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The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law.It deals with the case of linear elastic materials.Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used.
The first two scale factors of the coordinate system are independent of the last coordinate: ∂h 1 / ∂x 3 = ∂h 2 / ∂x 3 = 0, otherwise extra terms appear. The stream function has some useful properties: Since −∇ 2 ψ = ∇ × (∇ × ψ) = ∇ × u, the vorticity of the flow is just the negative of the Laplacian of ...
The constitutive relation is expressed as a linear first-order differential equation: = + ˙ This model represents a solid undergoing reversible, viscoelastic strain. Upon application of a constant stress, the material deforms at a decreasing rate, asymptotically approaching the steady-state strain.
The concept of a continuum underlies the mathematical framework for studying large-scale forces and deformations in materials. Although materials are composed of discrete atoms and molecules, separated by empty space or microscopic cracks and crystallographic defects, physical phenomena can often be modeled by considering a substance distributed throughout some region of space.
This is constitutive equation is also called the Newtonian law of viscosity. Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state .
For finite element structural analysis in time steps, it is advantageous to convert the constitutive law to a rate-type form. This may be achieved by approximating C g ( θ ) {\displaystyle C_{g}(\theta )} with a Kelvin chain model (or the associated relaxation function with a Maxwell chain model).
This explains the duality in Darcy's law as a governing equation and a defining equation for absolute permeability. The non-linearity of the material derivative in balance equations in general, and the complexities of Cauchy's momentum equation and Navier-Stokes equation makes the basic equations in classical mechanics exposed to establishing ...
This simple model is the basis for the "law of the wall", which is a surprisingly accurate model for wall-bounded, attached (not separated) flow fields with small pressure gradients. More general turbulence models have evolved over time, with most modern turbulence models given by field equations similar to the Navier–Stokes equations .