13.3 Constitutive equations for various materials

Constitutive equations describe how materials behave under different conditions. They're crucial for understanding how stuff reacts to forces and deformations. This topic covers various material types, from simple elastic to complex fluids.

We'll look at models for elastic, hyperelastic, viscoelastic, and plastic materials. We'll also dive into fluid mechanics, exploring key equations and applications. This knowledge is essential for tackling real-world engineering problems.

Elastic and Hyperelastic Materials

Characteristics and Behavior of Elastic Materials

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• Elastic materials return to original shape after deformation when stress is removed
• Exhibit linear relationship between stress and strain up to elastic limit
• Obey Hooke's law $\sigma = E\epsilon$ where $\sigma$ represents stress, $E$ represents Young's modulus, and $\epsilon$ represents strain
• Young's modulus measures material stiffness, varies for different materials (steel, rubber, wood)
• Poisson's ratio $\nu$ describes lateral contraction when material is stretched longitudinally
• Common elastic materials include metals, ceramics, and some polymers

Hyperelastic Material Models and Applications

• Hyperelastic materials exhibit nonlinear stress-strain relationship
• Can undergo large deformations without permanent changes
• Neo-Hookean model describes incompressible, isotropic hyperelastic materials
  • Strain energy density function: $W = \frac{\mu}{2}(I_1 - 3)$
  • $\mu$ represents shear modulus, $I_1$ is first invariant of left Cauchy-Green deformation tensor
• Mooney-Rivlin model extends Neo-Hookean model for better accuracy
  • Strain energy density function: $W = C_1(I_1 - 3) + C_2(I_2 - 3)$
  • $C_1$ and $C_2$ are material constants, $I_2$ is second invariant of left Cauchy-Green deformation tensor
• Applications include modeling rubber, soft tissues, and elastomers

Viscoelastic and Plastic Materials

Viscoelastic Material Behavior and Models

• Viscoelastic materials exhibit both viscous and elastic characteristics
• Time-dependent response to applied stress or strain
• Creep occurs under constant stress, stress relaxation under constant strain
• Maxwell model represents viscoelastic fluids using spring and dashpot in series
• Kelvin-Voigt model represents viscoelastic solids using spring and dashpot in parallel
• Standard Linear Solid model combines Maxwell and Kelvin-Voigt models for improved accuracy
• Applications include polymers, biological tissues, and some metals at high temperatures

Plastic Deformation and Yield Criteria

• Plastic materials undergo permanent deformation when stress exceeds yield strength
• Elastic-plastic behavior includes both elastic and plastic regions
• Yield criteria determine when plastic deformation begins
  • Von Mises yield criterion based on distortion energy theory
  • Tresca yield criterion based on maximum shear stress theory
• Strain hardening occurs as material becomes stronger during plastic deformation
• Bauschinger effect describes changes in yield strength during cyclic loading
• Applications include metal forming processes, structural analysis, and material selection for engineering design

Fluid Mechanics

Fundamental Concepts and Governing Equations

• Fluid mechanics studies behavior of liquids and gases under various conditions
• Fluids classified as Newtonian (linear stress-strain rate relationship) or non-Newtonian
• Continuity equation expresses conservation of mass in fluid flow
• Momentum equation derived from Newton's second law applied to fluid element
• Energy equation based on first law of thermodynamics for fluid systems
• Navier-Stokes equations describe motion of viscous fluid substances
  • Consist of continuity equation and momentum equations
  • For incompressible Newtonian fluids: $\rho(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla\mathbf{u}) = -\nabla p + \mu\nabla^2\mathbf{u} + \mathbf{f}$
  • $\rho$ represents fluid density, $\mathbf{u}$ velocity vector, $p$ pressure, $\mu$ dynamic viscosity, $\mathbf{f}$ body forces

Applications and Specialized Areas in Fluid Mechanics

• Aerodynamics studies air flow around objects (aircraft, vehicles)
• Hydrodynamics focuses on liquid flow (ship design, hydraulic systems)
• Computational Fluid Dynamics (CFD) uses numerical methods to solve fluid flow problems
• Microfluidics deals with behavior of fluids at microscale (lab-on-a-chip devices)
• Multiphase flow involves simultaneous flow of materials with different phases or chemical properties
• Rheology studies flow of complex fluids with both solid and liquid characteristics (blood, polymers)