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MechanicalFemPhysics
The GeMA Mechanical FEM Physics Plugin
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MechanicalFemPhysics
The GeMA Mechanical FEM Physics Plugin
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The yield function of Modified Mohr-Coulomb with based as a function of the invariants of stress tensor and desviatoric stress tensor (Zienkiewicz and Nayak, 1972), which includes rounding at the edges (Abbo et al, 2011) and C2 continuous of the yield surface. The cap is introduced as a Heaviside function as proposed by Pelessone (1987) that the only depends of the hydrostatic pressure and it operates as penalty function in the Mohr-Coulomb surface.
This model takes into account the effect of dilatency through the use of non-associated plasticity. The plastification criterion of this model is in agreement with the criterion of rupture, not hardening occurring during the plastic flow. The expression of the yield surface is expressed in terms of three stress invariants \( (p,J_2,\theta) \) as follows:
\begin{eqnarray*}F=\sqrt{{{J}_{2}}{{K}^{2}}\left( \theta \right)+{{\left( a\sin \phi \right)}^{2}}}-{{F}_{s}}\left( p \right){{F}_{cap}}\left( p \right)=0 \end{eqnarray*}
where :
\begin{eqnarray*}p = -\frac{1}{3}I_1 \end{eqnarray*}
\begin{eqnarray*}\theta =\frac{1}{3}{{\sin }^{-1}}\left( -\frac{3\sqrt{3}{{J}_{3}}}{2{{J}_{2}}^{3/2}} \right),\quad \theta \in \left[ -\frac{\pi }{6},\frac{\pi }{6} \right]\end{eqnarray*}
\begin{eqnarray*}F_s(p) = p\sin\phi + c \cos\phi \end{eqnarray*}
The cap function \( F_s(p) \) is expressed as:
\begin{eqnarray*}{{F}_{cap}}\left( p \right)=1-H\left[ p-{{p}_{a}} \right]{{\left( \frac{p-{{p}_{a}}}{R{{F}_{s}}\left( {{p}_{a}} \right)} \right)}^{2}} \end{eqnarray*}
such that
The C2 continuous rounding parameter of the octahedral plane, as proposed by Abbo et al (2011), is calculated according to:
\begin{eqnarray*} K(\theta) = \Bigg\{ \begin{matrix} A+B\sin 3\theta +C{{\sin }^{2}}3\theta &|\theta| > \theta_t \\ \cos \theta -\frac{1}{\sqrt{3}}\sin \phi \sin \theta & \left| \theta \right|\le {{\theta }_{T}} \\ \end{matrix} \end{eqnarray*}
The term \( \theta_t \) is the transition angle and the constants \( A\), \( B\) and \( C\) are defined as follows:
\begin{eqnarray*} A=-\frac{1}{\sqrt{3}}\sin \phi \left\langle \theta \right\rangle \sin {{\theta }_{T}}-B\left\langle \theta \right\rangle \sin 3{{\theta }_{T}}-C\left\langle \theta \right\rangle {{\sin }^{2}}3{{\theta }_{T}}+\cos {{\theta }_{T}}\end{eqnarray*}
\begin{eqnarray*} B=\frac{\left\langle \theta \right\rangle \sin 6{{\theta }_{T}}\left( \cos {{\theta }_{T}}-\frac{1}{\sqrt{3}}\sin \phi \left\langle \theta \right\rangle \sin {{\theta }_{T}} \right)-6\cos 6{{\theta }_{T}}\left( \left\langle \theta \right\rangle \sin {{\theta }_{T}}+\frac{1}{\sqrt{3}}\sin \phi \left\langle \theta \right\rangle \cos {{\theta }_{T}} \right)}{18{{\cos }^{3}}3{{\theta }_{T}}}\end{eqnarray*}
\begin{eqnarray*}C=\frac{\cos 6{{\theta }_{T}}\left( \cos {{\theta }_{T}}-\frac{1}{\sqrt{3}}\sin \phi \left\langle \theta \right\rangle \sin {{\theta }_{T}} \right)-3\left\langle \theta \right\rangle \sin 3{{\theta }_{T}}\left( \left\langle \theta \right\rangle \sin {{\theta }_{T}}+\frac{1}{\sqrt{3}}\sin \phi \left\langle \theta \right\rangle \cos {{\theta }_{T}} \right)}{18{{\cos }^{3}}3{{\theta }_{T}}} \end{eqnarray*}
The yield surface of Mohr-Coulomb with cap requires the following properties:
| Property | Description | Type | Def. Unit | required |
|---|---|---|---|---|
| E | Elastic modulus | Scalar | kPa | Yes |
| nu | Poisson's ratio | Scalar | – | Yes |
| Coh | Cohesion | Scalar | kPa | Yes |
| Phi | Angle of internal friction | Scalar | degree | Yes |
| Psi | Angle of dilatation | Scalar | degree | Yes |
| Pa | Cap variable | Scalar | kPa | Yes |
| R | Cap eccentricity | Scalar | – | Yes |
As way to use the Mohr-Coulomb with cap, the mechanical material type ('materialM') have to bet set as 'capMohrCoulomb' as shown in the example below:
Example:
The Mechanical Fem physics uses internally the international system of units for its calculations, as can be seen above by the default units for properties and attributes. The whole unit system can be replaced through the unitSystem physics attribute, as described at the gemaFemProcessCommonFemPhysicsOptions page.
When making the substitution, the units on the following table should be replaced by the desired units, forming a coherent unit system. The unit system of the material should be also replaced.
1.8.15