In physics, **deformation** is the continuum mechanics transformation of a body from a *reference* configuration to a *current* configuration.^{[1]} A configuration is a set containing the positions of all particles of the body.

A deformation may be caused by external loads,^{[2]} body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

**Strain** is a description of deformation in terms of *relative* displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called **elastic deformations**. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is **plastic deformation**, which occurs in material bodies after stresses have attained a certain threshold value known as the *elastic limit* or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is **viscous deformation**, which is the irreversible part of viscoelastic deformation.

In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

Strain is a measure of deformation representing the displacement between particles in the body relative to a reference length.

A general deformation of a body can be expressed in the form **x** = * F*(

We could, for example, define strain to be

where **I** is the identity tensor.
Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation.^{[3]}

A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the *normal strain*, and the amount of distortion associated with the sliding of plane layers over each other is the *shear strain*, within a deforming body.^{[4]} This could be applied by elongation, shortening, or volume changes, or angular distortion.^{[5]}

The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the *normal strain*, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the *shear strain*, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.

If there is an increase in length of the material line, the normal strain is called *tensile strain*, otherwise, if there is reduction or compression in the length of the material line, it is called *compressive strain*.

Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:

- Finite strain theory, also called
*large strain theory*,*large deformation theory*, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. - Infinitesimal strain theory, also called
*small strain theory*,*small deformation theory*,*small displacement theory*, or*small displacement-gradient theory*where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting elastic behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel. *Large-displacement*or*large-rotation theory*, which assumes small strains but large rotations and displacements.

In each of these theories the strain is then defined differently. The *engineering strain* is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g. elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%,^{[6]} thus other more complex definitions of strain are required, such as *stretch*, *logarithmic strain*, *Green strain*, and *Almansi strain*.

The **Cauchy strain** or **engineering strain** is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The *engineering normal strain* or *engineering extensional strain* or *nominal strain* e of a material line element or fiber axially loaded is expressed as the change in length Δ*L* per unit of the original length L of the line element or fibers. The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have

where e is the *engineering normal strain*, L is the original length of the fiber and A deformation may be caused by external loads,^{[2]} body forces (such as gravity or electromagnetic forces), or changes in temperature, moisture content, or chemical reactions, etc.

**Strain** is a description of deformation in terms of *relative* displacement of particles in the body that excludes rigid-body motions. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.

In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed are called **elastic deformations**. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations remain even after stresses have been removed. One type of irreversible deformation is **plastic deformation**, which occurs in material bodies after stresses have attained a certain threshold value known as the *elastic limit* or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Another type of irreversible deformation is **viscous deformation**, which is the irreversible part of viscoelastic deformation.

In the case of elastic deformations, the response function linking strain to the deforming stress is the compliance tensor of the material.

Strain is a measure of deformation representing the displacement between particles in the body relative to a reference length.

A general deformation of a body can be expressed in the form **x** = * F*(

We could, for example, define strain to be

where **I** is the identity tensor.
Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation.^{[3]}

A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the *normal strain*, and the amount of distortion associated with the sliding of plane layers over each other is the *shear strain*, within a deforming body.^{x = F(X) where X is the reference position of material points in the body. Such a measure does not distinguish between rigid body motions (translations and rotations) and changes in shape (and size) of the body. A deformation has units of length.
}

We could, for example, define strain to be

where **I** is the identity tensor.
Hence strains are dimensionless and are usually expressed as a decimal fraction, a percentage or in parts-per notation. Strains measure how much a given deformation differs locally from a rigid-body deformation.^{[3]}

A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the *normal strain*, and the amount of distortion associated with the sliding of plane layers over each other is the *shear strain*, within a deforming body.^{[4]} This could be applied by elongation, shortening, or volume changes, or angular distortion.^{[5]}

The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the *normal strain*, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the *shear strain*, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.

If there is an increase in length of the material line, the normal strain is called *tensile strain*, otherwise, if there is reduction or compression in the length of the material line, it is called *compressive strain*.

Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:

- Finite strain theory, also called
*large strain theory*,*large deformation theory*, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. - Infinitesimal strain theory, also called
*small strain theory<*A strain is in general a tensor quantity. Physical insight into strains can be gained by observing that a given strain can be decomposed into normal and shear components. The amount of stretch or compression along material line elements or fibers is the

*normal strain*, and the amount of distortion associated with the sliding of plane layers over each other is the*shear strain*, within a deforming body.^{[4]}This could be applied by elongation, shortening, or volume changes, or angular distortion.^{[5]}The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the

*normal strain*, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the*shear strain*, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.If there is an increase in length of the material line, the normal strain is called

*tensile strain*, otherwise, if there is reduction or compression in the length of the material line, it is called*compressive strain*.Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories:

- Finite strain theory, also called
*large strain theory*,*large deformation theory*, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%,^{[6]}thus other more complex definitions of strain are required, such as*stretch*,*logarithmic strain*,*Green strain*, and*Almansi strain*.#### Engineering strain

The

**Cauchy strain**or**engineering strain**is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The*engineering normal strain*or*engineering extensional strain*or*nominal strain*e of a material line element or fiber axially loaded is expressed as the change in length Δ*L*per unit of the original length L of the line element or fibers. The normal strain is positive if the material fibers are stretched and negative if they are compressed. Thus, we have

- Finite strain theory, also called