Difference Between Modulus of Elasticity and Modulus of Rigidity

Main Difference – Modulus of Elasticity vs. Modulus of Rigidity

Modulus of elasticity and modulus of rigidity are two numbers that are used by materials engineers to describe how a material gets deformed. The main difference between modulus of elasticity and modulus of rigidity is that the modulus of elasticity describes how a material gets deformed when a force is applied at right angles to a surface of an object, causing the material to elongate or shorten while the modulus of rigidity describes how a material gets deformed when a force is applied parallel to a surface of an object, causing one of the surfaces to become shifted with respect to another surface of the same object.

What is the Modulus of Elasticity

Modulus of elasticity (Young modulus) is a number that describes the ratio of stress to strain in an object that is being deformed by a force that is perpendicular to a surface of an object. The stress of a material is the deforming force per unit area. For instance, the figure below shows an object that becomes elongated as a result of a tensile force on it. In this case, the stress (\sigma) is given by:

\sigma=\frac{F}{A}

Since the deforming force acts at right angles to the face of the object, the stress is often called normal stress.

Difference Between Shear Stress and Tensile Stress - Tensile_Stress

Tensile stress from a force acting at right angles to a surface.

The strain is the fractional change in length of the object. Suppose the object had a length x_0 before the deforming force acted on it, and if the object gets elongated by a length \Delta x under the deforming force, then the strain (\varepsilon) is given by:

\varepsilon=\frac{\Delta x}{x_0}

The modulus of elasticity (E) is then given by:

E=\frac{\sigma}{\varepsilon}

What is the Modulus of Rigidity

Modulus of rigidity (shear modulus) is a number that gives the shear stress acting on a material per unit area. Here, the deforming force acts parallel to a face of the object, causing one face to become displaced with respect to another face. This is depicted below:

Difference Between Shear Stress and Tensile Stress - Shear_stress

Shear stress from a force parallel to the surface.

So, shear stress (\sigma) is given as:

\sigma=\frac{F}{A}

This equation has the same form as the equation for normal stress, the difference is in the way the force acts.

The shear strain (\varepsilon) is defined as the ratio of relative displacement between the surfaces to the separation between the surfaces. Here,

\varepsilon=\frac{\Delta x}{l}

Once again the shear modulus (G) is the ratio between shear stress and shear strain:

G=\frac{\sigma}{\varepsilon}

Relationship between Modulus of Elasticity and Modulus of Rigidity

Modulus of elasticity (E) and the modulus of rigidity (G) are related by the following equation:

E=2G\left( 1+\nu\right)

Here, \nu represents a number called Poisson’s ratio given to the particular material. When the material is being stretched in one direction, it gets shortened in a perpendicular direction. In the direction that the material becomes elongated, the axial strain (\varepsilon_a) is defined as the fractional increase in the length. In the direction that the material shortens, the transverse strain (\varepsilon_t) gives the fractional reduction in length. The diagram below illustrates these changes in shape:

Difference Between Modulus of Elasticity and Modulus of Rigidity - Poisson's_Ratio_Illustration

Defining Poisson’s ratio

In this diagram, the axial strain is:

\varepsilon_a=\frac{x'-x_0}{x_0}

The transverse strain is:

\varepsilon_t=\frac{y'-y_0}{y_0}

Note that since the object shortens in the directiopn perpendicular to the force, \varepsilon_t<0. Poisson’s ratio (\nu) is defined as:

\nu=-\frac{\varepsilon_t}{\varepsilon_a}

The minus sign has been introduced to ensure that \nu takes a positive value.

Difference Between Modulus of Elasticity and Modulus of Rigidity

Direction of Force

Modulus of elasticity is used to calculate the deformation of an object when a deforming force acts at right angles to a surface of the object.

Modulus of rigidity is used to calculate deformations when a deforming force acts parallel to the surface of an object.

Change in Shape

Where modulus of elasticity is calculated, the object under the deforming force either gets lengthened or shortened.

Where modulus of rigidity is calculated, one of the surfaces of the object becomes displaced with respect to another surface.

Relative Size

For most materials, the modulus of elasticity is larger than the modulus of rigidity. The exceptions to this rule are the so-called “auxetic” materials which have negative Poisson’s ratios, but these materials are less common.

About the Author: Nipun


Related pages


difference between consonance and alliterationdictionary gazedifferentiate between exons and intronsmorphemes psychologyvulcanization definitionexample of pathetic fallacydicot and monocot stemskanchivaram sareedeoxyribose sugar definitionelk vs caribouwhat is one difference between polyploidy and aneuploidyhow are spores different from seedswhat is the relationship between a monomer and a polymeranalog multimeter diagramidioms and proverbhow does interphase prepare cells for mitosisnoun clause relative clausechamfered definitionsemicolons vs commasmeaning of dawn to duskdistinguish between rotation and revolutionpoems and sonnetsdna prokaryotic or eukaryoticdefinition of melting and boiling pointbroccoli taxonomypolar and nonpolar definitionliterary examples of mooddefine digital multimeterstructure of flagella and ciliawhat is the difference between mass number and atomic numberneurotransmitters and behaviordifferences between a phrase and a clausedifferences between dna and rna nucleotidesdifference stromboli and calzonefunctions of rough and smooth erdifference between eubacteria and cyanobacteriapatronize defineliteral figurative language examplesdifference between endoparasites and ectoparasitesnucleotide nucleoside differencevaporization definition chemistryhomogenous and heterogenous mixturesc pap vs bi papis baking soda the same as bicarbscallion green onion differencedefinition unicellularchemical formula for guar gumspell madamexample of intermolecular hydrogen bondingmajor landforms and water bodies of the earthdifference between modernism and postmodernismunisexual flowers namesdifference between bicarb and baking sodadifference between diffusion and osmosisopen and closed syllables examplesmalaprop examplesretrovirus examples2d heart echounemployment rate calculation methodgustatory imagery examplespoly ethylene-propylenedefinition of karyokinesisconnective tissue proper definitionexamples of polysemesdifference between glucose and sucrosedefinition of monomersseries resonance and parallel resonancewhat does bewilderment meanthe difference between a wasp and a hornetmarginal analysis microeconomicsnames of metallic mineralssyntax grammar definitiondefinition of blank verse in literaturedefine super egodifference between digital and analog modulationketose testunits of kinematic viscositybrass copper difference