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ELASTICITY, ISOTROPIC, ANISOTROPIC, HOOKEAN

Isotropic materials are those that have the same value for a given property in all directions.Anisotropic materials are those that have different values for a given property in different directions.

Types of deformation:Permanent, Recoverable:

Elastic response    Viscous response         Plastic response

Derived response models:

Mechanical combinations of elastic, viscous and plastic response

Elastic Response

Strain is recoverable instantaneously

Response to stress is instantaneous

The relationship between stress and strain is linear

For a normal stress, s , producing an extension, e , the relationship is given by the equation: s = Ee

where the constant of proportionality, E, is call Young's modulus. This equation is called is Hooke's Law

Note that E and all of the other elastic moduli described below have the units of stress, because strain is dimensionless

Mechanical Analog for

Hookian Elastic Materials

Extension of a spring, note that:

strain-time indicates instantaneous response to stress

strain is also instantaneously recoverable

stress-strain relationship is linear

Other Elastic Responses

For dilation, the constant of proportionality is called the bulk modulus, K, where: s H = K (D V/V)

For a shear stress, s S , the constant of proportionality between shear stress and shear strain (g) is called the shear modulus, G, where: s S = Gc

(The shear modulus is sometimes called the modulus of rigidity)

Poisson's ratio (n ) is another property of an elastic material and is equal to the ratio of lateral contraction to longitudinal extension during tensional loading

Relationships among elastic constants:

G = E/2(1+n ) = 3K(1-2n )/2(1+n )

Viscous Response

Newton's concept of fluid viscosity: fluid material

Laminar flow occurs in the fluid between rigid plates as the plates slide over one another

The fluid exerts an internal frictional resistance to the movement which is called the viscosity of the fluid

Because of this resistance, force or shear stress must be continually applied to keep the plate moving

For such fluids, the strain is proportional to the stress and the time it acts, and is inversely proportional to the viscosity, h , of the fluid

Viscous Response

The strain is proportional to the stress and the time it acts, and is inversely proportional to the viscosity, h , of the fluid

This relationship is given by the following equations for normal stress and shear stress:

e = s t/h or s = h e /t s = h de /dt

similarly s s = h dg /dt

Note that the longer the stress is applied, the greater the strain for a given stress.

Note that the shorter the time involved, the greater the stress required to produce a given strain.

Viscous Response

A material that exhibits this property is said to be a Newtonian fluid

In a Newtonian fluid, the viscosity is constant for all rates of the application of stress or rates of strain - Figure (a) below

However, there are fluids whose viscosity varies as a function of strain rate these are called non-newtonian fluids - Figure (b) below

Most highly viscous fluids are non-newtonian

The change in viscosity as a function of strain rate is generally attributed to structural changes within the material

It may be that non-newtonian viscous behavior is a close approximation to that of real rocks at depth

Geological Implications of Viscous Behavior

The slower the strain rate the lower the stress required for deformation.

For very low strain rates (e.g. over geologic time), low stresses may be able to deform even very viscous fluids. This has major geological implications

Mechanical Analog of

Viscous Response

A mechanical analog for viscous behavior is the dashpot, a loose piston moving through a cylindrical tube whose walls are lubricated by some fluid. e.g. a hydraulic cylinder on a rowing machine. Note that:

Strain is time dependent (the longer the time, the more the strain)

(contrast this with elastic behavior)

The stress-strain rate relationship is linear

Plastic Response

An ideal plastic body does not yield until some critical stress (s c) is reached.

Most materials that approach being plastic exhibit elastic properties below this point so this critical stress is at the elastic limit or yield point of the material

Mechanical analog - a sliding mass (where s c is analogous to frictional resistance)

Plastic Response

Beyond the yield point a plastic material strains continuously and permanently

The plastic material flows but is different from viscous materials since it has some fundamental strength (the critical stress) which the viscous fluid does not have

Also, in ideally plastic materials, the strain takes place in localized regions where the critical stress has been reached, whereas, ideally viscous materials show deformation throughout the material wherever a deviatoric stress is present

Elastico-Plastic Response

As noted above, most materials that display plastic behavior display elastic behavior below the critical stress

Such materials are said to be elasticoplastic (sometimes called a Prandtl modal) and can be characterized with an analog model of a spring attached to a sliding mass

The mass requires a critical stress to begin moving against friction, but a certain amount of elastic strain is sustained by the spring before the critical stress is reached

Once the mass begins to move (strain) the applied stress has exceeded the critical stress and a constant stress keeps it moving (straining)

This is an example of a derived response model - one that includes a combination of one or more of the simple or fundamental models (elastic, plastic, viscous) in either series or parallel combinations

Derived Response Models

Most real materials do not respond to stress in a manner exactly like any of the three elementary responses

In many cases, combinations of these models do approach the behavior of real rock materials

Other derived response models

Plasticoviscous reponse

Firmoviscous response

Elasticoviscous response

Plasticoviscous Response (Bingham model)

Analog dashpot plus a sliding mass in parallel

Visco-Elastic Response

(Kelvin model)

Also called a firmoviscous response

Analog spring and dashpot in parallel

The dashpot causes the elastic response to be delayed so that the elastic (recoverable) part of the response is time dependent both on loading and unloading

Stressstrain relationship is linear with no fundamental strength for the material

Elasticoviscous Response (Maxwell model)

Analog spring plus dashpot in series

Response is instantaneous and strain may increase infinitely

Recovery of the elastic strain is instantaneous, but the viscous strain is not recovered