# Elastic Modulus

## Static Modulus

An elastic modulus (also known as modulus of elasticity) is the unit of measurement of an object’s or substance’s resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region:[1] A stiffer material will have a higher elastic modulus.

An elastic modulus has the form δ = stress/strain {displaystyle delta {stackrel {text{def}}{=}} {frac {text{stress}}{text{strain}}}}where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter. Since strain is a dimensionless quantity, the units of δ{displaystyle delta } will be the same as the units of stress.[2]

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:

1. 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 of tensile stress to tensile strain. It is often referred to simply as the elastic modulus.
2. The shear modulusor modulus of rigidity (G or {displaystyle mu ,}μ Lamé second parameter) 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 shear modulus is part of the derivation of viscosity.
3. 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, and is the inverse of compressibility. The bulk modulus is an extension of Young’s modulus to three dimensions.

Two other elastic moduli are Lamé’s first parameterλ, and P-wave modulus, M, as used in table of modulus comparisons given below references.

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 at the end of page.

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 for this group is always zero.

In some texts, the modulus of elasticity is referred to as the elastic constant, while the inverse quantity is referred to as elastic modulus.

## Dynamic modulus

Dynamic modulus (sometimes complex modulus[3]) is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free or forced vibration tests, in shear, compression, or elongation). It is a property of viscoelastic materials.

Viscoelasticity is studied using dynamic mechanical analysis where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.

• In purely elasticmaterials the stress and strain occur in phase, so that the response of one occurs simultaneously with the other.
• In purely viscousmaterials, there is a phase difference between stress and strain, where strain lags stress by a 90 degree ({displaystyle pi /2}π/2 radian) phase lag.
• Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.[4]

The storage and loss modulus in viscoelastic materials measure the stored energy, representing the elastic portion, and the energy dissipated as heat, representing the viscous portion.[4]

The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the tan δ {displaystyle tan delta } (cf. loss tangent), which provides a measure of damping in the material. {displaystyle tan delta } tan δ can also be visualized as the tangent of the phase angle (δ{displaystyle delta }) between the storage and loss modulus.

The Elastic Modulus of a broad variety of samples can be measured with the Linseis TMA (Thermonechanical Analyzer). The measurements can be done in the static or dynamic mode.
Various instrument models are available with a broad variety of accessories: The TMA PT 1000 for the temperature range -150 to 1000 °C, The TMA PT 1600 for temperatures from -150 to 1600 and the Kryo TMA which can measure down to 4 Kelvin (-269.15 °C).
The TMA can measure in both directions- compression and tension. More information can be found in the product section.

1. Askeland, Donald R.; Phulé, Pradeep P. (2006). The science and engineering of materials(5th ed.). Cengage Learning. p. 198. ISBN 978-0-534-55396-8.
2. 2.Beer, Ferdinand P.; Johnston, E. Russell; Dewolf, John; Mazurek, David (2009). Mechanics of Materials. McGraw Hill. p. 56ISBN978-0-07-015389-9.
3. The Open University (UK), 2000. T838 Design and Manufacture with Polymers: Solid
properties and design, page 30. Milton Keynes: The Open University.
4. Meyers and Chawla (1999): “Mechanical Behavior of Materials,” 98-103.