# Explained: Elastic Behavior of Materials

Young’s modulus (or elastic modulus), E, captures the intrinsic stiffness of a material when it elongates elastically, and it is defined as the proportionality between normal stress and normal strain when the material is subjected to normal stress in one direction.
To predict deformation and failure we need a mathematical relation between stress and strain. Here we consider the simplest relation, appropriate to materials that exhibit elastic behavior.

1. We usually design parts or structures to have only elastic deformations: when loads are removed, the body returns to its initial size and shape. A material is said to deform elastically if it substantially recovers its initial shape and size when loads are removed. Most materials respond elastically, at least when the deformations, or more precisely the strains, are very small, for example below 0.001 or 0.01, depending on the material.
During elastic deformation, there is usually no permanent internal change in the material. After loads are removed, one usually cannot detect that loads were ever applied. Furthermore, in most engineering applications, we want parts to retain their shape when loads are applied and removed. For these reasons, parts to be used repeatedly are designed to deform elastically under the loads they regularly experience.

2. Elastic deformations are proportional to the load. In addition, the stress is found to increase in proportion to the strain for most materials that deform elastically, at least at small strains. This is termed linear elastic behavior. For most engineering materials, stress is approximately a linear function of strain, starting from zero strain. For some biological materials or fabrics, initial elongation initially occurs with little increase in stress, although eventually the stress increases in proportion to strain.

3. Represent a material that undergoes linear elastic deformation mathematically with a linear stress–strain equation. To make quantitative predictions of the deformation of parts, we need a mathematical equation that captures the stress–strain relation for elastic deformation. The most commonly used equation idealizes the stress to be precisely proportional to the strain, starting from zero strain.
Young’s modulus (or elastic modulus), E, is defined as the proportionality between normal stress and strain when the material is linear elastic. E depends on the material. Since strain is unitless, E has the same units as stress (e.g., Pa or psi).

σ = Eε

4. The stiffer the material, the greater is E. Stiffness refers to the force necessary to achieve a given deformation. Stiffness of a part depends on its material and geometry. Because the stress and strain are defined to capture the intensity of force and deformation in an element independent of its size, E quantifies the intrinsic stiffness of a material, irrespective of its size. A material with a larger E is stiffer, because the same strain (ε) requires a larger stress (σ).

5. E varies by a factor of 105, even over commonly used materials, and it strongly affects a designer’s choice of material. E has been measured and tabulated for thousands of materials. This table shows typical values for a few materials. Notice the wide range over which the elastic stiffness can vary. Over commonly-used materials, such as steel and rubber, E is seen to vary by a factor of 105. 6. Materials that are elongated by a tensile stress in one direction, contract in the transverse (perpendicular) directions. When tensile stress acts in one direction, most materials contract in the perpendicular or transverse direction. This contraction is evident in highly flexible materials, such as rubber. Likewise, with a compressive stress in one direction, most materials expand in the transverse direction. It is important to note that we are considering uniaxial normal stress, which acts in one direction. There is no tensile or compressive stress acting in the horizontal direction on the lateral faces. But, there is strain in the horizontal direction.

7. Normal strain in the transverse direction is similarly defined as change in the transverse length divided by its initial length. Imagine measuring the change in length parallel to and perpendicular to the tensile axis with extensometers.
Say the initial lengths parallel and perpendicular to the tensile axis are L and Lt, respectively. Under tension, the axial length L changes by 𝛿(lengthens so 𝛿>0), and the transverse length Lt changes by 𝛿t (shortens so 𝛿t<0).
From lengths and elongations we find axial (ε) and transverse (εt) strains:

ε = 𝛿/L ; ε = 𝛿t/Lt

ε > 0 (Tensile)

εt < 0 (Compressive) 8. For a material that deforms elastically, the transverse strain is proportional to the axial strain.
If we monitor the changes in both lengths L and Lt during a single tensile test, we can find how the transverse strain changes as the axial strain increases.When the material is elastic, the transverse strain is found to change essentially in proportion to the axial strain.
Poisson ratio, υ, is defined as the proportionality between the transverse and axial strain magnitudes when a linear elastic material is subjected to uniaxial normal stress. υ depends on the material. Since ε and εt are unitless, υ must be unitless:

εt = -υε

9. The Poisson ratio varies over a narrow range and rarely affects design. Though it is more difficult to measure, has been tabulated for a number of materials. Generally, has values within a narrow range (generally in 0<υ<0.5). This table indicates typical values of for commonly used types of materials. usually has a minor effect on design, and is not as routinely measured as is E.

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