@f[
\nu = \frac{\lambda}{2(\lambda+\mu)},
@f]
-where $\lambda,\mu$ are the Lam\'e constants of the material.
+where $\lambda,\mu$ are the Lamé constants of the material.
Physical constraints indicate that $-1\le \nu\le \frac 12$ (the condition
also follows from mathematical stability considerations). If $\nu$
approaches $\frac 12$, then the material becomes incompressible. In that
// in the form $C_{ijkl} = \mu (\delta_{ik} \delta_{jl} + \delta_{il}
// \delta_{jk}) + \lambda \delta_{ij} \delta_{kl}$. This tensor maps
// symmetric tensor of rank 2 to symmetric tensors of rank 2. A function
- // implementing its creation for given values of the Lame constants $\lambda$
- // and $\mu$ is straightforward:
+ // implementing its creation for given values of the Lamé constants
+ // $\lambda$ and $\mu$ is straightforward:
template <int dim>
SymmetricTensor<4, dim> get_stress_strain_tensor(const double lambda,
const double mu)
// tensor. Note that in more elaborate programs, this will probably be a
// member variable of some class instead, or a function that returns the
// stress-strain relationship depending on other input. For example in
- // damage theory models, the Lame constants are considered a function of the
- // prior stress/strain history of a point. Conversely, in plasticity the
- // form of the stress-strain tensor is modified if the material has reached
- // the yield stress in a certain point, and possibly also depending on its
- // prior history.
+ // damage theory models, the Lamé constants are considered a function
+ // of the prior stress/strain history of a point. Conversely, in plasticity
+ // the form of the stress-strain tensor is modified if the material has
+ // reached the yield stress in a certain point, and possibly also depending on
+ // its prior history.
//
// In the present program, however, we assume that the material is
// completely elastic and linear, and a constant stress-strain tensor is
// Now for the implementation of the main class. First, we initialize the
// stress-strain tensor, which we have declared as a static const
- // variable. We chose Lame constants that are appropriate for steel:
+ // variable. We chose Lamé constants that are appropriate for steel:
template <int dim>
const SymmetricTensor<4, dim> TopLevel<dim>::stress_strain_tensor =
get_stress_strain_tensor<dim>(/*lambda = */ 9.695e10,
the fourth-order compliance tensor $A$; $\varepsilon^p$ provides the plastic
component of the strain to ensure that the stress does not exceed the yield
stress. We will only consider isotropic
-materials for which $A$ can be expressed in terms of the Lam\'e moduli
+materials for which $A$ can be expressed in terms of the Lamé moduli
$\lambda$ and $\mu$ or alternatively in terms of the bulk modulus
$\kappa$ and $\mu$.
The second equation is the force balance; we will here
// The constructor of the ConstitutiveLaw class sets the required material
// parameter for our deformable body. Material parameters for elastic
// isotropic media can be defined in a variety of ways, such as the pair $E,
- // \nu$ (elastic modulus and Poisson's number), using the Lame parameters
+ // \nu$ (elastic modulus and Poisson's number), using the Lamé
+ // parameters
// $\lambda,mu$ or several other commonly used conventions. Here, the
// constructor takes a description of material parameters in the form of
// $E,\nu$, but since this turns out to these are not the coefficients that
\underbrace{\kappa [ \mathcal{G}(J) ] }_{\Psi_{\textrm{vol}}(J)}
+ \underbrace{\bigl[c_1 [ \overline{I}_1 - 3] \bigr]}_{\Psi_{\text{iso}}(\overline{\mathbf{b}})} \, ,
@f]
-where $\kappa \dealcoloneq \lambda + 2/3 \mu$ is the bulk modulus ($\lambda$ and $\mu$ are the Lame parameters)
+where $\kappa \dealcoloneq \lambda + 2/3 \mu$ is the bulk modulus ($\lambda$ and $\mu$ are the Lamé parameters)
and $\overline{I}_1 \dealcoloneq \textrm{tr}\ \overline{\mathbf{b}}$.
The function $\mathcal{G}(J)$ is required to be strictly convex and satisfy the condition $\mathcal{G}(1) = 0$,
among others, see Holzapfel (2001) for further details.
// 1 - 2\textrm{ln}\; \widetilde{J} ]$, where $\kappa \dealcoloneq \lambda +
// 2/3 \mu$ is the <a href="http://en.wikipedia.org/wiki/Bulk_modulus">bulk
// modulus</a> and $\lambda$ is <a
- // href="http://en.wikipedia.org/wiki/Lam%C3%A9_parameters">Lame's first
- // parameter</a>.
+ // href="http://en.wikipedia.org/wiki/Lam%C3%A9_parameters">Lamé's
+ // first parameter</a>.
//
// The following class will be used to characterize the material we work with,
// and provides a central point that one would need to modify if one were to
* // This example is sufficiently complex to warrant the use of AD to,
* // at the very least, verify an unassisted implementation.
* const double mu_e = 10; // Shear modulus
- * const double lambda_e = 15; // Lame parameter
+ * const double lambda_e = 15; // Lamé parameter
* const double mu_0 = 4*M_PI*1e-7; // Magnetic permeability constant
* const double mu_r = 5; // Relative magnetic permeability
*
* finite element field and each component of the error estimator will then be
* weighted by the respective component in this coefficient function. In the
* other case, when all components have the same meaning (for example the
- * displacements in Lame's equations of elasticity), you can specify a scalar
- * coefficient which will then be used for all components.
+ * displacements in Lamé's equations of elasticity), you can specify a
+ * scalar coefficient which will then be used for all components.
*
*
* <h3>Boundary values</h3>
projects / dealii.git / commitdiff