MatE 22 Module 4.1 - Concepts of Stress and Strain PDF

MatE 22 Structure-Property Relationship of Materials II Concepts of Stress and Strain MODULE 4.1 Lecture slides adapted from previous MatE 22 classes (Ma’am Tiff Lao) Learning Outcomes Identify the different types of loading Define stress and strain Differentiate engineering and true 𝜎 − 𝜀...

MatE 22 Structure-Property Relationship of Materials II Concepts of Stress and Strain MODULE 4.1 Lecture slides adapted from previous MatE 22 classes (Ma’am Tiff Lao) Learning Outcomes Identify the different types of loading Define stress and strain Differentiate engineering and true 𝜎 − 𝜀 Review the mechanical properties solved using engineering true 𝜎 − 𝜀 curve MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Mechanical Property materials response to an applied load or force Learning Outcomes Types of Loading Affected by these factors: tensile Stress and Strain nature of applied load/force compressive Engineering vs True 𝜎 − 𝜀 application time shear Mechanical Properties Involved in 𝜎 − 𝜀 curve temperature MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Types of Loading/Force Learning Outcomes Tensile Load -force pulls apart or stretches a material Types of Loading Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝜎 − 𝜀 curve Compressive Load -force causes a material to deform to occupy a smaller volume Shear Load -force tending to cause deformation of a material by slippage along a plane/s parallel to it MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Stress internal distribution of forces within a body that balance and Learning Outcomes react to the loads applied to it Types of Loading Stress and Strain 𝐹 Engineering vs True 𝜎 − 𝜀 𝜎= 𝐴 Area of applied load Mechanical Properties Involved in 𝜎 − 𝜀 curve Unit of measure: 𝑁Τ𝑚2 = 𝑃𝑎 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Stress internal distribution of forces within a body that balance and Learning Outcomes react to the loads applied to it Types of Loading Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝜎 − 𝜀 curve MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Strain response of a system to an applied stress Learning Outcomes Types of Loading unitless but is often left in the unsimplified form, such as 𝑖𝑛Τ 𝑖𝑛 Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝜎 − 𝜀 curve MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Stress-Strain Curve Diagram representing the relationship between stress and Learning Outcomes strain of a given material Types of Loading Mechanical properties, e.g., Young’s Modulus, are obtained Stress and Strain using it Engineering vs True 𝝈 − 𝜺 Has two (2) types: Mechanical Properties Involved in 𝜎 − 𝜀 curve Engineering stress-strain diagram True stress-strain diagram MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Engineering Stress vs. True Stress Engineering Stress True Stress Instantaneous force divided by the Instantaneous force divided by original cross-sectional area instantaneous area 𝑭𝒕 𝑭𝒕 𝝈= 𝝈𝑻 = 𝑨𝟎 𝑨𝒊 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Engineering Strain vs. True Strain Engineering Strain True Strain Change in length divided by the Natural logarithm of the original length instantaneous gauge length of a specimen 𝑙𝑓 −𝑙𝑜 𝞮= 𝒍 𝒅𝒍 𝑙𝑜 𝞮𝑻 = ‫𝒍 𝒍׬‬ 𝒐 𝒍𝒊 𝞮𝑻 = 𝒍𝒏 𝒍𝒐 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  elastic region plastic region (temporary deformation) (permanent deformation) MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  eng’g true 𝑭𝒕 𝑭𝒕 True and Engineering 𝜎 − 𝜀 curves 𝝈= 𝝈𝑻 = coincide at the elastic region. 𝑨𝟎 𝑨𝒊 𝐴𝑜 = 𝐴𝑖 Hooke’s Law 𝜎 = 𝐸𝜀 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  eng’g true 𝑭𝒕 𝑭𝒕 𝝈= 𝝈𝑻 = True and Engineering 𝜎 − 𝜀 curves start to deviate 𝑨𝟎 𝑨𝒊 because material starts to deform permanently. 𝐴𝑜 > 𝐴𝑖 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  eng’g true Deviation widens because of strain-hardening occurring 𝑭𝒕 𝑭𝒕 𝝈= 𝝈𝑻 = at decreasing cross-sectional area. 𝐴 ≫ 𝐴 𝑨𝟎 𝑨𝒊 𝑜 𝑖 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  eng’g true For some metals, 𝜎𝑇 can be approximated using the constants 𝑭𝒕 𝑭𝒕 𝝈= 𝝈𝑻 = for strain hardening: 𝑛 𝑨𝟎 𝑨𝒊 𝜎𝑇 = 𝐾𝜀𝑇 NOTE: Equation applies ONLY from the beginning of plastic flow to the point where material begins to neck down 𝜎𝑇 = 𝐾𝜀𝑇𝑛 K = strength coefficient = value of σ when є = 1.0 n = strain hardening exponent = measure of material’s resistance to necking MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  eng’g true Deviation is at the largest upon onset of necking 𝑭𝒕 𝑭𝒕 𝐴𝑜 ≫> 𝐴𝑖 𝝈= 𝝈𝑻 = because of strain-hardening localized at a small area. 𝑨𝟎 𝑨𝒊 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  𝐹 𝜎𝑇 = Learning Outcomes 𝐴𝑖 Types of Loading Stress and Strain Conservation of volume: 𝐴𝑖 𝑙𝑖 = 𝐴0 𝑙0 Engineering vs True 𝝈 − 𝜺 𝐹𝑙𝑖 Mechanical Properties Involved in 𝜎 − 𝜀 𝜎𝑇 = curve 𝐴0 𝑙0 𝐹 𝑙𝑖 𝜎𝑇 = 𝐴0 𝑙0 𝑙0 + ∆𝑙 𝜎𝑇 = 𝜎 𝑙0 𝜎𝑇 = 𝜎 1 + 𝜀 Relationship of Engineering and True 𝜎 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  𝑙𝑖 𝜀𝑇 = 𝑙𝑛 𝑙0 Learning Outcomes But we know that: Types of Loading Stress and Strain ∆𝑙 = 𝑙𝑖 − 𝑙𝑜 Engineering vs True 𝝈 − 𝜺 Mechanical Properties Involved in 𝜎 − 𝜀 𝐿𝑖 curve 𝑙0 + ∆𝑙 𝜀𝑇 = 𝑙𝑛 𝑙0 Relationship of 𝜀𝑇 = 𝑙𝑛 1 + 𝜀 Engineering and True 𝜀 MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Mathematical Relationships of Learning Outcomes Engineering and True 𝜎 − ε Types of Loading Stress and Strain 𝞼𝑻 = 𝞼(𝟏 + 𝞮) Engineering vs True 𝝈 − 𝜺 Mechanical Properties Involved in 𝜎 − 𝜀 curve 𝞮𝑻 = 𝒍𝒏(𝟏 + 𝞮) Note: Valid only up to the onset of necking! Beyond, compute from actual values Due to non-uniform deformation, significant strain hardening, plastic instability MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez Learning Outcomes Why do we still use Engineering Stress-Strain Curve Types of Loading instead of True Stress-Strain Curve? Stress and Strain Engineering vs True 𝝈 − 𝜺 Easy to use Mechanical Properties Involved in 𝜎 − 𝜀 curve Mechanical properties can be solved even by just using Eng’g 𝜎 − 𝜀 curve. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez Mechanical Properties Disclaimer: only those that can be obtained from Engineering 𝜎 − 𝜀 curve MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Moduli of Elasticity and Rigidity Governed by Hooke’s Law Learning Outcomes Types of Loading Modulus of Elasticity, E tension Stress and Strain 𝞼 = 𝐸𝞮 Engineering vs True 𝜎 − 𝜀 Where Mechanical Properties Involved in 𝝈 − 𝜺 E = Young’s Modulus or curve Modulus of Elasticity (same units as 𝞼, N/mm2) Represents the material's stiffness or resistance to elastic deformation. A higher modulus of elasticity indicates a stiffer material. Crucial in applications where rigidity and dimensional stability are important, such as structural components, machine parts, and springs. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Moduli of Elasticity and Rigidity Governed by Hooke’s Law Learning Outcomes Types of Loading Modulus of Rigidity, G shear Stress and Strain 𝝉 = 𝑮𝜸  Engineering vs True 𝜎 − 𝜀 Where G 𝞽 = shear stress 1  Mechanical Properties Involved in 𝝈 − 𝜺 𝞬 = shear strain /2 curve G = Shear Modulus (same units as τ, N/mm2) /2 -  /2 /2 Components like shafts, axles, and gears are subjected to torsional stresses. A high modulus of rigidity is essential to prevent excessive twisting and deformation. The modulus of rigidity is a key property for designing composite materials, as it influences their shear strength and stiffness. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez Learning Outcomes Proportional Limit Types of Loading the point wherein the departure Stress and Strain from the linearity of the stress- strain curve starts Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve Yielding the phenomenon wherein transition from elastic to plastic deformation occurs MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Yield Strength, 𝜎Y taken as either: Learning Outcomes Types of Loading Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve Yield strength is a measure of the material's resistance to permanent deformation. It is essential in applications where the material must withstand loads without permanent deformation, such as in structural components and machine parts. 0.2% strain-offset the average of the lower yield (if elastic-plastic transition is strength not well defined) MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Ultimate Tensile Strength 𝜎𝑈𝑇𝑆 or UTS, or TS Learning Outcomes Types of Loading the stress at the maximum of the engineering stress-strain curve Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve UTS is a measure of the material's strength and is important in applications where the material must withstand high loads without breaking, such as in cables, wires, and structural components. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Fracture Stress, 𝜎𝐹 Also called Breaking Stress Learning Outcomes Types of Loading the stress at the point of breaking or fracture Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Ductility degree of plastic deformation that has been sustained at Learning Outcomes fracture Types of Loading expressed as percent elongation or percent area reduction Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve 𝑳𝒇 − 𝑳𝒐 %𝑬𝑳 = 𝒙𝟏𝟎𝟎 𝑳𝒐 𝑨𝒇 − 𝑨𝒐 %𝑹𝑨 = 𝒙𝟏𝟎𝟎 𝑨𝒐 Ductility is important in applications where the material needs to be formed or shaped, such as in sheet metal forming, wire drawing, and forging. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Modulus of Resilience, Ur the strain energy per unit volume required to stress a Learning Outcomes material from an unloaded state up to the point of yielding. Types of Loading Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve 𝞮𝒀 𝞼𝟐𝒀 This is the ability of a material to store elastic 𝑼𝑹 = ‫𝞮𝒅𝞼 𝟎׬‬ = 𝟐𝑬 energy and release it upon unloading. Resilience is important in applications where the material must absorb and release energy repeatedly, such as in springs and vibration dampers. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Toughness a measure of the ability of a material to absorb energy up to Learning Outcomes fracture Types of Loading the area under the stress–strain curve up to the point of Stress and Strain fracture Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve Toughness is important in applications where the material must withstand impact or shock loads, such as in armor plating, crash barriers, and machine components subjected to dynamic loads. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Poisson’s ratio, 𝜈 ratio of the lateral and axial strains Learning Outcomes Types of Loading (-) sign shows that lateral strain is in opposite sense to longitudinal strain Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve Anisotropic materials: Poisson's ratio can vary depending on the direction of loading in anisotropic materials like wood or fiber-reinforced composites. Nonlinear behavior: Poisson's ratio may not be constant at high stress levels or for materials exhibiting nonlinear behavior. MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Poisson’s ratio, 𝜈 ratio of the lateral and axial strains Learning Outcomes Types of Loading (-) sign shows that lateral strain is in opposite sense to longitudinal strain Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve Is it possible to have negative values for Poisson’s ratio? MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez  Poisson’s ratio, 𝜈 ratio of the lateral and axial strains Learning Outcomes Types of Loading (-) sign shows that lateral strain is in opposite sense to longitudinal strain Stress and Strain Engineering vs True 𝜎 − 𝜀 Mechanical Properties Involved in 𝝈 − 𝜺 curve Auxetic Materials MatE 22 - Structure-Property Relationship of Materials II | JABNarvaez Thank You! References: Lecture adapted from previous MatE 22 classes (Maam Tiff Lao) MatE 22 Module 4 – Stresses and Deformation Callister, Jr., W. D. (2013). Materials Science and Engineering - An Introduction. New York: John Wiley & Sons, Inc. Askeland, D. R., Fulay, P. P., & Wright, W. J. (2016). The Science and Engineering of Materials 7th Ed. Stamford: Cengage Learning. MATE 22 - STRUCTURE-PROPERTY RELATIONSHIP OF MATERIALS II | JABNARVAEZ