Willer Academy
Shear Modulus (Modulus of Rigidity)
Shear Modulus (G) is defined as the ratio of shear stress to shear strain within the elastic limit.
It measures a material's resistance to shearing deformation (shape change without volume change).
Formula: G = Shear Stress / Shear Strain = (F/A) / θ
Where θ is the angle of shear (in radians)
SI Unit: N/m² or Pascal (Pa)
Examples:
- Steel: 8.0 × 10¹⁰ N/m²
- Copper: 4.2 × 10¹⁰ N/m²
- Aluminum: 2.5 × 10¹⁰ N/m²
- Rubber: 1.0 × 10⁶ N/m²
कर्तन मापांक (G) को प्रत्यास्थ सीमा के भीतर कर्तन प्रतिबल और कर्तन विकृति के अनुपात के रूप में परिभाषित किया जाता है।
यह किसी पदार्थ की कर्तन विरूपण के प्रतिरोध को मापता है (आयतन परिवर्तन के बिना आकार परिवर्तन)।
सूत्र: G = कर्तन प्रतिबल / कर्तन विकृति = (F/A) / θ
जहाँ θ कर्तन कोण है (रेडियन में)
SI इकाई: N/m² या पास्कल (Pa)
उदाहरण:
- स्टील: 8.0 × 10¹⁰ N/m²
- तांबा: 4.2 × 10¹⁰ N/m²
- एल्युमीनियम: 2.5 × 10¹⁰ N/m²
- रबर: 1.0 × 10⁶ N/m²
Shear Modulus Formula
Where: F = Tangential force, L = Length, A = Cross-sectional area, Δx = Lateral displacement
Real-Life Experiment
Materials: Book, ruler, two identical thick rubber bands
Procedure:
- Place a heavy book on a table
- Attach two rubber bands to opposite sides of the book
- Pull the rubber bands in opposite directions horizontally
- Measure the lateral displacement and calculate shear modulus
Observation: The book undergoes shear deformation. The resistance to this deformation depends on the material properties.
Key Points
- Shear modulus is also called modulus of rigidity
- For most materials, G ≈ Y/3 (where Y is Young's modulus)
- Liquids and gases have zero shear modulus as they cannot sustain shear stress
- Shear modulus is important in designing shafts, beams, and structural elements
Bulk Modulus
Bulk Modulus (K) is defined as the ratio of hydraulic stress to volumetric strain within the elastic limit.
It measures a material's resistance to uniform compression (volume change under pressure).
Formula: K = - (ΔP) / (ΔV/V) = -V (ΔP/ΔV)
The negative sign indicates that volume decreases when pressure increases
SI Unit: N/m² or Pascal (Pa)
Examples:
- Diamond: 4.4 × 10¹¹ N/m²
- Steel: 1.6 × 10¹¹ N/m²
- Water: 2.2 × 10⁹ N/m²
- Air: 1.0 × 10⁵ N/m²
आयतन मापांक (K) को प्रत्यास्थ सीमा के भीतर हाइड्रोलिक प्रतिबल और आयतन विकृति के अनुपात के रूप में परिभाषित किया जाता है।
यह किसी पदार्थ की एकसमान संपीड़न के प्रतिरोध को मापता है (दबाव में आयतन परिवर्तन)।
सूत्र: K = - (ΔP) / (ΔV/V) = -V (ΔP/ΔV)
नकारात्मक चिह्न इंगित करता है कि दबाव बढ़ने पर आयतन घटता है
SI इकाई: N/m² या पास्कल (Pa)
उदाहरण:
- हीरा: 4.4 × 10¹¹ N/m²
- स्टील: 1.6 × 10¹¹ N/m²
- पानी: 2.2 × 10⁹ N/m²
- वायु: 1.0 × 10⁵ N/m²
Bulk Modulus Formula
Where: ΔP = Change in pressure, V = Original volume, ΔV = Change in volume
Real-Life Experiment
Materials: Syringe, water, oil, rubber stopper
Procedure:
- Fill a syringe with water and seal it with a rubber stopper
- Apply pressure to the plunger and note how much it moves
- Repeat with oil and compare the compressibility
- Calculate bulk modulus from the measurements
Observation: Water is less compressible than oil, showing higher bulk modulus. Gases are much more compressible than liquids.
Key Points
- Bulk modulus is the reciprocal of compressibility
- Solids have the highest bulk modulus, followed by liquids, then gases
- For ideal gases, bulk modulus equals pressure (K = P)
- Bulk modulus is important in hydraulic systems and deep-sea applications
Poisson's Ratio
Poisson's Ratio (σ) is defined as the ratio of lateral strain to longitudinal strain when a material is stretched.
It measures how much a material expands (or contracts) in directions perpendicular to the direction of compression (or stretching).
Formula: σ = - (Lateral Strain) / (Longitudinal Strain)
The negative sign indicates that lateral strain is opposite to longitudinal strain
It is a dimensionless quantity (no unit)
Examples:
- Cork: 0.0 (no lateral expansion)
- Steel: 0.28 - 0.30
- Rubber: 0.49 - 0.50 (almost incompressible)
- Concrete: 0.1 - 0.2
प्वासों अनुपात (σ) को पार्श्व विकृति और अनुदैर्ध्य विकृति के अनुपात के रूप में परिभाषित किया जाता है जब किसी पदार्थ को खींचा जाता है।
यह मापता है कि कोई पदार्थ संपीड़न (या खिंचाव) की दिशा के लंबवत दिशाओं में कितना फैलता (या सिकुड़ता) है।
सूत्र: σ = - (पार्श्व विकृति) / (अनुदैर्ध्य विकृति)
नकारात्मक चिह्न इंगित करता है कि पार्श्व विकृति अनुदैर्ध्य विकृति के विपरीत है
यह एक विमाहीन मात्रा है (कोई इकाई नहीं)
उदाहरण:
- कॉर्क: 0.0 (कोई पार्श्व विस्तार नहीं)
- स्टील: 0.28 - 0.30
- रबर: 0.49 - 0.50 (लगभग असंपीड्य)
- कंक्रीट: 0.1 - 0.2
Poisson's Ratio Formula
Where: ΔD = Change in diameter, D = Original diameter, ΔL = Change in length, L = Original length
Real-Life Experiment
Materials: Rubber band, ruler, marker
Procedure:
- Mark two points on a rubber band to measure its diameter
- Stretch the rubber band and measure the new length and diameter
- Calculate longitudinal strain and lateral strain
- Determine Poisson's ratio using the formula
Observation: As the rubber band stretches (longitudinal strain increases), its diameter decreases (lateral strain is negative). The ratio is close to 0.5 for rubber.
Key Points
- The theoretical limits for Poisson's ratio are -1 to 0.5
- Most materials have Poisson's ratio between 0.0 and 0.5
- Auxetic materials have negative Poisson's ratio (they expand laterally when stretched)
- For an incompressible material, Poisson's ratio is exactly 0.5
Potential Energy in Stretched Wire
When a wire is stretched, work is done against the interatomic forces. This work is stored in the wire as elastic potential energy.
The potential energy stored per unit volume of the wire is equal to the area under the stress-strain curve.
For a wire obeying Hooke's Law:
Potential Energy (U) = ½ × Stress × Strain × Volume
Or: U = ½ × (F × ΔL)
Where F is the applied force and ΔL is the extension
This energy is released when the wire returns to its original shape.
जब किसी तार को खींचा जाता है, तो अंतर-परमाण्विक बलों के विरुद्ध कार्य किया जाता है। यह कार्य तार में प्रत्यास्थ स्थितिज ऊर्जा के रूप में संचित हो जाता है।
तार के प्रति इकाई आयतन में संचित स्थितिज ऊर्जा प्रतिबल-विकृति वक्र के अंतर्गत क्षेत्र के बराबर होती है।
हुक के नियम का पालन करने वाले तार के लिए:
स्थितिज ऊर्जा (U) = ½ × प्रतिबल × विकृति × आयतन
या: U = ½ × (F × ΔL)
जहाँ F लगाया गया बल है और ΔL विस्तार है
यह ऊर्जा तब मुक्त होती है जब तार अपने मूल आकार में लौटता है।
Potential Energy Formulas
Real-Life Experiment
Materials: Rubber band, small weight, ruler
Procedure:
- Stretch a rubber band and attach a small weight to it
- Release the weight and observe how high it bounces
- Measure the extension and calculate potential energy stored
- Compare with the kinetic energy of the bouncing weight
Observation: The height the weight bounces is proportional to the potential energy stored in the stretched rubber band.
Key Points
- Elastic potential energy is the energy stored when a material is deformed elastically
- This energy is recoverable when the material returns to its original shape
- The area under the stress-strain curve represents work done per unit volume
- This concept is used in springs, rubber bands, and other elastic devices
Applications of Elasticity
Real-World Applications
Construction Engineering
Selection of materials with appropriate Young's modulus for buildings, bridges, and infrastructure to ensure safety and durability.
Automotive Industry
Design of springs, shock absorbers, and chassis components using knowledge of elasticity and energy storage.
Aerospace Engineering
Use of materials with high strength-to-weight ratio and appropriate elastic properties for aircraft and spacecraft.
Medical Devices
Design of orthopedic implants, dental braces, and surgical instruments considering biological tissue elasticity.
Sports Equipment
Design of tennis rackets, golf clubs, and running shoes using principles of elasticity for optimal performance.
Musical Instruments
Selection of materials for strings, membranes, and soundboards based on their elastic properties for desired acoustic characteristics.
The study of elasticity has numerous practical applications across various fields:
- Metallurgy: Determining the quality and purity of metals
- Geology: Studying the behavior of Earth's crust during earthquakes
- Civil Engineering: Designing structures that can withstand loads and environmental stresses
- Biomechanics: Understanding the mechanical properties of bones, muscles, and tissues
- Material Science: Developing new materials with tailored elastic properties
प्रत्यास्थता का अध्ययन विभिन्न क्षेत्रों में कई व्यावहारिक अनुप्रयोग हैं:
- धातुकर्म: धातुओं की गुणवत्ता और शुद्धता का निर्धारण
- भूविज्ञान: भूकंप के दौरान पृथ्वी की पपड़ी के व्यवहार का अध्ययन
- सिविल इंजीनियरिंग: ऐसी संरचनाओं को डिजाइन करना जो भार और पर्यावरणीय तनावों का सामना कर सकें
- बायोमैकेनिक्स: हड्डियों, मांसपेशियों और ऊतकों के यांत्रिक गुणों को समझना
- सामग्री विज्ञान: अनुकूलित प्रत्यास्थ गुणों वाली नई सामग्रियों का विकास
Important Formulas & Relationships
Complete Formula Sheet - Part 2
Important Relationships Between Elastic Constants
- Y = 3K(1 - 2σ)
- Y = 2G(1 + σ)
- 9/Y = 3/G + 1/K
- σ = (3K - 2G) / (6K + 2G)
- K = Y / [3(1 - 2σ)]
- G = Y / [2(1 + σ)]
Verification Experiment
Objective: Verify the relationship between Young's modulus, shear modulus, and Poisson's ratio
Materials: Steel wire, weights, measuring instruments
Procedure:
- Measure Young's modulus using standard method
- Measure shear modulus using torsion pendulum
- Calculate Poisson's ratio using the relationship Y = 2G(1 + σ)
- Compare with direct measurement of Poisson's ratio
Observation: The calculated and measured values of Poisson's ratio should be approximately equal, verifying the relationship between elastic constants.
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