Dynamic modulus (sometimes complex modulus
[The Open University (UK), 2000. ''T838 Design and Manufacture with Polymers: Solid properties and design'', page 30. Milton Keynes: The Open University.]) 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
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
materials.
Viscoelastic stress–strain phase-lag
Viscoelasticity
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly wi ...
is studied using
dynamic mechanical analysis
Dynamic mechanical analysis (abbreviated DMA) is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a subst ...
where an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured.
*In purely
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
materials the stress and strain occur in
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
, so that the response of one occurs simultaneously with the other.
*In purely
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
materials, there is a
phase difference
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
between stress and strain, where strain lags stress by a 90 degree (
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
) phase lag.
*Viscoelastic materials exhibit behavior somewhere in between that of purely viscous and purely elastic materials, exhibiting some phase lag in strain.
[Meyers and Chawla (1999): "Mechanical Behavior of Materials," 98-103.]
Stress and strain in a viscoelastic material can be represented using the following expressions:
*Strain:
*Stress:
[
where
: where is frequency of strain oscillation,
: is time,
: is phase lag between stress and strain.
The stress relaxation modulus is the ratio of the stress remaining at time after a step strain was applied at time :
,
which is the time-dependent generalization of ]Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
.
For visco-elastic solids, converges to the equilibrium shear modulus:
:.
The fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of the shear relaxation modulus is (see below).
Storage and loss modulus
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.[ The tensile storage and loss moduli are defined as follows:
*Storage:
*Loss: ][
Similarly we also define shear storage and shear loss moduli, and .
Complex variables can be used to express the moduli and as follows:
:
: ][
where is the ]imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
.
Ratio between loss and storage modulus
The ratio of the loss modulus to storage modulus in a viscoelastic material is defined as the , (cf. loss tangent
Dielectric loss quantifies a dielectric material's inherent dissipation of electromagnetic energy (e.g. heat). It can be parameterized in terms of either the loss angle ''δ'' or the corresponding loss tangent tan ''δ''. Both refer to the ...
), which provides a measure of damping in the material. can also be visualized as the tangent of the phase angle () between the storage and loss modulus.
Tensile:
Shear:
For a material with a greater than 1, the energy-dissipating, viscous component of the complex modulus prevails.
See also
*Dynamic mechanical analysis
Dynamic mechanical analysis (abbreviated DMA) is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers
A polymer (; Greek '' poly-'', "many" + ''-mer'', "part")
is a subst ...
* Elastic 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 ...
* Palierne equation
References
Physical quantities
Solid mechanics
Non-Newtonian fluids