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continuum mechanics Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
, the infinitesimal strain theory is a mathematical approach to the description of the
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
of a solid body in which the displacements of the material
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object which can be described by several physical property, physical or chemical property, chemical ...
s are assumed to be much smaller (indeed,
infinitesimally In mathematics, an infinitesimal number is a quantity that is closer to 0, zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century New Latin, Modern Latin coinage ''infinitesimus'', which ori ...
smaller) than any relevant dimension of the body; so that its geometry and the constitutive properties of the material (such as
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and
stiffness Stiffness is the extent to which an object resists deformation in response to an applied force. The complementary concept is flexibility or pliability: the more flexible an object is, the less stiff it is. Calculations The stiffness, k, of a b ...
) at each point of space can be assumed to be unchanged by the deformation. With this assumption, the equations of continuum mechanics are considerably simplified. This approach may also be called small deformation theory, small displacement theory, or small displacement-gradient theory. It is contrasted with the
finite strain theory In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
where the opposite assumption is made. The infinitesimal strain theory is commonly adopted in civil and mechanical engineering for the
stress analysis Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
of structures built from relatively stiff
elastic Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics. Elastic may also refer to: Alternative name * Rubber band, ring-shaped band of rubber used to hold objects togeth ...
materials like
concrete Concrete is a composite material composed of fine and coarse aggregate bonded together with a fluid cement (cement paste) that hardens (cures) over time. Concrete is the second-most-used substance in the world after water, and is the most wi ...
and
steel Steel is an alloy made up of iron with added carbon to improve its strength and fracture resistance compared to other forms of iron. Many other elements may be present or added. Stainless steels that are corrosion- and oxidation-resistant ty ...
, since a common goal in the design of such structures is to minimize their deformation under typical loads. However, this approximation demands caution in the case of thin flexible bodies, such as rods, plates, and shells which are susceptible to significant rotations, thus making the results unreliable.


Infinitesimal strain tensor

For ''infinitesimal deformations'' of a continuum body, in which the displacement gradient (2nd order tensor) is small compared to unity, i.e. \, \nabla \mathbf u\, \ll 1 , it is possible to perform a ''geometric linearization'' of any one of the (infinitely many possible) strain tensors used in finite strain theory, e.g. the Lagrangian strain tensor \mathbf E, and the Eulerian strain tensor \mathbf e. In such a linearization, the non-linear or second-order terms of the finite strain tensor are neglected. Thus we have \mathbf E = \frac \left(\nabla_\mathbf u + (\nabla_\mathbf u)^T + (\nabla_\mathbf u)^T\nabla_\mathbf u\right)\approx \frac\left(\nabla_\mathbf u + (\nabla_\mathbf u)^T\right) or E_= \frac \left(\frac +\frac+ \frac \frac\right)\approx \frac\left(\frac+\frac\right) and \mathbf e =\frac \left(\nabla_\mathbf u + (\nabla_\mathbf u)^T - \nabla_\mathbf u(\nabla_\mathbf u)^T\right)\approx \frac\left(\nabla_\mathbf u + (\nabla_\mathbf u)^T\right) or e_=\frac \left(\frac +\frac -\frac \frac\right)\approx \frac\left(\frac +\frac\right) This linearization implies that the Lagrangian description and the Eulerian description are approximately the same as there is little difference in the material and spatial coordinates of a given material point in the continuum. Therefore, the material displacement gradient components and the spatial displacement gradient components are approximately equal. Thus we have \mathbf E \approx \mathbf e \approx \boldsymbol \varepsilon = \frac\left((\nabla\mathbf u)^T + \nabla\mathbf u\right) or E_\approx e_\approx\varepsilon_ = \frac \left(u_+u_\right) where \varepsilon_ are the components of the ''infinitesimal strain tensor'' \boldsymbol \varepsilon, also called ''Cauchy's strain tensor'', ''linear strain tensor'', or ''small strain tensor''. \begin \varepsilon_ &= \frac\left(u_+u_\right) \\ &= \begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \end \\ &= \begin \frac & \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) & \frac \\ \end \end or using different notation: \begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \end = \begin \frac & \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac & \frac \left(\frac+\frac\right) \\ \frac \left(\frac+\frac\right) & \frac \left(\frac+\frac\right) & \frac \\ \end Furthermore, since the
deformation gradient In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
can be expressed as \boldsymbol = \boldsymbol\mathbf + \boldsymbol where \boldsymbol is the second-order identity tensor, we have \boldsymbol\varepsilon = \frac \left(\boldsymbol^T+\boldsymbol\right)-\boldsymbol Also, from the general expression for the Lagrangian and Eulerian finite strain tensors we have \begin \mathbf E_& =\frac (\mathbf U^-\boldsymbol) = \frac \boldsymbol^T\boldsymbol)^m - \boldsymbol\approx \frac ^m - \boldsymbolapprox \boldsymbol\\ \mathbf e_& = \frac (\mathbf V^-\boldsymbol)= \frac \boldsymbol\boldsymbol^T)^m - \boldsymbolapprox \boldsymbol \end


Geometric derivation

Consider a two-dimensional deformation of an infinitesimal rectangular material element with dimensions dx by dy (Figure 1), which after deformation, takes the form of a rhombus. From the geometry of Figure 1 we have \begin \overline &= \sqrt \\ &= dx\sqrt \\ \end For very small displacement gradients, i.e., \, \nabla \mathbf u\, \ll 1 , we have \overline \approx dx + \frac dx The
normal strain In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
in the x-direction of the rectangular element is defined by \varepsilon_x = \frac and knowing that \overline = dx, we have \varepsilon_x = \frac Similarly, the normal strain in the and becomes \varepsilon_y = \frac \quad , \qquad \varepsilon_z = \frac The engineering shear strain, or the change in angle between two originally orthogonal material lines, in this case line \overline and \overline , is defined as \gamma_= \alpha + \beta From the geometry of Figure 1 we have \tan \alpha = \frac = \frac \quad , \qquad \tan \beta=\frac=\frac For small rotations, i.e., \alpha and \beta are \ll 1 we have \tan \alpha \approx \alpha \quad , \qquad \tan \beta \approx \beta and, again, for small displacement gradients, we have \alpha=\frac \quad , \qquad \beta=\frac thus \gamma_= \alpha + \beta = \frac + \frac By interchanging x and y and u_x and u_y, it can be shown that \gamma_ = \gamma_. Similarly, for the y-z and x-z planes, we have \gamma_ = \gamma_ = \frac + \frac \quad , \qquad \gamma_ = \gamma_ = \frac + \frac It can be seen that the tensorial shear strain components of the infinitesimal strain tensor can then be expressed using the engineering strain definition, as \begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \end = \begin \varepsilon_ & \gamma_/2 & \gamma_/2 \\ \gamma_/2 & \varepsilon_ & \gamma_/2 \\ \gamma_/2 & \gamma_/2 & \varepsilon_ \\ \end


Physical interpretation

From
finite strain theory In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
we have d\mathbf^2 - d\mathbf^2 = d\mathbf X \cdot 2\mathbf E \cdot d\mathbf X \quad\text\quad (dx)^2 - (dX)^2 = 2E_\,dX_K\,dX_L For infinitesimal strains then we have d\mathbf^2 - d\mathbf^2 = d\mathbf X \cdot 2\mathbf \cdot d\mathbf X \quad\text\quad (dx)^2 - (dX)^2 = 2\varepsilon_\,dX_K\,dX_L Dividing by (dX)^2 we have \frac\frac=2\varepsilon_\frac\frac For small deformations we assume that dx \approx dX, thus the second term of the left hand side becomes: \frac \approx 2. Then we have \frac = \varepsilon_N_iN_j = \mathbf N \cdot \boldsymbol \varepsilon \cdot \mathbf N where N_i=\frac, is the unit vector in the direction of d\mathbf X, and the left-hand-side expression is the
normal strain In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
e_ in the direction of \mathbf N. For the particular case of \mathbf N in the X_1 direction, i.e., \mathbf N = \mathbf I_1, we have e_=\mathbf I_1 \cdot \boldsymbol \varepsilon \cdot \mathbf I_1 = \varepsilon_. Similarly, for \mathbf N=\mathbf I_2 and \mathbf N=\mathbf I_3 we can find the normal strains \varepsilon_ and \varepsilon_, respectively. Therefore, the diagonal elements of the infinitesimal strain tensor are the normal strains in the coordinate directions.


Strain transformation rules

If we choose an orthonormal coordinate system (\mathbf_1,\mathbf_2,\mathbf_3) we can write the tensor in terms of components with respect to those base vectors as \boldsymbol = \sum_^3 \sum_^3 \varepsilon_ \mathbf_i\otimes\mathbf_j In matrix form, \underline = \begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \end We can easily choose to use another orthonormal coordinate system (\hat_1,\hat_2,\hat_3) instead. In that case the components of the tensor are different, say \boldsymbol = \sum_^3 \sum_^3 \hat_ \hat_i\otimes\hat_j \quad \implies \quad \underline = \begin \hat_ & \hat_ & \hat_ \\ \hat_ & \hat_ & \hat_ \\ \hat_ & \hat_ & \hat_ \end The components of the strain in the two coordinate systems are related by \hat_ = \ell_~\ell_~\varepsilon_ where the
Einstein summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
for repeated indices has been used and \ell_ = \hat_i\cdot_j. In matrix form \underline = \underline ~\underline~ \underline^T or \begin \hat_ & \hat_ & \hat_ \\ \hat_ & \hat_ & \hat_ \\ \hat_ & \hat_ & \hat_ \end = \begin \ell_ & \ell_ & \ell_ \\ \ell_ & \ell_ & \ell_ \\ \ell_ & \ell_ & \ell_ \end \begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \end \begin \ell_ & \ell_ & \ell_ \\ \ell_ & \ell_ & \ell_ \\ \ell_ & \ell_ & \ell_ \end^T


Strain invariants

Certain operations on the strain tensor give the same result without regard to which orthonormal coordinate system is used to represent the components of strain. The results of these operations are called strain invariants. The most commonly used strain invariants are \begin I_1 & = \mathrm(\boldsymbol) \\ I_2 & = \tfrac\ \\ I_3 & = \det(\boldsymbol) \end In terms of components \begin I_1 & = \varepsilon_ + \varepsilon_ + \varepsilon_ \\ I_2 & = \varepsilon_\varepsilon_ + \varepsilon_\varepsilon_ + \varepsilon_\varepsilon_ - \varepsilon_^2 - \varepsilon_^2 - \varepsilon_^2 \\ I_3 & = \varepsilon_(\varepsilon_\varepsilon_ - \varepsilon_^2) - \varepsilon_(\varepsilon_\varepsilon_-\varepsilon_\varepsilon_) + \varepsilon_(\varepsilon_\varepsilon_-\varepsilon_\varepsilon_) \end


Principal strains

It can be shown that it is possible to find a coordinate system (\mathbf_1,\mathbf_2,\mathbf_3) in which the components of the strain tensor are \underline = \begin \varepsilon_ & 0 & 0 \\ 0 & \varepsilon_ & 0 \\ 0 & 0 & \varepsilon_ \end \quad \implies \quad \boldsymbol = \varepsilon_ \mathbf_1\otimes\mathbf_1 + \varepsilon_ \mathbf_2\otimes\mathbf_2 + \varepsilon_ \mathbf_3\otimes\mathbf_3 The components of the strain tensor in the (\mathbf_1,\mathbf_2,\mathbf_3) coordinate system are called the principal strains and the directions \mathbf_i are called the directions of principal strain. Since there are no shear strain components in this coordinate system, the principal strains represent the maximum and minimum stretches of an elemental volume. If we are given the components of the strain tensor in an arbitrary orthonormal coordinate system, we can find the principal strains using an
eigenvalue decomposition In linear algebra, eigendecomposition is the factorization of a matrix into a canonical form, whereby the matrix is represented in terms of its eigenvalues and eigenvectors. Only diagonalizable matrices can be factorized in this way. When the matr ...
determined by solving the system of equations (\underline - \varepsilon_i~\underline)~\mathbf_i = \underline This system of equations is equivalent to finding the vector \mathbf_i along which the strain tensor becomes a pure stretch with no shear component.


Volumetric strain

The ''dilatation'' (the relative variation of the volume) is the first strain invariant or
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album) Other uses in arts and entertainment * ''Trace'' ...
of the tensor: \delta=\frac = I_1 = \varepsilon_ + \varepsilon_ + \varepsilon_ Actually, if we consider a cube with an edge length ''a'', it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions a \cdot (1 + \varepsilon_) \times a \cdot (1 + \varepsilon_) \times a \cdot (1 + \varepsilon_) and ''V''0 = ''a''3, thus \frac = \frac as we consider small deformations, 1 \gg \varepsilon_ \gg \varepsilon_ \cdot \varepsilon_ \gg \varepsilon_ \cdot \varepsilon_ \cdot \varepsilon_ therefore the formula. In case of pure shear, we can see that there is no change of the volume.


Strain deviator tensor

The infinitesimal strain tensor \varepsilon_, similarly to the
Cauchy stress tensor In continuum mechanics, the Cauchy stress tensor \boldsymbol\sigma, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components \sigma_ that complete ...
, can be expressed as the sum of two other tensors: # a mean strain tensor or volumetric strain tensor or spherical strain tensor, \varepsilon_M\delta_, related to dilation or volume change; and # a deviatoric component called the strain deviator tensor, \varepsilon'_, related to distortion. \varepsilon_= \varepsilon'_ + \varepsilon_M\delta_ where \varepsilon_M is the mean strain given by \varepsilon_M = \frac = \frac = \tfracI^e_1 The deviatoric strain tensor can be obtained by subtracting the mean strain tensor from the infinitesimal strain tensor: \begin \ \varepsilon'_ &= \varepsilon_ - \frac\delta_ \\ \begin \varepsilon'_ & \varepsilon'_ & \varepsilon'_ \\ \varepsilon'_ & \varepsilon'_ & \varepsilon'_ \\ \varepsilon'_ & \varepsilon'_ & \varepsilon'_ \\ \end &=\begin \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_ \\ \end - \begin \varepsilon_M & 0 & 0 \\ 0 & \varepsilon_M & 0 \\ 0 & 0 & \varepsilon_M \\ \end \\ &=\begin \varepsilon_-\varepsilon_M & \varepsilon_ & \varepsilon_ \\ \varepsilon_ & \varepsilon_-\varepsilon_M & \varepsilon_ \\ \varepsilon_ & \varepsilon_ & \varepsilon_-\varepsilon_M \\ \end \\ \end


Octahedral strains

Let (\mathbf_1, \mathbf_2, \mathbf_3) be the directions of the three principal strains. An octahedral plane is one whose normal makes equal angles with the three principal directions. The engineering
shear strain In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
on an octahedral plane is called the octahedral shear strain and is given by \gamma_ = \tfrac\sqrt where \varepsilon_1, \varepsilon_2, \varepsilon_3 are the principal strains. The
normal strain In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
on an octahedral plane is given by \varepsilon_ = \tfrac(\varepsilon_1 + \varepsilon_2 + \varepsilon_3)


Equivalent strain

A scalar quantity called the equivalent strain, or the
von Mises Mises or von Mises may refer to: * Ludwig von Mises, an Austrian-American economist of the Austrian School, older brother of Richard von Mises ** Mises Institute, or the Ludwig von Mises Institute for Austrian Economics, named after Ludwig von ...
equivalent strain, is often used to describe the state of strain in solids. Several definitions of equivalent strain can be found in the literature. A definition that is commonly used in the literature on
plasticity Plasticity may refer to: Science * Plasticity (physics), in engineering and physics, the propensity of a solid material to undergo permanent deformation under load * Neuroplasticity, in neuroscience, how entire brain structures, and the brain it ...
is \varepsilon_ = \sqrt = \sqrt ~;~~ \boldsymbol^ = \boldsymbol - \tfrac\mathrm(\boldsymbol)~\boldsymbol This quantity is work conjugate to the equivalent stress defined as \sigma_ = \sqrt


Compatibility equations

For prescribed strain components \varepsilon_ the strain tensor equation u_+u_= 2 \varepsilon_ represents a system of six differential equations for the determination of three displacements components u_i, giving an over-determined system. Thus, a solution does not generally exist for an arbitrary choice of strain components. Therefore, some restrictions, named ''compatibility equations'', are imposed upon the strain components. With the addition of the three compatibility equations the number of independent equations are reduced to three, matching the number of unknown displacement components. These constraints on the strain tensor were discovered by
Saint-Venant Saint-Venant ( vls, Papingem) is a commune in the Pas-de-Calais department (administrative division) in the Hauts-de-France region of France. Geography Saint-Venant is situated some northwest of Béthune and west of Lille, at the junction o ...
, and are called the " Saint Venant compatibility equations". The compatibility functions serve to assure a single-valued continuous displacement function u_i. If the elastic medium is visualised as a set of infinitesimal cubes in the unstrained state, after the medium is strained, an arbitrary strain tensor may not yield a situation in which the distorted cubes still fit together without overlapping. In index notation, the compatibility equations are expressed as \varepsilon_+\varepsilon_-\varepsilon_-\varepsilon_=0 In engineering notation, * \frac + \frac = 2 \frac * \frac + \frac = 2 \frac * \frac + \frac = 2 \frac * \frac = \frac \left ( -\frac + \frac + \frac\right) * \frac = \frac \left ( \frac - \frac + \frac\right) * \frac = \frac \left ( \frac + \frac - \frac\right)


Special cases


Plane strain

In real engineering components,
stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
(and strain) are 3-D
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
s but in prismatic structures such as a long metal billet, the length of the structure is much greater than the other two dimensions. The strains associated with length, i.e., the normal strain \varepsilon_ and the shear strains \varepsilon_ and \varepsilon_ (if the length is the 3-direction) are constrained by nearby material and are small compared to the ''cross-sectional strains''. Plane strain is then an acceptable approximation. The
strain tensor In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
for plane strain is written as: \underline = \begin \varepsilon_ & \varepsilon_ & 0 \\ \varepsilon_ & \varepsilon_ & 0 \\ 0 & 0 & 0 \end in which the double underline indicates a second order
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
. This strain state is called ''plane strain''. The corresponding stress tensor is: \underline = \begin \sigma_ & \sigma_ & 0 \\ \sigma_ & \sigma_ & 0 \\ 0 & 0 & \sigma_ \end in which the non-zero \sigma_ is needed to maintain the constraint \epsilon_ = 0. This stress term can be temporarily removed from the analysis to leave only the in-plane terms, effectively reducing the 3-D problem to a much simpler 2-D problem.


Antiplane strain

Antiplane strain is another special state of strain that can occur in a body, for instance in a region close to a
screw dislocation In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sl ...
. The
strain tensor In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
for antiplane strain is given by \underline = \begin 0 & 0 & \varepsilon_ \\ 0 & 0 & \varepsilon_\\ \varepsilon_ & \varepsilon_ & 0 \end


Infinitesimal rotation tensor

The infinitesimal strain tensor is defined as \boldsymbol = \frac boldsymbol\mathbf + (\boldsymbol\mathbf)^T/math> Therefore the displacement gradient can be expressed as \boldsymbol\mathbf = \boldsymbol + \boldsymbol where \boldsymbol := \frac boldsymbol\mathbf - (\boldsymbol\mathbf)^T/math> The quantity \boldsymbol is the infinitesimal rotation tensor. This tensor is skew symmetric. For infinitesimal deformations the scalar components of \boldsymbol satisfy the condition , \omega_, \ll 1. Note that the displacement gradient is small only if both the strain tensor and the rotation tensor are infinitesimal.


The axial vector

A skew symmetric second-order tensor has three independent scalar components. These three components are used to define an axial vector, \mathbf, as follows \omega_ = -\epsilon_~w_k ~;~~ w_i = -\tfrac~\epsilon_~\omega_ where \epsilon_ is the
permutation symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
. In matrix form \underline = \begin 0 & -w_3 & w_2 \\ w_3 & 0 & -w_1 \\ -w_2 & w_1 & 0\end ~;~~ \underline = \begin w_1 \\ w_2 \\ w_3 \end The axial vector is also called the infinitesimal rotation vector. The rotation vector is related to the displacement gradient by the relation \mathbf = \tfrac~ \boldsymbol \times \mathbf In index notation w_i = \tfrac~\epsilon_~u_ If \lVert\boldsymbol\rVert \ll 1 and \boldsymbol = \boldsymbol then the material undergoes an approximate rigid body rotation of magnitude , \mathbf, around the vector \mathbf.


Relation between the strain tensor and the rotation vector

Given a continuous, single-valued displacement field \mathbf and the corresponding infinitesimal strain tensor \boldsymbol, we have (see
Tensor derivative (continuum mechanics) The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly i ...
) \boldsymbol\times\boldsymbol = e_~\varepsilon_~\mathbf_k\otimes\mathbf_l = \tfrac~e_~
_ + u_ U or u, is the twenty-first and sixth-to-last Letter (alphabet), letter and fifth vowel letter of the Latin alphabet, used in the English alphabet, modern English alphabet, the alphabets of other western European languages and others worldwide ...
\mathbf_k\otimes\mathbf_l Since a change in the order of differentiation does not change the result, u_ = u_. Therefore e_ u_ = (e_+e_) u_ + (e_+e_) u_ + (e_ + e_) u_ = 0 Also \tfrac~e_~u_ = \left(\tfrac~e_~u_\right)_ = \left(\tfrac ~ e_~u_\right)_ = w_ Hence \boldsymbol \times \boldsymbol = w_~\mathbf_k\otimes\mathbf_l = \boldsymbol\mathbf


Relation between rotation tensor and rotation vector

From an important identity regarding the curl of a tensor we know that for a continuous, single-valued displacement field \mathbf, \boldsymbol\times(\boldsymbol\mathbf) = \boldsymbol. Since \boldsymbol\mathbf = \boldsymbol + \boldsymbol we have \boldsymbol\times\boldsymbol = -\boldsymbol\times\boldsymbol = - \boldsymbol \mathbf.


Strain tensor in cylindrical coordinates

In
cylindrical polar coordinates A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
(r, \theta, z), the displacement vector can be written as \mathbf = u_r~\mathbf_r + u_\theta~\mathbf_\theta + u_z~\mathbf_z The components of the strain tensor in a cylindrical coordinate system are given by: \begin \varepsilon_ & = \cfrac \\ \varepsilon_ & = \cfrac\left(\cfrac + u_r\right) \\ \varepsilon_ & = \cfrac \\ \varepsilon_ & = \cfrac \left(\cfrac\cfrac + \cfrac - \cfrac\right) \\ \varepsilon_ & = \cfrac \left(\cfrac + \cfrac \cfrac\right) \\ \varepsilon_ & = \cfrac \left(\cfrac + \cfrac\right) \end


Strain tensor in spherical coordinates

In
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
(r, \theta, \phi), the displacement vector can be written as \mathbf = u_r~\mathbf_r + u_\theta~\mathbf_\theta + u_\phi~\mathbf_\phi The components of the strain tensor in a spherical coordinate system are given by \begin \varepsilon_ & = \cfrac \\ \varepsilon_ & = \cfrac\left(\cfrac + u_r\right) \\ \varepsilon_ & = \cfrac\left(\cfrac + u_r\sin\theta + u_\theta\cos\theta\right)\\ \varepsilon_ & = \cfrac\left(\cfrac\cfrac + \cfrac- \cfrac\right) \\ \varepsilon_ & = \cfrac\left(\cfrac\cfrac + \cfrac - u_\phi\cot\theta\right) \\ \varepsilon_ & = \cfrac\left(\cfrac\cfrac + \cfrac - \cfrac\right) \end


See also

*
Deformation (mechanics) In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
*
Compatibility (mechanics) In continuum mechanics, a compatible deformation (or strain) tensor field in a body is that ''unique'' tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. Compatibility is the study of th ...
*
Stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
*
Strain gauge A strain gauge (also spelled strain gage) is a device used to measure strain on an object. Invented by Edward E. Simmons and Arthur C. Ruge in 1938, the most common type of strain gauge consists of an insulating flexible backing which supports ...
*
Stress–strain curve In engineering and materials science, a stress–strain curve for a material gives the relationship between stress and strain. It is obtained by gradually applying load to a test coupon and measuring the deformation, from which the stress and ...
*
Hooke's law In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring (device), spring by some distance () Proportionality (mathematics)#Direct_proportionality, scales linearly with respect to that ...
*
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
*
Finite strain theory In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
*
Strain rate In materials science, strain rate is the change in strain (deformation) of a material with respect to time. The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change wi ...
*
Plane stress In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analys ...
*
Digital image correlation Digital image correlation and tracking is an optical method that employs tracking and image registration techniques for accurate 2D and 3D measurements of changes in images. This method is often used to measure full-field displacement and strains ...


References


External links

{{DEFAULTSORT:Infinitesimal Strain Theory Physical quantities Elasticity (physics) Materials science Solid mechanics Mechanics