Introduction to Stress and Strain PDF

# Different Types of Force (Load) on a Component ## 'Stress' and 'Strain' ### 'Stress' - When a material is subjected to an external force, a RESISTING FORCE is set up within the component. - **The Internal Resisting Force per Unit Area acting on a component or intensity of the forces distributed...

# Different Types of Force (Load) on a Component ## 'Stress' and 'Strain' ### 'Stress' - When a material is subjected to an external force, a RESISTING FORCE is set up within the component. - **The Internal Resisting Force per Unit Area acting on a component or intensity of the forces distributed over a given section is called 'Stress'.** - Formula: $σ = \frac{P}{A}$ - **Units:** P is expressed in Newton (N) and A, original area, in square meters (m²), the stress σ will be expresses in N/m². This unit is called Pascal (Pa). # Various Types of Stresses 1. Simple or direct stress - Tension - Compression - Shear 2. Indirect stress - Bending - Torsion 3. Combined stress: Any possible combination of types 1 and 2. ## (A) Direct Stresses - Simple stress is often called direct stress because it develops under direct loading conditions. - Simple tension and simple compression occur when the applied force, called load, is in line with the axis of the member (axial loading) ### 1. Tensile Stress ($σ_T$) - When a component is subjected to Tensile loading, stress developed within the material is called Compressive Stress. ### 2. Compressive Stress ($σ_C$) - When a component is subjected to compressive loading, stress developed within the material is called Compressive Stress. ### 3. Shear Stress (τ) - When equal, parallel, and opposite forces tend to cause a surface to slide relative to the adjacent surface. ## (B) Indirect Stresses - In certain loading situations, stresses developed are not simple stresses. ### 1. Bending Stress - When a member is subjected to the load, which is acting perpendicular to the axis of member. - This loading will cause bending of the member, resulting in deformation of member, and development of bending stresses within the member. ### 2. Torsional Shear Stress (τ) - When a member is subjected to the twisting loading, Torsional shear stresses will developed within the member. # 'Strain' - Strain is the measure of the deformation produced in a member by the stress. ### Tensile Strain ($ε_T$) - The elongation per unit length is known as tensile strain. - Formula: $ε_T = \frac{ΔL}{L_0}$ . - It is engineering strain. ### Compressive Strain ($ε_C$) - If the applied force is compressive then the reduction of length per unit length is known as compressive strain. - Formula: $ε_C = \frac{(-ΔL)}{L_0}$. ### Shear Strain - In case of shear loading, a shear strain will be produced, which is measured by angle through which body distorts. - Formula: $ε_s = \frac{NN'}{NP} = tan(φ)$ - Since, φ is very small. ### Volumetric Strain - **Defined as**: Ratio of change in volume to the original volume of the body. - Formula: $ε_ν = \frac{Change in volume }{Original volume} = \frac{δV}{V}$ # 'Engineering Stress' vs. 'True Stress' ## 'Engineering' Stress - It is the ratio of resisting force to the original (initial) cross-sectional area of the component. - Formula: $Engineering Stress = s = \frac{P}{A_0}$ - Where, - P = Load - $A_0$ = Initial (Original) cross-sectional area ## 'True' Stress - It is the ratio of resisting force to the Instantaneous Cross-sectional area of the component. - Formula: $True Stress = σ = \frac{P}{A_i}$ - $A_i$ = Instantaneous cross-sectional area # 'Engineering Strain' vs. 'True Strain' ## 'Engineering' Strain - Ratio of Change in length to original length. - Formula: $ε_e = \frac{ΔL}{L_0}$ ## 'True' Strain - In this, change in length is referred to the instantaneous gage length, rather than original gage length. - Formula: $ε = Σ\frac{L₁-L₀}{L₀} + \frac{L₂-L₁}{L₁} + \frac{L₃-L₂}{L₂} + ...$ - Formula: $ε = \int^L_{L₀} \frac{dL}{L} = ln\frac{L}{L₀}$ # Converting Engineering Stress to True Stress - $σ = s(1+ε)$ # Converting Engineering Strain to True Strain - $ε = ln(1+ε)$ # Different Types of Mechanical Testing of a material - Uni-axial Tensile Test - Compression Test - Impact Test - Fatigue Test - Hardness Test - Torsion Test - Bending Test # 1. Uni-axial Tensile Test - **Test Specimen:** A diagram of the test specimen is shown - **Universal Testing Machine (UTM):** - **Test Standard - ASTM E08:** - **Stress vs. Strain Curve:** A diagram is shown of a Stress vs. Strain Curve. This diagram includes the following information: - Ultimate tensile strength - Fracture strength - Yield strength - Young's Modulus - Elastic Deformation - Plastic Strain - Total Strain - Necking # Stress vs. Strain Curves for Various Types of Ferrous and Non-ferrous Materials # Stress vs. Strain Curves of Ductile Materials - Ductile materials are those materials that have large plastic deformation before failure. - **Example of Ductile materials:** Mild Steel, Aluminium, Copper, etc. # Engineering Stress vs. Engineering Strain Curve of Mild Steel during Tensile Testing 1. **Limit of Proportionality (Point A)**: It is the limiting value of the stress up to which stress is proportional to strain. - **Hook's law** is followed within this limit. - Formula: E = Stress / Strain 2. **Elastic Limit (Point B)** : This point is slightly beyond the 'Limit of Proportionality'. The material remains elastic up to point B and returns to its original shape when the load is removed. - Stress at point B is called as elastic limit. 3. **Upper Yield Point (Point C)**: This is the stress at which, the load starts reducing and the extension increases. - This phenomenon is called 'Yielding' of material. - At this stage strain is about 0.125% and stress is about 250 N/mm2. 4. **Lower Yield Point (Point D)** : At this stage the stress remains same but strain increases for some time. 5. **Ultimate Stress (Point E)**: This is the maximum stress the material can resist. - At this stage cross-sectional area at a particular section starts reducing very fast. This is called neck formation. (Necking) - After this stage load resisted and hence the stress developed starts reducing. - This stress is about 370-400 N/mm2 for Mild Steel. 6. **Breaking Point OR Fracture Point (Point F)**: The stress at which finally the specimen fails is called 'Breaking Point'. - At this point, strain is 20 to 25%. # Comparison of Engineering and the True Stress-Strain Curves - The true stress-strain curve is also known as the flow curve. - True stress-strain curve gives a true indication of deformation characteristics because it is based on the instantaneous dimension of the specimen. - In engineering stress-strain curve, stress drops down after necking since it is based on the original area. - In true stress-strain curve, the stress however increases after necking since the crosssectional area of the specimen decreases rapidly after necking. # Engineering Stress vs. Engineering Strain Curve of Aluminium during Tensile Testing 2. This curve has No obvious yield point. - Yield stress can be determined by offset method. - **Note**: Aluminium alloy is also ductile, because it has plasticity (large permanent deformation) before failure. - Other ductile materials include Copper, Nickel, Bronze, etc. # Offset Method to Determine Yield Stress - At 0.2% strain (0.002) draw a line parallel to linear part. - It cuts stress-strain diagram at A, which is defined as yield stress. # Stress-Strain Diagram for Brittle Materials - Brittle materials are those materials that have very less plastic deformation before failure. - In Brittle materials, Failure is sudden. - **Example of brittle materials:** Cast Iron, Concrete, Glass, Ceramics, Stone, etc. # Stress-Strain Diagram for Brittle Materials (eg. Cast Iron) 3. Brittle materials fail only after a little elongation after the proportional limit (point A) is exceeded and doesn't exhibit significant plasticity as ductile materials. - No Necking. # Stress-Strain Diagrams of Various Ferrous Alloys and Al (after superimposed) 4. **Heat treated Alloy steel** - **Alloy Steel** - **Carbon Steel (mild steel)** - **Cast iron** - **Cast iron (Gray)** - **Al** # Stress-Strain Diagram of Rubber (Elastic Material) - Rubber is not ductile, because it doesn't give permanent deformation and returns to original configuration upon release of load. # Stress-Strain Diagram of Rubber (Elastic Material) 5. **Linear relationship between stress and strain upto relatively large strain (as high as 0.1 or 0.2).** - **Beyond that behavior is non-linear and depends on type of rubber.** # Stress-Strain Diagram of Other Materials # Stress-Strain Diagram of Other Materials 6. - For Linear Elastic Material - For Rigid Material - For Linear Elastic - Perfectly Plastic Material - For Linear Elastic - Linear Hardening Plastic Material # Various Mechanical Properties Determined from Tension Test - Modulus of Elasticity (or) Young's Modulus (E) - Yield Strength (YS) - Ultimate Tensile Strength (UTS) - Fracture Stress - Ductility - Toughness - Resilience # Modulus of Elasticity (or) Young's Modulus (E) - It is the Slope of linear elastic region of stress-strain curve within Proportionality limit. - Formula: $E = \frac{σ}{ε}$ # Yield Strength (YS) - It s the stress required to produce a small specific amount of permanent deformation. - The offset yield strength can be determined by the stress corresponding to the intersection of the stress-strain curve and a line parallel to the elastic line offset by a strain of 0.2 or 0.1%. (€ = 0.002 or 0.001). # Tensile Strength (or) Ultimate Tensile Strength (UTS) - It is the maximum load (Pmax) divided by the original cross-sectional area (Ao) of the specimen. # Fracture Stress - It is the stress at the point of Fracture. # Ductility Parameters - **% Elongation, € = \frac{L-L_0}{L_0} X 100** - **% Reduction in Cross-section area q = \frac{A-A_0}{A_0} X 100** # Toughness - It represents Amount of energy absorbed by material TILL FRACTURE. - It is represented by the area under stress-strain curve. # Modulus of Resilience - It is the amount of energy absorbed WITHIN ELASTIC REGIME. # Stress-Strain Diagram During Compression Loading # Stress-Strain Diagram During Compression Loading - For ductile material linear regime remains same as in tension - With increasing load, specimen takes barrel shape, finally flattens out to provide great resistance to further shortening - For brittle materials bulging doesn't occur. Material actually breaks. However, ultimate compressive stress is much higher than ultimate tensile stress. -