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Continuum mechanics

In continuum mechanics, deformation or strain is the change in the metric properties of a continuous body B in the displacement from an initial placement κ0(B) to a final placement κ(B). A change in the metric properties means that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If all the curves do not change length, it is said that a rigid body displacement occurred.

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 fulfill 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.

Different equivalent choices may be made for the expression of a strain field depending on whether it is defined in the initial or in the final placement 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 elastic 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, which remain even after stresses have been removed, are called plastic deformations. Such deformations occur 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.

Deformation is measured in units of length.



Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body. It measures how much a given displacement differs locally from a rigid-body displacement (Jacob Lubliner). Strain defines the amount of stretch or compression along a material line elements or fibers, the normal strain, and the amount of distortion associated with the sliding of plane layers over each other, the 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. This could be applied by elongation, shortening, or volume changes, or angular distortion.[1]

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.


Strain measures

Depending on the amount of strain, or 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 behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel.
  3. 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% (David Rees page 41), 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 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 or negative if they are compressed. Thus, we have

\ e=\frac{\Delta L}{L}=\frac{\ell -L}{L}

where 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.

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.

\ \lambda=\frac{\ell}{L}

The extension ratio is related to the engineering strain by

\ e=\frac{\ell-L}{L}=\lambda-1

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 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.

The logarithmic strain ε, also called natural strain, true strain or Hencky strain. Considering an incremental strain (Ludwik)

\ \delta \varepsilon=\frac{\delta \ell}{\ell}

the logarithmic strain is obtained by integrating this incremental strain:

\ \begin{align} \int\delta \varepsilon &=\int_{L}^{\ell}\frac{\delta \ell}{\ell}\ \varepsilon&=\ln\left(\frac{\ell}{L}\right)=\ln \lambda \ &=\ln(1+e) \ &=e-e^2/2+e^3/3- \cdots \ \end{align}

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 (David Rees).

The Green strain is defined as

\ \varepsilon_G=\frac{1}{2}\left(\frac{\ell^2-L^2}{L^2}\right)=\frac{1}{2}(\lambda^2-1)

The Green strain is addressed in more detail in the article on finite strain theory.

The Euler-Almansi strain is defined as

\ \varepsilon_E=\frac{1}{2}\left(\frac{\ell^2-L^2}{\ell^2}\right)=\frac{1}{2}\left(1-\frac{1}{\lambda^2}\right)

The Euler-Almansi strain is addressed in more detail in the finite strain theory.

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 {nowrap|1=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 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.


Figure 1. Motion of a continuum body.

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 \ \kappa_0(\mathcal B) to a current or deformed configuration \ \kappa_t(\mathcal 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) = uiei in the Lagrangian description, or U(x, t) = UJEJ 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

\ \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

\ \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 αJi are the direction cosines between the material and spatial coordinate systems with unit vectors EJ and ei, respectively. Thus

\ \mathbf E_J \cdot \mathbf e_i = \alpha_{Ji}=\alpha_{iJ}

and the relationship between ui and UJ is then given by

\ u_i=\alpha_{iJ}U_J \qquad \text{or} \qquad U_J=\alpha_{Ji}u_i

Knowing that

\ \mathbf e_i = \alpha_{iJ}\mathbf E_J


\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 b = 0, and the direction cosines become Kronecker deltas:

\ \mathbf E_J \cdot \mathbf e_i = \delta_{Ji}=\delta_{iJ}.

Thus, we have

\ \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

or in terms of the spatial coordinates as

\ \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

Displacement gradient tensor

The partial differentiation of the displacement vector with respect to the material coordinates yields the material displacement gradient tensor \ \mathbf u\nabla_{\mathbf X}. Thus we have:

\ \begin{align} \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{align} \qquad \text{or} \qquad \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}

where \ \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 \ \mathbf U\nabla_{\mathbf x}. Thus we have,

\ \begin{align} \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{align} \qquad \text{or} \qquad \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}


  1. ^ "Earth."Encyclopædia Britannica from Encyclopædia Britannica 2006 Ultimate Reference Suite DVD .[2009].


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