3.4. Materials

Springs and Deformation

• Deformation Process

  • A spring or wire requires equal and opposite forces for shaping.

  • Tensile Forces: Act away from the center of the spring, causing extension.

  • Compressive Forces: Act toward the center, resulting in compression.

Hooke’s Law

• Definition: Describes the behavior of materials under tension and compression within their elastic limit.

• Proportionality: The force applied (F) is directly proportional to the extension (x) of the material.

• Formula:

  • F ∝ x

  • F = kx (where k is the force constant).

• Force Constant (k):

  • Measure of stiffness; larger values indicate stiffer materials.

  • Measured in Nm⁻¹ and applies only within the elastic limit.

Force-Extension Graph for a Spring

• Segment Up to Point A:

  • Represents elastic deformation (returns to original shape).

  • The gradient here represents the force constant (k).

• Post Point A:

  • Material undergoes plastic deformation (permanent change).

Force-Extension Graphs for Different Materials

• Metal Wire:

  • Obeys Hooke’s law until elastic limit; shows elastic and then plastic deformation.

• Rubber:

  • Does not obey Hooke’s law; exhibits hysteresis loop (area between loading and unloading curves).

  • Energy lost is converted to thermal energy.

• Polyethene:

  • Exhibits plastic deformation and does not obey Hooke’s law.

Investigating Force-Extension Characteristics

• Setup:

  • Material suspended with clamp and boss; meter ruler used for measurements.

  • Standard masses applied to the material.

• Error Reduction:

  • Measure extension at eye-level with a set square for accuracy.

• Gradient Calculation:

  • Need slope of straight section of force-extension graph to determine force constant.

• Springs in Series and Parallel:

  • Series:

    • 1/k_total = 1/k1 + 1/k2 + 1/k3

  • Parallel:

    • k_total = k1 + k2 + ... + kn

Mechanical Properties of Materials

Work Done and Elastic Potential Energy

• Elastic Deformation:

  • Work done on materials during elastic deformation is stored as elastic potential energy (E).

• Calculation:

  • Area under force-extension graph; for triangles, E = 1/2 * base * height.

  • Using F = kx, E can also be expressed as E = 1/2 kx².

• Plastic Deformation:

  • Work is taken to rearrange atoms; stored energy as elastic potential is minimal.

Stress and Strain

• Definitions:

  • Tensile Stress (σ): Force per unit area (σ = F/A), units in Nm⁻² (Pa).

  • Tensile Strain (ε): Extension or compression relative to original length (ε = x/L), unitless.

• Young's Modulus (E):

  • Defined as stress/strain; measures material's stiffness; independent of shape.

  • E = σ/ε

Young's Modulus Determination Procedure

1. Measure wire diameter using a micrometer (average multiple readings).

2. Clamp wire; apply tension using a pulley system.

3. Record masses and corresponding wire extensions.

4. Calculate stress/strain, plot on graph, and find the gradient for Young's modulus.

Ultimate Tensile Strength (UTS)

• Stress-Strain Graph Analysis:

  • Point P = limit of proportionality where Hooke's law applies.

  • Point E = elastic limit, beyond which plastic deformation occurs.

  • Yield Points Y1 and Y2 indicate rapid elongation.

  • UTS: Maximum stress before fracture; high for strong materials.

Material Behavior in Stress-Strain Graphs

• Brittle Materials (e.g., glass):

  • No plastic deformation; material breaks at the breakpoint.

• Elastic Materials (e.g., rubber):

  • No plastic deformation; unload strain curve differs from loading; thermal energy loss exists.

• Ductile Materials:

  • Experience both elastic and plastic deformation; can stretch or be hammered into shapes.