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Elasticity and Seismic Waves

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Stress and Strain

Stress is defined as force per unit area.

Strain is defined as the amount of deformation anobject experiences compared to its original size andshape.

When a force is applied to a material, it deforms: stress

induces strain The SI unit for stress is the Pascal (symbol Pa), which is

equivalent to one Newton (force) per square meter(unit area), that is N/m2. In Imperial units, stress ismeasured in pound-force per square inch, which is

abbreviated as psi. Strains are dimensionless and are usually expressed as

a decimal fraction or a percentage.

For some materials, displacement is reversible = elastic

materials

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Compression. Stress that acts to shorten an object.

Tension. Stress that acts to lengthen an object.

NormalStress. Stress that acts perpendicular to a surface. Can be

either compressional or tensional. ShearStress that acts parallel to a surface. It can cause one object

to slide over another. It also tends to deform originally rectangular

objects into parallelograms. The most general definition is that

shear acts to change the angles in an object.

HydrostaticStress (usually compressional) that is uniform in all

directions. Stress in the earth is nearly hydrostatic. The term for

uniform stress in the earth is lithostatic.

DirectedStress. Stress that varies with direction. Stress under a

stone slab is directed; there is a force in one direction but nocounteracting forces perpendicular to it.

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LongitudinalorLinearStrain. Strain that changes thelength of a line without changing its direction. Can beeither compressional or tensional.

Compressionstrain. Longitudinal strain that shortens anobject.

Tension. Longitudinal strain that lengthens an object.

ShearStrain that changes the angles of an object. Shearcauses lines to rotate.

HomogeneousStrain. Uniform strain. Straight lines inthe original object remain straight. Parallel lines remainparallel. Circles deform to ellipses. Note that this

definition rules out folding, since an originally straightlayer has to remain straight.

InhomogeneousStrain. How real geology behaves.Deformation varies from place to place. Lines may bend

and do not necessarily remain parallel.

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Stress-strain relation

Strain is proportional to stress = Hookes law

Stress-strain relation: Elastic domain:

Stress-strain relation is linear

Hookes law applies

Beyond elastic domain: Initial shape not recovered when stress is removed

Plastic deformation

Eventually stress > strength of material => failure

Failure can occur within the elastic domain = brittlebehavior

Strain as a function of time under stress: Elastic = no permanent strain

Plastic = permanent strain

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Elastic Modulus

An elastic modulus, or modulus of elasticity, isthe mathematical description of an object orsubstance's tendency to be deformed elastically(i.e., non-permanently) when a force is applied to

it.

where lambda () is the elastic modulus. If stressis measured in Pascal, since strain is adimensionless quantity, then the units of arePascal as well.

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Young's modulus (E) describes tensile elasticity, or the

tendency of an object to deform along an axis when opposing

forces are applied along that axis; it is defined as the ratio oftensile stress to tensile strain. It is often referred to simply as

the elastic modulus.

The shear modulus or modulus of rigidity (G or ) describes

an object's tendency to shear (the deformation of shape at

constant volume) when acted upon by opposing forces; it is

defined as shear stress over shear strain.

The bulk modulus (K) describes volumetric elasticity, or the

tendency of an object to deform in all directions when

uniformly loaded in all directions; it is defined as volumetric

stress over volumetric strain.

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Poissons ratio

When an isotropic elastic object is subject toelongation or compressive stress z a strain z

results in the direction of the stress. At the

same time a strain arises in the transversedirections, x , y. Ifz is an elongation, then

x = y are compressive. The ratio

= x/z = y/z

is called Poissons ratio.