Stress and Strain

stress strain deformation mechanics
Last updated: 2025-01-18

Stress and Strain

Surface interactions between continuous media are best described in terms of mechanical stress and strain. These fundamental quantities characterize how forces are distributed through materials and how materials deform in response.

Mechanical Stress

Stress (σ, sigma) is defined as force per unit area:

σ = f/A

where:

  • σ = stress
  • f = force
  • A = area over which force is distributed

Physical Meaning

Stress measures how intensely a force is distributed over a region. The same total force produces different effects depending on the area:

Example 1: High heels on a floor A woman weighing less than a man can apply more stress to a floor if she wears narrower heels because her weight is distributed over a smaller area.

Example 2: Sharp versus dull objects Sharp objects can penetrate hard barriers more easily than dull objects with the same force because stress is concentrated over a smaller area.

Types of Stress

Stresses can be oriented perpendicular to a surface, tangential (parallel) to a surface, or a combination of both.

Stress types Figure 2.2: (a) Hand pushing top of a table at an angle; (b) The corresponding normal (perpendicular) stress and tangential (shear) stress.

Perpendicular Stresses

Stresses acting perpendicular (normal) to a surface:

Tensile Stress:

  • Points away from the surface
  • Tends to pull material apart
  • Example: Stress in a rope supporting a weight

Compressional Stress:

  • Points toward the surface
  • Tends to push material together
  • Example: Stress in a column supporting a building

Pressure:

  • The magnitude of a compressional stress
  • Commonly used quantity in voice science
  • Will be discussed further in Chapter 3

Tangential Stresses

Shear Stress:

  • All tangential stresses are called shear stresses
  • Act parallel to the surface
  • Tend to cause sliding of adjacent layers

Examples:

  • Wind resistance along the side of a car (shear stress)
  • Wind resistance against the front bumper (pressure)
  • Stress in a connecting rod that flexes (shear stress)
  • Stress in a straight connecting rod under tension (tensile stress)

Combined Stresses

In real situations, stresses are typically combinations:

  • A hand pushing a table at an angle (Figure 2.2a) produces both perpendicular and tangential components (Figure 2.2b)
  • Vocal fold tissue during vibration experiences tension, compression, and shear simultaneously

Strain

When stress is applied to any surface of a continuous medium, a deformation results, unless the medium is infinitely stiff.

Strain (ε, epsilon) measures normalized elongation:

ε = (L - L₀)/L₀

where:

  • ε = strain
  • L = stressed (deformed) length
  • L₀ = unstressed (rest) length

Key Properties of Strain

Dimensionless Quantity:

  • Strain is a ratio of lengths
  • Has no units
  • Often expressed as percentage

Examples:

  • Strain of 0.2 = 20% elongation over rest length
  • Strain of -0.1 = 10% contraction over rest length
  • Strain of 0 = no deformation

Sign Convention:

  • Positive strain: Elongation (material lengthens)
  • Negative strain: Contraction (material shortens)

Coupled Deformations

A deformation in one dimension typically results in opposite deformation in another dimension:

Examples:

  • Elongating a rubber band contracts its thickness
  • Active contraction of a muscle along its length increases its cross-section
  • Stretching vocal fold tissue longitudinally reduces its thickness

This coupling occurs because materials resist volume change. When elongated in one direction, they contract in perpendicular directions to approximately preserve volume.

Volumetric Deformation

Expansion and Compression

If deformation is applied uniformly over the body to change its entire volume:

Expansion:

  • Increase in volume
  • In acoustic terms: rarefaction
  • Example: Air expanding as pressure decreases

Compression:

  • Decrease in volume
  • In acoustic terms: condensation
  • Example: Air compressing as pressure increases

Volumetric Strain:

ε_vol = ΔV/V₀

where:

  • ΔV = change in volume
  • V₀ = original volume

Incompressibility

Sometimes volume is completely conserved during deformation—the medium is incompressible:

Characteristics:

  • Volume remains constant
  • Elongation in one direction requires contraction in another
  • Strain in different directions must sum appropriately

Examples:

  • Liquids are nearly incompressible
  • Solids are nearly incompressible
  • Biological soft tissues are approximately incompressible

Important Note: Incompressibility is an idealization. Some small volume change always occurs with any deformation, but it serves as a reasonable approximation for many biological materials, particularly soft tissues of the human body.

Stress-Strain Relationships

The relationship between stress and strain defines material behavior:

Linear Elastic:

  • Stress proportional to strain
  • Hooke’s law applies
  • Example: Steel at small deformations

Nonlinear Elastic:

  • Stress not proportional to strain
  • Material still returns to original shape when stress removed
  • Example: Rubber, biological tissues

Viscoelastic:

  • Time-dependent relationship
  • Both elastic and viscous components
  • Example: Most biological tissues

Plastic:

  • Permanent deformation remains after stress removal
  • Material does not return to original shape
  • Example: Clay, metals beyond yield point

Practical Examples

Example 1: Vocal Fold Tissue

During phonation, vocal fold tissue experiences:

Tensile stress:

  • Longitudinal stretching from muscle contraction
  • Determines vocal fold length and tension

Compressional stress:

  • From collision with opposite fold
  • From aerodynamic pressure

Shear stress:

  • Between tissue layers with different velocities
  • During mucosal wave propagation

Strain:

  • Can reach 30-40% during vibration
  • Nonlinear response (tissue stiffens with increasing strain)
  • Viscoelastic behavior (time-dependent)

Example 2: Hand Clapping

The stinging sensation after prolonged clapping results from:

Impact stress:

  • Large force distributed over hand area
  • Compressional stress perpendicular to palm
  • Repeated application causes cumulative effect

Tissue response:

  • Skin and subcutaneous tissue deform
  • Viscoelastic properties mean some energy dissipates as heat
  • Repeated stress can cause inflammation

This analogy helps understand vocal fold collision during phonation.

Multi-Dimensional Stress and Strain

Stress Tensor

In three dimensions, stress at a point is not a simple scalar or vector—it’s a tensor:

  • Nine components (three normal, six shear)
  • Describes stress on all faces of an infinitesimal cube
  • Only six components are independent (symmetry)

Components:

  • σ_xx, σ_yy, σ_zz: Normal stresses
  • σ_xy, σ_xz, σ_yz: Shear stresses (with symmetric counterparts)

Strain Tensor

Similarly, strain is a tensor in three dimensions:

  • Describes deformation in all directions
  • Includes elongation/contraction and angular distortion
  • Related to displacement gradients

Why This Matters

For simple cases (one-dimensional elongation), stress and strain are scalars. For complex biological tissues:

  • Multiple stress components act simultaneously
  • Deformation occurs in all directions
  • Tensor description is necessary for accurate analysis
  • Computer models use tensor formulations

Summary

Stress and strain are fundamental quantities in continuum mechanics:

Stress:

  • Force per unit area
  • Can be tensile, compressional (pressure), or shear
  • Measured in Pascals (Pa) or kilopascals (kPa)

Strain:

  • Normalized deformation
  • Dimensionless (often expressed as percentage)
  • Can be elongation, contraction, or volumetric change

Key Relationships:

  • Deformation in one dimension typically produces opposite deformation in perpendicular directions
  • Biological tissues are approximately incompressible
  • Three-dimensional stress and strain are tensor quantities

For vocal fold tissue:

  • Multiple stress types act simultaneously during vibration
  • Strains can be large (30-40%)
  • Relationship between stress and strain is nonlinear and time-dependent
  • Understanding these quantities is essential for analyzing phonation mechanics

Key Takeaways

  • ✅ Stress is force per unit area, measuring intensity of force distribution
  • ✅ Three types of stress: tensile (pulling), compressional (pressure), and shear (tangential)
  • ✅ Strain is normalized deformation, dimensionless, often expressed as percentage
  • ✅ Deformation in one direction typically produces opposite deformation in perpendicular directions
  • ✅ Biological tissues are approximately incompressible (volume preserved)
  • ✅ Vocal fold tissue experiences multiple stress types simultaneously during phonation

Further Reading

  1. Fung, Y. C. (1981). Biomechanics: Mechanical properties of living tissues. New York: Springer-Verlag.
  2. Perlman, A. L., Titze, I. R., & Cooper, D. S. (1984). Elasticity of canine vocal fold tissue. Journal of Speech and Hearing Research, 27, 212-219.
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