1. Strain

Strain represents the displacement between particles in the body relative to a reference length.

Deformation of a body is expressed in the form x = F(X) where X is the reference position of material points of 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 [math]\displaystyle{ \boldsymbol{\varepsilon} \doteq \cfrac{\partial}{\partial\mathbf{X}}\left(\mathbf{x} - \mathbf{X}\right) = \boldsymbol{F}'- \boldsymbol{I}, }[/math] 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.^[1]

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.^[2] This could be applied by elongation, shortening, or volume changes, or angular distortion.^[3]

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.

1.1. Strain Measures

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%,^[4] thus other more complex definitions of strain are required, such as stretch, logarithmic strain, Green strain, and Almansi strain.

Engineering Strain

Engineering strain, also known as Cauchy strain, is expressed as the ratio of total deformation to the initial dimension of the material body on which forces are 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 ΔL 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 [math]\displaystyle{ e=\frac{\Delta L}{L} = \frac{l -L}{L} }[/math] where e is the engineering normal strain, L is the original length of the fiber and l is the final length of the fiber. Measures of strain are often expressed in parts per million or microstrains.

The true shear strain is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The engineering shear strain is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application which sometimes makes it easier to calculate.

Stretch Ratio

The stretch ratio or extension 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 l and the initial length L of the material line. [math]\displaystyle{ \lambda = \frac{l}{L} }[/math]

The extension ratio is approximately related to the engineering strain by [math]\displaystyle{ e = \frac{l-L}{L} = \lambda - 1 }[/math]

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

The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastomers, 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.

True Strain

The logarithmic strain ε, also called, true strain or Hencky strain.^[5] Considering an incremental strain (Ludwik) [math]\displaystyle{ \delta \varepsilon = \frac{\delta l}{l} }[/math] the logarithmic strain is obtained by integrating this incremental strain: [math]\displaystyle{ \begin{align} \int\delta \varepsilon &= \int_L^l \frac{\delta l}{l} \\ \varepsilon &= \ln\left(\frac{l}{L}\right) = \ln (\lambda) \\ &= \ln(1+e) \\ &= e - \frac{e^2}{2} + \frac{e^3}{3} - \cdots \end{align} }[/math] where 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 the influence of the strain path.^[2]

Green Strain

The Green strain is defined as: [math]\displaystyle{ \varepsilon_G = \tfrac{1}{2} \left(\frac{l^2-L^2}{L^2}\right) = \tfrac{1}{2} (\lambda^2-1) }[/math]

Almansi Strain

The Euler-Almansi strain is defined as [math]\displaystyle{ \varepsilon_E = \tfrac{1}{2} \left(\frac{l^2-L^2}{l^2}\right) = \tfrac{1}{2} \left(1-\frac{1}{\lambda^2}\right) }[/math]

1.2. Normal and Shear Strain

Two-dimensional geometric deformation of an infinitesimal material element. https://handwiki.org/wiki/index.php?curid=2024745

Two-dimensional geometric deformation of an infinitesimal material element. https://handwiki.org/wiki/index.php?curid=2024745

Strains are classified as either normal or shear. A normal strain is perpendicular to the face of an element, and a shear strain is parallel to it. These definitions are consistent with those of normal stress and shear stress.

Normal Strain

For an isotropic material that obeys Hooke's law, a normal stress will cause a normal strain. Normal strains produce dilations.

Consider a two-dimensional, infinitesimal, rectangular material element with dimensions dx × dy, which, after deformation, takes the form of a rhombus. The deformation is described by the displacement field u. From the geometry of the adjacent figure we have [math]\displaystyle{ \mathrm{length}(AB) = dx }[/math] and [math]\displaystyle{ \begin{align} \mathrm{length}(ab) &= \sqrt{\left(dx+\frac{\partial u_x}{\partial x}dx \right)^2 + \left( \frac{\partial u_y}{\partial x}dx \right)^2} \\ &= \sqrt{dx^2\left(1+\frac{\partial u_x}{\partial x} \right)^2 + dx^2\left( \frac{\partial u_y}{\partial x} \right)^2} \\ &= dx~\sqrt{\left(1+\frac{\partial u_x}{\partial x} \right)^2 + \left( \frac{\partial u_y}{\partial x} \right)^2} \end{align} }[/math] For very small displacement gradients the square of the derivative of [math]\displaystyle{ u_y }[/math] are negligible and we have [math]\displaystyle{ \mathrm{length}(ab) \approx dx \left(1+\frac{\partial u_x}{\partial x}\right) = dx + \frac{\partial u_x}{\partial x} dx }[/math] The normal strain in the x-direction of the rectangular element is defined by [math]\displaystyle{ \varepsilon_x = \frac{\text{extension}}{\text{original length}} = \frac{\mathrm{length}(ab) - \mathrm{length}(AB)}{\mathrm{length}(AB)} = \frac{\partial u_x}{\partial x} }[/math] Similarly, the normal strain in the y- and z-directions becomes [math]\displaystyle{ \varepsilon_y = \frac{\partial u_y}{\partial y} \quad , \qquad \varepsilon_z = \frac{\partial u_z}{\partial z} }[/math]

Shear Strain

The engineering shear strain (γ_xy) is defined as the change in angle between lines AC and AB. Therefore, [math]\displaystyle{ \gamma_{xy} = \alpha + \beta }[/math]

From the geometry of the figure, we have [math]\displaystyle{ \begin{align} \tan \alpha & = \frac{\tfrac{\partial u_y}{\partial x} dx}{dx + \tfrac{\partial u_x}{\partial x}dx} = \frac{\tfrac{\partial u_y}{\partial x}}{1 + \tfrac{\partial u_x}{\partial x}} \\ \tan \beta & = \frac{\tfrac{\partial u_x}{\partial y}dy}{dy+\tfrac{\partial u_y}{\partial y}dy}=\frac{\tfrac{\partial u_x}{\partial y}}{1+\tfrac{\partial u_y}{\partial y}} \end{align} }[/math] For small displacement gradients we have [math]\displaystyle{ \frac{\partial u_x}{\partial x} \ll 1 ~;~~ \frac{\partial u_y}{\partial y} \ll 1 }[/math] For small rotations, i.e. α and β are ≪ 1 we have tan α ≈ α, tan β ≈ β. Therefore, [math]\displaystyle{ \alpha \approx \frac{\partial u_y}{\partial x} ~;~~ \beta \approx \frac{\partial u_x}{\partial y} }[/math] thus [math]\displaystyle{ \gamma_{xy} = \alpha + \beta = \frac{\partial u_y}{\partial x} + \frac{\partial u_x}{\partial y} }[/math] By interchanging x and y and u_x and u_y, it can be shown that γ_xy = γ_yx.

Similarly, for the yz- and xz-planes, we have [math]\displaystyle{ \gamma_{yz} = \gamma_{zy} = \frac{\partial u_y}{\partial z} + \frac{\partial u_z}{\partial y} \quad , \qquad \gamma_{zx} = \gamma_{xz} = \frac{\partial u_z}{\partial x} + \frac{\partial u_x}{\partial z} }[/math]

The tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, γ, as [math]\displaystyle{ \underline{\underline{\boldsymbol{\varepsilon}}} = \begin{bmatrix} \varepsilon_{xx} & \varepsilon_{xy} & \varepsilon_{xz} \\ \varepsilon_{yx} & \varepsilon_{yy} & \varepsilon_{yz} \\ \varepsilon_{zx} & \varepsilon_{zy} & \varepsilon_{zz} \\ \end{bmatrix} = \begin{bmatrix} \varepsilon_{xx} & \tfrac 1 2\gamma_{xy} & \tfrac12\gamma_{xz} \\ \tfrac 1 2\gamma_{yx} & \varepsilon_{yy} & \tfrac 1 2\gamma_{yz} \\ \tfrac12\gamma_{zx} & \tfrac 1 2 \gamma_{zy} & \varepsilon_{zz} \\ \end{bmatrix} }[/math]

1.3. Metric Tensor

A strain field associated with a displacement is defined, at any point, by the change in length of the tangent vectors representing the speeds of arbitrarily parametrized curves passing through that point. A basic geometric result, due to Fréchet, von Neumann and Jordan, states that, if the lengths of the tangent vectors fulfil the axioms of a norm and the parallelogram law, then the length of a vector is the square root of the value of the quadratic form associated, by the polarization formula, with a positive definite bilinear map called the metric tensor.

2. Description of Deformation

Deformation is the change in the metric properties of a continuous body, meaning that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If none of the curves changes length, it is said that a rigid body displacement occurred.

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 be one the body actually will ever occupy. Often, the configuration at t = 0 is considered the reference configuration, κ₀(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 analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest.

The components Xi of the position vector 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 xi of the position vector x of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the spatial 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 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.

2.1. Affine Deformation

A deformation is called an affine deformation if it can be described by an affine transformation. Such a transformation is composed of a linear transformation (such as rotation, shear, extension and compression) and a rigid body translation. Affine deformations are also called homogeneous deformations.^[6]

Therefore, an affine deformation has the form [math]\displaystyle{ \mathbf{x}(\mathbf{X},t) = \boldsymbol{F}(t) \cdot \mathbf{X} + \mathbf{c}(t) }[/math] where x is the position of a point in the deformed configuration, X is the position in a reference configuration, t is a time-like parameter, F is the linear transformer and c is the translation. In matrix form, where the components are with respect to an orthonormal basis, [math]\displaystyle{ \begin{bmatrix} x_1(X_1, X_2, X_3, t) \\ x_2(X_1, X_2, X_3, t) \\ x_3(X_1, X_2, X_3, t) \end{bmatrix} = \begin{bmatrix} F_{11}(t) & F_{12}(t) & F_{13}(t) \\ F_{21}(t) & F_{22}(t) & F_{23}(t) \\ F_{31}(t) & F_{32}(t) & F_{33}(t) \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} + \begin{bmatrix} c_1(t) \\ c_2(t) \\ c_3(t) \end{bmatrix} }[/math]

The above deformation becomes non-affine or inhomogeneous if F = F(X,t) or c = c(X,t).

2.2. Rigid Body Motion

A rigid body motion is a special affine deformation that does not involve any shear, extension or compression. The transformation matrix F is proper orthogonal in order to allow rotations but no reflections.

A rigid body motion can be described by [math]\displaystyle{ \mathbf{x}(\mathbf{X},t) = \boldsymbol{Q}(t)\cdot\mathbf{X} + \mathbf{c}(t) }[/math] where [math]\displaystyle{ \boldsymbol{Q}\cdot\boldsymbol{Q}^T = \boldsymbol{Q}^T \cdot \boldsymbol{Q} = \boldsymbol{\mathit{1}} }[/math] In matrix form, [math]\displaystyle{ \begin{bmatrix} x_1(X_1, X_2, X_3, t) \\ x_2(X_1, X_2, X_3, t) \\ x_3(X_1, X_2, X_3, t) \end{bmatrix} = \begin{bmatrix} Q_{11}(t) & Q_{12}(t) & Q_{13}(t) \\ Q_{21}(t) & Q_{22}(t) & Q_{23}(t) \\ Q_{31}(t) & Q_{32}(t) & Q_{33}(t) \end{bmatrix} \begin{bmatrix} X_1 \\ X_2 \\ X_3 \end{bmatrix} + \begin{bmatrix} c_1(t) \\ c_2(t) \\ c_3(t) \end{bmatrix} }[/math]

3. Displacement

Figure 1. Motion of a continuum body. https://handwiki.org/wiki/index.php?curid=2085219

Figure 1. Motion of a continuum body. https://handwiki.org/wiki/index.php?curid=2085219

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 consists 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 κ₀(B) to a current or deformed configuration κ_t(B) (Figure 1).

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

The vector joining the positions of a particle P in the undeformed configuration and deformed configuration is called the displacement vector u(X,t) = uie_i in the Lagrangian description, or U(x,t) = UJE_J in the Eulerian description.

A displacement 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 [math]\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 }[/math] or in terms of the spatial coordinates as [math]\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 }[/math] where α_Ji are the direction cosines between the material and spatial coordinate systems with unit vectors E_J and e_i, respectively. Thus [math]\displaystyle{ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji} = \alpha_{iJ} }[/math] and the relationship between u_i and U_J is then given by [math]\displaystyle{ u_i = \alpha_{iJ} U_J \qquad \text{or} \qquad U_J = \alpha_{Ji} u_i }[/math]

Knowing that [math]\displaystyle{ \mathbf e_i = \alpha_{iJ} \mathbf E_J }[/math] then [math]\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) }[/math]

It is common to superimpose the coordinate systems for the undeformed and deformed configurations, which results in b = 0, and the direction cosines become Kronecker deltas: [math]\displaystyle{ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji} = \delta_{iJ} }[/math]

Thus, we have [math]\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 = x_i - X_i }[/math] or in terms of the spatial coordinates as [math]\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 = x_J - X_J }[/math]

Displacement Gradient Tensor

The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor ∇_Xu. Thus we have: [math]\displaystyle{ \begin{align} \mathbf{u}(\mathbf{X},t) & = \mathbf{x}(\mathbf{X},t) - \mathbf{X} \\ \nabla_\mathbf{X}\mathbf{u} & = \nabla_\mathbf{X} \mathbf{x} - \mathbf{I} \\ \nabla_\mathbf{X}\mathbf{u} & = \mathbf{F} - \mathbf{I} \end{align} }[/math] or [math]\displaystyle{ \begin{align} u_i & = x_i - \delta_{iJ} X_J = x_i - X_i\\ \frac{\partial u_i}{\partial X_K} & = \frac{\partial x_i}{\partial X_K} - \delta_{iK} \end{align} }[/math] where 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 ∇_xU. Thus we have, [math]\displaystyle{ \begin{align} \mathbf U(\mathbf x,t) &= \mathbf x - \mathbf X(\mathbf x,t) \\ \nabla_{\mathbf x} \mathbf U &= \mathbf I - \nabla_{\mathbf x} \mathbf X \\ \nabla_{\mathbf x} \mathbf U &= \mathbf I -\mathbf F^{-1} \end{align} }[/math] or [math]\displaystyle{ \begin{align} U_J& = \delta_{Ji}x_i-X_J =x_J - X_J\\ \frac{\partial U_J}{\partial x_k} &= \delta_{Jk} - \frac{\partial X_J}{\partial x_k} \end{align} }[/math]

4. Examples of Deformations

Homogeneous (or affine) deformations are useful in elucidating the behavior of materials. Some homogeneous deformations of interest are

• uniform extension
• pure dilation
• equibiaxial tension
• simple shear
• pure shear

Plane deformations are also of interest, particularly in the experimental context.

Plane Deformation

A plane deformation, also called plane strain, is one where the deformation is restricted to one of the planes in the reference configuration. If the deformation is restricted to the plane described by the basis vectors e₁, e₂, the deformation gradient has the form [math]\displaystyle{ \boldsymbol{F} = F_{11} \mathbf{e}_1 \otimes \mathbf{e}_1 + F_{12} \mathbf{e}_1 \otimes \mathbf{e}_2 + F_{21} \mathbf{e}_2 \otimes \mathbf{e}_1 + F_{22} \mathbf{e}_2 \otimes \mathbf{e}_2 + \mathbf{e}_3 \otimes \mathbf{e}_3 }[/math] In matrix form, [math]\displaystyle{ \boldsymbol{F} = \begin{bmatrix} F_{11} & F_{12} & 0 \\ F_{21} & F_{22} & 0 \\ 0 & 0 & 1 \end{bmatrix} }[/math] From the polar decomposition theorem, the deformation gradient, up to a change of coordinates, can be decomposed into a stretch and a rotation. Since all the deformation is in a plane, we can write^[6] [math]\displaystyle{ \boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U} = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ -\sin\theta & \cos\theta & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} \lambda_1 & 0 & 0 \\ 0 & \lambda_2 & 0 \\ 0 & 0 & 1 \end{bmatrix} }[/math] where θ is the angle of rotation and λ₁, λ₂ are the principal stretches.

Isochoric Plane Deformation

If the deformation is isochoric (volume preserving) then det(F) = 1 and we have [math]\displaystyle{ F_{11} F_{22} - F_{12} F_{21} = 1 }[/math] Alternatively, [math]\displaystyle{ \lambda_1 \lambda_2 = 1 }[/math]

Simple Shear

A simple shear deformation is defined as an isochoric plane deformation in which there is a set of line elements with a given reference orientation that do not change length and orientation during the deformation.^[6]

If e₁ is the fixed reference orientation in which line elements do not deform during the deformation then λ₁ = 1 and F·e₁ = e₁. Therefore, [math]\displaystyle{ F_{11}\mathbf{e}_1 + F_{21}\mathbf{e}_2 = \mathbf{e}_1 \quad \implies \quad F_{11} = 1 ~;~~ F_{21} = 0 }[/math] Since the deformation is isochoric, [math]\displaystyle{ F_{11} F_{22} - F_{12} F_{21} = 1 \quad \implies \quad F_{22} = 1 }[/math] Define [math]\displaystyle{ \gamma := F_{12} }[/math] Then, the deformation gradient in simple shear can be expressed as [math]\displaystyle{ \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} }[/math] Now, [math]\displaystyle{ \boldsymbol{F}\cdot\mathbf{e}_2 = F_{12}\mathbf{e}_1 + F_{22}\mathbf{e}_2 = \gamma\mathbf{e}_1 + \mathbf{e}_2 \quad \implies \quad \boldsymbol{F} \cdot (\mathbf{e}_2 \otimes \mathbf{e}_2) = \gamma \mathbf{e}_1\otimes \mathbf{e}_2 + \mathbf{e}_2 \otimes\mathbf{e}_2 }[/math] Since [math]\displaystyle{ \mathbf{e}_i \otimes \mathbf{e}_i = \boldsymbol{\mathit{1}} }[/math] we can also write the deformation gradient as [math]\displaystyle{ \boldsymbol{F} = \boldsymbol{\mathit{1}} + \gamma\mathbf{e}_1 \otimes \mathbf{e}_2 }[/math]