Young's modulus

Usage

Young's modulus enables the calculation of the change in the dimension of a bar made of an isotropic elastic material under tensile or compressive loads. For instance, it predicts how much a material sample extends under tension or shortens under compression. The Young's modulus directly applies to cases of uniaxial stress; that is, tensile or compressive stress in one direction and no stress in the other directions. Young's modulus is also used in order to predict the deflection that will occur in a statically determinate beam when a load is applied at a point in between the beam's supports.

Other elastic calculations usually require the use of one additional elastic property, such as the shear modulus ${\displaystyle G}$, bulk modulus ${\displaystyle K}$, and Poisson's ratio ${\displaystyle \nu }$. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For example, calculating physical properties of cancerous skin tissue, has been measured and found to be a Poisson's ratio of 0.43±0.12 and an average Young's modulus of 52 KPa. Defining the elastic properties of skin may become the first step in turning elasticity into a clinical tool. For homogeneous isotropic materials simple relations exist between elastic constants that allow calculating them all as long as two are known: $${\displaystyle \begin{array}{cccccc}E& =G(2+2\nu )& & =K(3-6\nu )& & =\frac{9KG}{3K+G}\\ G& =\frac{E}{2+2\nu }& & =\frac{K(3-6\nu )}{2+2\nu }& & =\frac{E}{3-\frac{E}{3K}}\\ K& =\frac{G(2+2\nu )}{3-6\nu }& & =\frac{E}{3-6\nu }& & =\frac{EG}{9G-3E}\\ \nu & =\frac{E}{2G}-1& & =\frac{1}{2}-\frac{E}{6K}& & =\frac{3K-2G}{2(3K+G)}\end{array}}$$

Linear versus non-linear

Young's modulus represents the factor of proportionality in Hooke's law, which relates the stress and the strain. However, Hooke's law is only valid under the assumption of an elastic and linear response. Any real material will eventually fail and break when stretched over a very large distance or with a very large force; however, all solid materials exhibit nearly Hookean behavior for small enough strains or stresses. If the range over which Hooke's law is valid is large enough compared to the typical stress that one expects to apply to the material, the material is said to be linear. Otherwise (if the typical stress one would apply is outside the linear range), the material is said to be non-linear.

Steel, carbon fiber and glass among others are usually considered linear materials, while other materials such as rubber and soils are non-linear. However, this is not an absolute classification: if very small stresses or strains are applied to a non-linear material, the response will be linear, but if very high stress or strain is applied to a linear material, the linear theory will not be enough. For example, as the linear theory implies reversibility, it would be absurd to use the linear theory to describe the failure of a steel bridge under a high load; although steel is a linear material for most applications, it is not in such a case of catastrophic failure.

In solid mechanics, the slope of the stress–strain curve at any point is called the tangent modulus. It can be experimentally determined from the slope of a stress–strain curve created during tensile tests conducted on a sample of the material.

Directional materials

Young's modulus is not always the same in all orientations of a material. Most metals and ceramics, along with many other materials, are isotropic, and their mechanical properties are the same in all orientations. However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional. These materials then become anisotropic, and Young's modulus will change depending on the direction of the force vector. Anisotropy can be seen in many composites as well. For example, carbon fiber has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain). Other such materials include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures.

Temperature dependence

The Young's modulus of metals varies with the temperature and can be realized through the change in the interatomic bonding of the atoms, and hence its change is found to be dependent on the change in the work function of the metal. Although classically, this change is predicted through fitting and without a clear underlying mechanism (for example, the Watchman's formula), the Rahemi-Li model demonstrates how the change in the electron work function leads to change in the Young's modulus of metals and predicts this variation with calculable parameters, using the generalization of the Lennard-Jones potential to solids. In general, as the temperature increases, the Young's modulus decreases via ${\displaystyle E(T)=\beta (\phi (T){)}^6}$ where the electron work function varies with the temperature as ${\displaystyle \phi (T)={\phi }_0-\gamma \frac{(k_{B}T{)}^2}{{\phi }_0}}$ and ${\displaystyle \gamma }$ is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). ${\displaystyle {\phi }_0}$ is the electron work function at T=0 and ${\displaystyle \beta }$ is constant throughout the change. Additionally, the Rahemi–Li model links the temperature dependence of Young’s modulus in metals to changes in the electron work function, providing a theoretical basis using a generalized Lennard-Jones potential.

Calculation

Young's modulus is calculated by dividing the tensile stress, ${\displaystyle \sigma (\epsilon )}$, by the engineering extensional strain, ${\displaystyle \epsilon }$, in the elastic (initial, linear) portion of the physical stress–strain curve:

$${\displaystyle E\equiv \frac{\sigma (\epsilon )}{\epsilon }=\frac{F/A}{\mathrm{\Delta }L/L_0}=\frac{F{L}_0}{A\mathrm{\Delta }L}}$$ where

• ${\displaystyle E}$ is the Young's modulus (modulus of elasticity);
• ${\displaystyle F}$ is the force exerted on an object under tension;
• ${\displaystyle A}$ is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force;
• ${\displaystyle \mathrm{\Delta }L}$ is the amount by which the length of the object changes (${\displaystyle \mathrm{\Delta }L}$ is positive if the material is stretched, and negative when the material is compressed);
• ${\displaystyle L_0}$ is the original length of the object.

An equivalent definition of Young's modulus is $${\displaystyle E=\frac{L_0}{A}{\frac{d^{2}U}{d{L}^2}|}_{L_0}}$$ where ${\displaystyle U}$ is material's energy.

Force exerted by stretched or contracted material

Young's modulus of a material can be used to calculate the force it exerts under specific strain.

${\displaystyle F=\frac{EA\mathrm{\Delta }L}{L_0}}$

where ${\displaystyle F}$ is the force exerted by the material when contracted or stretched by ${\displaystyle \mathrm{\Delta }L}$.

Hooke's law for a stretched wire can be derived from this formula:

${\displaystyle F=\left(\frac{EA}{L_0}\right)\mathrm{\Delta }L=kx}$

where it comes in saturation

${\displaystyle k\equiv \frac{EA}{L_0}}$ and ${\displaystyle x\equiv \mathrm{\Delta }L.}$

Note that the elasticity of coiled springs comes from shear modulus, not Young's modulus. When a spring is stretched, its wire's length doesn't change, but its shape does. This is why only the shear modulus of elasticity is involved in the stretching of a spring.

Elastic potential energy

The elastic potential energy stored in a linear elastic material is given by the integral of the Hooke's law:

${\displaystyle U_e=\int kxdx=\frac{1}{2}k{x}^2.}$

now by explicating the intensive variables:

${\displaystyle U_e=\int \frac{EA\mathrm{\Delta }L}{L_0}d\mathrm{\Delta }L=\frac{EA}{L_0}\int \mathrm{\Delta }Ld\mathrm{\Delta }L=\frac{EA{\mathrm{\Delta }L}^2}{2L_0}}$

This means that the elastic potential energy density (that is, per unit volume) is given by:

${\displaystyle \frac{U_e}{A{L}_0}=\frac{E{\mathrm{\Delta }L}^2}{2L_0^2}=\frac{1}{2}\times \frac{E\mathrm{\Delta }L}{L_0}\times \frac{\mathrm{\Delta }L}{L_0}=\frac{1}{2}\times \sigma (\epsilon )\times \epsilon }$

or, in simple notation, for a linear elastic material: $u_e(\epsilon )=\int E\epsilon d\epsilon =\frac{1}{2}E{\epsilon }^2$, since the strain is defined $\epsilon \equiv \frac{\mathrm{\Delta }L}{L_0}$.

In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds, and the elastic energy is not a quadratic function of the strain:

${\displaystyle u_e(\epsilon )=\int E(\epsilon )\epsilon d\epsilon \ne \frac{1}{2}E{\epsilon }^2}$

Examples

Young's modulus can vary somewhat due to differences in sample composition and test method. The rate of deformation has the greatest impact on the data collected, especially in polymers. The values here are approximate and only meant for relative comparison.