Buche-Silberstein model

class BucheSilberstein(method, nondimensional_critical_extension, nondimensional_link_stiffness, number_of_links)

The Buche-Silberstein hyperelastic damage constitutive model.

1. Helmholtz method for both the Helmholtz free energy and the equilibrium distribution.

2. Gibbs-Legendre method for both the Helmholtz free energy and the equilibrium distribution.

3. Gibbs-Legendre for the Helmholtz free energy and a Gaussian equilibrium distribution.

nondimensional_critical_extension

The nondimensional critical extension which irreversibly breaks chains.

nondimensional_link_stiffness

The nondimensional stiffness of each link in a chain.

number_of_links

The number of links in a chain.

uniaxial_tension(stretch)

The nondimensional Cauchy stress as a function of stretch in uniaxial tension,

\[\beta\sigma_{11}(t)/n = 2\pi\int_0^\infty r\,dr\int_0^\infty dz\,P^\mathrm{eq}(\gamma_0)\Theta(\gamma_0; t)\,\frac{\eta(\gamma)}{\gamma}\,\left(2z^2-r^2\right),\]

where \(\gamma=\sqrt{z^2+r^2}\) and \(\gamma_0(t)=\sqrt{z^2/F_{11}^2(t)+r^2F_{11}(t)}\). The chain damage function is

\[\begin{split}\Theta(\gamma_0; t) = \begin{cases} 1, & {}_{(t)}\gamma(s) \leq \gamma_\mathrm{c}~~\forall s\in[0, t], \\ 0, & \mathrm{otherwise}, \end{cases}\end{split}\]

where \({}_{(t)}\gamma(s)=\sqrt{z^2F_{11}^2(s)/F_{11}^2(t)+r^2F_{11}(t)/F_{11}(s)}\).

Parameters:

stretch (numpy.ndarray) – The applied stretch history \(F_{11}(t)\).

Returns:

• (numpy.ndarray) - The nondimensional Cauchy stress \(\beta\sigma_{11}(t)/n\).

• (numpy.ndarray) - The total probability of intact chains \(P^\mathrm{tot}(t)\).

Return type:

tuple