2 1 2 3 i 1 1 2 2 3 3 v 2 v Dot product: multiply like components add up results q w MathReview-22
Physical Meaning of the Dot Product The dot product can also be expressed in terms of physical or geometric features of the two vectors. v q v w v w cosq w The dot product of two perpendicular vectors is zero. The dot product of two parallel vectors is the product of their magnitudes. MathReview-23
Physical Meaning of the Dot Product The dot product also gives the projection of one vector onto another. v q x w x v cosq v w w extract information: how much of v is in w direction? MathReview-24
A Note on Unit Vectors The vectors e 1, e 2, and e 3 are all of unit length and are all perpendicular to each other. Then x 3 e 3 e 2 x 2 ei ej ij 1 0 ij i j i j Kronecker delta e 1 x 1 MathReview-25
v w v 1 e 1 v 2 e 2 v 3 e 3 w 1 e 1 w 2 e 2 w 3 e 3 v 1 w 1 e 1 e 1 v 1 w 2 e 1 e 2 v 1 w 3 e 1 e 3 v 2 w 1 e 2 e 1 v 2 w 2 e 2 e 2 v 2 w 3 e 2 e 3 v 3 w 1 e 3 e 1 v 3 w 2 e 3 e 2 v 3 w 3 e 3 e 3 v 1 w 1 v 2 w 2 v 3 w 3 MathReview-26
v w v 1 e 1 v 2 e 2 v 3 e 3 w 1 e 1 w 2 e 2 w 3 e 3 v 1 w 1 e 1 e 1 v 1 w 2 e 1 e 2 v 1 w 3 e 1 e 3 1 v 2 w 1 e 2 e 1 v 2 w 2 e 2 e 2 v 2 w 3 e 2 e 3 1 v 3 w 1 e 3 e 1 v 3 w 2 e 3 e 2 v 3 w 3 e 3 e 3 1 v 1 w 1 v 2 w 2 v 3 w 3 MathReview-27
Vector Einstein Notation v w v e v e v e w e w e w e 1 1 2 2 3 3 1 1 2 2 3 3 3 3 i i j j i1 j1 v i i j j vw i j i j vw i j ij vw i e v i e w e e e w e v w v w v w 1 1 2 2 3 3 Einstein summation convention a notational convention that implies summation, thus achieving notational brevity. When an index variable appears twice in a single term it implies summation of that term over all the values of the index. MathReview-28
Recap: Scalars and Vectors Some things are naturally represented by vectors (velocity, force, displacement) Gradient, : a special vector Direction of change Operation adds directionality (scalar) = vector (vector) = TENSOR Multiplication Dot product (reduce complexity): vector vector = scalar Cross product (maintain complexity): vector vector = vector Dyadic product (increase complexity): vector vector = TENSOR MathReview-29
------------- BREAK ------------- Keep info, use in equations. MathReview-30
Tensors A TENSOR contains information about two different directions simultaneously. Example: force acting on a surface the stress tensor tells what force (which has direction) acts on a surface facing in a specific direction. f n Example: the gradient of velocity tells how the velocity vector changes when one moves in some direction (not necessarily the direction of the velocity). MathReview-31
Ways to build a Tensor: Gradient, : a special vector Direction of change Operation adds directionality (scalar) = vector (vector) = TENSOR Multiplication Dot product (reduce complexity): vector vector = scalar Cross product (maintain complexity): vector vector = vector Dyadic product (increase complexity): vector vector = TENSOR MathReview-32
Gradient of Velocity: two directions The velocity gradient has two sets of directional information: The direction of the velocity (black) The direction of the observer (orange or green) Note that the change in velocity with respect to position depends on the direction of the change. MathReview-33
Gradient of Velocity: two directions The velocity gradient has two sets of directional information: The direction of the velocity (black) The direction of the observer in each (orange direction or green) Note that the change in velocity with respect to position depends on the direction of the change. Gradient = vector of slopes Gradient of Vector Field: must consider 3 components, each with slopes in 3 directions; 3x3=9 things MathReview-34
Rheology Short Course: Math Review Sunday, June 19, 2016 Given: simple shear flow velocity field Find: velocity gradient tensor Solution: I only have velocity in x-direction It changes linearly with y-coordinate location It cares about y-location in the goo Example_SimpleShear_Rheology Short Course Math Review Page 1
Tensor Notation and the Dyad Product The dyad product is normally written simply by writing the two vectors next to each other (e.g., e 1 e 2 ) We will often express a tensor as a matrix. We will often refer to the components by double subscript. T 1e e 3e e 0e e 2e e 7e e 4e e 1 1 1 2 1 3 2 1 2 2 2 3 5e e 6e e 9e e i 3 1 3 2 3 3 j T ee ij i j 1 3 0 T11 T12 T13 2 7 4 T T T 21 22 23 5 6 9 T31 T32 T 33 Note: Tensors, like vectors, are added componentwise Dyad product style bookkeeping of tensor information Linear algebra (matrix) style bookkeeping of tensor information MathReview-35
Some terms The transpose T T of a matrix (or tensor) T is defined by [T T ] ij = T ji. T11 T12 T13 T11 T21 T31 T T T21 T22 T 23 T12 T22 T T 32 T T T T T T 31 32 33 13 23 33 A matrix (or tensor) is said to be symmetric if it is equal to its own transpose, i.e. T T T or TT ij ji MathReview-36
Tensor-Vector Multiplication For this course, when we multiply TENSORS we will always preserve or reduce complexity. We will use dot products. The same argument about unit vectors can be used to define the dot product of a tensor and a vector by using the dyadic notation. For example, and e 1 e 2 e 2 = e 1 (e 2 e 2 ) = e 1 e 1 e 2 e 1 = e 1 (e 2 e 1 ) = 0. Note that the operation is not symmetric: e 1 e 2 e 2 e 2 e 1 e 2.. MathReview-37
T v T v i ij j j Matrix multiplication (linear algebra) T T T 11 12 13 v 1 T v T 21 T 22 T 23 v 2 T T T 31 32 33 v 3 MathReview-38
Einstein notation 3 3 3 T v T e e v e ij i j k k i1 j1 k 1 T ij i j k k Tv ij k i j k Tv ij k i jk Tv e e ij j i 3 3 e e e e i1 j1 v e Tv e e ij j i MathReview-39
Exercise 1 2 1 1 If T= -1-1 -1 and = -2 v, T v=? 0 0 3 1 MathReview-40
Matrix-Matrix Multiplication 3 3 3 3 T A A e e B e e ij i j kl k l i1 j1 k 1 l1 A e e ij i j kl k l AB ij kl i j k l AB ij kl i jk l AB e e e ee ij jl i l 3 3 3 B i1 j1 l1 e e e e e AB ee ij jl i l MathReview-41
Exercise cosq sinq 0 T If F= sinq cosq 0, B= F F? 0 0 1 MathReview-42
Important Note Matrix-vector multiplication is not necessarily symmetric... T v v T Linear algebra tells us that v T = T T v Thus, if T is symmetric (i.e., T ij = T ji ), we can be sloppy and get away with it. MathReview-43
The Dot Product as a Machine Vector/Vector The dot product extracts information of reduced complexity Recall: The gradient vector tells us the change of a scalar field in each coordinate direction Now: we may want to know how the scalar field varies in any given direction, n: direction goes in (n) p Dot product: how much in this direction? vector vector = scalar n p p n Extract out the gradient in that given direction, n MathReview-44
A Field and Its Gradient e.g. pressure: black = high white = low Look at the point marked with the red circle. If you moved to the left, the pressure would increase (n left p > 0). If you moved to the right, it would decrease (n right p < 0). If you moved up or down, it would not change (n up p = 0). MathReview-45
The Dot Product as a Machine Vector/Tensor direction goes in (n) vector TENSOR = vector T n T t n Dot product: how much in this direction? (or on this surface) Extract out information in that given direction (here n represents the surface of interest) MathReview-46
Recall: Tensor Notation T T T T T T T T T T 11 12 13 21 22 23 31 32 33 3 3 i1 j1 e e T i j ij i surface identification j direction of stress MathReview-47
The Dot Product as a Machine Vector/Tensor Suppose that we want to know the traction (vector) that a moving fluid exerts on a surface (defined by n), t n. The stress tensor T keeps track of all stresses in all directions. Use the dot product as a machine to extract out the traction on a given surface with normal direction, n t n n T 3 t e T T e 1 1 1i i1 T 3 i1 e t i i i MathReview-48
One Step Beyond Suppose that we want to know the amount of traction acting in the normal direction (scalar). Extract this information (scalar) from the traction vector with another dot product t nt n nt n n MathReview-49
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What have we learned? The three fundamental classes are Scalars (magnitude but no direction) vectors (magnitude and one direction) TENSORS (magnitude* and two directions) A scalar, such as temperature or pressure, Is entirely defined by a single number A vector, such as pressure gradient (p) or velocity (v), Can always be decomposed into three components in three cardinal directions Addition: add the respective components The magnitude of a vector is equal to the square root of the sum of the squares of the components (Pythagorean theorem) MathReview-53
What have we learned? A TENSOR, such as velocity gradient (v) or stress (T), Can be decomposed into nine components Is often expressed as a matrix in terms of those components Can be added to another tensor by adding the respective components. Vectors and tensors do the book-keeping in the mathematically appropriate way Dot products are mathematical machines to extract information vector vector = scalar vector TENSOR = vector MathReview-54
Recall objective Objective: help you survive the mathematical parts of the course involving vectors, tensors, and their operations. Approach: Big idea: keep info, use in equations Review details: notation, gradient, products. Key Question: What happens when I push on it? (solid, liquid, things in-between) Neo-Hookean elastic solid Newtonian viscous fluid σ GB - I σγ stress tensor strain tensor (so-called Finger tensor) rate-of-strain tensor (notation [2D] also used) MathReview-55