Elasticity

Introduction to Elasticity

Elasticity is the study of how solid materials deform and return to their original shape when subjected to external forces. Unlike rigid bodies, which do not change shape under force, real materials can stretch, compress, twist, or bend. This unit focuses on the behavior of solids under various types of stress and strain, and introduces key material properties such as Young's modulus, shear modulus, and bulk modulus.

• Rigid Body: An object that does not change shape when acted upon by a force.

• Non-Rigid Body: An object that deforms under force.

• Phases of Matter: Solids (definite shape and volume), liquids (definite volume, no definite shape), gases (no definite shape or volume).

• Compressibility: The ability to decrease the volume of a material by applying pressure.

Stress and Strain

The Concept of Stress

Stress quantifies the internal force per unit area within a material when an external force is applied. It is analogous to pressure, and both share the same units.

• Definition:

• Units: N.m-2 (Pascal, Pa)

• Tensile Stress: Force pulls material apart (stretching).

• Compressive Stress: Force pushes material together (compression).

• Shear Stress: Force applied parallel to the surface, causing layers to slide.

• Volume (Hydrostatic) Stress: Uniform pressure applied over the entire surface, changing the volume.

The Concept of Strain

Strain measures the degree of deformation in a material, expressed as a fractional change in a dimension (length, angle, or volume).

• Tensile/Compressive Strain:

• Shear Strain:

• Volume Strain:

• Dimensionless: Strain has no units, as it is a ratio of lengths or volumes.

Worked Examples

• Calculating Tensile Stress: For a steel wire bearing a load of 20 kg and cross-sectional area m2: N Pa

• Calculating Tensile Strain: For a wire cable of original length 40.0 m, increased by 4.0 cm:

Twisting and Shear

• Twisting: Shear stress can cause twisting in cylindrical objects. Shear strain for twisting: (where is in radians, is radius, is length)

Volume Stress and Strain

• Hydrostatic Stress:

• Volume Strain:

Geometric Formulas

Common shapes and their formulas:

Hooke's Law and Elastic Moduli

Hooke's Law

Within the elastic limit, the extension (strain) of a material is proportional to the applied load (stress). This relationship is known as Hooke's Law.

• Mathematical Form:

• Elastic Limit: Maximum stress a material can withstand without permanent deformation.

Young's Modulus (Y)

Young's modulus quantifies the stiffness of a material under tensile or compressive stress.

• Definition:

• Formula:

• Units: N.m-2 (Pa)

Shear Modulus (G)

Shear modulus measures a material's response to shear stress.

• Definition:

• Units: N.m-2 (Pa)

Bulk Modulus (B) and Compressibility (k)

Bulk modulus describes a material's resistance to uniform compression.

• Definition:

• Compressibility: (units: m2.N-1)

Stress-Strain Curves and Material Behavior

Stress vs Strain Curve for Ductile Metals

The stress-strain curve illustrates how a material responds to increasing stress. Key points:

• Proportional Limit (a): Stress is proportional to strain (Hooke's law applies).

• Elastic Limit (b): Maximum stress before permanent deformation.

• Plastic Region: Beyond elastic limit, material deforms permanently.

• Fracture Point (d): Material breaks.

• Ductile Materials: Exhibit significant plastic deformation before fracture (e.g., copper).

• Brittle Materials: Fracture soon after elastic limit (e.g., tungsten).

Elastic Energy

Work done in stretching a material within the elastic limit is stored as elastic potential energy.

• Work Done:

• For a rod:

• Elastic Energy: , where

Stress-Strain Curve for Elastomers

Elastomers (e.g., rubber) can stretch much more than metals and return to their original shape, but do not obey Hooke's law. Their stress-strain curve forms a hysteresis loop, indicating energy loss as heat during stretching and relaxing.

• Hysteresis: The path of increasing and decreasing strain differs; energy is lost as heat.

• Example: Stretching and relaxing a rubber band warms it due to energy dissipation.

Applications and Observations

Material Design and Geometry

• Hollow Cylinders: Have nearly the same strength as solid cylinders but are much lighter. Used in bone structure and engineering.

• Load Distribution: Geometry can distribute applied loads, combining tensile and shear stresses for structural efficiency.

• Reinforced Concrete: Steel (high Young's modulus) is used to reinforce concrete in buildings and bridges.

Temperature and Material Properties

• Elastic Moduli Dependence: Values of Young's modulus and shear modulus are determined at standard temperature and pressure. High temperatures can significantly reduce material strength (e.g., steel columns in the Twin Towers disaster).

Summary and Learning Goals

• Understand definitions and differences between compressive, tensile, shear, and volume stress and strain.

• Identify types of stress in various situations.

• Apply Hooke's law and select the appropriate elastic constant (Y, G, or B).

• Interpret stress-strain curves for ductile, brittle, and elastomeric materials.

• Understand hysteresis and its implications for energy dissipation.

• Recognize the role of geometry in material strength and load distribution.

Practice Problems

Worked examples and self-test questions are provided to reinforce concepts, including calculations of stress, strain, elastic moduli, and analysis of structural scenarios.

Key Equations

• — Stress

• — Tensile/Compressive Strain

• — Shear Strain

• — Volume Strain

• — Young's Modulus

• — Shear Modulus

• — Bulk Modulus

• — Compressibility

• — Elastic Energy

Additional info: The notes include practical examples, applications in engineering and biology, and emphasize the importance of material properties in real-world scenarios.