Constitutive Equations

Constitutive equations are mathematical relations between kinetic (force, stress, moment) and kinematic (displacement, strain, strain-rate) specific to each materials.

The equations are formulated to distinguish between a variety of materials system, like

• Rocks

• Wood

• Soil

• Metals (Al, Mg, Hg, steel..)

• Ceramics 

• Glass

• Polymers

• Composites (any useful combination)

• Liquid (oil, water, tar)

• Gases (reactive, air)

The material behavior depends on

• Temperature

  • Very low

  • Room Temperature
  • Very High

• Environment

  • Corrosive

  • Reactive
  • Moisture

• Rates of Loading

  • Very slow

  • Normal
  • Very High

• Duration (time)

  • Very long (geological, Billions of years)

  • Normal (short, seconds)
  • Very fast (milli, micro, nano, pico, femto seconds)

The material characteristics depends on the spatial scale of interest

• a few nanometers (atomic interactions, dislocations ...)
• a few micrometers (grains, grain boundary,...)
• a few mm (size of a test specimen)
• a few meters (size of a products)
• a few Km (size of a large thunderstorm cell, earthquack zones)

With these various factors determining the response of the medium (material), we still need some mathematical descriptions to idealize the behavior of materials. It is impossible to build one single set of equations. However we strive to formulate their behavior under a subset of those conditions. That is the job of the materials engineers. We describe the material behavior to be used in analysis, design of products and hopefully in the development of materials themselves.

Hence constitutive equations are mathematical idealization of the behavior of materials under a set of given conditions.