Constitutive Equations

constitutive-equations Hookes-law elasticity material-properties
Last updated: 2025-01-18

Constitutive Equations

The type of deformation a substance undergoes is governed by the internal forces of attraction or repulsion between molecules. Strain is the macroscopic end product of many internal dynamic interactions. The final macroscopic relation between applied stress and resulting strain is called the constitutive equation.

What is a Constitutive Equation?

A constitutive equation is a mathematical relationship that:

  • Connects stress to strain (or strain rate)
  • Defines material behavior
  • Captures internal molecular interactions at the macroscopic level
  • Is specific to each material or class of materials

The name “constitutive” reflects that these equations describe what constitutes the material—its fundamental mechanical character.

Hooke’s Law: Linear Elasticity

One of the simplest constitutive equations is Hooke’s law (named after British scientist Robert Hooke, 1635-1703), which governs the deformation of linear elastic solids.

The Equation

σ = Eε

where:

  • σ = stress
  • ε = strain
  • E = modulus of elasticity

Alternative names for E:

  • Modulus of elasticity
  • Young’s modulus (after British scientist Thomas Young, 1773-1829)
  • Elastic modulus

Meaning of Proportionality

Hooke’s law states that stress is proportional to strain:

  • If strain doubles, stress doubles
  • If strain triples, stress triples
  • The ratio σ/ε remains constant (equals E)

Physical Interpretation of E

The modulus of elasticity E represents:

  • Material stiffness: Higher E means stiffer material
  • Resistance to deformation: More force needed to produce same strain
  • Slope of stress-strain curve: For linear elastic materials

Examples of E values:

  • Steel: ~200 GPa (very stiff)
  • Bone: ~15-20 GPa (stiff)
  • Rubber: ~0.01-0.1 GPa (compliant)
  • Vocal fold tissue: ~0.001-0.040 GPa (very compliant)

When Hooke’s Law Applies

Hooke’s law is valid when:

  • Deformations are small
  • Material returns to original shape when stress removed (elastic)
  • Stress-strain relationship is linear

Examples:

  • Steel springs at small deformations
  • Guitar strings
  • Rubber bands at small extensions

When Hooke’s Law Fails

Many materials, especially biological ones, do not follow Hooke’s law:

  • Large deformations produce nonlinear response
  • Stress-strain curve bends
  • Stiffness changes with deformation level

Nonlinear Elasticity

Often stress is not proportional to strain, but the material nevertheless returns to its original configuration when stress is removed. The material is then elastic, but not linearly elastic.

Characteristics

Still elastic:

  • Material returns to original shape
  • No permanent deformation
  • Energy stored during loading is recovered during unloading

But nonlinear:

  • Stress-strain curve is not a straight line
  • Often shows upward curvature (stiffening with deformation)
  • E is not constant but varies with strain level

Mathematical Forms

The constitutive equation for nonlinearly elastic materials is more complex than Hooke’s law. Various forms exist:

Polynomial:

σ = E₁ε + E₂ε² + E₃ε³ + ...

Exponential:

σ = A(e^(Bε) - 1)

Power law:

σ = Kε^n

where E₁, E₂, A, B, K, n are material constants determined experimentally.

Biological Tissues

Most biological tissues exhibit nonlinear elasticity:

  • Vocal fold tissue: Stiffens progressively with elongation
  • Blood vessels: Compliant at low pressure, stiffen at high pressure
  • Skin: Easy to stretch initially, becomes increasingly resistant

This nonlinearity serves important physiological functions:

  • Prevents excessive deformation
  • Provides stability at large strains
  • Allows wide range of functional deformations

Ideal Gas Law

Another simple constitutive equation describes equilibrium state of an ideal gas:

PV = nRT

where:

  • P = absolute pressure
  • V = volume
  • n = number of moles of gas
  • R = universal gas constant
  • T = absolute temperature

Application to Voice Science

For constant temperature, a change in pressure (ΔP) relates to a change in volume (ΔV):

ΔP = -P(ΔV/V)

Interpretation:

  • The ratio ΔV/V is the volumetric strain
  • The pressure P is analogous to the elastic modulus
  • The negative sign indicates that pressure increase causes volume decrease

Relevance:

  • Small pressure changes above and below atmospheric pressure occur in speech
  • This relation describes how air compresses and expands in the vocal tract
  • Will be important in Chapter 3 on respiration

Beyond Simple Constitutive Equations

Three-Dimensional Complexity

For real materials, constitutive equations become more complex:

Multiple dimensions:

  • Stress and strain are tensors, not scalars
  • Elongation in one direction produces contraction in others
  • Shear deformations couple with normal deformations

Anisotropy:

  • Properties differ in different directions
  • Example: Vocal fold tissue is stiffer along fibers than across fibers
  • Requires different elastic moduli for different directions

Time dependence:

  • Many materials exhibit viscous as well as elastic behavior
  • Constitutive equations must include time derivatives
  • Called viscoelastic materials (covered in next section)

General Form

A general constitutive equation for three-dimensional, anisotropic, viscoelastic materials can be quite complex:

σᵢⱼ = f(εₖₗ, dεₖₗ/dt, history, temperature, ...)

where:

  • i,j,k,l range over three spatial directions
  • Function f can be very complicated
  • May depend on deformation history
  • Temperature and other factors may influence behavior

Determining Constitutive Equations

Experimental Approach

Constitutive equations are determined through:

1. Mechanical testing:

  • Apply known stresses or strains
  • Measure resulting deformations or forces
  • Plot stress-strain curves

2. Curve fitting:

  • Choose mathematical form (linear, polynomial, exponential, etc.)
  • Fit parameters to experimental data
  • Validate with independent tests

3. Multi-dimensional testing:

  • Test in multiple directions
  • Apply combined loading (tension + shear)
  • Determine all material constants

Theoretical Approach

For some materials, constitutive equations can be derived from:

  • Molecular structure
  • Thermodynamic principles
  • Statistical mechanics

This approach is more advanced but provides deeper understanding.

Examples in Voice Science

Steel String

A steel string on a guitar follows Hooke’s law very well:

  • Linear stress-strain relationship
  • Constant E (modulus of elasticity)
  • No time dependence
  • Simple to model and predict

Vocal Fold Tissue

Vocal fold tissue has complex constitutive behavior:

  • Nonlinear: Stiffens with increasing strain
  • Viscoelastic: Time-dependent response
  • Anisotropic: Different properties along and across fibers
  • Layered: Each layer has different properties

The constitutive equation must account for all these factors, making analysis challenging.

Summary

Constitutive equations are fundamental to continuum mechanics, relating stress to strain and defining material behavior:

Hooke’s Law (σ = Eε):

  • Simplest constitutive equation
  • Valid for linear elastic materials
  • E is the elastic modulus (stiffness)
  • Limited applicability to biological tissues

Nonlinear Elasticity:

  • Stress not proportional to strain
  • Common in biological materials
  • More complex mathematical forms
  • Still elastic (returns to original shape)

Ideal Gas Law:

  • Constitutive equation for gases
  • Relates pressure to volume
  • Relevant for air in vocal tract

Complex Constitutive Equations:

  • Three-dimensional tensor formulations
  • Include anisotropy, viscoelasticity
  • Determined experimentally
  • Necessary for accurate modeling of vocal fold tissue

Understanding constitutive equations enables:

  • Prediction of tissue behavior under stress
  • Design of mechanical models of phonation
  • Interpretation of experimental measurements
  • Development of treatments for voice disorders

Key Takeaways

  • ✅ Constitutive equations relate stress to strain, defining material behavior
  • ✅ Hooke’s law (σ = Eε) describes linear elastic materials with constant stiffness E
  • ✅ Many biological tissues are nonlinearly elastic—stress-strain relationship is curved
  • ✅ Ideal gas law is a constitutive equation relating pressure to volume
  • ✅ Real materials often require complex constitutive equations including anisotropy and time dependence
  • ✅ Constitutive equations are determined experimentally through mechanical testing

Further Reading

  1. Fung, Y. C. (1981). Biomechanics: Mechanical properties of living tissues. New York: Springer-Verlag.
  2. Alipour-Haghighi, F., & Titze, I. R. (1991). Elastic models of vocal fold tissues. Journal of the Acoustical Society of America, 90, 1326-1331.
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