What is Strain?
The Strain definition is - “The result of a material's deformation due to stress is called strain”.
A Mathematical Definition of What is Strain?
According to the Mathematical Strain definition “The ratio of the change in length to the original length is strain”.
Strain = \[\epsilon =\frac{\text{change in length}}{\text{original length}}= \frac{\delta l}{l}\]
Deformation
Deformation is the transition of a body from a reference configuration to a current configuration in continuum mechanics. A configuration is a collection of all the locations of the body's particles.
External loads, body forces such as gravity or electromagnetic forces, changes in temperature, moisture content, or chemical reactions, among other things, may cause deformation.
A deformation field occurs in a continuous body as a result of a stress field caused by applied forces or as a result of changes in the temperature field within the body.
Constitutive equations explain the relationship between stresses and induced strains.
Elastic deformations are deformations that recover after the stress field has been removed. The continuum fully restores its original structure in this situation.
Irreversible deformations, on the other hand, persist even after stresses have been removed. Plastic deformation is one type of irreversible deformation that occurs in material bodies after stresses reach a certain threshold value known as the elastic limit or yield stress, and is caused by slip or dislocation mechanisms at the atomic level. Viscous deformation, which is the irreversible component of viscoelastic deformation, is another type of irreversible deformation.
The compliance tensor of the material is the response function relating strain to the deforming stress in the elastic deformations.
What is Strain in Physics?
Strain is a deformation measurement that represents the displacement of particles in the body in relation to a reference length. A strain is a tensor quantity.
Mechanical strain is a geometric deformation measure that represents the relative displacement of particles within a material body.
Observing that a given strain can be decomposed into normal and shear components can provide physical insight into strains.
Normal strains occur when deformations are applied perpendicular to the cross-section. Inside a deforming body, the normal strain is the amount of stretch or tension along with material line elements or fibres.
Shear strains occur when deformations are applied parallel to the cross-section. The shear strain is the amount of distortion associated with the sliding of plane layers over each other.
Elongation, shortening, volume changes, or angular distortion could all be used to achieve strain mechanics.
The stress state of a material location of the continuum body is defined as the totality of all the variations of the length of the lines or fibres, the normal strain, through which this point passes and also all the changes of the angle between lines initially perpendicular, from that point on to each other, and the shear strain. But the normal components of strain and shear in a number of three perpendicular directions are sufficient to know each other.
The normal strain is called tensile strain if the length of the material line increases; otherwise, it is called compressive strain if the length of the material line decreases or compresses.
Strain Measurement
The deformation analysis is divided into three deformation theories according to the amount of strain or local deformation:
Finite strain theory is a deformation theory that both rotations and strains are arbitrarily large. In this case, the configurations of the continuum are undeformed and deformed and a distinct distinction must be drawn. This is usually the case of elastomers, materials that deform plastically and other fluids and soft biological tissue.
Small strain theory, also known as small deformation theory, small displacement theory, or small displacement-gradient theory, is a type of infinitesimal strain theory in which the strains and rotations are both small. The undeformed and deformed body configurations can be considered to be similar in this situation. The infinitesimal strain principle is applied to the study of deformations in elastic structures, such as those used in mechanical and civil engineering applications.
Small strains but large rotations and displacements are assumed in large-displacement or large-rotation theory.
Engineering Strain
The most common definition for materials used in mechanical and structural engineering that are subjected to very small deformations is engineering strain.
The ratio of total deformation to the initial dimension of the material body in which the forces are applied is known as the Cauchy strain or engineering strain.
The change in length ΔL per unit of the original length L of a material line element or fibre axially loaded is expressed as the engineering normal strain or engineering extensional strain, or nominal strain e. If the material fibres are stretched, the normal strain is positive; if they are compressed, the normal strain is negative.
e = \[\frac{\Delta L}{L}=\frac{l-L}{L}\]
Where e is the engineering strain, ΔL is the change in original length and L is the original length.
The engineering shear strain is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application and is defined as the tangent of that angle.
The stretch ratio, also known as the extension ratio is a measure of a differential line element's extensional or normal strain, which can be described in either the undeformed or deformed configuration. It is defined as the ratio of the material line's final length l to its initial length L.
The stretch ratio is given by the formula,
\[\lambda =\frac{l}{L}\]
Where λ is the stretch ratio and L is the length
Relating stretch ratio with engineering strain,
e = \[\frac{\Delta L}{L}=\frac{l-L}{L}=\lambda -1\]
The stretch ratio is used to evaluate materials that can withstand significant deformations, such as elastomers, which can withstand stretch ratios of 3 or 4 before failing. Traditional engineering materials, such as concrete or steel, on the other hand, fail at much lower stretch ratios.
True Strain
The natural log of the quotient of current length over original length equals true strain or logarithmic strain.
The true strain is obtained by integrating the incremental strain. True strain is given as follows:
\[\epsilon = ln\left ( 1+e \right )\]
Where ε is the true strain and e is the engineering strain.
As deformation occurs in a sequence of increments, the true strain provides the correct measure of the final strain, taking into account the strain path's effect.
Normal Strain
The reaction of a solid to the application of a normal force, which is a force that is perpendicular to the object's cross-sectional field, is referred to as a normal strain.
Tensile and compressive strains are the two types of normal strains. The tensile strain occurs when the applied force causes the material to stretch or elongate. The strain is known as compressive if the applied force causes the material to shorten or compress.
Normal stress can induce a normal strain in an isotropic material that obeys Hooke's law. Dilations are produced by normal strains.
The normal strain in a three dimensional, infinitesimal, rectangular material element with dimensions dx, dy, dz.
\[\epsilon _{x}=\frac{\partial u_{x}}{\partial x}\]
\[\epsilon _{y}=\frac{\partial u_{y}}{\partial y}\]
\[\epsilon _{z}=\frac{\partial u_{z}}{\partial z}\]
Where \[\epsilon _{x}, \epsilon _{y}, \epsilon _{z}\]are the normal strains in the x-axis, y-axis, and z-axis respectively.
ux, uy, and uz are the displacement in the x-axis, y-axis, and z-axis respectively.
Shear Strain
Shear strain is the ratio of the change in length perpendicular to the axes of the member due to shear stress to its original length. Shear stress is stress that runs parallel to the structural member's cross-section.
The shear strain is a three dimensional, infinitesimal, rectangular material element with dimensions dx, dy, dz.
\[\gamma _{xy}=\gamma _{yx}=\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}\]
\[\gamma _{yz}=\gamma _{zy}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\]
\[\gamma _{zx}=\gamma _{xz}=\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z}\]
Where \[\gamma _{xy}, \gamma _{yx}, \gamma _{yz}, \gamma _{zy}, \gamma _{zx} \text{and} \gamma _{xz}\] are the shear strain in x-y plane, y-x plane, y-z plane, z-y plane, z-x plane and x-z plane respectively.
ux, uy, and uz are the displacement in the x-axis, y-axis, and z-axis respectively.
Conclusion
Strain is a number that defines the relative deformation or change in shape and size of elastic, plastic, and fluid materials under applied forces in physical sciences and engineering. Deformation, expressed as strain, occurs in the substance as the particles that make up the material, such as molecules, atoms, and ions, are slightly displaced from their natural positions.
FAQs on Strain Mechanics
1. What is a strain?
Strain is a deformation measurement that represents the displacement of particles in the body in relation to a reference length. A strain is a tensor quantity.
2. What is normal strain and shear strain?
Normal strains occur when deformations are applied perpendicular to the cross-section. Inside a deforming body, the normal strain is the amount of stretch or tension along with material line elements or fibres.
Shear strains occur when deformations are applied parallel to the cross-section. The shear strain is the amount of distortion associated with the sliding of plane layers over each other.
3. What is the normal strain and shear strain formulas for a three dimensional, infinitesimal, rectangular material element?
The normal strain in a three dimensional, infinitesimal, rectangular material element with dimensions dx, dy, dz.
\[\epsilon _{x}=\frac{\partial u_{x}}{\partial x}\]
\[\epsilon _{y}=\frac{\partial u_{y}}{\partial y}\]
\[\epsilon _{z}=\frac{\partial u_{z}}{\partial z}\]
Where \[\epsilon _{x}, \epsilon _{y}, \epsilon _{z}\]are the normal strains in the x-axis, y-axis, and z-axis respectively.
ux, uy, and uz are the displacement in the x-axis, y-axis, and z-axis respectively.
The shear strain is a three dimensional, infinitesimal, rectangular material element with dimensions dx, dy, dz.
\[\gamma _{xy}=\gamma _{yx}=\frac{\partial u_{y}}{\partial x}+\frac{\partial u_{x}}{\partial y}\]
\[\gamma _{yz}=\gamma _{zy}=\frac{\partial u_{y}}{\partial z}+\frac{\partial u_{z}}{\partial y}\]
\[\gamma _{zx}=\gamma _{xz}=\frac{\partial u_{z}}{\partial x}+\frac{\partial u_{x}}{\partial z}\]
Where\[\gamma _{xy},\gamma _{yx},\gamma _{yz},\gamma _{zy},\gamma _{zx},\gamma _{xz}\] are the shear strain in x-y plane, y-x plane, y-z plane, z-y plane, z-x plane and x-z plane respectively.
ux, uy and uz are the displacement in the x-axis, y-axis, and z-axis respectively.
