Youngs modulus

Youngs modulus , Youngs modulus , or modulus of elasticity of stress, is a mechanical property that measures the tensile stiffness of a solid material. It measures the relationship between tensile strain (force per unit area) and axial strain (proportional deformation) in a material’s linear elastic field and is determined using the formula:

E \sigma \varepsilon
{\displaystyle E={\frac {\sigma }{\varepsilon }}}

Young’s moduli are usually so large that they are expressed in gigapascals (GPA) and not in pascals .

Although Youngs modulus is named after the 19th century British scientist Thomas Young , the concept was developed by Leonhard Euler in 1727. The first experiments to use the concept of Youngs modulus in its current form were in 1782 by the Italian scientist Giordano Riccati , dating Young’s work by 25 years. [2] The word modulus is derived from the Latin root modus meaning measure .

Definition

Linear elasticity

A solid material will undergo elastic deformation when a small load is applied to it in compression or expansion. Elastic deformation is reversible (the material returns to its original shape once the load is removed).

At near-zero stress and strain, the stress–strain curve is linear , and the relationship between stress and strain is described by Hooke’s law which states that stress is proportional to strain. The coefficient of proportionality is Youngs modulus. The higher the modulus, the more stress is required to produce the same amount of stress; An ideal rigid body will have infinite Youngs modulus. In contrast, a very soft material such as a fluid, would deform without force, and would have zero Youngs modulus.

Many materials are linear and not elastic beyond a small amount of deformation.

Pay attention

Material hardness should not be confused with these properties:

• Strength : The maximum amount of stress a material can withstand while in the elastic (reversible) deformation regime;
• Geometric rigidity: a global characteristic of a body that depends on its shape, not just on the local properties of the material; For example, an I-beam has a higher bending stiffness than a rod of the same material for a given mass per length;
• Hardness : The relative resistance of a material surface to penetration by a rigid body;
• Toughness : The amount of energy that a material can absorb before fracture.

Use

Youngs modulus enables the calculation of the change in amplitude of a bar made of an isotropic elastic material under tensile or compressive loads . For example, it predicts how much a sample of material expands under tension or shrinks under compression. Youngs modulus applies directly to cases of axial stress; i.e. tensile or compressive stress in one direction and no stress in the other directions. Youngs modulus is also used to predict the deflection that occurs in a statically fixed beam when a load is applied to a point between the beam’s supports.

Other elastic calculations usually require the use of an additional elastic property, such as the shear modulus , bulk modulus , and Poisson’s ratio . Any two of these parameters are sufficient to fully describe the elasticity in an isotropic material. For homogeneous isotropic materials there exist simple relationships between elastic constants that allow them to be calculated as long as two are known: G K v

E = 2G (1+ \nu) = 3K (1-2 \nu). \,

Linear vs non-linear

Youngs modulus represents the factor of proportionality in Hooke’s law , which relates stress and strain. However, Hooke’s law is valid only under the assumption of elastic and linear response. Any real material will eventually fail and break when stretched over great distances or with too great a force; However all solids exhibit almost Hookean behavior for small enough strains or stresses. If the range over which Hooke’s law is valid is much larger than the specific stress applied to the material, the material is said to be linear. Otherwise (if the normal stress applied is outside the linear limit) the material is said to be non-linear.

Steel , carbon fiber and glass are generally considered to be linear materials, while other materials such as rubber and clay are non-linear . However, this is not a complete classification: if very small stresses or strains are applied to a non-linear material, the response will be linear, but if a lot of stress or strain is applied to the linear material, the linear principle will not. Sufficient. For example, as linear theory implies reversibility , it would be absurd to use linear theory to describe the failure of a steel bridge under high loads; Although steel is a linear material for most applications, it is not the case with catastrophic failure.

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

Directional material

Youngs modulus is not always the same in all orientations of a material. Most metals and ceramics, as with many other materials, are isotropic , and their mechanical properties are similar in all orientations. However, metals and ceramics can be treated with some impurities, and metals can be mechanically worked to make their grain structures directional. These materials then become anisotropic , and Youngs modulus will change depending on the direction of the force vector. [3] Anisotropy can also be observed in many composites. For example, when a force is loaded parallel to the fiber (along the grain), the Youngs modulus in carbon fiber is very high (very hard). Wood among other such materialsand reinforced concrete. Engineers can use this directional phenomenon to their advantage in building structures.

Temperature dependence

The Youngs modulus of metals varies with temperature and can be realized through changes in the inter-atomic bonding of atoms and hence its change is found to depend on the change in work function of the metal. Although classically, this change is predicted through fitting and without an explicit underlying mechanism (for example, Watchman’s formula), the Rahimi–Lee model [4] shows how the electron work function changes from Youngs modulus of metals changes and predicts this variation with countable parameters, using the generalization of the Lenard–Jones potential for solids. In general, as the temperature increases, Youngs modulus decreases where the electron work varies with the work temperature .

 .{\displaystyle E(T)=\beta (\varphi (T))^{6}}{\displaystyle \varphi (T)=\varphi _{0}-\gamma {\frac {(k_{B}T)^{2}}{\varphi _{0}}}}\gamma

is a countable physical property that is dependent on the crystal structure (eg, bcc, fcc). T=0 and . But the electron work function is constant during the transformation.

\varphi _{0}\beta

Calculation

Youngs modulus E , can be calculated by dividing the tensile stress, , by the engineering extensor stress, in the elastic (initial, linear) part of the physical stress–strain curve:

{\displaystyle \sigma (\varepsilon )}\varepsilon
{\displaystyle E\equiv {\frac {\sigma (\varepsilon )}{\varepsilon }}={\frac {F/A}{\Delta L/L_{0}}}={\frac {FL_{0}}{A\,\Delta L}}}

E Young’s modulus is (modulus of elasticity)

F The force exerted on an object under tension is;

A is the actual cross-sectional area, which is equal to the area of ​​the cross-section perpendicular to the applied force;

\Delta L

is the amount by which the length of the object changes ( positive if the material is stretched, and negative when the material is compressed);

\Delta L


Lo is the original length of the object.

force exerted by a material that is stretched or contracted

The Young’s modulus of a material can be used to calculate the force it exerts under specific stress.

{\displaystyle F={\frac {EA\,\Delta L}{L_{0}}}}

where is the force exerted by the material when it is contracted or stretched.

F\Delta L


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

{\displaystyle F=\left({\frac {EA}{L_{0}}}\right)\,\Delta L=kx\,}

where it comes in saturation

{\displaystyle k\equiv {\frac {EA}{L_{0}}}\,}~~And~~x\equiv \Delta L.\,

But note that the elasticity of coiled springs comes from the shear modulus, not Young’s modulus. citation needed ]

Elastic potential energy

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

U_ {e} = \int {kx} \, dx = {\frac {1} {2}} kx ^ {2}.

Now by explaining the intensive variable:

{\displaystyle U_ {e} = \int {\frac {EA \, \Delta L} {L_ {0}}} \, d \Delta L = {\frac {EA} {L_ {0}}} \int \Delta L \, d \Delta L = {\frac {EA \, {\Delta L} ^ {2}} {2L_ {0}}}}

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

{\displaystyle {\frac {U_{e}}{AL_{0}}}={\frac {E\,{\Delta L}^{2}}{2L_{0}^{2}}}}

Or, in simpler notation, for a linear elastic material: , since the strain is defined as .

u_ {e} (\varepsilon) = \int {E \, \varepsilon} \, d \varepsilon = {\frac {1} {2}} E {\varepsilon} ^ {2}\varepsilon \equiv {\frac {\Delta L}{L_{0}}}

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

{\displaystyle u_ {e} (\varepsilon) = \int E (\varepsilon) \, \varepsilon \, d \varepsilon \neq {\frac {1} {2}} E \varepsilon ^ {2}}

Estimated value

Young’s modulus may vary slightly 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 are for relative comparison only.