Elasticity I

Ali K. Abdel-Fattah

Elasticity• In physics, elasticity is a physical property of materials which return to their original shape after

they are deformed.

• When an elastic material is deformed due to an external force, it experiences internal forces that oppose the deformation and restore it to its original state if the external force is removed.

• There are various elastic moduli, such as Young's modulus, the shear modulus, and the bulk modulus, all of which are measures of the inherent stiffness of a material as a resistance to deformation under an applied load. The various moduli apply to different kinds of deformation.

• The elasticity of materials is described by a stress-strain curve, which shows the relation between stress (the average restorative internal force per unit area) and strain (the relative deformation).

• To understand the propagation of elastic waves we need to describe kinematically the deformation of our medium and the resulting forces (stress). The relation between deformation and stress is governed by elastic constants.

Stress

• In continuum mechanics, stress is a measure of the internal forces acting within a deformable body.

• Quantitatively, stress is a measure of the average force per unit area of a surface within the body on which internal forces act.

• These internal forces arise as a reaction to external forces applied to the body.

Stress

• Forces acting on elements of the continuum are of two types:

Body forces (e.g. gravity) Surface forces (Traction), exerted by external forces applied to the boundary of the body. Traction is the force per unit area acting on a surface within the continuum and can be defined by:

Stress

• Triaxial Stress– Stresses act along three orthogonal axes,

perpendicular to faces of solid, e.g. Stretching a bar.

• Axial Stress– Stresses act in one direction only

• Pressure– Forces act equally in all directions perpendicular to

faces of a body, e.g. Pressure on a cube in water

Stress• Surface force (Traction), can be decomposed in to two

perpendicular components: normal and shear (tangential) stresses.

– Normal stresses act normally to the surface of the body

• Change in volume• Associated with P wave radiation• Three components of stress (xx & yy & zz)

– Shear stresses act tangentially to the surface of the body

• No change in volume• Change in shape• Fluids and air can not support shear stresses• Associated with S wave radiation• Six components of stresses (xy & xz & yx&

yz& zx& zy)

Shear angle

Stress Tensor• Stress is a symmetric tensor quantity

consists of nine components. Normal and shear stress components are simply defined by the diagonal and off-diagonal elements of the tensor:

• Sense of stress:– Positive normal stress tension;

stresses towards the exterior– Negative normal stress

compression; stresses towards the interior

Stress units of force per area(1 N/m2 = 1 Pa)

Principals, spherical and deviatoric stresses

• Principal stresses: normal to surface with no shear tractions. Principal stresses are the normal stresses on these surfaces. Principal stress axes are eigenvectors of the stress tensor and principal stresses are the eigenvalues.

• The stress tensor can be decomposed into tensors; spherical and deviatoric stress tensors.– Spherical stress tensor, is the average of the summation of the

principal stresses (the mean normal stress), [(xx+yy+ zz)/3]. Hydrostatic Pressure is a special state of stress where the stresses are equal in all directions, normal stresses with zero shear stress, occurs in fluids (liquids/gasses), e.g. air pressure, causes only volumetric changes

– Deviatoric stress tensor, stress tensor minus the mean normal stress (1/3 of the trace of the stress tensor). It is important because within the Earth large compressive stresses can mask the stresses from tectonic forces.

Problem 1

• Determine the deviator stress values for the stress tensors

100

0106

0610

ij

320

294

0412

ij

Strain

• Strain is the deformation caused in the body, and is expressed as the ratio of change in length or volume to its original length or volume

The symmetric strain tensor describes deformation in a body due to differential motion in the body;

=

zzzyzx

yzyyyx

xzxyxx

eee

eee

eee

Strain

• There are two basic types of strain:

– Normal strain,

• A normal strain results from a normal stress. Deforms a square into a rectangle. Angles between sides remain unchanged.

• In which the deformation is perpendicular to the faces of a given volume element (i.e., normal to the faces). The examination of the figure should convince you that the displacement field is divergent that means there is a change in volume and the curl or rotation is zero that means no change in shape.

– Shear strain,

• A shear strain results from a shear stress. Deforms a square into a lozenge. Angles between sides change.

• In which the deformation is parallel to the surfaces. In this case it is the divergence that is zero that means no change in volume, while the rotation or curl is non-zero that means there is a change in shape.

Strain

• The change in volume of a material is known as a dilatation which is given by the summation of the diagonal terms of the strain tensor.

• The change in shape of a material is known as rotation or curl which is described by the off-diagonal terms of the strain tensor.

Hooke’s Law• Stress is proportional to strain:

– At low to moderate strains: Hooke’s Law applies and a solid body is said to behave elastically, i.e. will return to original form when stress removed.

– At high strains: the elastic limit is exceeded and a body deforms in a plastic or ductile manner: it is unable to return to its original shape, being permanently strained, or damaged.

– At very high strains: a solid will fracture, e.g. in earthquake faulting. Constant of proportionality is called the modulus, and is ratio of stress to strain, e.g. Young’s modulus in triaxial strain.