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

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
 is applied to it. The elastic modulus of an object is defined as the slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 of its stress-strain curve
Stress-strain curve

File:Metal yield.svgDuring testing of a material sample, the stress?strain curve is a graphical representation of the relationship between Stress , derived from measuring the load applied on the sample, and Strain , derived from measuring the deformation of the sample, i.e....
 in the elastic deformation region:

where ? (lambda) is the elastic modulus; stress
Stress (physics)

In continuum mechanics, stress is a measure of the average amount of force exerted per unit area. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces....
 is the force causing the deformation divided by the area to which the force is applied; and strain
Strain (materials science)

In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal Deformation s of a Continuum mechanics....
 is the ratio of the change caused by the stress to the original state of the object.






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

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
 is applied to it. The elastic modulus of an object is defined as the slope
Slope

Slope is used to describe the steepness, incline, gradient, or grade of a line . A higher slope value indicates a steeper incline. The slope is defined as the ratio of the "rise" divided by the "run" between two points on a line, or in other words, the ratio of the altitude change to the horizontal distance between any two point...
 of its stress-strain curve
Stress-strain curve

File:Metal yield.svgDuring testing of a material sample, the stress?strain curve is a graphical representation of the relationship between Stress , derived from measuring the load applied on the sample, and Strain , derived from measuring the deformation of the sample, i.e....
 in the elastic deformation region:

where ? (lambda) is the elastic modulus; stress
Stress (physics)

In continuum mechanics, stress is a measure of the average amount of force exerted per unit area. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces....
 is the force causing the deformation divided by the area to which the force is applied; and strain
Strain (materials science)

In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal Deformation s of a Continuum mechanics....
 is the ratio of the change caused by the stress to the original state of the object. If stress is measured in pascal
Pascal (unit)

The pascal is the SI derived unit of pressure, stress , Young's modulus and tensile strength. It is a measure of force per unit area i.e. equivalent to one newton per square meter or one joule per cubic meter....
s, since strain is a unitless ratio, then the units of ? are pascals as well. An alternative definition is that the elastic modulus is the stress required to cause a sample of the material to double in length. This is not realistic for most materials because the value is far greater than the yield stress of the material or the point where elongation becomes nonlinear, but some may find this definition more intuitive.

Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are

  • Young's modulus
    Young's modulus

    In solid mechanics, Young's modulus is a measure of the stiffness of an isotropic elastic material. It is also known as the Young modulus, modulus of elasticity, elastic modulus or tensile modulus....
     (E) describes tensile elasticity
    Elasticity (physics)

    In physics, elasticity is the physical property of a material when it deforms under stress , but returns to its original shape when the stress is removed....
    , 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 of tensile stress to tensile strain
    Strain (materials science)

    In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal Deformation s of a Continuum mechanics....
    . It is often referred to simply as the elastic modulus.
  • The shear modulus
    Shear modulus

    In materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or ?, is defined as the ratio of shear stress to the shear strain:...
     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
    Shear stress

    File:Shear stress.JPGA shear stress, denoted , is defined as a stress which is applied parallel or tangent to a face of a material, as opposed to a normal stress which is applied perpendicularly....
     over shear strain
    Shear strain

    Shear strain is a strain that acts parallel to the surface of a material that it is acting on. Normal strain, in contrast, acts perpendicular to the surface....
    . The shear modulus is part of the derivation of viscosity
    Viscosity

    Viscosity is a measure of the Drag of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness"....
    .
  • The bulk modulus
    Bulk modulus

    The bulk modulus of a substance measures the substance's resistance to uniform compression. It is defined as the pressure increase needed to cause a given relative decrease in volume....
     (K) describes volumetric elasticity, or the tendency of an object's volume to deform when under pressure; it is defined as volumetric stress
    Stress (physics)

    In continuum mechanics, stress is a measure of the average amount of force exerted per unit area. It is a measure of the intensity of the total internal forces acting within a body across imaginary internal surfaces, as a reaction to external applied forces and body forces....
     over volumetric strain, and is the inverse of compressibility
    Compressibility

    In thermodynamics and fluid mechanics, compressibility is a Measure of the relative volume change of a fluid or solid as a response to a pressure change....
    . The bulk modulus is an extension of Young's modulus to three dimensions.


Three other elastic moduli are Poisson's ratio
Poisson's ratio

Poisson's ratio , named after Simeon Poisson, is the ratio of the contraction or transverse strain , to the extension or axial strain .When a sample cube of a materials is stretched in one direction, it tends to contract in the other two directions perpendicular to the direction of stretch....
, Lamé's first parameter, and P-wave modulus
P-wave modulus

In linear elasticity, the P-wave modulus , also known as the longitudinal modulus, is one of the elastic moduli available to describe isotropic homogeneous materials....
.

Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below.

Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus
Young's modulus

In solid mechanics, Young's modulus is a measure of the stiffness of an isotropic elastic material. It is also known as the Young modulus, modulus of elasticity, elastic modulus or tensile modulus....
 is always zero.

See also

  • Stiffness
    Stiffness

    Stiffness is the resistance of an Elasticity body to deformation by an applied force. It is an intensive and extensive properties....
  • Elastic limit
  • Elasticity (physics)
    Elasticity (physics)

    In physics, elasticity is the physical property of a material when it deforms under stress , but returns to its original shape when the stress is removed....
  • Impulse excitation technique
    Impulse excitation technique

    PracticalThe impulse excitation technique is a nondestructive test method that uses natural frequency, dimensions and mass of a test-piece to determine Young's modulus, Shear modulus, Poisson's ratio and damping coefficient....
  • Tensile strength
    Tensile strength

    Tensile strength , or is the Stress at which a material breaks or permanently deforms. Tensile strength is an Intensive and extensive properties and, consequently, does not depend on the size of the test specimen....
  • Elastic wave
  • Dynamic modulus
    Dynamic modulus

    Dynamic modulus is the ratio of stress to strain under vibratory conditions . It is a property of viscoelastic materials....
  • Transverse isotropy
    Transverse isotropy

    A transversely isotropic material is symmetry about an axis that is normal to a plane of isotropy. This transverse plane has infinite planes of symmetry and thus, within this plane, the material properties are same in all directions....
  • Flexural modulus
    Flexural modulus

    The flexural modulus is the ratio of stress to Strain in flexural deformation, or the tendency for a materal to bend. It is an intensive property....