Flexural modulus

{{Short description|Intensive property in mechanics}}

In mechanics, the flexural modulus, bending modulus,{{citation

|author1=Zweben, C. |author2=W. S. Smith |author3=M. W. Wardle |name-list-style=amp | title = Test methods for fiber tensile strength, composite flexural modulus, and properties of fabric-reinforced laminates

| journal = Composite Materials: Testing and Design (Fifth Conference)

| publisher = ASTM International

| year = 1979|pages=228–228–35 |doi=10.1520/STP36912S |isbn=978-0-8031-4495-8 }} or modulus of rigidity{{citation

| author1=Hibbeler, R. C. | title = Mechanics of Materials | pages = 108-110 | publisher = Pearson | location = Hoboken, NJ | year = 2015 | edition = 10th | isbn=978-0-1343-1965-0 }} 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.{{citation

| title = D790-03: Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials

| publisher = ASTM International

| location = West Conshohocken, PA

| year = 2003}} The flexural modulus defined using the 2-point (cantilever) and 3-point bend tests assumes a linear stress strain response.{{Cite book|last=Askeland|first=Donald R.|url=https://www.worldcat.org/oclc/903959750|title=The science and engineering of materials|others=Wright, Wendelin J.|year=2016|isbn=978-1-305-07676-1|edition=Seventh|location=Boston, MA|pages=200|oclc=903959750}}

File:Flexural modulus measurement.png

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:

:$$

E_{\mathrm{flex}} = \frac {L^3 F}{4 w h^3 d}

From elastic beam theory

:$d\; =\; \backslash frac\; \{L^3\; F\}\{48\; I\; E\; \}$

and for rectangular beam

: $I\; =\; \backslash frac\{1\}\{12\}wh^3$

thus $E\_\{\backslash mathrm\{flex\}\}\; =\; E$ (Elastic modulus)

For very small strains in isotropic materials – like glass, metal or polymer – flexural or bending modulus of elasticity is equivalent to the tensile modulus (Young's modulus) or compressive modulus of elasticity. However, in anisotropic materials, for example wood, these values may not be equivalent. Moreover, composite materials like fiber-reinforced polymers{{Cite book |last=Tsai |first=S. W. |title=Composite Materials, Testing and Design |date=December 1979 |publisher=ASTM |isbn=9780803103078 |pages=247}} or biological tissues{{Cite journal |last1=Chahine |first1=Nadeen O. |last2=Wang |first2=Christopher C-B. |last3=Hung |first3=Clark T. |last4=Ateshian |first4=Gerard A. |title=Anisotropic strain-dependent material properties of bovine articular cartilage in the transitional range from tension to compression |date=August 2004 |journal=Journal of Biomechanics |volume=37 |issue=8 |pages=1251–1261 |doi=10.1016/j.jbiomech.2003.12.008 |issn=0021-9290 |pmc=2819725 |pmid=15212931}} are inhomogeneous combinations of two or more materials, each with different material properties, therefore their tensile, compressive, and flexural moduli usually are not equivalent.