Heat Treatment of Steel

Heat Treatment of Steel

• Heat treatment involves heating and cooling metals to alter their physical and mechanical properties without changing the object's shape.

Purposes of Heat Treatment

• It can increase hardness, relieve stress, improve machinability, soften metal, improve wear resistance, refine grain size, improve ductility and improve overall toughness.

Theory of Heat Treatment

• Properties of steel change with temperature.
• When heated, the atoms become more mobile allowing them to rearrange.
• Crystal structure depends on temperature and steel composition.
• Ferrite has a body-centered cubic (BCC) structure, making it soft and ductile.
• Austenite has a face-centered cubic (FCC) structure, harder and stronger than ferrite.
• Martensite has a body-centered tetragonal (BCT) structure, very hard and brittle.
• Cementite is a compound of iron and carbon (Fe3C).
• Rapid cooling (quenching) can trap non-equilibrium crystal structures, like martensite.
• Slow cooling (annealing) allows atoms to rearrange into stable structures like ferrite.

Effects of Heat Treatment

• Effects include increased hardness, strength, ductility, and toughness.
• It also leads to improved machinability and wear resistance.
• Stress is reduced and grain size is refined.

Types of Heat Treatment Processes

• Annealing: Metal heated to a specific temperature and cooled slowly to soften it, relieve stress, improve machinability and refine grain size.
• Normalizing: Metal heated above its upper critical temperature and cooled in air to refine grain size, improve machinability, and reduce stress.
• Hardening: Metal heated above its upper critical temperature, then rapidly cooled (quenched) in water or oil to increase hardness and strength.
• Tempering: Hardened metal heated below its lower critical temperature to reduce brittleness and improve toughness.
• Case Hardening: Hardens the surface of a metal part while leaving the core soft, enhancing wear resistance.

Methods of Quenching

• Water Quenching: very effective for hardening but may cause cracking.
• Oil Quenching: less effective than water but less likely to cause cracking.
• Air Quenching: least effective but poses the lowest risk of cracking.
• Brine Quenching: more aggressive water quenching for achieving maximum hardness in certain alloys.

Applications of Heat Treatment

• Used in manufacturing tools and dies.
• Employed in automotive, aerospace, construction, medical, and machinery parts manufacturing.

Matrices

• A matrix is a rectangular array of numbers, either real or complex.
• Denoted as $A = \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \ a_{21} & a_{22} & \cdots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix}$.

Basic Matrix Definitions

• An $m \times n$ matrix has $m$ rows and $n$ columns.
• $a_{ij}$ represents the element in the $i$-th row and $j$-th column.
• A row vector is a $1 \times n$ matrix.
• A column vector is an $m \times 1$ matrix.
• A square matrix has equal number of rows and columns ($m = n$).
• Diagonal elements are $a_{11}, a_{22}, \dots, a_{nn}$.
• The trace of a matrix is the sum of its diagonal elements: $\text{tr}(A) = \sum_{i=1}^n a_{ii}$.
• An upper triangular matrix has $a_{ij} = 0$ for $i > j$.
• A lower triangular matrix has $a_{ij} = 0$ for $i < j$.
• A diagonal matrix has $a_{ij} = 0$ for $i \neq j$.
• The identity matrix is a diagonal matrix with all diagonal elements equal to 1.
• In the identity matrix, $I = \begin{bmatrix} 1 & 0 & \cdots & 0 \ 0 & 1 & \cdots & 0 \ \vdots & \vdots & \ddots & \vdots \ 0 & 0 & \cdots & 1 \end{bmatrix}$.
• A zero matrix has all elements equal to zero.

Matrix Operations

• Addition: $C = A + B$, where $c_{ij} = a_{ij} + b_{ij}$ (matrices $A$ and $B$ must have the same dimensions).
• Scalar Multiplication: $B = \alpha A$, where $b_{ij} = \alpha a_{ij}$.
• Matrix Multiplication: $C = AB$, where $c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}$ ($A$ is $m \times n$, $B$ is $n \times p$, $C$ is $m \times p$).
• Transpose: $B = A^T$, where $b_{ij} = a_{ji}$.
• Conjugate Transpose (Hermitian Transpose): $B = A^H$, where $b_{ij} = \overline{a_{ji}}$.
• For a real matrix $A$, $A^H = A^T$.

Special Matrices

• Symmetric Matrix: $A^T = A$.
• Hermitian Matrix: $A^H = A$.
• Skew-Symmetric Matrix: $A^T = -A$.
• Skew-Hermitian Matrix: $A^H = -A$.
• Orthogonal Matrix: $A^T = A^{-1}$ (for real matrices).
• Unitary Matrix: $A^H = A^{-1}$ (for complex matrices).
• Normal Matrix: $A A^H = A^H A$.
• Positive Definite Matrix: $x^H A x > 0$ for all non-zero vectors $x$.
• Positive Semi-Definite Matrix: $x^H A x \ge 0$ for all vectors $x$.

Properties of Matrix Operations

• $(A + B)^T = A^T + B^T$
• $(\alpha A)^T = \alpha A^T$
• $(AB)^T = B^T A^T$
• $(A + B)^H = A^H + B^H$
• $(\alpha A)^H = \overline{\alpha} A^H$
• $(AB)^H = B^H A^H$
• $(A^{-1})^{-1} = A$
• $(A^T)^{-1} = (A^{-1})^T$
• $(A^H)^{-1} = (A^{-1})^H$

Determinant of a Matrix

• The determinant is a scalar value computed from the elements of a square matrix.
• For a $2 \times 2$ matrix: $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, $\det(A) = ad - bc$.
• For a $3 \times 3$ matrix: $A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}$, $\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$.

Properties of Determinants

• $\det(A^T) = \det(A)$
• $\det(AB) = \det(A) \det(B)$
• $\det(A^{-1}) = \frac{1}{\det(A)}$
• If $A$ has a row or column of zeros, $\det(A) = 0$.
• If $A$ has two identical rows or columns, $\det(A) = 0$.
• If $A$ is upper or lower triangular, $\det(A)$ is the product of the diagonal elements.

Singular and Non-Singular Matrices

• A square matrix $A$ is singular if $\det(A) = 0$.
• A square matrix $A$ is non-singular if $\det(A) \neq 0$.

Algorithmic Complexity

• Algorithmic complexity measures the resources needed to run an algorithm, usually in terms of input size.
• Landau notation (Big O notation) describes the asymptotic behavior of a function.

Landau Notation Definition

• $f(n) = O(g(n))$ if there are constants $c > 0$ and $n_0 \in \mathbb{N}$ such that $\forall n \geq n_0, \quad f(n) \leq c \cdot g(n)$.
• $f(n) = O(g(n))$ means $f(n)$ grows no faster than $g(n)$ as $n$ approaches infinity.

Landau Notation Properties

• $O(f(n)) + O(g(n)) = O(\max(f(n), g(n)))$
• $O(c \cdot f(n)) = O(f(n))$ where c is constant
• $O(f(n)) \cdot O(g(n)) = O(f(n) \cdot g(n))$

Complexity Examples

• $O(1)$: constant complexity
• $O(\log n)$: logarithmic complexity
• $O(n)$: linear complexity
• $O(n \log n)$: quasi-linear complexity
• $O(n^2)$: quadratic complexity
• $O(n^3)$: cubic complexity
• $O(2^n)$: exponential complexity
• $O(n!)$: factorial complexity

Classical Sorting Algorithms

• Sorting algorithms are methods used to organize data in a specific order.

Insertion Sort

• Scan the array from left to right and insert each element into its correct position within the sorted part of the array to the left.
• Worst case complexity: $O(n^2)$.
• Best case complexity: $O(n)$.
• Average complexity: $O(n^2)$.

Selection Sort

• Find the smallest element in the array and exchange it with the first element, then repeat the process with the rest of the array.
• Worst case complexity: $O(n^2)$.
• Best case complexity: $O(n^2)$.
• Average complexity: $O(n^2)$.

Bubble Sort

• Repeatedly compare adjacent pairs and swap them if they are in the wrong order, iterating through the array until it is sorted.
• Worst case complexity: $O(n^2)$.
• Best case complexity: $O(n)$.
• Average complexity: $O(n^2)$.

Quicksort

• Uses a "divide and conquer" approach.
• Choose a pivot element, partition the array into two sub-arrays (elements less than and greater than the pivot), and recursively sort the sub-arrays.
• Worst case complexity: $O(n^2)$.
• Best case complexity: $O(n \log n)$.
• Average complexity: $O(n \log n)$.

Mergesort

• Uses a "divide and conquer" approach.
• Divide the array into two halves, recursively sort the halves, and then merge the sorted halves into a single sorted array.
• Worst case complexity: $O(n \log n)$.
• Best case complexity: $O(n \log n)$.
• Average complexity: $O(n \log n)$.

Sorting Algorithm Comparison

Bernoulli's Principle

• An increase in fluid's speed occurs simultaneously with a decrease in pressure or a decrease in fluid's potential energy.

Key Concepts

• Pressure: Force exerted per unit area.
• Kinetic Energy: Energy possessed by an object due to its motion.
• Potential Energy: Energy possessed by an object due to its position, internal stresses, charge, etc.

Bernoulli's Equation

• $P + \frac{1}{2} \rho v^2 + \rho g h = constant$
• P is static pressure
• $\rho$ is fluid density
• v is fluid velocity
• g is acceleration due to gravity
• h is elevation

Bernoulli's Terms

• Static Pressure ($P$): Pressure exerted by fluid while not moving.
• Dynamic Pressure ($\frac{1}{2} \rho v^2$): Pressure from the kinetic energy of the fluid.
• Hydrostatic Pressure ($\rho g h$): Pressure exerted by fluid due to gravity.

Application of Bernoulli's Principle: Airplane Wing

• Airplane wing shape causes air above to move faster than air below.
• Faster air above creates lower pressure than slower air below.
• Pressure difference generates lift, allowing the plane to fly.

Application of Bernoulli's Principle: Venturi Meter

• Venturi meter measures flow rate of fluid in a pipe.
• Constricted section speeds up fluid, decreases pressure.
• Pressure difference determines flow rate.

Functions of One Variable

• Definition 2.1.1: A function from set $E$ to $F$ associates each element of $E$ with a unique element of $F$.
• $f: E \rightarrow F$
• $x \mapsto f(x)$
• $E$: domain of definition of $f$.
• $F$: codomain.
• $f(x)$: image of $x$ by $f$.

Basic Function Definitions

• Graph: set of ordered pairs $(x, f(x))$ for all $x \in E$.
• $\operatorname{Gr}(f)={(x, f(x)) \mid x \in E} \subset E \times F$
• Image: set of all values attained by $f$.
• $\operatorname{Im}(f)={f(x) \mid x \in E} \subset F$
• Inverse Image: If $B \subset F$, the inverse image of $B$ by $f$ is the set of all elements in $E$ whose image lies in $B$.
• $f^{-1}(B)={x \in E \mid f(x) \in B} \subset E$

Function Example

• Exemple 2.1.1:
• $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto x^2$.
• $\operatorname{Gr}(f)={(x,x^2) \mid x \in \mathbb{R}}$.
• $\operatorname{Im}(f)=\mathbb{R}^+=[0,+\infty[$.
• $f^{-1}({4})={-2,2}$.
• $f^{-1}([0,9])=[-3,3]$.

Types of Functions

• Injective: Every element of $F$ has at most one antecedent in $E$.
• $\forall x_1, x_2 \in E, f(x_1)=f(x_2) \Rightarrow x_1 = x_2$.
• Surjective: Every element of $F$ has at least one antecedent in $E$.
• $\forall y \in F, \exists x \in E \mid f(x)=y$.
• Bijective: Every element of $F$ has exactly one antecedent in $E$.
• $\forall y \in F, \exists! x \in E \mid f(x)=y$.
• $f$ is bijective $\Leftrightarrow$ $f$ is injective and surjective.

Additional Remarks

• If $f: E \rightarrow F$ is bijective, there exists an inverse function $f^{-1}: F \rightarrow E$ such that $f^{-1}(f(x))=x$ for all $x \in E$ and $f(f^{-1}(y))=y$ for all $y \in F$.

Example Function Classifications

• Exemple 2.1.2:
• $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto x^2$ is neither injective nor surjective.
• $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+, x \mapsto x^2$ is bijective.