1. Concept of Stress and Strain

  • Rigid body is an idealized model, but in reality, objects deform under forces.
  • Stress ( ( \sigma ) ): Measures the force per unit area causing deformation.
  • Strain ( ( \varepsilon ) ): Measures the relative deformation in response to stress.
  • Elastic modulus (Young’s modulus, ( E ) ): Defines the relationship between stress and strain.

Stress in a Prismatic Bar (Under Tension)

For a prismatic bar with axial force ( P ) and cross-sectional area ( A ):

[ \sigma = \frac{P}{A} ]

Strain in a Prismatic Bar

For a bar of original length ( L ) that undergoes elongation ( \delta ):

[ \varepsilon = \frac{\delta}{L} ]

Assumptions:

  1. Uniform deformation.
  2. Loads act through the centroid.
  3. The material is homogeneous.

2. Stress-Strain Diagram

  • Elastic region: The material returns to its original shape when stress is removed.

  • Plastic region: Permanent deformation occurs after yielding.

  • Proportional limit (A): The region where stress and strain are proportional.

In solid mechanics, the proportional limit is the region where stress and strain have a linear relationship. This follows Hooke’s Law, which states:

[ \sigma = E \varepsilon ]

where:

  • ( \sigma ) = stress (force per unit area, measured in Pascals, Pa)
  • ( \varepsilon ) = strain (dimensionless, ratio of deformation to original length)

  • ( E ) = Young’s modulus (elastic modulus, measures material stiffness, in Pascals)

  • Young’s modulus (( E )) is a material property that defines how much a material deforms under stress.
  • A higher ( E ) means the material is stiffer (harder to deform).
  • A lower ( E ) means the material is more flexible.

For example:

  • Steel has a high ( E ) → it resists deformation.

  • Rubber has a low ( E ) → it stretches easily.

  • The proportional limit (Point A) is the point on the stress-strain curve where stress and strain are still proportional.
  • Before reaching this point, the material fully obeys Hooke’s Law.
  • Beyond this point, the stress-strain relationship becomes nonlinear, and permanent deformation may begin.
  • Engineering stress: Uses the original cross-sectional area.
  • True stress: Uses the updated cross-section.

3. Linear Elasticity and Hooke’s Law

Hooke’s Law (Axial Loading)

For an axially loaded member:

[ \sigma = E \varepsilon ]

Using force equilibrium and displacement relations:

[ \delta = \frac{PL}{EA} ]

where:

  • ( E ) = Young’s modulus
  • ( A ) = Cross-sectional area
  • ( L ) = Length
  • ( P ) = Applied load

[ P = \frac{EA}{L} \delta ]

which resembles Hooke’s law for a spring: ( F = kx ).

Derivation and Explanation of Axial Deformation Formula

When an axial force ( P ) is applied to a prismatic bar (a bar with uniform cross-section), the bar stretches or compresses. The amount of elongation or shortening is given by:

[ \delta = \frac{PL}{EA} ]

where:

  • ( \delta ) = Axial deformation (elongation or compression)
  • ( P ) = Axial force applied
  • ( L ) = Original length of the bar
  • ( E ) = Young’s modulus (material stiffness)
  • ( A ) = Cross-sectional area of the bar

1. Force Equilibrium and Stress-Strain Relations

(1) Stress Definition

Stress ( \sigma ) in the bar is given by:

[ \sigma = \frac{P}{A} ]

Since Hooke’s Law states that stress and strain are proportional in the elastic region:

[ \sigma = E \varepsilon ]

where:

  • ( \varepsilon ) = strain = relative deformation.

(2) Strain Definition

Strain is defined as:

[ \varepsilon = \frac{\delta}{L} ]

Substituting ( \sigma = \frac{P}{A} ) into Hooke’s Law:

[ \frac{P}{A} = E \varepsilon ]

Since ( \varepsilon = \frac{\delta}{L} ), we substitute:

[ \frac{P}{A} = E \frac{\delta}{L} ]

Rearranging for ( \delta ):

[ \delta = \frac{PL}{EA} ]

which is the axial deformation formula.

2. Rearranging the Formula

The equation can also be rewritten in terms of force:

[ P = \frac{EA}{L} \delta ]

This equation shows that:

  • The restoring force ( P ) in an axially loaded bar acts like a spring, where the term ( \frac{EA}{L} ) represents the stiffness of the bar.

  • The form of the equation is similar to Hooke’s Law for a spring:

    [ F = kx ]

    where:

    • ( k = \frac{EA}{L} ) (stiffness of the bar),
    • ( x = \delta ) (displacement),
    • ( F = P ) (applied force).

Thus, an elastic bar under axial load behaves like a linear elastic spring.

3. Interpretation

✅ What This Equation Tells Us

  • Larger ( P ) → Larger ( \delta ): More force leads to more elongation.
  • Larger ( L ) → Larger ( \delta ): Longer bars deform more for the same force.
  • Larger ( E ) or ( A ) → Smaller ( \delta ): A stiffer material (high ( E )) or a thicker bar (large ( A )) deforms less.

❌ When This Formula Does NOT Work

  • Material is beyond the elastic limit (plastic deformation occurs).
  • Cross-section is not uniform (requires integration).
  • Non-axial loading (torsion or bending is involved).

Poisson’s Ratio: Definition, Concept, and Applications

1. What is Poisson’s Ratio?

When a material is stretched in one direction, it contracts in the perpendicular directions. Similarly, when compressed, it expands laterally.

This behavior is quantified by Poisson’s ratio ( \nu ), which is defined as:

[ \nu = -\frac{\text{Lateral strain}}{\text{Axial strain}} ]

where:

  • Lateral strain ( \varepsilon_{\text{lat}} = \frac{\Delta d}{d} ) (change in width/diameter per unit width)
  • Axial strain ( \varepsilon_{\text{ax}} = \frac{\Delta L}{L} ) (change in length per unit length)

The negative sign ensures ( \nu ) is positive, since stretching (positive axial strain) causes contraction (negative lateral strain), and vice versa.

2. Interpretation of Poisson’s Ratio

  • Higher Poisson’s ratio (( \nu \approx 0.5 )): Material is highly compressible laterally (e.g., rubber).
  • Lower Poisson’s ratio (( \nu \approx 0 )): Material shows almost no lateral change (e.g., cork).
  • Negative Poisson’s ratio (( \nu < 0 )): The material expands laterally when stretched, called an auxetic material.

Common Values of Poisson’s Ratio

| Material | Poisson’s Ratio (( \nu )) | |————-|——————| | Rubber | 0.49 - 0.50 | | Steel | 0.27 - 0.30 | | Aluminum | 0.33 | | Concrete | 0.15 - 0.20 | | Cork | 0.0 (no lateral contraction) | | Auxetic Materials | Negative ( \nu ) |

3. Relationship with Elastic Moduli

Poisson’s ratio connects different elastic properties of a material.

(1) Shear Modulus and Young’s Modulus

The shear modulus ( G ) (modulus of rigidity) is related to Young’s modulus ( E ) by:

[ G = \frac{E}{2(1+\nu)} ]

  • If ( \nu ) is high, ( G ) is lower → material resists shear deformation less.
  • If ( \nu ) is low, ( G ) is higher → material resists shear more.

(2) Bulk Modulus and Young’s Modulus

The bulk modulus ( K ) (resistance to uniform compression) is:

[ K = \frac{E}{3(1 - 2\nu)} ]

  • If ( \nu \approx 0.5 ), bulk modulus is very high, meaning material is nearly incompressible (like rubber).
  • If ( \nu \approx 0 ), bulk modulus is smaller, meaning material compresses more easily (like foam or cork).

4. Physical Meaning of Poisson’s Ratio

(1) Stretching a Rubber Band

  • When you pull a rubber band, it becomes thinner.
  • Rubber has a high Poisson’s ratio (~0.5), meaning it contracts significantly in width when stretched.

(2) Compressing a Sponge

  • A sponge compresses easily in all directions.
  • Its Poisson’s ratio is low, meaning lateral contraction is minimal.

(3) Cork vs. Rubber

  • Cork has ( \nu \approx 0 ), meaning it does not contract laterally.
  • That’s why wine corks are easy to insert into a bottle—they don’t expand sideways when pushed in.

5. Special Case: Auxetic Materials (( \nu < 0 ))

  • Auxetic materials expand laterally when stretched (opposite of normal materials).
  • They have applications in biomechanics, flexible armor, and shock absorption.
  • Example: Some foams and engineered structures exhibit negative Poisson’s ratio.

Shear and Torsion in Solid Mechanics

1. Shear Stress and Shear Strain

Shear stress occurs when a force is applied parallel to a surface rather than perpendicular (as in normal stress). This type of stress deforms the material by causing adjacent layers to slide over each other.

(1) Shear Stress ( \tau )

Shear stress is defined as:

[ \tau = \frac{F}{A} ]

where:

  • ( \tau ) = Shear stress (Pa)
  • ( F ) = Applied force parallel to the surface (N)
  • ( A ) = Area on which the force acts (m²)

(2) Shear Strain ( \gamma )

Shear strain is the deformation due to shear stress, defined as:

[ \gamma = \frac{\Delta x}{h} = \tan \theta \approx \theta ]

where:

  • ( \gamma ) = Shear strain (dimensionless)
  • ( \Delta x ) = Lateral displacement (m)
  • ( h ) = Height of the material (m)
  • ( \theta ) = Angle of deformation in radians

2. Hooke’s Law for Shear

Similar to Hooke’s Law for normal stress, the shear stress-strain relationship is:

[ \tau = G \gamma ]

where:

  • ( G ) = Shear modulus (modulus of rigidity) (Pa)
  • ( \gamma ) = Shear strain

Relationship Between Young’s Modulus ( E ) and Shear Modulus ( G )

The shear modulus ( G ) is related to Young’s modulus ( E ) and Poisson’s ratio ( \nu ) by:

[ G = \frac{E}{2(1 + \nu)} ]

  • If ( G ) is high, the material resists shearing (e.g., steel).
  • If ( G ) is low, the material deforms easily in shear (e.g., rubber).

3. Torsion in Circular Shafts

Torsion occurs when a circular shaft is twisted due to a torque ( T ). This generates shear stress inside the shaft.

(1) Shear Strain in Torsion

The shear strain ( \gamma ) at a distance ( r ) from the center of the shaft is:

[ \gamma = \frac{\phi r}{L} ]

where:

  • ( \phi ) = Angle of twist (radians)
  • ( r ) = Radial distance from the center (m)
  • ( L ) = Length of the shaft (m)

(2) Shear Stress in Torsion

Using Hooke’s Law:

[ \tau = G \gamma ]

Substituting ( \gamma ):

[ \tau = G \frac{\phi r}{L} ]

(3) Torque and Polar Moment of Inertia

The total torque ( T ) in the shaft is given by:

[ T = G I_p \frac{\phi}{L} ]

where:

  • ( I_p ) = Polar moment of inertia, which depends on the cross-sectional area:

    [ I_p = \int_A r^2 dA ]

For a solid circular shaft of radius ( R ):

[ I_p = \frac{\pi R^4}{2} ]

For a hollow circular shaft (outer radius ( R_o ), inner radius ( R_i )):

[ I_p = \frac{\pi}{2} (R_o^4 - R_i^4) ]

4. Key Insights

✅ Shear and Torsion in Materials

  • Shear stress is caused by forces parallel to a surface.
  • Torsion is caused by rotational forces in circular shafts.
  • Both follow Hooke’s Law, with shear stress proportional to strain.

✅ Material Properties Affect Shear and Torsion

  • High ( G ) (e.g., steel) → Resists shear and torsion.
  • Low ( G ) (e.g., rubber) → Easily deforms under shear forces.

✅ Torsion Design Considerations

  • Larger radius ( R ) → Higher resistance to torsion (( I_p \propto R^4 )).
  • Hollow shafts are more efficient than solid shafts (high strength with less weight).

Understanding shear stress and torsion is essential for designing mechanical components like shafts, gears, and structural beams! 🚀

4. Strain Energy

Stored Energy in an Elastic Bar

For a bar subjected to axial force ( P ):

[ U = \int_0^{\delta} P \, d\delta = \frac{1}{2} P \delta ]

Using ( \delta = \frac{PL}{EA} ):

[ U = \frac{P^2 L}{2EA} ]

Strain Energy Density

[ u = \frac{U}{V} = \frac{1}{2} \sigma \varepsilon ]

For a bar with volume ( V = AL ):

[ U = \frac{1}{2} \frac{P^2 L}{EA} = \frac{1}{2} \frac{\sigma^2}{E} A L ]

5. Stress Concentration

  • Stress concentration occurs due to geometric discontinuities (holes, sharp corners, etc.).
  • Saint-Venant’s principle: Stresses become uniform at a distance from the load application point.
  • Maximum stress ( \sigma_{\text{max}} ) is higher than average stress.

[ \sigma_{\text{max}} = K_t \sigma_{\text{avg}} ]

where ( K_t ) is the stress concentration factor.

6. Allowable Stress and Factor of Safety

To prevent failure, we define the factor of safety (SF):

[ SF = \frac{\text{actual strength}}{\text{required strength}} ]

For yielding:

[ SF = \frac{\sigma_y}{\sigma_{\text{actual}}} ]

where ( \sigma_y ) is the yield stress.

Design Considerations:

  • ( SF > 1 ) ensures safety.
  • Higher ( SF ) is used for critical structures (e.g., bridges, aircraft).

7. Summary

✅ Key Concepts

  1. Stress and Strain: Fundamental properties of material deformation.
  2. Hooke’s Law: Describes linear elasticity.
  3. Poisson’s Ratio: Relationship between axial and lateral strain.
  4. Strain Energy: Energy stored in elastic deformation.
  5. Stress Concentration: Localized stress increase due to discontinuities.
  6. Allowable Stress & Safety Factor: Ensures safe structural design.