Hooke's law


Hooke's law
Hooke's law models the properties of springs for small changes in length
Prof. Walter Lewin explains Hooke's law, in MIT's Course 8.01[1]
A test of Hooke's law (MIT OCW)[2]

In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Hooke's law in simple terms says that strain is directly proportional to stress.

Mathematically, Hooke's law states that

 \mathbf{F}=-k\mathbf{x},

where

x is the displacement of the spring's end from its equilibrium position (a distance, in SI units: meters);
F is the restoring force exerted by the spring on that end (in SI units: N or kg·m/s2); and
k is a constant called the rate or spring constant (in SI units: N/m or kg/s2).

When this holds, the behavior is said to be linear. If shown on a graph, the line should show a direct variation. There is a negative sign on the right hand side of the equation because the restoring force always acts in the opposite direction of the displacement (for example, when a spring is stretched to the left, it pulls back to the right).

Hooke's law is named after the 17th century British physicist Robert Hooke. He first stated this law in 1660 as a Latin anagram,[3] whose solution he published in 1678 as Ut tensio, sic vis, meaning, "As the extension, so the force".

Contents

General application to elastic materials

Hooke's law describes how far the spring will stretch under a specific force

Objects that quickly regain their original shape after being deformed by a force, with the molecules or atoms of their material returning to the initial state of stable equilibrium, often obey Hooke's law.

We may view a rod of any elastic material as a linear spring. The rod has length L and cross-sectional area A. Its extension (strain) is linearly proportional to its tensile stress σ, by a constant factor, the inverse of its modulus of elasticity, E, hence,

σ = Eε

or

\Delta L = \frac{F}{E A} L = \frac{\sigma}{E} L.

Hooke's law only holds for some materials under certain loading conditions. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic range (i.e., for stresses below the yield strength). For some other materials, such as aluminium, Hooke's law is only valid for a portion of the elastic range. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible.

Rubber is generally regarded as a "non-hookean" material because its elasticity is stress dependent and sensitive to temperature and loading rate.

Applications of the law include spring operated weighing machines, stress analysis and modelling of materials.

The spring equation

Stress–strain curve for low-carbon steel. Hooke's law is only valid for the portion of the curve between the origin and the yield point(2).
1. Ultimate strength
2. Yield strength – corresponds to yield point
3. Rupture
4. Strain hardening region
5. Necking region
A: (F/A0)
B: True stress (F/A)

The most commonly encountered form of Hooke's law is probably the spring equation, which relates the force exerted by a spring to the distance it is stretched by a spring constant, k, measured in force per length.

F=-kx\,

The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement. It is called a "restoring force", as it tends to restore the system to equilibrium. The potential energy stored in a spring is given by

 PE={1\over2}kx^2

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of force over displacement. (Note that potential energy of a spring is always non-negative.)

This potential can be visualized as a parabola on the U-x plane. As the spring is stretched in the positive x-direction, the potential energy increases (the same thing happens as the spring is compressed). The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position (x = 0). The tendency for the spring is to therefore decrease its potential energy by returning to its equilibrium (unstretched) position, just as a ball rolls downhill to decrease its gravitational potential energy.

If a mass m is attached to the end of such a spring, the system becomes a harmonic oscillator. It will oscillate with a natural frequency given either as an angular frequency

\omega =  \sqrt{k \over m}

or as a natural frequency

f = {1 \over 2 \pi} \sqrt{k \over m}.

This idealized description of spring mechanics works as long as the mass of the spring is very small compared to the mass m, there is no significant friction on the system, and the spring is not overextended beyond its natural range (which can deform it permanently).


Multiple springs

When two springs are attached to a mass and compressed, the following table compares values of the springs.

Comparison In Parallel In Series
SpringsInParallel.svg
SpringsInSeries.svg
Equivalent
spring constant
k_{eq} = k_1 + k_2 \, \frac{1}{k_{eq}} =  \frac{1}{k_1} + \frac{1}{k_2} \,
Compressed
distance
x_1 = x_2 \, \frac{x_1}{x_2} = \frac{k_2}{k_1} \,
Energy
stored
\frac{E_1}{E_2} = \frac{k_1}{k_2} \, \frac{E_1}{E_2} = \frac{k_2}{k_1} \,

Derivation

Tensor expression of Hooke's Law

Note: the Einstein summation convention of summing on repeated indices is used below.


When working with a three-dimensional stress state, a 4th order tensor \mathsf{c} (c_{ijk\ell}) containing 81 elastic coefficients must be defined to link the stress tensor \boldsymbol{\sigma}ij) and the strain tensor \boldsymbol{\epsilon} (\epsilon_{k\ell}).

\boldsymbol{\sigma} = \mathsf{c}:\boldsymbol{\epsilon} ~.

Expressed in terms of components with respect to an orthonormal basis, the generalized form of Hooke's law is written as (using the summation convention)

\sigma_{ij} = c_{ijk\ell}~\epsilon_{k\ell}

The tensor \mathsf{c} is called the stiffness tensor or the elasticity tensor. Due to the symmetry of the stress tensor, strain tensor, and stiffness tensor, only 21 elastic coefficients are independent. As stress is measured in units of pressure and strain is dimensionless, the entries of c_{ijk\ell} are also in units of pressure.

The expression for generalized Hooke's law can be inverted to get a relation for the strain in terms of stress:

 \boldsymbol{\epsilon} = \mathsf{s}:\boldsymbol{\sigma} \qquad {\rm or} \qquad
      \epsilon_{ij} = s_{ijk\ell}~\sigma_{k\ell} ~.

The tensor \mathsf{s} is called the compliance tensor.

Generalization for the case of large deformations is provided by models of neo-Hookean solids and Mooney-Rivlin solids.

Isotropic materials

(see viscosity for an analogous development for viscous fluids.)

Isotropic materials are characterized by properties which are independent of direction in space. Physical equations involving isotropic materials must therefore be independent of the coordinate system chosen to represent them. The strain tensor is a symmetric tensor. Since the trace of any tensor is independent of any coordinate system, the most complete coordinate-free decomposition of a symmetric tensor is to represent it as the sum of a constant tensor and a traceless symmetric tensor.[4]:Ch. 10 Thus:


  \varepsilon_{ij} = \left(\tfrac{1}{3}\varepsilon_{kk}\delta_{ij}\right) + 
         \left(\varepsilon_{ij}-\tfrac{1}{3}\varepsilon_{kk}\delta_{ij}\right)

where δij is the Kronecker delta. In direct tensor notation


   \boldsymbol{\varepsilon} = \mathrm{vol}(\boldsymbol{\varepsilon}) +  
        \mathrm{dev}(\boldsymbol{\varepsilon}) ~;~~ 
     \mathrm{vol}(\boldsymbol{\varepsilon}) := \tfrac{1}{3}~\mathrm{tr}(\boldsymbol{\varepsilon})~\mathbf{I} ~;~~
     \mathrm{dev}(\boldsymbol{\varepsilon}) := \boldsymbol{\varepsilon} - \mathrm{vol}(\boldsymbol{\varepsilon})

where \mathbf{I} is the second-order identity tensor. The first term on the right is the constant tensor, also known as the volumetric strain tensor, and the second term is the traceless symmetric tensor, also known as the deviatoric strain tensor or shear tensor.

The most general form of Hooke's law for isotropic materials may now be written as a linear combination of these two tensors:


  \sigma_{ij}=3K\left(\tfrac{1}{3}\varepsilon_{kk}\delta_{ij}\right)
       +2G\left(\varepsilon_{ij}-\tfrac{1}{3}\varepsilon_{kk}\delta_{ij}\right)\,~;~~
  \boldsymbol{\sigma} = 3K~\mathrm{vol}(\boldsymbol{\varepsilon}) + 2G~\mathrm{dev}(\boldsymbol{\varepsilon})

where K is the bulk modulus and G is the shear modulus.

Using the relationships between the elastic moduli, these equations may also be expressed in various other ways. A common form of Hooke's law for isotropic materials, expressed in direct tensor notation, is [5]


   \boldsymbol{\sigma} = \lambda~\mathrm{tr}(\boldsymbol{\varepsilon})~\mathbf{I} + 2\mu~\boldsymbol{\varepsilon}
     = \mathsf{c}:\boldsymbol{\varepsilon} ~;~~ \mathsf{c} = \lambda~\mathbf{I}\otimes\mathbf{I} + 2\mu~\mathsf{I}

where λ: = K − 2 / 3G and μ: = G are the Lamé constants, \mathbf{I} is the second-order identity tensor, and \mathsf{I} is the symmetric part of the fourth-order identity tensor. In terms of components with respect to a Cartesian basis,


   \sigma_{ij} = \lambda~\varepsilon_{kk}~\delta_{ij} + 2\mu~\varepsilon_{ij} = c_{ijk\ell}~\varepsilon_{k\ell} ~;~~ c_{ijk\ell} = \lambda~\delta_{ij}~\delta_{k\ell} + \mu~(\delta_{ik}~\delta_{j\ell} + \delta_{i\ell}~\delta_{jk})

The inverse relationship is[6]


   \boldsymbol{\varepsilon} = \tfrac{1}{2\mu}~\boldsymbol{\sigma} - \tfrac{\lambda}{2\mu(3\lambda+2\mu)}~\mathrm{tr}(\boldsymbol{\sigma})~\mathbf{I} = \tfrac{1}{2G}~\boldsymbol{\sigma} + \left(\tfrac{1}{9K} - \tfrac{1}{6G}\right)~\mathrm{tr}(\boldsymbol{\sigma})~\mathbf{I}

Therefore the compliance tensor in the relation \boldsymbol{\varepsilon} = \mathsf{s}:\boldsymbol{\sigma} is


   \mathsf{s} = - \tfrac{\lambda}{2\mu(3\lambda+2\mu)}~\mathbf{I}\otimes\mathbf{I} + \tfrac{1}{2\mu}~\mathsf{I}
      = \left(\tfrac{1}{9K} - \tfrac{1}{6G}\right)~\mathbf{I}\otimes\mathbf{I} + \tfrac{1}{2G}~\mathsf{I}

In terms of Young's modulus and Poisson's ratio, Hooke's law for isotropic materials can then be expressed as


   \boldsymbol{\varepsilon} = \tfrac{1}{E}~\boldsymbol{\sigma} - \tfrac{\nu}{E}\left[\mathrm{tr}(\boldsymbol{\sigma})~\mathbf{I} - \boldsymbol{\sigma}\right]

This is the form in which the strain is expressed in terms of the stress tensor in engineering. The expression in expanded form is


  \begin{align}
  \varepsilon_{11} & = \tfrac{1}{E}\left[ \sigma_{11} - \nu(\sigma_{22}+\sigma_{33}) \right] \\
  \varepsilon_{22} & = \tfrac{1}{E}\left[\sigma_{22} - \nu(\sigma_{11}+\sigma_{33}) \right] \\
  \varepsilon_{33} & = \tfrac{1}{E}\left[\sigma_{33} - \nu(\sigma_{11}+\sigma_{22}) \right] \\
  \varepsilon_{12} & = \tfrac{1}{2G}~\sigma_{12} ~;~~
  \varepsilon_{13} = \tfrac{1}{2G}~\sigma_{13} ~;~~
  \varepsilon_{23} = \tfrac{1}{2G}~\sigma_{23}
  \end{align}

where E is the modulus of elasticity and ν is Poisson's ratio. (See 3-D elasticity).

In matrix form, Hooke's law for isotropic materials can be written as


   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix} = 
   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \gamma_{23} \\ \gamma_{31} \\ \gamma_{12} \end{bmatrix} = 
   \cfrac{1}{E}
   \begin{bmatrix} 1 & -\nu & -\nu & 0 & 0 & 0 \\
                   -\nu & 1 & -\nu & 0 & 0 & 0 \\
                   -\nu & -\nu & 1 & 0 & 0 & 0 \\
                   0 & 0 & 0 & 2(1+\nu) & 0 & 0 \\
                   0 & 0 & 0 & 0 & 2(1+\nu) & 0 \\
                   0 & 0 & 0 & 0 & 0 & 2(1+\nu) \end{bmatrix}
    \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix}

where γij: = 2εij is the engineering shear strain. The inverse relation may be written as


   \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix}
   = \cfrac{E}{(1+\nu)(1-2\nu)}
   \begin{bmatrix} 1-\nu & \nu & \nu & 0 & 0 & 0 \\
                   \nu & 1-\nu & \nu & 0 & 0 & 0 \\
                   \nu & \nu & 1-\nu & 0 & 0 & 0 \\
                   0 & 0 & 0 & (1-2\nu)/2 & 0 & 0 \\
                   0 & 0 & 0 & 0 & (1-2\nu)/2 & 0 \\
                   0 & 0 & 0 & 0 & 0 & (1-2\nu)/2 \end{bmatrix}
    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix}

which expression can be simplified thanks to the Lamé constants :


   \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix}
   = 
   \begin{bmatrix} 2\mu+\lambda & \lambda & \lambda & 0 & 0 & 0 \\
                   \lambda & 2\mu+\lambda & \lambda & 0 & 0 & 0 \\
                   \lambda & \lambda & 2\mu+\lambda & 0 & 0 & 0 \\
                   0 & 0 & 0 & \mu & 0 & 0 \\
                   0 & 0 & 0 & 0 & \mu & 0 \\
                   0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix}
    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix}

Plane stress Hooke's law

Under plane stress conditions σ33 = σ31 = σ13 = σ32 = σ23 = 0. In that case Hooke's law takes the form


   \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ 2\varepsilon_{12} \end{bmatrix} = \cfrac{1}{E}
   \begin{bmatrix} 1 & -\nu & 0 \\
                   -\nu & 1 & 0 \\
                    0 & 0 & 2(1+\nu) \end{bmatrix}
    \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}

The inverse relation is usually written in the reduced form


   \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}
   = \cfrac{E}{1-\nu^2}
   \begin{bmatrix} 1 & \nu & 0 \\
                   \nu & 1 & 0 \\
                   0 & 0 & \cfrac{1-\nu}{2} \end{bmatrix}
    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ 2\varepsilon_{12} \end{bmatrix}

Anisotropic materials

The symmetry of the Cauchy stress tensor (\sigma_{ij} = \sigma_{ji}\,) and the generalized Hooke's laws (\sigma_{ij} = c_{ijk\ell}~ \epsilon_{k\ell}) implies that c_{ijk\ell} = c_{jik\ell}\,. Similarly, the symmetry of the infinitesimal strain tensor implies that c_{ijk\ell} = c_{ij\ell k}\,. These symmetries are called the minor symmetries of the stiffness tensor (\mathsf{c}).

If in addition, since the displacement gradient and the Cauchy stress are work conjugate, the stress-strain relation can be derived from a strain energy density functional (U), then


 \sigma_{ij} = \cfrac{\partial U}{\partial \epsilon_{ij}} \quad \implies \quad
c_{ijk\ell} =  \cfrac{\partial^2 U}{\partial \epsilon_{ij}\partial \epsilon_{k\ell}}~.

The arbitrariness of the order of differentiation implies that c_{ijk\ell} = c_{k\ell ij}\,. These are called the major symmetries of the stiffness tensor. The major and minor symmetries indicate that the stiffness tensor has only 21 independent components.

Matrix representation (stiffness tensor)

It is often useful to express the anisotropic form of Hooke's law in matrix notation, also called Voigt notation. To do this we take advantage of the symmetry of the stress and strain tensors and express them as six-dimensional vectors in an orthonormal coordinate system (\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3) as


   [\boldsymbol{\sigma}] = \begin{bmatrix}\sigma_{11}\\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix} \equiv
\begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} ~;~~
[\boldsymbol{\epsilon}] = \begin{bmatrix}\epsilon_{11}\\ \epsilon_{22} \\ \epsilon_{33} \\ 2\epsilon_{23} \\ 2\epsilon_{31} \\ 2\epsilon_{12} \end{bmatrix} \equiv
\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_3 \\ \epsilon_4 \\ \epsilon_5 \\ \epsilon_6 \end{bmatrix}

Then the stiffness tensor (\mathsf{c}) can be expressed as


   [\mathsf{C}] = \begin{bmatrix} c_{1111} & c_{1122} & c_{1133} & c_{1123} & c_{1131} & c_{1112} \\
      c_{2211} & c_{2222} & c_{2233} & c_{2223} & c_{2231} & c_{2212} \\
c_{3311} & c_{3322} & c_{3333} & c_{3323} & c_{3331} & c_{3312} \\
c_{2311} & c_{2322} & c_{2333} & c_{2323} & c_{2331} & c_{2312} \\
c_{3111} & c_{3122} & c_{3133} & c_{3123} & c_{3131} & c_{3112} \\
c_{1211} & c_{1222} & c_{1233} & c_{1223} & c_{1231} & c_{1212} 
     \end{bmatrix} \equiv \begin{bmatrix}
  C_{11} & C_{12} & C_{13} & C_{14} & C_{15} & C_{16} \\
C_{12} & C_{22} & C_{23} & C_{24} & C_{25} & C_{26} \\
C_{13} & C_{23} & C_{33} & C_{34} & C_{35} & C_{36} \\
C_{14} & C_{24} & C_{34} & C_{44} & C_{45} & C_{46} \\
C_{15} & C_{25} & C_{35} & C_{45} & C_{55} & C_{56} \\
C_{16} & C_{26} & C_{36} & C_{46} & C_{56} & C_{66} \end{bmatrix}

and Hooke's law is written as


   [\boldsymbol{\sigma}] = [\mathsf{C}][\boldsymbol{\epsilon}] \qquad \text{or} \qquad \sigma_i = C_{ij} \epsilon_j ~.

Similarly the compliance tensor (\mathsf{s}) can be written as


   [\mathsf{S}] = \begin{bmatrix}
s_{1111} & s_{1122} & s_{1133} & 2s_{1123} & 2s_{1131} & 2s_{1112} \\
s_{2211} & s_{2222} & s_{2233} & 2s_{2223} & 2s_{2231} & 2s_{2212} \\
s_{3311} & s_{3322} & s_{3333} & 2s_{3323} & 2s_{3331} & 2s_{3312} \\
2s_{2311} & 2s_{2322} & 2s_{2333} & 4s_{2323} & 4s_{2331} & 4s_{2312} \\
2s_{3111} & 2s_{3122} & 2s_{3133} & 4s_{3123} & 4s_{3131} & 4s_{3112} \\
2s_{1211} & 2s_{1222} & 2s_{1233} & 4s_{1223} & 4s_{1231} & 4s_{1212} 
 \end{bmatrix} \equiv \begin{bmatrix} 
S_{11} & S_{12} & S_{13} & S_{14} & S_{15} & S_{16} \\
S_{12} & S_{22} & S_{23} & S_{24} & S_{25} & S_{26} \\
S_{13} & S_{23} & S_{33} & S_{34} & S_{35} & S_{36} \\
S_{14} & S_{24} & S_{34} & S_{44} & S_{45} & S_{46} \\
S_{15} & S_{25} & S_{35} & S_{45} & S_{55} & S_{56} \\
S_{16} & S_{26} & S_{36} & S_{46} & S_{56} & S_{66} \end{bmatrix}

Change of coordinate system

If a linear elastic material is rotated from a reference configuration to another, then the material is symmetric with respect to the rotation if the components of the stiffness tensor in the rotated configuration are related to the components in the reference configuration by the relation[7]


   c_{pqrs} = l_{pi}~l_{qj}~l_{rk}~l_{s\ell}~c_{ijk\ell}

where lab are the components of an orthogonal rotation matrix [L]. The same relation also holds for inversions.

In matrix notation, if the transformed basis (rotated or inverted) is related to the reference basis by


   [\mathbf{e}_i'] = [L][\mathbf{e}_i]

then


   C_{ij}~\epsilon_i~\epsilon_j = C_{ij}'~\epsilon'_i~\epsilon'_j ~.

In addition, if the material is symmetric with respect to the transformation [L] then


   C_{ij} = C'_{ij} \quad \implies \quad C_{ij}~(\epsilon_i~\epsilon_j - \epsilon'_i~\epsilon'_j) = 0 ~.

Orthotropic materials

Orthotropic materials have three orthogonal planes of symmetry. If the basis vectors (\mathbf{e}_1,\mathbf{e}_2,\mathbf{e}_3) are normals to the planes of symmetry then the coordinate transformation relations imply that


\begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} = 
\begin{bmatrix}
  C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\
C_{12} & C_{22} & C_{23} & 0 & 0 & 0 \\
C_{13} & C_{23} & C_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & C_{44} & 0 & 0 \\
0 & 0 & 0 & 0 & C_{55} & 0 \\
0 & 0 & 0 & 0 & 0 & C_{66} \end{bmatrix}
\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_3 \\ \epsilon_4 \\ \epsilon_5 \\ \epsilon_6 \end{bmatrix}

The inverse of this relation is commonly written as[8]


  \begin{bmatrix}
    \epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
  \end{bmatrix}
  = \begin{bmatrix}
    \tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm zx}}{E_{\rm z}} & 0 & 0 & 0 \\
    -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm zy}}{E_{\rm z}} & 0 & 0 & 0 \\
    -\tfrac{\nu_{\rm xz}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm z}} & 0 & 0 & 0 \\
    0 & 0 & 0 & \tfrac{1}{G_{\rm yz}} & 0 & 0 \\
    0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm zx}} & 0 \\
    0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
    \end{bmatrix}
  \begin{bmatrix}
    \sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
  \end{bmatrix}

where

{E}_{\rm i}\, is the Young's modulus along axis i
G_{\rm ij}\, is the shear modulus in direction j on the plane whose normal is in direction i
\nu_{\rm ij}\, is the Poisson's ratio that corresponds to a contraction in direction j when an extension is applied in direction i.

Under plane stress conditions, σzz = σzx = σyz = 0, Hooke's law for an orthotropic material takes the form


   \begin{bmatrix}\varepsilon_{\rm xx} \\ \varepsilon_{\rm yy} \\ 2\varepsilon_{\rm xy} \end{bmatrix} = 
   \begin{bmatrix} \frac{1}{E_{\rm x}} & -\frac{\nu_{\rm yx}}{E_{\rm y}} & 0 \\
                   -\frac{\nu_{\rm xy}}{E_{\rm x}} & \frac{1}{E_{\rm y}} & 0 \\
                    0 & 0 & \frac{1}{G_{\rm xy}} \end{bmatrix}
    \begin{bmatrix}\sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm xy} \end{bmatrix} \,.

The inverse relation is


   \begin{bmatrix}\sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm xy} \end{bmatrix}
   = \cfrac{1}{1-\nu_{\rm xy}\nu_{\rm yx}}
   \begin{bmatrix} E_{\rm x} & \nu_{\rm yx}E_{\rm x} & 0 \\
                   \nu_{\rm xy}E_{\rm y} & E_{\rm y} & 0 \\
                   0 & 0 & G_{\rm xy}(1-\nu_{\rm xy}\nu_{\rm yx}) \end{bmatrix}
    \begin{bmatrix}\varepsilon_{\rm xx} \\ \varepsilon_{\rm yy} \\ 2\varepsilon_{\rm xy} \end{bmatrix} \,.

The transposed form of the above stiffness matrix is also often used.

Transversely isotropic materials

A transversely isotropic material is symmetric with respect to a rotation about an axis of symmetry. For such a material, if \mathbf{e}_3 is the axis of symmetry, Hooke's law can be expressed as


\begin{bmatrix} \sigma_1 \\ \sigma_2 \\ \sigma_3 \\ \sigma_4 \\ \sigma_5 \\ \sigma_6 \end{bmatrix} = 
\begin{bmatrix}
  C_{11} & C_{12} & C_{13} & 0 & 0 & 0 \\
C_{12} & C_{11} & C_{13} & 0 & 0 & 0 \\
C_{13} & C_{13} & C_{33} & 0 & 0 & 0 \\
0 & 0 & 0 & C_{44} & 0 & 0 \\
0 & 0 & 0 & 0 & C_{44} & 0 \\
0 & 0 & 0 & 0 & 0 & \tfrac{1}{2}(C_{11}-C_{12}) \end{bmatrix}
\begin{bmatrix} \epsilon_1 \\ \epsilon_2 \\ \epsilon_3 \\ \epsilon_4 \\ \epsilon_5 \\ \epsilon_6 \end{bmatrix}

More frequently, the x \equiv \mathbf{e}_1 axis is taken to be the axis of symmetry and the inverse Hooke's law is written as [9]


  \begin{bmatrix}
    \epsilon_{{\rm xx}} \\ \epsilon_{\rm yy} \\ \epsilon_{\rm zz} \\ 2\epsilon_{\rm yz} \\ 2\epsilon_{\rm zx} \\ 2\epsilon_{\rm xy}
  \end{bmatrix}
  = \begin{bmatrix}
    \tfrac{1}{E_{\rm x}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & - \tfrac{\nu_{\rm yx}}{E_{\rm y}} & 0 & 0 & 0 \\
    -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & \tfrac{1}{E_{\rm y}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & 0 & 0 & 0 \\
    -\tfrac{\nu_{\rm xy}}{E_{\rm x}} & - \tfrac{\nu_{\rm yz}}{E_{\rm y}} & \tfrac{1}{E_{\rm y}} & 0 & 0 & 0 \\
    0 & 0 & 0 & \tfrac{2(1+\nu_{\rm yz})}{E_{\rm y}} & 0 & 0 \\
    0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} & 0 \\
    0 & 0 & 0 & 0 & 0 & \tfrac{1}{G_{\rm xy}} \\
    \end{bmatrix}
  \begin{bmatrix}
    \sigma_{\rm xx} \\ \sigma_{\rm yy} \\ \sigma_{\rm zz} \\ \sigma_{\rm yz} \\ \sigma_{\rm zx} \\ \sigma_{\rm xy}
  \end{bmatrix}

Thermodynamic basis of Hooke's law

Linear deformations of elastic materials can be approximated as adiabatic. Under these conditions and for quasistatic processes the first law of thermodynamics for a deformed body can be expressed as


   \delta W = \delta U\,

where δU is the increase in internal energy and δW is the work done by external forces. The work can be split into two terms


   \delta W = \delta W_s + \delta W_b\,

where δWs is the work done by surface forces while δWb is the work done by body forces. If \delta\mathbf{u} is a variation of the displacement field \mathbf{u} in the body, then the two external work terms can be expressed as


   \delta W_s = \int_{\partial\Omega} \mathbf{t}\cdot\delta\mathbf{u}~{\rm dS} ~;~~
   \delta W_b = \int_{\Omega} \mathbf{b}\cdot\delta\mathbf{u}~{\rm dV}

where \mathbf{t} is the surface traction vector, \mathbf{b} is the body force vector, \Omega\, represents the body and \partial\Omega represents its surface. Using the relation between the Cauchy stress and the surface traction, \mathbf{t} = \mathbf{n}\cdot\boldsymbol{\sigma} (where \mathbf{n} is the unit outward normal to \partial\Omega), we have


   \delta W = \delta U = \int_{\partial\Omega} (\mathbf{n}\cdot\boldsymbol{\sigma})\cdot\delta\mathbf{u}~{\rm dS} + \int_{\Omega} \mathbf{b}\cdot\delta\mathbf{u}~{\rm dV}

Converting the surface integral into a volume integral via the divergence theorem gives


   \delta U = \int_{\Omega} [\boldsymbol{\nabla}\cdot(\boldsymbol{\sigma}\cdot\delta\mathbf{u}) + \mathbf{b}\cdot\delta\mathbf{u}]~ {\rm dV} ~.

Using the symmetry of the Cauchy stress and the identity


\boldsymbol{\nabla}\cdot(\boldsymbol{A}\cdot\mathbf{b}) = (\boldsymbol{\nabla}\cdot\boldsymbol{A})\cdot\mathbf{b}+
\tfrac{1}{2}[\boldsymbol{A}^T:\boldsymbol{\nabla}\mathbf{b}+
             \boldsymbol{A}:(\boldsymbol{\nabla}\mathbf{b})^T]

we have


   \delta U = \int_{\Omega} [\boldsymbol{\sigma}:
\tfrac{1}{2}\{\boldsymbol{\nabla}\delta\mathbf{u}+(\boldsymbol{\nabla}\delta\mathbf{u})^T\} + \{\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}+\mathbf{b}\}\cdot\delta\mathbf{u}]~{\rm dV} ~.

From the definition of strain and from the equations of equilibrium we have


   \delta\boldsymbol{\epsilon} = \tfrac{1}{2}[\boldsymbol{\nabla}\delta\mathbf{u}+(\boldsymbol{\nabla}\delta\mathbf{u})^T] ~;~~
\boldsymbol{\nabla}\cdot\boldsymbol{\sigma}+\mathbf{b}=\mathbf{0} ~.

Hence we can write


   \delta U = \int_{\Omega} \boldsymbol{\sigma}:\delta\boldsymbol{\epsilon}~{\rm dV}

and therefore the variation in the internal energy density is given by


   \delta U_0 = \boldsymbol{\sigma}:\delta\boldsymbol{\epsilon} ~.

An elastic material is defined as one in which the total internal energy is equal to the potential energy of the internal forces (also called the elastic strain energy). Therefore the internal energy density is a function of the strains, U_0 = U_0(\boldsymbol{\epsilon}) and the variation of the internal energy can be expressed as


   \delta U_0 = \cfrac{\partial U_0}{\partial\boldsymbol{\epsilon}}:\delta\boldsymbol{\epsilon} ~.

Since the variation of strain is arbitrary, the stress-strain relation of an elastic material is given by


   \boldsymbol{\sigma} = \cfrac{\partial U_0}{\partial\boldsymbol{\epsilon}}~.

For a linear elastic material, the quantity \partial U_0/\partial\boldsymbol{\epsilon} is a linear function of \boldsymbol{\epsilon}, and can therefore be expressed as


   \boldsymbol{\sigma} = \mathsf{c}:\boldsymbol{\epsilon}

where \mathsf{c} is a fourth-order tensor of material constants, also called the stiffness tensor. We can see why \mathsf{c} must be a fourth-order tensor by noting that, for a linear elastic material,


  \cfrac{\partial}{\partial\boldsymbol{\epsilon}}[\boldsymbol{\sigma}(\boldsymbol{\epsilon})] = \text{constant} = \mathsf{c} \,.

In index notation


   \cfrac{\partial\sigma_{ij}}{\partial\epsilon_{k\ell}} = \text{constant} = c_{ijk\ell} \,.

Clearly, the right hand side constant requires four indices and is a fourth-order quantity. We can also see that this quantity must be a tensor because it is a linear transformation that takes the strain tensor to the stress tensor. We can also show that the constant obeys the tensor transformation rules for fourth order tensors.

See also

Notes

  1. ^ Walter Lewin (1 October 1999) (in English) (ogg). Hook's Law, Simple Harmonic Oscillator. MIT Course 8.01: Classical Mechanics, Lecture 10. (videotape). Cambridge, MA USA: MIT OCW. Event occurs at 1:21–10:10. http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-10/. Retrieved 23 December 2010. "...arguably the most important equation in all of Physics." 
  2. ^ Walter Lewin (1 October 1999) (in English) (ogg). Hook's Law, Simple Harmonic Oscillator. MIT Course 8.01: Classical Mechanics, Lecture 10. (videotape). Cambridge, MA USA: MIT OCW. Event occurs at 10:10–16:33. http://ocw.mit.edu/courses/physics/8-01-physics-i-classical-mechanics-fall-1999/video-lectures/lecture-10/. Retrieved 23 December 2010. 
  3. ^ The anagram was "ceiiinosssttuv", [1]; cf. the anagram for the Catenary, which appeared in the preceding paragraph.
  4. ^ Symon, Keith (1971). Mechanics. Addison-Wesley, Reading, MA. ISBN 0-201-07392-7. 
  5. ^ Simo, J. C. and Hughes, T. J. R., 1998, Computational Inelasticity, Springer.
  6. ^ Milton, G. W., 2002, Theory of Composites, Cambridge University Press.
  7. ^ Slaughter, W. S., 2002, The Linearized Theory of Elasticity, Birkhauser
  8. ^ Boresi, A. P, Schmidt, R. J. and Sidebottom, O. M., 1993, Advanced Mechanics of Materials, Wiley.
  9. ^ Tan, S. C., 1994, Stress Concentrations in Laminated Composites, Technomic Publishing Company, Lancaster, PA.

References

  • A.C. Ugural, S.K. Fenster, Advanced Strength and Applied Elasticity, 4th ed

External links

Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these, thus given any two, any other of the elastic moduli can be calculated according to these formulas.
(\lambda,\,G) (E,\,G) (K,\,\lambda) (K,\,G) (\lambda,\,\nu) (G,\,\nu) (E,\,\nu) (K,\, \nu) (K,\,E) (M,\,G)
K=\, \lambda+ \tfrac{2G}{3} \tfrac{EG}{3(3G-E)} \tfrac{\lambda(1+\nu)}{3\nu} \tfrac{2G(1+\nu)}{3(1-2\nu)} \tfrac{E}{3(1-2\nu)} M - \tfrac{4G}{3}
E=\, \tfrac{G(3\lambda + 2G)}{\lambda + G} \tfrac{9K(K-\lambda)}{3K-\lambda} \tfrac{9KG}{3K+G} \tfrac{\lambda(1+\nu)(1-2\nu)}{\nu} 2G(1+\nu)\, 3K(1-2\nu)\, \tfrac{G(3M-4G)}{M-G}
\lambda=\, \tfrac{G(E-2G)}{3G-E} K-\tfrac{2G}{3} \tfrac{2 G \nu}{1-2\nu} \tfrac{E\nu}{(1+\nu)(1-2\nu)} \tfrac{3K\nu}{1+\nu} \tfrac{3K(3K-E)}{9K-E} M - 2G\,
G=\, \tfrac{3(K-\lambda)}{2} \tfrac{\lambda(1-2\nu)}{2\nu} \tfrac{E}{2(1+\nu)} \tfrac{3K(1-2\nu)}{2(1+\nu)} \tfrac{3KE}{9K-E}
\nu=\, \tfrac{\lambda}{2(\lambda + G)} \tfrac{E}{2G}-1 \tfrac{\lambda}{3K-\lambda} \tfrac{3K-2G}{2(3K+G)} \tfrac{3K-E}{6K} \tfrac{M - 2G}{2M - 2G}
M=\, \lambda+2G\, \tfrac{G(4G-E)}{3G-E} 3K-2\lambda\, K+\tfrac{4G}{3} \tfrac{\lambda(1-\nu)}{\nu} \tfrac{2G(1-\nu)}{1-2\nu} \tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)} \tfrac{3K(1-\nu)}{1+\nu} \tfrac{3K(3K+E)}{9K-E}

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