# Flexural modulus

In mechanics, the flexural modulus or bending modulus[1] is an intensive property that is computed as the ratio of stress to strain in flexural deformation, or the tendency for a material to resist bending. It is determined from the slope of a stress-strain curve produced by a flexural test (such as the ASTM D790), and uses units of force per area.[2] The flexural modulus defined using the 3-point bend test assumes a linear stress strain response.[3]

Flexural modulus measurement

For a 3-point test of a rectangular beam behaving as an isotropic linear material, where w and h are the width and height of the beam, I is the second moment of area of the beam's cross-section, L is the distance between the two outer supports, and d is the deflection due to the load F applied at the middle of the beam, the flexural modulus:[4]

${\displaystyle E_{\mathrm {flex} }={\frac {L^{3}F}{4wh^{3}d}}}$

From elastic beam theory

${\displaystyle d={\frac {L^{3}F}{48IE}}}$

and for rectangular beam

${\displaystyle I={\frac {1}{12}}wh^{3}}$

thus ${\displaystyle E_{\mathrm {flex} }=E}$ (Elastic modulus)

Ideally, flexural or bending modulus of elasticity is equivalent to the tensile modulus (Young's modulus) or compressive modulus of elasticity. In reality, these values may be different, especially for polymers which are often viscoelastic (time dependent) materials.[3] Equivalence of the flexural modulus with Young's modulus also assumes equivalent compressive and tensile moduli as bend specimens have both tensile and compressive strain. Polymers in particular often have drastically different compressive and tensile moduli for the same material.[5]

## References

1. ^ Zweben, C.; W. S. Smith & M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International: 228–228–35, doi:10.1520/STP36912S, ISBN 978-0-8031-4495-8
2. ^ D790-03: Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials, West Conshohocken, PA: ASTM International, 2003
3. ^ a b Askeland, Donald R. (2016). The science and engineering of materials. Wright, Wendelin J. (Seventh ed.). Boston, MA. p. 200. ISBN 978-1-305-07676-1. OCLC 903959750.
4. ^ Zweben, C.; W. S. Smith & M. W. Wardle (1979), "Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates", Composite Materials: Testing and Design (Fifth Conference), ASTM International: 228–228–35, doi:10.1520/STP36912S, ISBN 978-0-8031-4495-8
5. ^ Tsai, S. W. (December 1979). Composite Materials, Testing and Design. ASTM. p. 247. ISBN 9780803103078.