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Linear Elasticity

This chapter covers the constitutive law (Hooke's law) for linear elastic materials. For details on model selection and input specifications, see the functions section 03_material.

Phenomenon (Linear Elastic Response)

A linear elastic body is a material that, in a range where the strain under load is sufficiently small, exhibits a linear relationship between stress and strain and a reversible, history-independent response in which all strain returns to its original state upon unloading. Unlike plasticity or creep, it has no internal variables representing the state (such as plastic strain or viscous strain), and the stress is uniquely determined solely by the current strain.

In the small-deformation range, the Cauchy stress \(\boldsymbol{\sigma}\) and infinitesimal strain \(\boldsymbol{\varepsilon}\) are used; in the finite-deformation range (problems that include large rotations but in which the strain itself is small), the second Piola-Kirchhoff stress \(\boldsymbol{S}\) and Green-Lagrange strain \(\boldsymbol{E}\) are used (St.Venant-Kirchhoff material).

Constitutive Law

Isotropic Linear Elastic Material

In the small-deformation range, isotropic Hooke's law is expressed using the Lamé constants \(\lambda, \mu\) as

\[ \boldsymbol{\sigma} = \lambda\, \mathrm{tr}(\boldsymbol{\varepsilon})\, \boldsymbol{I} + 2\mu\, \boldsymbol{\varepsilon} \]

In component form,

\[ C_{ijkl} = \lambda\, \delta_{ij}\delta_{kl} + \mu\, \bigl( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} \bigr) \]

and double contraction with the strain gives \(\sigma_{ij} = C_{ijkl}\,\varepsilon_{kl}\). The Lamé constants are related to Young's modulus \(E\) and Poisson's ratio \(\nu\) as follows:

\[ \lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}, \qquad \mu = \frac{E}{2(1+\nu)}. \]

In Voigt notation, the relationship between the strain vector \(\hat{\varepsilon}\) and stress vector \(\hat{\sigma}\) is written as \(\hat{\sigma} = D\, \hat{\varepsilon}\), and the three-dimensional material matrix \(D\) is

\[ D = \begin{bmatrix} \lambda + 2\mu & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda + 2\mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix} \]

This matrix representation is used in the implementation.

Orthotropic Linear Elastic Material

For materials whose elastic constants differ by principal-axis direction, an orthotropic model with nine independent constants is used: Young's moduli in the principal-axis directions \(E_1, E_2, E_3\), Poisson's ratios between the principal axes \(\nu_{12}, \nu_{23}, \nu_{31}\), and shear moduli \(G_{12}, G_{23}, G_{31}\). For the explicit form of the \(D\) matrix, refer to standard references.

Finite-Deformation Range: St.Venant-Kirchhoff Material

For problems that include large rotations but in which the strain itself is small, the St.Venant-Kirchhoff material is used as a linear elastic law relating the second Piola-Kirchhoff stress \(\boldsymbol{S}\) and Green-Lagrange strain \(\boldsymbol{E}\):

\[ \boldsymbol{S} = \boldsymbol{C} : \boldsymbol{E}, \qquad C_{ijkl} = \lambda\, \delta_{ij}\delta_{kl} + \mu\, \bigl( \delta_{ik}\delta_{jl} + \delta_{il}\delta_{jk} \bigr). \]

The definitions of the Lamé constants \(\lambda, \mu\) are the same as in the small-deformation case. Note, however, that because the definitions of the stress-strain pair differ, this is a different constitutive law from Hooke's law for small deformation.

Hypoelastic Material

In the Updated Lagrange method, a hypoelastic material is used that adopts the linear elastic law between the Jaumann rate of the relative Kirchhoff stress tensor \(\hat{\boldsymbol{\sigma}}^{\nabla J}\) and the rate-of-deformation tensor \(\boldsymbol{D}\):

\[ \hat{\boldsymbol{\sigma}}^{\nabla J} = \boldsymbol{C} : \boldsymbol{D} \]

Here again, the definition of \(\boldsymbol{C}\) in terms of the Lamé constants is the same as above, but because the stress-strain pair differs, this is a different constitutive law. For details of how \(\boldsymbol{\sigma}_{t_{n+1}}\) is updated through the time-integration procedure (forward Euler integration and central-difference approximation), see tmptexdocs/11b_continuum_mechanics.tex.

Reduction for Two-Dimensional Analysis and Shell Elements

Plane Stress (\(\sigma_{33} = 0\))

For plane stress, representing loading of thin plates and shells, \(\sigma_{33} = \sigma_{13} = \sigma_{23} = 0\) is assumed, and \(D\) with the out-of-plane strain component \(\varepsilon_{33}\) eliminated algebraically is used:

\[ D = \frac{E}{1 - \nu^2} \begin{bmatrix} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \dfrac{1-\nu}{2} \end{bmatrix}. \]

Plane Strain (\(\varepsilon_{33} = 0\))

For plane strain, representing the cross section of a long structure constrained in the thickness direction, \(\varepsilon_{33} = \varepsilon_{13} = \varepsilon_{23} = 0\) is assumed:

\[ D = \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & 0 \\ \nu & 1-\nu & 0 \\ 0 & 0 & \dfrac{1-2\nu}{2} \end{bmatrix}. \]

Axisymmetric

In axisymmetric analysis, which treats problems that are uniform in the \(\theta\) direction in cylindrical coordinates \((r, \theta, z)\), the four components \(\boldsymbol{\sigma} = (\sigma_{rr}, \sigma_{\theta\theta}, \sigma_{zz}, \sigma_{rz})^T\) and \(\boldsymbol{\varepsilon} = (\varepsilon_{rr}, \varepsilon_{\theta\theta}, \varepsilon_{zz}, 2\varepsilon_{rz})^T\) are retained, and

\[ D = \frac{E}{(1+\nu)(1-2\nu)} \begin{bmatrix} 1-\nu & \nu & \nu & 0 \\ \nu & 1-\nu & \nu & 0 \\ \nu & \nu & 1-\nu & 0 \\ 0 & 0 & 0 & \dfrac{1-2\nu}{2} \end{bmatrix} \]

is obtained.

Shell Elements (Plane Stress + Transverse Shear)

For shell elements, in-plane stress is treated as plane stress, and for the transverse shear strain components \((2\varepsilon_{13}, 2\varepsilon_{23})\), \(\kappa\, G\) is used with the shear correction factor \(\kappa\) (generally \(\kappa = 5/6\)). For the detailed formulation of each element, see Advanced Element Formulations. The specification in the functions section 03_material that "shell elements support linear elasticity only" is based on this simplification of treating the in-plane state as plane stress.

Temperature Dependence

Young's modulus \(E\), Poisson's ratio \(\nu\), and coefficient of linear expansion \(\alpha\) are generally specified as functions of temperature \(T\). In FrontISTR, values are entered for a finite sequence of temperature points \(T_1 < T_2 < \cdots < T_n\), and the temperature \(T\) during calculation is evaluated by linear interpolation between adjacent points. For temperatures outside the range (\(T < T_1\) or \(T > T_n\)), the value at the nearest endpoint, \(T_1\) or \(T_n\), is used for extrapolation (constant extrapolation).

For the temperature dependence of thermal and heat-conduction properties (density, specific heat, and thermal conductivity), see Thermal Properties. The interpolation rules are common to these properties.