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Deformation (physics)

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Deformation is the change in shape and/or size of a continuum body after it undergoes a displacement between an initial or undeformed configuration ${\displaystyle \ \kappa _{0}({\mathcal {B}})}$, at time ${\displaystyle \ t=0}$, and a current or deformed configuration ${\displaystyle \ \kappa _{t}({\mathcal {B}})}$, at the current time ${\displaystyle \ t}$.

In general, the displacement of a continuum body has two components: a rigid-body displacement component and a deformation component. If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the the hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero i.e. the distance between particles remains unchanged, then there is no deformation and a rigid-body displacement is said to have occurred.

Deformations results from stresses within the continuum induced by external forces or due to changes in its temperature. The relation between stresses and induced strains is expressed by constitutive equations, e.g. Hooke's law for linear elastic materials.

Deformation is measured in units of length.

Strain is the geometrical representation of deformation representing the relative displacement between particles in the material body, i.e. a measure of how much a given displacement differs locally from a rigid-body displacement (Jaboc Lubliner). Strain defines the amount of stretch or compression, i.e. normal strain, and distortion, i.e. shear strain, within a deforming body (David Rees). Strain is a dimensionless quantity, which can be expressed as a decimal fraction, a percentage or in parts-per notation.

Strain

Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body, i.e. a measure of how much a given displacement differs locally from a rigid-body displacement (Jaboc Lubliner). Strain defines the amount of stretch or compression, i.e. normal strain, and distortion, i.e. shear strain, within a deforming body (David Rees). Strain is a dimensionless quantity, which can be expressed as a decimal fraction, a percentage or in parts-per notation.

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, i.e. 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, i.e. 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.

Strain measures

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

1. 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.
2. 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 behaviour, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.
3. Large-displacementorlarge-rotation theory, which assumes small strains but large rotations and displacements.

In each of this 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% (David Rees page 41), thus other more complexed definitions of strain are required, such as stretch, logarithmic strain, Green strain, and Almansi strain.

Cauchy or engineering strain

The 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 strainorengineering extensional strain ${\displaystyle \ e}$ of a material line element or fiber axially loaded is expressed as the change in length ${\displaystyle \ \Delta L}$ per unit of the original length ${\displaystyle \ L}$ of the line element or fibers. The normal strain is positive if the material fibers are stretched or negative if they are compressed. Thus, we have

${\displaystyle \ e={\frac {\Delta L}{L}}={\frac {l-L}{L}}}$

where ${\displaystyle \ l}$ is the final length of the fiber.

The engineering shear strain is defined as the change in the angle between two material line elements initially perpendicular to each other in the undeformed or initial configuration.

Stretch

The stretch ratioorextension ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration. It is defined as the ratio between the final length ${\displaystyle \ l}$ and the initial length ${\displaystyle \ L}$ of the material line.

${\displaystyle \ \lambda ={\frac {l}{L}}}$

The extension ratio is related to the engineering strain by

${\displaystyle \ e={\frac {l-L}{L}}=\lambda -1}$

This equation implies that the normal strain is zero, i.e. no deformation, when the stretch is equal to unity.

The stretch ratio is used in the anlaysis of materials that exhibit large deformations, such as elastometers, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios, perhaps of the order of 1.001 (reference?)

Logarithmic strain

The logarithmic strain ${\displaystyle \ \varepsilon _{}}$, also called natural strain, true strainorHencky strain. Considering a incremental strain (Ludwik)

${\displaystyle \ \delta \varepsilon ={\frac {\delta L}{l}}}$

the logarithmic strain is obtained by integrating this incremental strain:

${\displaystyle \ {\begin{aligned}\int \delta \varepsilon &=\int _{L}^{l}{\frac {\delta L}{l}}\\\varepsilon &=\ln \left({\frac {l}{L}}\right)=\ln \lambda \\&=\ln(1+e)\\&=e-e^{2}/2+e^{3}/3-...\\\end{aligned}}}$

where ${\displaystyle \ e}$ is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account of the influence of the strain path (David Rees).

Green strain

The Green strain is defined as

${\displaystyle \ \varepsilon _{G}={\frac {1}{2}}\left({\frac {L^{2}-L_{0}^{2}}{L_{0}^{2}}}\right)={\frac {1}{2}}(\lambda ^{2}-1)}$

Euler-Almansi strain

The Euler-Almansi strain is defined as

${\displaystyle \ \varepsilon _{E}={\frac {1}{2}}\left({\frac {L^{2}-L_{0}^{2}}{L^{2}}}\right)={\frac {1}{2}}\left(1-{\frac {1}{\lambda ^{2}}}\right)}$

Description of deformation

It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not to be one the body actually will ever occupy. Often, the configuration at ${\displaystyle \ t=0}$ is considered the reference configuration, ${\displaystyle \ \kappa _{0}({\mathcal {B}})}$. The configuration at the current time t is the current configuration.

For deformation analysis, the reference configuration is identified as undeformed configuration, and the current configuration as deformed configuration. Additionally, time is not considered when analysing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no iterest.

The components ${\displaystyle \ X_{i}}$ of the position vector ${\displaystyle \ \mathbf {X} }$ of a particle in the reference configuration, taken with respect to the reference coordinate system, are called the material or reference coordinates. On the other hand, the components ${\displaystyle \ x_{i}}$ of the position vector ${\displaystyle \ \mathbf {x} }$ of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the material or reference coordinates

There are two methods for analysing the deformation of a continuum. One description is made in terms of the material or referential coordinates, called material description or Lagrangian description. A second description is of deformation is made in terms of the spatial coordinates it is called the spatial description or Eulerian description.

There is continuity during deformation of a continuum body in the sense that:

• The material points forming a closed curve at any instant will always form a closed curve at any subsequent time.
• The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

Displacement

A change in the configuration of a continuum body results in a displacement. The displacement of a body has two components: a rigid-body displacement and a deformation. A rigid-body displacement consist of a simultaneous translation and rotation of the body without changing its shape or size. Deformation implies the change in shape and/or size of the body from an initial or undeformed configuration ${\displaystyle \ \kappa _{0}({\mathcal {B}})}$ to a current or deformed configuration ${\displaystyle \ \kappa _{t}({\mathcal {B}})}$ (Figure 2).

If after a displacement of the continuum there is a relative displacement between particles, a deformation has occurred. On the the hand, if after displacement of the continuum the relative displacement between particles in the current configuration is zero i.e. the distance between particles remains unchanged, then there is no deformation and a rigid-body displacement is said to have occurred.

The vector joining the positions of a particle ${\displaystyle \ P}$ in the undeformed configuration and deformed configuration is called the displacement vector ${\displaystyle \ \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}}$, in the Lagrangian description, or ${\displaystyle \ \mathbf {U} (\mathbf {x} ,t)=U_{J}\mathbf {E} _{J}}$, in the Eulerian description.

Adisplacement field is a vector field of all displacement vectors for all particles in the body, which relates the deformed configuration with the undeformed configuration. It is convenient to do the analysis of deformation or motion of a continuum body in terms of the displacement field, In general, the displacement field is expressed in terms of the material coordinates as

${\displaystyle \ \mathbf {u} (\mathbf {X} ,t)=\mathbf {b} (\mathbf {X} ,t)+\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=\alpha _{iJ}b_{J}+x_{i}-\alpha _{iJ}X_{J}}$

or in terms of the spatial coordinates as

${\displaystyle \ \mathbf {U} (\mathbf {x} ,t)=\mathbf {b} (\mathbf {x} ,t)+\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=b_{J}+\alpha _{Ji}x_{i}-X_{J}\,}$

where ${\displaystyle \ \alpha _{Ji}}$ are the direction cosines between the material and spatial coordinate systems with unit vectors ${\displaystyle \ \mathbf {E} _{J}}$ and ${\displaystyle \mathbf {e} _{i}}$, respectively. Thus

${\displaystyle \ \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\alpha _{Ji}=\alpha _{iJ}}$

and the relationship between ${\displaystyle \ u_{i}}$ and ${\displaystyle \ U_{J}}$ is then given by

${\displaystyle \ u_{i}=\alpha _{iJ}U_{J}\qquad {\text{or}}\qquad U_{J}=\alpha _{Ji}u_{i}}$

Knowing that

${\displaystyle \ \mathbf {e} _{i}=\alpha _{iJ}\mathbf {E} _{J}}$

then

${\displaystyle \mathbf {u} (\mathbf {X} ,t)=u_{i}\mathbf {e} _{i}=u_{i}(\alpha _{iJ}\mathbf {E} _{J})=U_{J}\mathbf {E} _{J}=\mathbf {U} (\mathbf {x} ,t)}$

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in ${\displaystyle \ \mathbf {b} =0}$, and the direction cosines become Kronecker deltas, i.e.

${\displaystyle \ \mathbf {E} _{J}\cdot \mathbf {e} _{i}=\delta _{Ji}=\delta _{iJ}}$

Thus, we have

${\displaystyle \ \mathbf {u} (\mathbf {X} ,t)=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \qquad {\text{or}}\qquad u_{i}=x_{i}-\delta _{iJ}X_{J}}$

or in terms of the spatial coordinates as

${\displaystyle \ \mathbf {U} (\mathbf {x} ,t)=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\qquad {\text{or}}\qquad U_{J}=\delta _{Ji}x_{i}-X_{J}}$

Displacement gradient tensor

The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor

${\displaystyle \ {\begin{aligned}\mathbf {u} (\mathbf {X} ,t)&=\mathbf {x} (\mathbf {X} ,t)-\mathbf {X} \\\mathbf {u} \nabla _{\mathbf {X} }&=\mathbf {x} \nabla _{\mathbf {X} }-\mathbf {I} \\\mathbf {u} \nabla _{\mathbf {X} }&=\mathbf {F} -\mathbf {I} \\\end{aligned}}\qquad {\text{or}}\qquad {\begin{aligned}u_{i}&=x_{i}-\delta _{iJ}X_{J}\\{\frac {\partial u_{i}}{\partial X_{K}}}&={\frac {\partial x_{i}}{\partial X_{K}}}-\delta _{iK}\\\end{aligned}}}$

where ${\displaystyle \ \mathbf {F} }$ is the deformation gradient tensor.

Similarly, the partial differentiation of the displacement vector with respect to the spatial coordinates yields the spatial displacement gradient tensor

${\displaystyle \ {\begin{aligned}\mathbf {U} (\mathbf {x} ,t)&=\mathbf {x} -\mathbf {X} (\mathbf {x} ,t)\\\mathbf {U} \nabla _{\mathbf {x} }&=\mathbf {I} -\mathbf {X} \nabla _{\mathbf {X} }\\\mathbf {U} \nabla _{\mathbf {x} }&=\mathbf {I} -\mathbf {F} ^{-1}\\\end{aligned}}\qquad {\text{or}}\qquad {\begin{aligned}U_{J}&=\delta _{Ji}x_{i}-X_{J}\\{\frac {\partial U_{J}}{\partial x_{i}}}&=\delta _{Ji}-{\frac {\partial X_{J}}{\partial x_{i}}}\\\end{aligned}}}$

Deformation gradient tensor

The material deformation gradient tensor ${\displaystyle \ \mathbf {F} (\mathbf {X} ,t)=F_{jK}\mathbf {e} _{j}\otimes \mathbf {I} _{K}}$ is a second-order tensor that represents the gradient of the mapping function or functional relation ${\displaystyle \ \chi (\mathbf {X} ,t)}$, which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector ${\displaystyle \ \mathbf {X} }$, i.e. deformation at neighbouring points, by transforming (linear transformation) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function ${\displaystyle \ \chi (\mathbf {X} ,t)}$, i.e differentiable functionof${\displaystyle \mathbf {X} }$ and time ${\displaystyle \ t}$, which implies that cracks and voids do not open or close during the deformation. Thus we have,

${\displaystyle {\begin{aligned}d\mathbf {x} &={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}\,d\mathbf {X} \\&=\nabla \chi (\mathbf {X} ,t)\,d\mathbf {X} \\&=\mathbf {F} (\mathbf {X} ,t)\,d\mathbf {X} \\\end{aligned}}\qquad {\text{or}}\qquad {\begin{aligned}dx_{j}&={\frac {\partial x_{j}}{\partial X_{K}}}\,dX_{K}\\dx_{j}&=F_{jK}\,dX_{K}\end{aligned}}}$

The deformation gradient tensor ${\displaystyle \ \mathbf {F} (\mathbf {X} ,t)=F_{jK}\mathbf {e} _{j}\otimes \mathbf {I} _{K}}$ is related to both the reference and current configuration, as seen by the unit vectors ${\displaystyle \ \mathbf {e} _{j}}$ and ${\displaystyle \ \mathbf {I} _{K}}$, therefore it is a two-point tensor.

Due to the assumption of continuity of ${\displaystyle \ \chi (\mathbf {X} ,t)}$, ${\displaystyle \ \mathbf {F} }$ has the inverse ${\displaystyle \ \mathbf {H} =\mathbf {F} ^{-1}}$, where ${\displaystyle \ \mathbf {H} }$ is the spatial deformation gradient. Then, by the implicit function theorem (Lubliner), the Jacobian determinant ${\displaystyle \ J(\mathbf {X} ,t)}$ must be nonsingular, i.e. ${\displaystyle \ J(\mathbf {X} ,t)=\det \mathbf {F} (\mathbf {X} ,t)\neq 0}$

Note: The notation and terminology used here was introduced in the "Non-Linear Field Theories of Mechanics” by C.Truesdell and myself (Walter Noll), published in 1965. I invented much of this notation and terminology, but I now realize that some of it is misleading and should be changed. For example “Deformation Gradient” should be replaced by “Transplacement Gradient”. A modern, frame-free and coordinate-free analysis of the mathematical concept of deformation can be found in the first four parts of my "Five Contributions to Natural Philosophy", published in 2005 on my website www.math.cmu.edu/~wn0g/noll.

Transformation of a surface and volume element

Polar decomposition of the deformation gradient tensor

The deformation gradient ${\displaystyle \mathbf {F} }$, like any second-order tensor, can be decomposed, using the polar decomposition theorem, into a product of two second-order tensors (Truesdell and Noll, 1965): an orthogonal tensor and a positive definite symmetric tensor, i.e.

${\displaystyle \mathbf {F} =\mathbf {R} \mathbf {U} =\mathbf {V} \mathbf {R} }$

where the tensor ${\displaystyle \mathbf {R} }$ is a proper orthogonal tensor, i.e. ${\displaystyle \ \mathbf {R} ^{-1}=\mathbf {R} ^{T}}$ and ${\displaystyle \det \mathbf {R} =+1}$, representing a rotation; the tensor ${\displaystyle \mathbf {U} }$ is the right stretch tensor; and ${\displaystyle \mathbf {V} }$ the left stretch tensor. The terms right and left means that they are to the right and left of the rotation tensor ${\displaystyle \mathbf {R} }$, respectively. ${\displaystyle \mathbf {U} }$ and ${\displaystyle \mathbf {V} }$ are both positive definite, i.e. ${\displaystyle \ \mathbf {x} \cdot \mathbf {U} \cdot \mathbf {x} \geq 0}$ and ${\displaystyle \ \mathbf {x} \cdot \mathbf {V} \cdot \mathbf {x} \geq 0}$, and symmetric tensors, i.e. ${\displaystyle \ \mathbf {U} =\mathbf {U} ^{T}}$ and ${\displaystyle \ \mathbf {V} =\mathbf {V} ^{T}}$, of second order.

This decomposition implies that the deformation of a line element ${\displaystyle \ d\mathbf {X} }$ in the undeformed configuration onto ${\displaystyle \ d\mathbf {x} }$ in the deformed configuration, i.e. ${\displaystyle d\mathbf {x} =\mathbf {F} \,d\mathbf {X} }$, may be obtained either by first stretching the element by ${\displaystyle \ \mathbf {U} }$, i.e. ${\displaystyle d\mathbf {x} '=\mathbf {U} \,d\mathbf {X} }$, followed by a rotation ${\displaystyle \ \mathbf {R} }$, i.e. ${\displaystyle \ d\mathbf {x} =\mathbf {R} \,d\mathbf {x} '}$; or equivalently, by applying a rigid rotation ${\displaystyle \ \mathbf {R} }$ first, i.e. ${\displaystyle d\mathbf {x} '=\mathbf {R} \,d\mathbf {x} }$, followed later by a stretching ${\displaystyle \ \mathbf {V} }$, i.e. ${\displaystyle \ d\mathbf {x} =\mathbf {V} \,d\mathbf {x} '}$ (See Figure).

It can be shown that,

${\displaystyle \ \mathbf {V} =\mathbf {R} \cdot \mathbf {U} \cdot \mathbf {R} ^{T}}$

so that ${\displaystyle \ \mathbf {U} }$ and ${\displaystyle \ \mathbf {V} }$ have the same eigenvalues or principal stretches, but different eigenvectorsorprincipal directions ${\displaystyle \mathbf {N} _{i}}$ and ${\displaystyle \mathbf {n} _{i}}$, respectively. The principal directions are related by

${\displaystyle \mathbf {n} _{i}=\mathbf {R} \mathbf {N} _{i}}$

This polar decomposition is unique as ${\displaystyle \ \mathbf {F} }$ is non-symmetric.

Deformation tensors

Several rotation-independent deformation tensors are used in mechanics. In solid mechanics, the most popular of these are the right and left Cauchy-Green deformation tensors. The Finger deformation tensor is mainly used in describing the motion of nonlinear fluids.^{[citation needed]}

Since a pure rotation should not induce any stresses in a deformable body, it is often convenient to use rotation-independent measures of deformation in continuum mechanics. As a rotation followed by its inverse rotation leads to no change (${\displaystyle \mathbf {R} \mathbf {R} ^{T}=\mathbf {R} ^{T}\mathbf {R} =\mathbf {1} }$) we can exclude the rotation by multiplying ${\displaystyle \mathbf {F} }$ by its transpose.

The Right Cauchy-Green deformation tensor

In 1839, George Green introduced a deformation tensor known as the right Cauchy-Green deformation tensororGreen's deformation tensor, defined as:

${\displaystyle \mathbf {C} =\mathbf {F} ^{T}\mathbf {F} =\mathbf {U} ^{2}\qquad {\text{or}}\qquad C_{ij}={\frac {\partial x_{k}}{\partial X_{I}}}{\frac {\partial x_{k}}{\partial X_{J}}}}$

Physically, the Cauchy-Green tensor gives us the square of local change in distances due to deformation, i.e. ${\displaystyle \ d\mathbf {x} ^{2}=d\mathbf {X} \cdot \mathbf {C} \cdot d\mathbf {X} }$

Invariants of ${\displaystyle \mathbf {C} }$ are often used in the expressions for strain energy density functions. The most commonly used invariants are

${\displaystyle {\begin{aligned}I_{1}^{C}&:={\text{tr}}(\mathbf {C} )=C_{ii}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}\\I_{2}^{C}&:={\tfrac {1}{2}}\left[{\text{tr}}(\mathbf {C} ^{2})-({\text{tr}}~\mathbf {C} )^{2}\right]={\tfrac {1}{2}}\left[C_{ik}C_{ki}-C_{jj}^{2}\right]=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}\\I_{3}^{C}&:=\det(\mathbf {C} )=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}\end{aligned}}}$

The Cauchy deformation tensor

Earlier in 1828, Augustin Louis Cauchy introduced a deformation tensor defined as (reference: Inelastic analysis of structures)

${\displaystyle \mathbf {c} =\mathbf {F} ^{-T}\mathbf {F} ^{-1}\qquad {\text{or}}\qquad c_{ij}={\frac {\partial X_{K}}{\partial x_{i}}}{\frac {\partial X_{K}}{\partial x_{j}}}}$