Elastic modulus - Citizendia

An elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically (i. e. , non-permanently) when a force is applied to it. In Physics, a force is whatever can cause an object with Mass to Accelerate. The elastic modulus of an object is defined as the slope of its stress-strain curve in the elastic deformation region:

$\lambda \ \stackrel{\text{def}}{=}\ \frac {\text{stress}} {\text{strain}}$

where λ (lambda) is the elastic modulus; stress is the force causing the deformation divided by the area to which the force is applied; and strain is the ratio of the change caused by the stress to the original state of the object. Slope is used to describe the steepness incline gradient or grade of a straight line. During testing of a material sample the stress–strain curve is a graphical representation of the relationship between stress, derived from measuring the load applied on the Stress is a measure of the average amount of Force exerted per unit Area. If stress is measured in pascals, since strain is a unitless ratio, then the units of λ are pascals as well. An alternative definition is that the elastic modulus is the stress required to cause a sample of the material to double in length. This is not realistic for most materials because the value is far greater than the yield stress of the material or the point where elongation becomes nonlinear, but some may find this definition more intuitive.

Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are

Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material A material is said to be elastic if it deforms under stress (e Stress is a measure of the average amount of Force exerted per unit Area. It is often referred to simply as the elastic modulus.
The shear modulus or modulus of rigidity (G or $μ$ ) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. In Materials science, shear modulus or modulus of rigidity, denoted by G, or sometimes S or μ, is defined as the ratio of Shear A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material Shear strain is a strain that acts parallel to the face of a material that it is acting on The shear modulus is part of the derivation of viscosity. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress.
The bulk modulus (K) describes volumetric elasticity, or the tendency of an object's volume to deform when under pressure; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. Stress is a measure of the average amount of Force exerted per unit Area. In Thermodynamics and Fluid mechanics, compressibility is a measure of the relative volume change of a Fluid or Solid as a response The bulk modulus is an extension of Young's modulus to three dimensions.

Three other elastic moduli are Poisson's ratio, Lamé's first parameter, and P-wave modulus. Poisson's ratio ( ν) named after Simeon Poisson, is the ratio of the relative contraction strain, or transverse strain (normal to In Linear elasticity, the Lamé parameters are the two parameters λ also called Lamé's first parameter. In Linear elasticity, the P-wave modulus M also known as the longitudinal modulus, is one of the elastic moduli available to describe isotropic homogeneous

Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Isotropy is uniformity in all directions Precise definitions depend on the subject area Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below.

Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. This also implies that Young's modulus is always zero. In Solid mechanics, Young's modulus (E is a measure of the Stiffness of an isotropic elastic material

Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
	$(\lambda,\,\mu)$	$(E,\,\mu)$	$(K,\,\lambda)$	$(K,\,\mu)$	$(\lambda,\,\nu)$	$(\mu,\,\nu)$	$(E,\,\nu)$	$(K,\, \nu)$	$(K,\,E)$	$(M,\,\mu)$
$K=\,$	$\lambda+ \frac{2\mu}{3}$	$\frac{E\mu}{3(3\mu-E)}$			$\lambda\frac{1+\nu}{3\nu}$	$\frac{2\mu(1+\nu)}{3(1-2\nu)}$	$\frac{E}{3(1-2\nu)}$			$M - \frac{8\mu}{3}$
$E=\,$	$\mu\frac{3\lambda + 2\mu}{\lambda + \mu}$		$9K\frac{K-\lambda}{3K-\lambda}$	$\frac{9K\mu}{3K+\mu}$	$\frac{\lambda(1+\nu)(1-2\nu)}{\nu}$	$2\mu(1+\nu)\,$		$3K(1-2\nu)\,$		$\mu\frac{3M-4\mu}{M-\mu}$
$\lambda=\,$		$\mu\frac{E-2\mu}{3\mu-E}$		$K-\frac{2\mu}{3}$		$\frac{2 \mu \nu}{1-2\nu}$	$\frac{E\nu}{(1+\nu)(1-2\nu)}$	$\frac{3K\nu}{1+\nu}$	$\frac{3K(3K-E)}{9K-E}$	$M - 2\mu\,$
$\mu=\,$			$3\frac{K-\lambda}{2}$		$\lambda\frac{1-2\nu}{2\nu}$		$\frac{E}{2+2\nu}$	$3K\frac{1-2\nu}{2+2\nu}$	$\frac{3KE}{9K-E}$
$\nu=\,$	$\frac{\lambda}{2(\lambda + \mu)}$	$\frac{E}{2\mu}-1$	$\frac{\lambda}{3K-\lambda}$	$\frac{3K-2\mu}{2(3K+\mu)}$					$\frac{3K-E}{6K}$	$\frac{M - 2\mu}{2M - 3\mu}$
$M=\,$	$\lambda+2\mu\,$	$\mu\frac{4\mu-E}{3\mu-E}$	$3K-2\lambda\,$	$K+\frac{4\mu}{3}$	$\lambda \frac{1-\nu}{\nu}$	$\mu\frac{2-2\nu}{1-2\nu}$	$E\frac{1-\nu}{(1+\nu)(1-2\nu)}$	$3K\frac{1-\nu}{1+\nu}$	$3K\frac{3K+E}{9K-E}$

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Stiffness is the resistance of an elastic body to Deformation by an applied Force. The yield strength or yield point of a Material is defined in Engineering and Materials science as the stress at which a material A material is said to be elastic if it deforms under stress (e Practical The impulse excitation technique is a Nondestructive test method that uses Natural frequency, dimensions and mass of a test-piece to determine Young's Tensile strength \sigma_{UTS} or S_U is the Stress at which a material breaks or permanently deforms Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions Dynamic modulus is the ratio of stress to strain under vibratory conditions (calculated from data obtained from either free or forced vibration tests in shear compression A transversely isotropic material is symmetric about an axis that is normal to a plane of Isotropy.

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