mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...

, the flexural modulus or bending modulus is an intensive property that is computed as the ratio of

stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...

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 ASTM International, formerly known as American Society for Testing and Materials, is an international standards organization that develops and publishes voluntary consensus technical standards for a wide range of materials, products, systems, an ...

D790), and uses units of force per area. The flexural modulus defined using the 2-point (cantilever) and 3-point bend tests assumes a linear stress strain response. 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 The second moment of area, or second area moment, or quadratic moment of area and also known as the area moment of inertia, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The ...

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_ = \frac

From elastic beam theory :

d = \frac

and for rectangular beam :

I = \fracwh^3

thus

E_ = E

(

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 ...

) For very small strains in isotropic materials like glass, metal or polymer, flexural or bending

modulus of elasticity 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 ...

is equivalent to the tensile modulus (

Young's modulus Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied ...

) or compressive modulus of elasticity. However, in anisotropic materials, for example wood, these values may not be equivalent. Moreover,

composite materials A composite material (also called a composition material or shortened to composite, which is the common name) is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or ...

like fiber-reinforced polymers or biological tissues are inhomogeneous combinations of two or more materials, each with different material properties, therefore their tensile, compressive, and flexural moduli usually are not equivalent.

Stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a ...

References

Materials science Elasticity (physics) {{materials-sci-stub

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