diff --git a/doc/dev-doc/manual/appendix/material-parameters.rst b/doc/dev-doc/manual/appendix/material-parameters.rst
index 4d0d11dfb..4ab43b393 100644
--- a/doc/dev-doc/manual/appendix/material-parameters.rst
+++ b/doc/dev-doc/manual/appendix/material-parameters.rst
@@ -1,4 +1,129 @@
 .. _app-material-parameters:
 
 Material Parameters
 ===================
+
+Linear elastic isotropic
+------------------------
+
+Keyword: :ref:`elastic <sect-smm-linear-elastic-isotropic>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``E``: (*Real*) Young's modulus
+- ``nu``: (*Real*) Poisson's ratio
+- ``Plane_stress``: (*bool*) Plane stress simplification (only 2D problems)
+
+
+Linear elastic anisotropic
+--------------------------
+
+Keyword: :ref:`elastic_anisotropic <sect-smm-linear-elastic-anisotropic>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``n1``: (*Vector<Real>*) Direction of main material axis
+- ``n2``: (*Vector<Real>*) Direction of second material axis
+- ``n3``: (*Vector<Real>*) Direction of third material axis
+- ``C..``: (*Real*) Coefficient ij of material tensor C (all the 36 values in
+  Voigt notation can be entered)
+- ``alpha``: (*Real*) Viscous propertion (default is 0)
+
+
+Linear elastic anisotropic
+------------------------
+
+Keyword: :ref:`elastic_orthotropic <sect-smm-linear-elastic-orthotropic>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``n1``: (*Vector<Real>*) Direction of main material axis
+- ``n2``: (*Vector<Real>*) Direction of second material axis (if applicable)
+- ``n3``: (*Vector<Real>*) Direction of third material axis (if applicable)
+- ``E1``: (*Real*) Young's modulus (n1)
+- ``E2``: (*Real*) Young's modulus (n2)
+- ``E3``: (*Real*) Young's modulus (n3)
+- ``nu1``: (*Real*) Poisson's ratio (n1)
+- ``nu2``: (*Real*) Poisson's ratio (n2)
+- ``nu3``: (*Real*) Poisson's ratio (n3)
+- ``G12``: (*Real*) Shear modulus (12)
+- ``G13``: (*Real*) Shear modulus (13)
+- ``G23``: (*Real*) Shear modulus (23)
+
+
+Neohookean (finite strains)
+---------------------------
+
+Keyword: :ref:`neohookean <sect-smm-cl-neohookean>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``E``: (*Real*) Young's modulus
+- ``nu``: (*Real*) Poisson's ratio
+- ``Plane_stress``: (*bool*) Plane stress simplification (only 2D problems)
+
+
+Standard linear solid
+---------------------
+
+Keyword: :ref:`sls_deviatoric <sect-smm-cl-sls>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``E``: (*Real*) Young's modulus
+- ``nu``: (*Real*) Poisson's ratio
+- ``Plane_stress``: (*bool*) Plane stress simplification (only 2D problems)
+- ``Eta``: (*Real*) Viscosity
+- ``Ev``: (*Real*) Stiffness of viscous element
+
+
+Elasto-plastic linear isotropic hardening
+-----------------------------------------
+
+Keyword: :ref:`plastic_linear_isotropic_hardening <sect-smm-cl-plastic>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``E``: (*Real*) Young's modulus
+- ``nu``: (*Real*) Poisson's ratio
+- ``h``: (*Real*) Hardening modulus
+- ``sigma_y``: (*Real*) Yield stress
+
+
+Marigo
+------
+
+Keyword: :ref:`marigo <sect-smm-cl-damage-marigo>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``E``: (*Real*) Young's modulus
+- ``nu``: (*Real*) Poisson's ratio
+- ``Plane_stress``: (*bool*) Plane stress simplification (only 2D problems)
+- ``Yd``: (*Random*) Hardening modulus
+- ``Sd``: (*Real*) Damage energy
+
+
+Mazars
+------
+
+Keyword: :ref:`mazars <sect-smm-cl-damage-mazars>`
+
+Parameters:
+
+- ``rho``: (*Real*) Density
+- ``E``: (*Real*) Young's modulus
+- ``nu``: (*Real*) Poisson's ratio
+- ``At``: (*Real*) Traction post-peak asymptotic value
+- ``Bt``: (*Real*) Traction decay shape
+- ``Ac``: (*Real*) Compression post-peak asymptotic value
+- ``Bc``: (*Real*) Compression decay shape
+- ``K0``: (*Real*) Damage threshold
+- ``beta``: (*Real*) Shear parameter
diff --git a/doc/dev-doc/manual/constitutive-laws.rst b/doc/dev-doc/manual/constitutive-laws.rst
index 1ae6c6c71..3416daffe 100644
--- a/doc/dev-doc/manual/constitutive-laws.rst
+++ b/doc/dev-doc/manual/constitutive-laws.rst
@@ -1,917 +1,925 @@
 .. _sect-smm-cl:
 
 Constitutive Laws
 -----------------
 
 In order to compute an element’s response to deformation, one needs to
 use an appropriate constitutive relationship. The constitutive law is
 used to compute the element’s stresses from the element’s strains.
 
 In the finite-element discretization, the constitutive formulation is
 applied to every quadrature point of each element. When the implicit
 formulation is used, the tangent matrix has to be computed.
 
 | The chosen materials for the simulation have to be specified in the
   mesh file or, as an alternative, they can be assigned using the at
   ``element_material`` vector. For
   every material assigned to the problem one has to specify the material
   characteristics (constitutive behavior and material properties) using
   the text input file (see :ref:`sect-io-material`).
 | In order to conveniently store values at each quadrature in a material point
   ``Akantu`` provides a special data structure, the at :cpp:class:`InternalField
   <akantu::InternalField>`. The internal fields are inheriting from the at
   :cpp:class:`ElementTypeMapArray <akantu::ElementTypeMapArray>`. Furthermore,
   it provides several functions for initialization, auto-resizing and auto
   removal of quadrature points.
 
 Sometimes it is also desired to generate random distributions of
 internal parameters. An example might be the critical stress at which
 the material fails. To generate such a field, in the text input file, a
 random quantity needs be added to the base value:
 
 All parameters are real numbers. For the uniform distribution, minimum
 and maximum values have to be specified. Random parameters are defined
 as a :math:`base` value to which we add a random number that follows the
 chosen distribution.
 
 The
 `Uniform <http://en.wikipedia.org/wiki/Uniform_distribution_(continuous)>`__
 distribution is gives a random values between in :math:`[min, max)`. The
 `Weibull <http://en.wikipedia.org/wiki/Weibull_distribution>`__
 distribution is characterized by the following cumulative distribution
 function:
 
 .. math:: F(x) = 1- e^{-\left({x/\lambda}\right)^m}
 
 which depends on :math:`m` and :math:`\lambda`, which are the shape
 parameter and the scale parameter. These random distributions are
 different each time the code is executed. In order to obtain always the
 same one, it possible to manually set the *seed* that is the number from
 which these pseudo-random distributions are created. This can be done by
 adding the following line to the input file *outside* the material
 parameters environments:
 
 .. code-block::
 
    seed = 1.0
 
 where the value 1.0 can be substituted with any number. Currently
 ``Akantu`` can reproduce always the same distribution when the seed is
 specified *only* in serial. The value of the *seed* can be also
 specified directly in the code (for instance in the main file) with the
 command:
 
 .. code-block::
 
    RandGenerator<Real>::seed(1.0)
 
 The same command, with empty brackets, can be used to check the value of
 the *seed* used in the simulation.
 
 The following sections describe the constitutive models implemented in
 ``Akantu``. In Appendix `7 <#app:material-parameters>`__ a summary of
 the parameters for all materials of ``Akantu`` is provided.
 
 Elastic
 ```````
 
 The elastic law is a commonly used constitutive relationship that can be
 used for a wide range of engineering materials (*e.g.*, metals,
 concrete, rock, wood, glass, rubber, etc.) provided that the strains
 remain small (*i.e.*, small deformation and stress lower than yield
 strength).
 
 The elastic laws are often expressed as
 :math:`\boldsymbol{\sigma} =
 \boldsymbol{C}:\boldsymbol{\varepsilon}` with
 where :math:`\boldsymbol{\sigma}` is the Cauchy stress
 tensor, :math:`\boldsymbol{\varepsilon}` represents the
 infinitesimal strain tensor and :math:`\boldsymbol{C}` is
 the elastic modulus tensor.
 
 .. _sect-smm-linear-elastic-isotropic:
 
 Linear isotropic
 ''''''''''''''''
 
 The linear isotropic elastic behavior is described by Hooke’s law, which
 states that the stress is linearly proportional to the applied strain
 (material behaves like an ideal spring), as illustrated in
-Figure `[fig:smm:cl:elastic] <#fig:smm:cl:elastic>`__.
+ :numref:`fig:smm:cl:el`.
+
+.. figure:: figures/cl/stress_strain_el.svg
+   :alt: Elastic strain-stress curve
+   :name: fig:smm:cl:el
+   :width: 60.0%
+
+   Stress-strain curve of elastic material and schematic representation of
+   Hooke's law, denoted as a spring.
 
 The equation that relates the strains to the displacements is: point)
 from the displacements as follows:
 
 .. math::
 
    \label{eqn:smm:strain_inf}
      \boldsymbol{\varepsilon} =
      \frac{1}{2} \left[ \nabla_0 \boldsymbol{u}+\nabla_0 \boldsymbol{u}^T \right]
 
 where :math:`\boldsymbol{\varepsilon}` represents the
 infinitesimal strain tensor,
 :math:`\nabla_{0}\boldsymbol{u}` the displacement gradient
 tensor according to the initial configuration. The constitutive equation
 for isotropic homogeneous media can be expressed as:
 
 .. math::
 
    \label{eqn:smm:material:constitutive_elastic}
      \boldsymbol{\sigma } =\lambda\mathrm{tr}(\boldsymbol{\varepsilon})\boldsymbol{I}+2 \mu\boldsymbol{\varepsilon}
 
 where :math:`\boldsymbol{\sigma}` is the Cauchy stress
 tensor (:math:`\lambda` and :math:`\mu` are the the first and second
 Lame’s coefficients).
 
 In Voigt notation this correspond to
 
 .. math::
 
    \begin{aligned}
      \left[\begin{array}{c}
          \sigma_{11}\\
          \sigma_{22}\\
          \sigma_{33}\\
          \sigma_{23}\\
          \sigma_{13}\\
          \sigma_{12}\\
        \end{array}\right]
      &= \frac{E}{(1+\nu)(1-2\nu)}\left[
        \begin{array}{cccccc}
          1-\nu & \nu   & \nu   & 0 & 0 & 0\\
          \nu   & 1-\nu & \nu   & 0 & 0 & 0\\
          \nu   & \nu   & 1-\nu & 0 & 0 & 0\\
          0     &  0    &  0    & \frac{1-2\nu}{2} & 0 & 0 \\
          0     &  0    &  0    & 0 & \frac{1-2\nu}{2} & 0 \\
          0     &  0    &  0    & 0 & 0 & \frac{1-2\nu}{2} \\
        \end{array}\right]
      \left[\begin{array}{c}
          \varepsilon_{11}\\
          \varepsilon_{22}\\
          \varepsilon_{33}\\
          2\varepsilon_{23}\\
          2\varepsilon_{13}\\
          2\varepsilon_{12}\\
        \end{array}\right]\end{aligned}
 
 .. _sect-smm-linear-elastic-anisotropic:
 
 Linear anisotropic
 ''''''''''''''''''
 
 This formulation is not sufficient to represent all elastic material
 behavior. Some materials have characteristic orientation that have to be
 taken into account. To represent this anisotropy a more general
 stress-strain law has to be used. For this we define the elastic modulus
 tensor as follow:
 
 .. math::
 
    \begin{aligned}
      \left[\begin{array}{c}
          \sigma_{11}\\
          \sigma_{22}\\
          \sigma_{33}\\
          \sigma_{23}\\
          \sigma_{13}\\
          \sigma_{12}\\
        \end{array}\right]
      &= \left[
        \begin{array}{cccccc}
          c_{11} & c_{12} & c_{13} & c_{14} & c_{15} & c_{16}\\
          c_{21} & c_{22} & c_{23} & c_{24} & c_{25} & c_{26}\\
          c_{31} & c_{32} & c_{33} & c_{34} & c_{35} & c_{36}\\
          c_{41} & c_{42} & c_{43} & c_{44} & c_{45} & c_{46}\\
          c_{51} & c_{52} & c_{53} & c_{54} & c_{55} & c_{56}\\
          c_{61} & c_{62} & c_{63} & c_{64} & c_{65} & c_{66}\\
        \end{array}\right]
      \left[\begin{array}{c}
          \varepsilon_{11}\\
          \varepsilon_{22}\\
          \varepsilon_{33}\\
          2\varepsilon_{23}\\
          2\varepsilon_{13}\\
          2\varepsilon_{12}\\
        \end{array}\right]\end{aligned}
 
 To simplify the writing of input files the :math:`\boldsymbol{C}` tensor
 is expressed in the material basis. And this basis as to be given too.
 This basis :math:`\Omega_{{\mathrm{mat}}}
 = \{\boldsymbol{n_1}, \boldsymbol{n_2}, \boldsymbol{n_3}\}`
 is used to define the rotation :math:`R_{ij} =
 \boldsymbol{n_j} . \boldsymbol{e_i}`. And
 :math:`\boldsymbol{C}` can be rotated in the global basis
 :math:`\Omega
 = \{\boldsymbol{e_1}, \boldsymbol{e_2}, \boldsymbol{e_3}\}`
 as follow:
 
 .. math::
 
    \begin{aligned}
    \boldsymbol{C}_{\Omega} &= \boldsymbol{R}_1 \boldsymbol{C}_{\Omega_{{\mathrm{mat}}}} \boldsymbol{R}_2\\
    \boldsymbol{R}_1  &= \left[
      \begin{array}{cccccc}
        R_{11} R_{11} & R_{12} R_{12} & R_{13} R_{13} & R_{12} R_{13} & R_{11} R_{13} & R_{11} R_{12}\\
        R_{21} R_{21} & R_{22} R_{22} & R_{23} R_{23} & R_{22} R_{23} & R_{21} R_{23} & R_{21} R_{22}\\
        R_{31} R_{31} & R_{32} R_{32} & R_{33} R_{33} & R_{32} R_{33} & R_{31} R_{33} & R_{31} R_{32}\\
        R_{21} R_{31} & R_{22} R_{32} & R_{23} R_{33} & R_{22} R_{33} & R_{21} R_{33} & R_{21} R_{32}\\
        R_{11} R_{31} & R_{12} R_{32} & R_{13} R_{33} & R_{12} R_{33} & R_{11} R_{33} & R_{11} R_{32}\\
        R_{11} R_{21} & R_{12} R_{22} & R_{13} R_{23} & R_{12} R_{23} & R_{11} R_{23} & R_{11} R_{22}\\
      \end{array}\right]\\
    \boldsymbol{R}_2  &= \left[
      \begin{array}{cccccc}
        R_{11} R_{11} & R_{21} R_{21} & R_{31} R_{31} & R_{21} R_{31} & R_{11} R_{31} & R_{11} R_{21}\\
        R_{12} R_{12} & R_{22} R_{22} & R_{32} R_{32} & R_{22} R_{32} & R_{12} R_{32} & R_{12} R_{22}\\
        R_{13} R_{13} & R_{23} R_{23} & R_{33} R_{33} & R_{23} R_{33} & R_{13} R_{33} & R_{13} R_{23}\\
        R_{12} R_{13} & R_{22} R_{23} & R_{32} R_{33} & R_{22} R_{33} & R_{12} R_{33} & R_{12} R_{23}\\
        R_{11} R_{13} & R_{21} R_{23} & R_{31} R_{33} & R_{21} R_{33} & R_{11} R_{33} & R_{11} R_{23}\\
        R_{11} R_{12} & R_{21} R_{22} & R_{31} R_{32} & R_{21} R_{32} & R_{11} R_{32} & R_{11} R_{22}\\
      \end{array}\right]\\\end{aligned}
 
 .. _sect-smm-linear-elastic-orthotropic:
 
 Linear orthotropic
 ''''''''''''''''''
 
 A particular case of anisotropy is when the material basis is orthogonal
 in which case the elastic modulus tensor can be simplified and rewritten
 in terms of 9 independents material parameters.
 
 .. math::
 
    \begin{aligned}
      \left[\begin{array}{c}
          \sigma_{11}\\
          \sigma_{22}\\
          \sigma_{33}\\
          \sigma_{23}\\
          \sigma_{13}\\
          \sigma_{12}\\
        \end{array}\right]
      &= \left[
        \begin{array}{cccccc}
          c_{11} & c_{12} & c_{13} &   0   &   0   &   0  \\
                & c_{22} & c_{23} &   0   &   0   &   0  \\
                &       & c_{33} &   0   &   0   &   0  \\
                &       &       & c_{44} &   0   &   0  \\
                &  \multicolumn{2}{l}{\text{sym.}}       &       & c_{55} &   0  \\
                &       &       &       &       & c_{66}\\
        \end{array}\right]
      \left[\begin{array}{c}
          \varepsilon_{11}\\
          \varepsilon_{22}\\
          \varepsilon_{33}\\
          2\varepsilon_{23}\\
          2\varepsilon_{13}\\
          2\varepsilon_{12}\\
        \end{array}\right]\end{aligned}
 
 .. math::
 
    \begin{aligned}
      c_{11} &= E_1 (1 - \nu_{23}\nu_{32})\Gamma \qquad c_{22} = E_2 (1 - \nu_{13}\nu_{31})\Gamma \qquad c_{33} = E_3 (1 - \nu_{12}\nu_{21})\Gamma\\
      c_{12} &= E_1 (\nu_{21} - \nu_{31}\nu_{23})\Gamma = E_2 (\nu_{12} - \nu_{32}\nu_{13})\Gamma\\
      c_{13} &= E_1 (\nu_{31} - \nu_{21}\nu_{32})\Gamma = E_2 (\nu_{13} - \nu_{21}\nu_{23})\Gamma\\
      c_{23} &= E_2 (\nu_{32} - \nu_{12}\nu_{31})\Gamma = E_3 (\nu_{23} - \nu_{21}\nu_{13})\Gamma\\
      c_{44} &= \mu_{23} \qquad  c_{55} = \mu_{13} \qquad  c_{66} = \mu_{12} \\
      \Gamma &= \frac{1}{1 - \nu_{12} \nu_{21} - \nu_{13} \nu_{31} - \nu_{32} \nu_{23} - 2 \nu_{21} \nu_{32} \nu_{13}}\end{aligned}
 
 The Poisson ratios follow the rule
 :math:`\nu_{ij} = \nu_{ji} E_i / E_j`.
 
 .. _sect-smm-cl-neohookean:
 
 Neo-Hookean
 '''''''''''
 
 The hyperelastic Neo-Hookean constitutive law results from an extension
 of the linear elastic relationship (Hooke’s Law) for large deformation.
 Thus, the model predicts nonlinear stress-strain behavior for bodies
 undergoing large deformations.
 
 .. figure:: figures/cl/stress_strain_neo.svg
    :alt: Neo-hookean Stress-strain curve.
    :name: fig:smm:cl:neo_hookean
    :width: 40.0%
 
    Neo-hookean Stress-strain curve.
 
 As illustrated in :numref:`fig:smm:cl:neo_hookean`, the behavior
 is initially linear and the mechanical behavior is very close to the
 corresponding linear elastic material. This constitutive relationship,
 which accounts for compressibility, is a modified version of the one
 proposed by Ronald Rivlin :cite:`Belytschko:2000`.
 
 The strain energy stored in the material is given by:
 
 .. math::
 
    \label{eqn:smm:constitutive:neohookean_potential}
      \Psi(\boldsymbol{C}) = \frac{1}{2}\lambda_0\left(\ln J\right)^2-\mu_0\ln J+\frac{1}{2}
      \mu_0\left(\mathrm{tr}(\boldsymbol{C})-3\right)
 
 where :math:`\lambda_0` and :math:`\mu_0` are, respectively, Lamé’s
 first parameter and the shear modulus at the initial configuration.
 :math:`J` is the jacobian of the deformation gradient
 (:math:`\boldsymbol{F}=\nabla_{\!\!\boldsymbol{X}}\boldsymbol{x}`):
 :math:`J=\text{det}(\boldsymbol{F})`. Finally
 :math:`\boldsymbol{C}` is the right Cauchy-Green
 deformation tensor.
 
 Since this kind of material is used for large deformation problems, a
 finite deformation framework should be used. Therefore, the Cauchy
 stress (:math:`\boldsymbol{\sigma}`) should be computed
 through the second Piola-Kirchhoff stress tensor
 :math:`\boldsymbol{S}`:
 
 .. math:: \boldsymbol{\sigma } = \frac{1}{J}\boldsymbol{F}\boldsymbol{S}\boldsymbol{F}^T
 
 Finally the second Piola-Kirchhoff stress tensor is given by:
 
 .. math::
 
    \boldsymbol{S}  = 2\frac{\partial\Psi}{\partial\boldsymbol{C}} = \lambda_0\ln J
      \boldsymbol{C}^{-1}+\mu_0\left(\boldsymbol{I}-\boldsymbol{C}^{-1}\right)
 
 The parameters to indicate in the material file are the same as those
 for the elastic case: ``E`` (Young’s modulus), ``nu`` (Poisson’s ratio).
 
 .. _sect-smm-cl-sls:
 
 Visco-Elasticity
 ''''''''''''''''
 
 Visco-elasticity is characterized by strain rate dependent behavior.
 Moreover, when such a material undergoes a deformation it dissipates
 energy. This dissipation results in a hysteresis loop in the
 stress-strain curve at every loading cycle (see
 :numref:`fig:smm:cl:visco-elastic:hyst`).
 In principle, it can be applied to many materials, since all materials
 exhibit a visco-elastic behavior if subjected to particular conditions
 (such as high temperatures).
 
 .. figure:: figures/cl/stress_strain_visco.svg
    :name:   fig:smm:cl:visco-elastic:hyst
    :align: center
    :width: 40.0%
 
    Characteristic stress-strain behavior of a visco-elastic material with hysteresis loop
 
 .. figure:: figures/cl/visco_elastic_law.svg
    :name:   fig:smm:cl:visco-elastic:model
    :align: center
    :width: 40.0%
 
    Schematic representation of the standard rheological linear solid visco-elastic model
 
 The standard rheological linear solid model (see Sections 10.2 and 10.3
 of :cite:`simo92`) has been implemented in ``Akantu``. This
 model results from the combination of a spring mounted in parallel with
 a spring and a dashpot connected in series, as illustrated in
 :numref:`fig:smm:cl:visco-elastic:model`.
 The advantage of this model is that it allows to account for creep or
 stress relaxation. The equation that relates the stress to the strain is
 (in 1D):
 
 .. math:: \frac{d\varepsilon(t)}{dt} = \left ( E + E_V \right ) ^ {-1} \cdot \left [ \frac{d\sigma(t)}{dt} + \frac{E_V}{\eta}\sigma(t) - \frac{EE_V}{\eta}\varepsilon(t) \right ]
 
 where :math:`\eta` is the viscosity. The equilibrium condition is unique and is
 attained in the limit, as :math:`t \to \infty`. At this stage, the response is
 elastic and depends on the Young’s modulus :math:`E`. The mandatory parameters
 for the material file are the following: ``rho`` (density), ``E`` (Young’s
 modulus), ``nu`` (Poisson’s ratio), ``Plane_Stress`` (if set to zero plane
 strain, otherwise plane stress), ``eta`` (dashpot viscosity) and ``Ev``
 (stiffness of the viscous element).
 
 Note that the current standard linear solid model is applied only on the
 deviatoric part of the strain tensor. The spheric part of the strain
 tensor affects the stress tensor like an linear elastic material.
 
 .. _sect-smm-cl-plastic:
 
 Plastic
 ```````
 
 Small-Deformation Plasticity
 ''''''''''''''''''''''''''''
 
 The small-deformation plasticity is a simple plasticity material
 formulation which accounts for the additive decomposition of strain into
 elastic and plastic strain components. This formulation is applicable to
 infinitesimal deformation where the additive decomposition of the strain
 is a valid approximation. In this formulation, plastic strain is a
 shearing process where hydrostatic stress has no contribution to
 plasticity and consequently plasticity does not lead to volume change.
 :numref:`fig:smm:cl:Lin-strain-hard` shows the linear strain
 hardening elasto-plastic behavior according to the additive
 decomposition of strain into the elastic and plastic parts in
 infinitesimal deformation as
 
 .. math::
 
    \boldsymbol{\varepsilon} &= \boldsymbol{\varepsilon}^e +\boldsymbol{\varepsilon}^p\\
    \boldsymbol{\sigma} &= 2G(\boldsymbol{\varepsilon}^e) + \lambda  \mathrm{tr}(\boldsymbol{\varepsilon}^e)\boldsymbol{I}
 
 .. figure:: figures/cl/isotropic_hardening_plasticity.svg
    :name:   fig:smm:cl:Lin-strain-hard
    :align: center
 
    Stress-strain curve for the small-deformation plasticity with linear isotropic hardening.
 
 In this class, the von Mises yield criterion is used. In the von Mises
 yield criterion, the yield is independent of the hydrostatic stress.
 Other yielding criteria such as Tresca and Gurson can be easily
 implemented in this class as well.
 
 In the von Mises yield criterion, the hydrostatic stresses have no
 effect on the plasticity and consequently the yielding occurs when a
 critical elastic shear energy is achieved.
 
 .. math::
 
    \label{eqn:smm:constitutive:von Mises}
      f = \sigma_{{\mathrm{eff}}} - \sigma_y = \left(\frac{3}{2} {\boldsymbol{\sigma}}^{{\mathrm{tr}}} : {\boldsymbol{\sigma}}^{{\mathrm{tr}}}\right)^\frac{1}{2}-\sigma_y (\boldsymbol{\varepsilon}^p)
 
 .. math::
 
    \label{eqn:smm:constitutive:yielding}
      f < 0 \quad \textrm{Elastic deformation,} \qquad f = 0 \quad  \textrm{Plastic deformation}
 
 where :math:`\sigma_y` is the yield strength of the material which can
 be function of plastic strain in case of hardening type of materials and
 :math:`{\boldsymbol{\sigma}}^{{\mathrm{tr}}}` is the
 deviatoric part of stress given by
 
 .. math::
 
    \label{eqn:smm:constitutive:deviatoric stress}
      {\boldsymbol{\sigma}}^{{\mathrm{tr}}}=\boldsymbol{\sigma} - \frac{1}{3} \mathrm{tr}(\boldsymbol{\sigma}) \boldsymbol{I}
 
 After yielding :math:`(f = 0)`, the normality hypothesis of plasticity
 determines the direction of plastic flow which is normal to the tangent
 to the yielding surface at the load point. Then, the tensorial form of
 the plastic constitutive equation using the von Mises yielding criterion
 (see equation 4.34) may be written as
 
 .. math::
 
    \label{eqn:smm:constitutive:plastic contitutive equation}
      \Delta {\boldsymbol{\varepsilon}}^p = \Delta p \frac {\partial{f}}{\partial{\boldsymbol{\sigma}}}=\frac{3}{2} \Delta p \frac{{\boldsymbol{\sigma}}^{{\mathrm{tr}}}}{\sigma_{{\mathrm{eff}}}}
 
 In these expressions, the direction of the plastic strain increment (or
 equivalently, plastic strain rate) is given by
 :math:`\frac{{\boldsymbol{\sigma}}^{{\mathrm{tr}}}}{\sigma_{{\mathrm{eff}}}}`
 while the magnitude is defined by the plastic multiplier
 :math:`\Delta p`. This can be obtained using the *consistency condition*
 which impose the requirement for the load point to remain on the
 yielding surface in the plastic regime.
 
 Here, we summarize the implementation procedures for the
 small-deformation plasticity with linear isotropic hardening:
 
 #. Compute the trial stress:
 
    .. math:: {\boldsymbol{\sigma}}^{{\mathrm{tr}}} = {\boldsymbol{\sigma}}_t + 2G\Delta \boldsymbol{\varepsilon} + \lambda \mathrm{tr}(\Delta \boldsymbol{\varepsilon})\boldsymbol{I}
 
 #. Check the Yielding criteria:
 
    .. math:: f = (\frac{3}{2} {\boldsymbol{\sigma}}^{{\mathrm{tr}}} : {\boldsymbol{\sigma}}^{{\mathrm{tr}}})^{1/2}-\sigma_y (\boldsymbol{\varepsilon}^p)
 
 #. Compute the Plastic multiplier:
 
    .. math::
 
       \begin{aligned}
           d \Delta p &= \frac{\sigma^{tr}_{eff} - 3G \Delta P^{(k)}- \sigma_y^{(k)}}{3G + h}\\
           \Delta p^{(k+1)} &= \Delta p^{(k)}+ d\Delta p\\
           \sigma_y^{(k+1)} &= (\sigma_y)_t+ h\Delta p
         \end{aligned}
 
 #. Compute the plastic strain increment:
 
    .. math:: \Delta {\boldsymbol{\varepsilon}}^p = \frac{3}{2} \Delta p \frac{{\boldsymbol{\sigma}}^{{\mathrm{tr}}}}{\sigma_{{\mathrm{eff}}}}
 
 #. Compute the stress increment:
 
    .. math:: {\Delta \boldsymbol{\sigma}} = 2G(\Delta \boldsymbol{\varepsilon}-\Delta \boldsymbol{\varepsilon}^p) + \lambda  \mathrm{tr}(\Delta \boldsymbol{\varepsilon}-\Delta \boldsymbol{\varepsilon}^p)\boldsymbol{I}
 
 #. Update the variables:
 
    .. math::
 
       \begin{aligned}
           {\boldsymbol{\varepsilon^p}} &= {\boldsymbol{\varepsilon}}^p_t+{\Delta {\boldsymbol{\varepsilon}}^p}\\
           {\boldsymbol{\sigma}} &= {\boldsymbol{\sigma}}_t+{\Delta \boldsymbol{\sigma}}
         \end{aligned}
 
 We use an implicit integration technique called *the radial return method* to
 obtain the plastic multiplier. This method has the advantage of being
 unconditionally stable, however, the accuracy remains dependent on the step
 size. The plastic parameters to indicate in the material file are:
 :math:`\sigma_y` (Yield stress) and ``h`` (Hardening modulus). In addition, the
 elastic parameters need to be defined as previously mentioned: ``E`` (Young’s
 modulus), ``nu`` (Poisson’s ratio).
 
 Damage
 ``````
 
 In the simplified case of a linear elastic and brittle material,
 isotropic damage can be represented by a scalar variable :math:`d`,
 which varies from :math:`0` to :math:`1` for no damage to fully broken
 material respectively. The stress-strain relationship then becomes:
 
 .. math:: \boldsymbol{\sigma} = (1-d)\, \boldsymbol{C}:\boldsymbol{\varepsilon}
 
 where :math:`\boldsymbol{\sigma}`,
 :math:`\boldsymbol{\varepsilon}` are the Cauchy stress and
 strain tensors, and :math:`\boldsymbol{C}` is the elastic
 stiffness tensor. This formulation relies on the definition of an
 evolution law for the damage variable. In ``Akantu``, many possibilities
 exist and they are listed below.
 
 .. _sect-smm-cl-damage-marigo:
 
 Marigo
 ''''''
 
 This damage evolution law is energy based as defined by Marigo
 :cite:`marigo81a`, :cite:`lemaitre96a`. It is an isotropic damage law.
 
 .. math::
 
    \begin{aligned}
      Y &= \frac{1}{2}\boldsymbol{\varepsilon}:\boldsymbol{C}:\boldsymbol{\varepsilon}\\
      F &= Y - Y_d - S d\\
      d &= \left\{
        \begin{array}{l l}
          \mathrm{min}\left(\frac{Y-Y_d}{S},\;1\right) & \mathrm{if}\; F > 0\\
          \mathrm{unchanged} & \mathrm{otherwise}
        \end{array}
      \right.\end{aligned}
 
 In this formulation, :math:`Y` is the strain energy release rate,
 :math:`Y_d` the rupture criterion and :math:`S` the damage energy. The
 non-local version of this damage evolution law is constructed by
 averaging the energy :math:`Y`.
 
 .. _sect-smm-cl-damage-mazars:
 
 Mazars
 ''''''
 
 This law introduced by Mazars :cite:`mazars84a` is a
 behavioral model to represent damage evolution in concrete. This model
 does not rely on the computation of the tangent stiffness, the damage is
 directly evaluated from the strain.
 
 The governing variable in this damage law is the equivalent strain
 :math:`\varepsilon_{{\mathrm{eq}}} =
 \sqrt{<\boldsymbol{\varepsilon}>_+:<\boldsymbol{\varepsilon}>_+}`,
 with :math:`<.>_+` the positive part of the tensor. This part is defined
 in the principal coordinates (I, II, III) as
 :math:`\varepsilon_{{\mathrm{eq}}} =
 \sqrt{<\boldsymbol{\varepsilon_I}>_+^2 + <\boldsymbol{\varepsilon_{II}}>_+^2 + <\boldsymbol{\varepsilon_{III}}>_+^2}`.
 The damage is defined as:
 
 .. math::
 
    \begin{aligned}
      D &= \alpha_t^\beta D_t + (1-\alpha_t)^\beta D_c\\
      D_t &= 1 - \frac{\kappa_0 (1- A_t)}{\varepsilon_{{\mathrm{eq}}}} - A_t \exp^{-B_t(\varepsilon_{{\mathrm{eq}}}-\kappa_0)}\\
      D_c &= 1 - \frac{\kappa_0 (1- A_c)}{\varepsilon_{{\mathrm{eq}}}} - A_c
      \exp^{-B_c(\varepsilon_{{\mathrm{eq}}}-\kappa_0)}\\
      \alpha_t &= \frac{\sum_{i=1}^3<\varepsilon_i>_+\varepsilon_{{\mathrm{nd}}\;i}}{\varepsilon_{{\mathrm{eq}}}^2}\end{aligned}
 
 With :math:`\kappa_0` the damage threshold, :math:`A_t` and :math:`B_t`
 the damage parameter in traction, :math:`A_c` and :math:`B_c` the damage
 parameter in compression, :math:`\beta` is the shear parameter.
 :math:`\alpha_t` is the coupling parameter between traction and
 compression, the :math:`\varepsilon_i` are the eigenstrain and the
 :math:`\varepsilon_{{\mathrm{nd}}\;i}` are the eigenvalues of the strain
 if the material were undamaged.
 
 The coefficients :math:`A` and :math:`B` are the post-peak asymptotic
 value and the decay shape parameters.
 
 .. _sect:smm:CLNL:
 
 Non-Local Constitutive Laws
 ```````````````````````````
 
 Continuum damage modeling of quasi-brittle materials undergo significant
 softening after the onset of damage. This fast growth of damage causes a loss of
 ellipticity of partial differential equations of equilibrium. Therefore, the
 numerical simulation results won't be objective anymore, because the dissipated
 energy will depend on mesh size used in the simulation. One way to avoid this
 effect is the use of non-local damage formulations. In this approach a local
 quantity such as the strain is replaced by its non-local average, where the size
 of the domain, over which the quantitiy is averaged, depends on the underlying
 material microstructure. ``Akantu`` provides non-local versions of many
 constitutive laws for damage. Examples are for instance the material
 :ref:`sect-smm-cl-damage-mazars` and the material
 :ref:`sect-smm-cl-damage-marigo`, that can be used in a non-local context. In
 order to use the corresponding non-local formulation the user has to define the
 non-local material he wishes to use in the text input file:
 
-.. code-block::
+.. code-block:: none
 
    material constitutive_law_non_local [
        name = material_name
        rho = $value$
        ...
    ]
 
 where ``constitutive_law_non_local`` is the name of the non-local constitutive law, *e.g.* `marigo_non_local`.
 In addition to the material the non-local neighborhood, that should be used for the averaging process needs to be defined in the material file as well:
 
-.. code-block::
+.. code-block:: none
 
   non_local neighborhood_name weight_function_type [
      radius = $value$
      ...
       weight_function weight_parameter [
         damage_limit = $value$
         ...
      ]
   ]
 
 for the non-local averaging, *e.g.* ``base_wf``, followed by the properties of the non-local neighborhood, such as the radius, and the weight function parameters. It is important to notice that the non-local neighborhood must have the same name as the material to which the neighborhood belongs!
 The following two sections list the non-local constitutive laws and different type of weight functions available in ``Akantu``.
 \subsection{Non-local constitutive laws}
 Let us consider a body having a volume :math:`V` and a boundary :math:`\Gamma`. The stress-strain relation for a non-local damage model can be described as follows:
 
 .. _eq:non-local-const:
  .. math:: \vec{\sigma} = (1-\bar{d}) \vec{D}:\epsilon
 
 with :math:`\vec{D}` the elastic moduli tensor, :math:`\sigma` the stress tensor, :math:`\epsilon` the strain tensor and :math:`\bar{d}` the non-local damage variable. Note that this stres-strain relationship is similar to the relationship defined in Damage model except :math:`\bar{d}`. The non-local damage model can be extended to the damage constitutive laws: :ref:`sect-smm-cl-damage-marigo` and :ref:`sect-smm-cl-damage-mazars`.
 
 The non-local damage variable :math:`\bar{d}` is defined as follows:
 
 .. _eq:non-local-const:
  .. math:: \bar{d}(\vec{x}) = \int_{V}W(\vec{x}, \vec{y}) d(\vec{y}) dV(\vec{y})
 
 with :math:`W(\vec{x},\vec{y})` the weight function which averages local damage variables to describe the non-local interactions. A list of available weight functions and its functionalities in \akantu are explained in the next section.
 
 Non-local weight functions
 ''''''''''''''''''''''''''
 
 The available weight functions in ``Akantu`` are follows:
 
  - ``base_weight_function``: This weight function averages local damage variables by using a bell-shape function on spatial dimensions.
  -  ``damaged_weight_function``: A linear-shape weight function is applied to average local damage variables. Its slope is determined by damage variables. For example, the damage variables for an element which is highly damaged are averaged over  large spatial dimension (linear function including a small slope).
  - ``remove_damaged_weight_function``: This weight function averages damage values by using a bell-shape function as  ``base_weight_function``, but excludes elements which are fully damaged.
  - ``remove_damaged_with_damage_rate_weight_function``: A bell-shape function is applied to average local damage variables for elements having small damage rates.
  - ``stress_based_weight_function``: Non local integral takes stress states, and use the states to construct weight function: an ellipsoid shape. Detailed explanations of this weight function are given in Giry et al. :cite:`giry13a`.
 
 
 
 .. _sec-cohesive-laws:
 
 Cohesive Constitutive laws
 ``````````````````````````
 
 .. _ssect-smm-cl-coh-snozzi:
 
 Linear Irreversible Law
 '''''''''''''''''''''''
 
 .. figure:: figures/cl/linear_cohesive_law.svg
    :alt: Irreversible cohesive laws for explicit simulations.
    :name: fig:smm:coh:linear_cohesive_law
    :align: center
    :width: 60.0%
 
    Irreversible cohesive laws for explicit simulations.
 
 
 `Akantu` includes the Snozzi-Molinari :cite:`snozzi_cohesive_2013`
 linear irreversible cohesive law (see
 :numref:`fig:smm:coh:linear_cohesive_law`). It is an extension to
 the Camacho-Ortiz :cite:`camacho_computational_1996` cohesive law in
 order to make dissipated fracture energy path-dependent. The concept
 of free potential energy is dropped and a new independent parameter
 :math:`\kappa` is introduced:
 
 .. math::
   \kappa = \frac{G_\mathrm{c, II}}{G_\mathrm{c, I}}
 
 
 where :math:`G_\mathrm{c, I}` and :math:`G_\mathrm{c, II}` are the
 necessary works of separation per unit area to open completely a
 cohesive zone under mode I and mode II, respectively. Their model yields to the
 following equation for cohesive tractions :math:`\vec{T}` in case of crack
 opening :math:`{\delta}`:
 
 .. math::
   \vec{T} = \left( \frac{\beta^2}{\kappa} \Delta_\mathrm{t} \vec{t} +
     \Delta_\mathrm{n} \vec{n} \right)
   \frac{\sigma_\mathrm{c}}{\delta}
   \left( 1- \frac{\delta}{\delta_\mathrm{c}} \right)
   = \hat{\vec T}\,
   \frac{\sigma_\mathrm{c}}{\delta}
   \left( 1- \frac{\delta}{\delta_\mathrm{c}} \right)
   :label: eq-smm-coh-tractions
 
 where :math:`\sigma_\mathrm{c}` is the material strength along the fracture,
 :math:`\delta_\mathrm{c}` the critical effective displacement after which
 cohesive tractions are zero (complete decohesion), :math:`\Delta_\mathrm{t}`
 and :math:`\Delta_\mathrm{n}` are the tangential and normal components of
 the opening displacement vector :math:`\vec{\Delta}`, respectively. The
 parameter :math:`\beta` is a weight that indicates how big the tangential
 opening contribution is. The effective opening displacement is:
 
 .. math::
    \delta = \sqrt{\frac{\beta^2}{\kappa^2} \Delta_\mathrm{t}^2 + \Delta_\mathrm{n}^2}
 
 In case of unloading or reloading :math:`\delta < \delta_\mathrm{max}`,
 tractions are calculated as:
 
 .. math::
    \begin{eqnarray}
    T_\mathrm{n} &= \Delta_\mathrm{n}\, \frac{\sigma_\mathrm{c}}{\delta_\mathrm{max}} \left( 1- \frac{\delta_\mathrm{max}}{\delta_\mathrm{c}} \right) \\
   T_\mathrm{t} &= \frac{\beta^2}{\kappa}\, \Delta_\mathrm{t}\, \frac{\sigma_\mathrm{c}}{\delta_\mathrm{max}} \left( 1- \frac{\delta_\mathrm{max}}{\delta_\mathrm{c}} \right)
    \end{eqnarray}
 
 so that they vary linearly between the origin and the maximum attained
 tractions. As shown in :numref:`fig:smm:coh:linear_cohesive_law`,
 in this law, the dissipated and reversible energies are:
 
 .. math::
    \begin{eqnarray}
    E_\mathrm{diss} &= \frac{1}{2} \sigma_\mathrm{c}\, \delta_\mathrm{max}\\[1ex]
    E_\mathrm{rev} &= \frac{1}{2} T\, \delta
    \end{eqnarray}
 
 Moreover, a damage parameter :math:`D` can be defined as:
 
 .. math::
   D = \min \left(
     \frac{\delta_\mathrm{max}}{\delta_\mathrm{c}},1 \right)
 
 which varies from 0 (undamaged condition) and 1 (fully
 damaged condition). This variable can only increase because damage is
 an irreversible process. A simple penalty contact model has been incorporated
 in the cohesive law so that normal tractions can be returned in
 case of compression:
 
 .. math::
   T_\mathrm{n} = \alpha \Delta_\mathrm{n} \quad\text{if}\quad
   \Delta_\mathrm{n}\quad <\quad 0
 
 where :math:`\alpha` is a stiffness parameter that defaults to zero. The
 relative contact energy is equivalent to reversible energy but in
 compression.
 
 The material name of the linear decreasing cohesive law is
 ``material_cohesive_linear`` and its parameters with their respective default
 values are:
 
 - ``sigma_c = 0``
 - ``delta_c = 0``
 - ``beta = 0``
 - ``G_c = 0``
 - ``kappa = 1``
 - ``penalty = 0``
 
 where ``G_c`` corresponds to :math:`G_\mathrm{c, I}`. A random number
 generator can be used to assign a random :math:`\sigma_\mathrm{c}` to each
 facet following a given distribution (see
 Section :ref:`sect-smm-cl`). Only one parameter between ``delta_c``
 and ``G_c`` has to be specified. For random :math:`\sigma_\mathrm{c}`
 distributions, the chosen parameter of these two is kept fixed and the
 other one is varied.
 
 The bi-linear constitutive law works exactly the same way as the linear
 one, except for the additional parameter ``delta_0`` that by
 default is zero. Two examples for the extrinsic and intrinsic cohesive
 elements and also an example to assign different properties to
 inter-granular and trans-granular cohesive elements can be found in
 the folder ``examples/cohesive_element/``.
 
 .. _ssect:smm:cl:coh-friction:
 
 Linear Cohesive Law with Friction
 '''''''''''''''''''''''''''''''''
 
 This law represents a variation of the linear irreversible cohesive of
 the previous section, which adds friction.  The friction behavior is
 approximated with an elasto-plastic law, which relates the friction
 force to the relative sliding between the two faces of the cohesive
 element.  The slope of the elastic branch is called
 ``penalty_for_friction``, and is defined by the user, together
 with the friction coefficient, as a material property.  The friction
 contribution evolves with the damage of the cohesive law: it is null
 when the damage is zero, and it becomes maximum when the damage is
 equal to one.  This is done by defining a current value of the
 friction coefficient (mu) that increases linearly with the damage, up
 to the value of the friction coefficient defined by the user.  The
 yielding plateau of the friction law is given by the product of the
 current friction coefficient and the local compression stress acting
 in the cohesive element.  Such an approach is equivalent to a
 node-to-node contact friction. Its accuracy is acceptable only for
 small displacements.
 
 The material name of the linear cohesive law with friction is
 ``material_cohesive_linear_friction``. Its additional parameters
 with respect to those of the linear cohesive law without friction,
 with the respective default values, are:
 
 - ``mu = 0``
 - ``penalty_for_friction = 0``
 
 .. _ssect:smm:cl:coh-fatigue:
 
 Linear Cohesive Law with Fatigue
 ''''''''''''''''''''''''''''''''
 
 This law represents a variation of the linear irreversible cohesive
 law of the previous section, that removes the hypothesis of elastic
 unloading-reloading cycles. With this law, some energy is dissipated
 also during unloading and reloading with hysteresis. The
 implementation follows the work of :cite:`nguyen2001`. During the
 unloading-reloading cycle, the traction increment is computed as
 
 .. math::
   \dot{T} =
   \begin{cases}
     K^- \, \dot{\delta} & \text{if $\dot{\delta} < 0$} \\
     K^+ \, \dot{\delta} & \text{if $\dot{\delta} > 0$} \\
   \end{cases}
 
 where :math:`\dot{\delta}` and :math:`\dot{T}` are respectively the effective
 opening displacement and the cohesive traction increments with respect
 to time, while :math:`K^-` and :math:`K^+` are respectively the unloading and
 reloading incremental stiffness. The unloading path is linear and
 results in an unloading stiffness
 
 .. math::
   K^- = \frac{T_\mathrm{max}}{\delta_\mathrm{max}}
 
 where :math:`T_\mathrm{max}` and :math:`\delta_\mathrm{max}` are the maximum
 cohesive traction and the effective opening displacement reached
 during the precedent loading phase. The unloading stiffness remains
 constant during the unloading phase. On the other hand the reloading
 stiffness increment :math:`\dot{K}^+` is calculated as
 
 .. math::
   \dot{K}^+ =
   \begin{cases}
     - K^+ \, \dot{\delta} / \delta_\mathrm{f} & \text{if $\dot{\delta}
       > 0$} \\
     \left( K^+ - K^- \right) \, \dot{\delta} / \delta_\mathrm{f} &
     \text{if $\dot{\delta}$ < $0$}
   \end{cases}
 
 where :math:`\delta_\mathrm{f}` is a material parameter (refer
 to :cite:`vocialta15` for more details). During unloading the stiffness
 :math:`K^+` tends to :math:`K^-`, while during reloading :math:`K^+` gets decreased at
 every time step. If the cohesive traction during reloading exceeds the
 upper limit given by equation :eq:`eq-smm-coh-tractions`, it is
 recomputed following the behavior of the linear decreasing cohesive
 law for crack opening.
 
 .. _ssect:smm:cl:coh-exponential:
 
 Exponential Cohesive Law
 '''''''''''''''''''''''''
 
 Ortiz and Pandolfi proposed this cohesive law in 1999 :cite:`ortiz1999`.  The
 traction-opening equation for this law is as follows:
 
 .. math::
    T = e \sigma_c \frac{\delta}{\delta_c}e^{-\delta/ \delta_c}
    :label: eq:exponential_law
 
-This equation is plotted in Figure :numref:`fig:smm:cl:ecl`. The term
+This equation is plotted in :numref:`fig:smm:cl:ecl`. The term
 :math:`\partial{\vec{T}}/ \partial{\delta}` after the necessary derivation
 can expressed as
 
 .. math::
    \frac{\partial{\vec{T}}} {\partial{\delta}} = \hat{\vec{T}} \otimes
    \frac                       {\partial{(T/\delta)}}{\partial{\delta}}
    \frac{\hat{\vec{T}}}{\delta}+ \frac{T}{\delta}  \left[ \beta^2 \mat{I} +
    \left(1-\beta^2\right) \left(\vec{n} \otimes \vec{n}\right)\right]
    :label: eq:tangent_cohesive
 
 where
 
 .. math::
   \frac{\partial{(T/ \delta)}}{\partial{\delta}} = \left\{\begin{array} {l l}
       -e  \frac{\sigma_c}{\delta_c^2  }e^{-\delta  /  \delta_c} &  \quad  \text{if}
       \delta \geq \delta_{max}\\
       0 & \quad \text{if} \delta < \delta_{max}, \delta_n > 0
     \end{array} \right.
 
 
 As regards the behavior in compression, two options are available:
 a contact penalty approach with stiffness following the formulation of
 the exponential law and a contact penalty approach with constant
 stiffness. In the second case, the stiffness is defined as a function
 of the tangent of the exponential law at the origin.
 
 .. figure:: figures/cl/cohesive_exponential.png
    :alt:    Exponential cohesive law
    :name:   fig:smm:cl:ecl
    :align: center
 
    Exponential cohesive law
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diff --git a/doc/dev-doc/manual/new-constitutive-laws.rst b/doc/dev-doc/manual/new-constitutive-laws.rst
index 04c94884e..f8925e0fe 100644
--- a/doc/dev-doc/manual/new-constitutive-laws.rst
+++ b/doc/dev-doc/manual/new-constitutive-laws.rst
@@ -1,375 +1,373 @@
 Adding a New Constitutive Law
 -----------------------------
 
 There are several constitutive laws in ``Akantu`` as described in the previous
 Section :ref:`sect-smm-cl`. It is also possible to use a user-defined material
 for the simulation. These materials are referred to as local materials since
 they are local to the example of the user and not part of the ``Akantu``
 library. To define a new local material, two files (``material_XXX.hh`` and
 ``material_XXX.cc``) have to be provided where ``XXX`` is the name of the new
 material. The header file ``material_XXX.hh`` defines the interface of your
 custom material. Its implementation is provided in the ``material_XXX.cc``. The
 new law must inherit from the :cpp:class:`Material <akantu::Material>` class or
 any other existing material class. It is therefore necessary to include the
 interface of the parent material in the header file of your local material and
-indicate the inheritance in the declaration of the class:
-
-.. code-block:: c++
+indicate the inheritance in the declaration of the class::
 
    auto & solver = model.getNonLinearSolver();
    solver.set("max_iterations", 1);
    solver.set("threshold", 1e-4);
    solver.set("convergence_type", SolveConvergenceCriteria::_residual);
 
    model.solveStep();
 
 
    /* ---------------------------------------------------------------------- */
    #include "material.hh"
    /* ---------------------------------------------------------------------- */
 
    #ifndef __AKANTU_MATERIAL_XXX_HH__
    #define __AKANTU_MATERIAL_XXX_HH__
 
    namespace akantu {
 
    class MaterialXXX : public Material {
 
    /// declare here the interface of your material
 
    };
 
 In the header file the user also needs to declare all the members of the new
 material. These include the parameters that a read from the
 material input file, as well as any other material parameters that will be
 computed during the simulation and internal variables.
 
 
 In the following the example of adding a new damage material will be
 presented. In this case the parameters in the material will consist of the
 Young's modulus, the Poisson coefficient, the resistance to damage and the
 damage threshold. The material will then from these values compute its Lamé
 coefficients and its bulk modulus. Furthermore, the user has to add a new
 internal variable ``damage`` in order to store the amount of damage at each
 quadrature point in each step of the simulation. For this specific material the
 member declaration inside the class will look as follows::
 
    class LocalMaterialDamage : public Material {
 
    /// declare constructors/destructors here
 
    /// declare methods and accessors here
 
      /* -------------------------------------------------------------------- */
      /* Class Members                                                        */
      /* -------------------------------------------------------------------- */
 
      AKANTU_GET_MACRO_BY_ELEMENT_TYPE_CONST(Damage, damage, Real);
    private:
 
      /// the young modulus
      Real E;
 
      /// Poisson coefficient
      Real nu;
 
      /// First Lame coefficient
      Real lambda;
 
      /// Second Lame coefficient (shear modulus)
      Real mu;
 
      /// resistance to damage
      Real Yd;
 
      /// damage threshold
      Real Sd;
 
      /// Bulk modulus
      Real kpa;
 
      /// damage internal variable
      InternalField<Real> damage;
 
    };
 
 In order to enable to print the material parameters at any point in
 the user's example file using the standard output stream by typing::
 
    for (UInt m = 0; m  < model.getNbMaterials(); ++m)
      std::cout << model.getMaterial(m) << std::endl;
 
 the standard output stream operator has to be redefined. This should be done at the end of the header file::
 
    class LocalMaterialDamage : public Material {
 
      /// declare here the interace of your material
 
    }:
    /* ---------------------------------------------------------------------- */
    /* inline functions                                                       */
    /* ---------------------------------------------------------------------- */
    /// standard output stream operator
    inline std::ostream & operator <<(std::ostream & stream, const LocalMaterialDamage & _this)
    {
      _this.printself(stream);
      return stream;
    }
 
 However, the user still needs to register the material parameters that
 should be printed out. The registration is done during the call of the
 constructor. Like all definitions the implementation of the
 constructor has to be written in the ``material_XXX.cc``
 file. However, the declaration has to be provided in the
 ``material_XXX.hh`` file::
 
    class LocalMaterialDamage : public Material {
      /* -------------------------------------------------------------------- */
      /* Constructors/Destructors                                             */
      /* -------------------------------------------------------------------- */
    public:
 
      LocalMaterialDamage(SolidMechanicsModel & model, const ID & id = "");
    };
 
 The user can now define the implementation of the constructor in the
 ``material_XXX.cc`` file::
 
    /* ---------------------------------------------------------------------- */
    #include "local_material_damage.hh"
    #include "solid_mechanics_model.hh"
 
    namespace akantu {
 
    /* ---------------------------------------------------------------------- */
    LocalMaterialDamage::LocalMaterialDamage(SolidMechanicsModel & model,
               const ID & id)  :
      Material(model, id),
      damage("damage", *this) {
      AKANTU_DEBUG_IN();
 
      this->registerParam("E", E, 0., _pat_parsable, "Young's modulus");
      this->registerParam("nu", nu, 0.5, _pat_parsable, "Poisson's ratio");
      this->registerParam("lambda", lambda, _pat_readable, "First Lame coefficient");
      this->registerParam("mu", mu, _pat_readable, "Second Lame coefficient");
      this->registerParam("kapa", kpa, _pat_readable, "Bulk coefficient");
      this->registerParam("Yd", Yd,   50., _pat_parsmod);
      this->registerParam("Sd", Sd, 5000., _pat_parsmod);
 
      damage.initialize(1);
 
      AKANTU_DEBUG_OUT();
    }
 
 During the intializer list the reference to the model and the material id are
 assigned and the constructor of the internal field is called. Inside the scope
 of the constructor the internal values have to be initialized and the
 parameters, that should be printed out, are registered with the function:
 ``registerParam``::
 
    void registerParam(name of the parameter (key in the material file),
           member variable,
           default value (optional parameter),
           access permissions,
           description);
 
 The available access permissions are as follows:
 - ``_pat_internal``: Parameter can only be output when the material is printed.
 - ``_pat_writable``: User can write into the parameter. The parameter is output when the material is printed.
 - ``_pat_readable``: User can read the parameter. The parameter is output when the material is printed.
 - ``_pat_modifiable``: Parameter is writable and readable.
 - ``_pat_parsable``: Parameter can be parsed, *i.e.* read from the input file.
 - ``_pat_parsmod``: Parameter is modifiable and parsable.
 
 In order to implement the new constitutive law the user needs to
 specify how the additional material parameters, that are not
 defined in the input material file, should be calculated. Furthermore,
 it has to be defined how stresses and the stable time step should be
 computed for the new local material. In the case of implicit
 simulations, in addition, the computation of the tangent stiffness needs
 to be defined. Therefore, the user needs to redefine the following
 functions of the parent material::
 
    void initMaterial();
 
    // for explicit and implicit simulations void
    computeStress(ElementType el_type, GhostType ghost_type = _not_ghost);
 
    // for implicit simulations
    void computeTangentStiffness(const ElementType & el_type,
               Array<Real> & tangent_matrix,
               GhostType ghost_type = _not_ghost);
 
    // for explicit and implicit simulations
    Real getStableTimeStep(Real h, const Element & element);
 
 In the following a detailed description of these functions is provided:
 
 - ``initMaterial``: This method is called after the material file is fully read
   and the elements corresponding to each material are assigned. Some of the
   frequently used constant parameters are calculated in this method. For
   example, the Lam\'{e} constants of elastic materials can be considered as such
   parameters.
 
 - ``computeStress``: In this method, the stresses are computed based on the
   constitutive law as a function of the strains of the quadrature points. For
   example, the stresses for the elastic material are calculated based on the
   following formula:
 
   .. math::
 
      \mat{\sigma }  =\lambda\mathrm{tr}(\mat{\varepsilon})\mat{I}+2 \mu \mat{\varepsilon}
 
   Therefore, this method contains a loop on all quadrature points assigned to
   the material using the two macros:
   ``MATERIAL_STRESS_QUADRATURE_POINT_LOOP_BEGIN`` and
   ``MATERIAL_STRESS_QUADRATURE_POINT_LOOP_END``
 
   .. code::
 
      MATERIAL_STRESS_QUADRATURE_POINT_LOOP_BEGIN(element_type);
 
      // sigma <- f(grad_u)
 
      MATERIAL_STRESS_QUADRATURE_POINT_LOOP_END;
 
   The strain vector in Akantu contains the values of :math:`\nabla \vec{u}`,
   i.e. it is really the *displacement gradient*,
 
 - ``computeTangentStiffness``: This method is called when the tangent to the
   stress-strain curve is desired (see Fig \ref {fig:smm:AL:K}). For example,
   it is called in the implicit solver when the stiffness matrix for the
   regular elements is assembled based on the following formula:
 
   .. math::
      \label{eqn:smm:constitutive_elasc} \mat{K }
      =\int{\mat{B^T}\mat{D(\varepsilon)}\mat{B}}
 
   Therefore, in this method, the ``tangent`` matrix (\mat{D}) is
   computed for a given strain.
 
   The ``tangent`` matrix is a :math:`4^{th}` order tensor which is stored as
   a matrix in Voigt notation.
 
   .. _fig:smm:AL:K:
   .. figure:: figures/tangent.svg
               :align: center
               :width: 60%
 
               Tangent to the stress-strain curve.
 
 ..
      \begin{figure}[!htb]
        \begin{center}
          \includegraphics[width=0.4\textwidth,keepaspectratio=true]{figures/tangent.pdf}
          \caption{Tangent to the stress-strain curve.}
          \label{fig:smm:AL:K}
        \end{center}
      \end{figure}
 
 - ``getCelerity``: The stability criterion of the explicit integration scheme
   depend on the fastest wave celerity~\eqref{eqn:smm:explicit:stabletime}. This
   celerity depend on the material, and therefore the value of this velocity
   should be defined in this method for each new material. By default, the
   fastest wave speed is the compressive wave whose celerity can be defined in ``getPushWaveSpeed``.
 
 Once the declaration and implementation of the new material has been
 completed, this material can be used in the user's example by including the header file::
 
    #include "material_XXX.hh"
 
 For existing materials, as mentioned in Section~\ref{sect:smm:CL}, by
 default, the materials are initialized inside the method
 ``initFull``. If a local material should be used instead, the
 initialization of the material has to be postponed until the local
 material is registered in the model. Therefore, the model is
 initialized with the boolean for skipping the material initialization
 equal to true::
 
    /// model initialization
    model.initFull(_analysis_method = _explicit_lumped_mass);
 
 Once the model has been initialized, the local material needs
 to be registered in the model::
 
    model.registerNewCustomMaterials<XXX>("name_of_local_material");
 
 Only at this point the material can be initialized::
 
    model.initMaterials();
 
 A full example for adding a new damage law can be found in
 ``examples/new_material``.
 
 Adding a New Non-Local Constitutive Law
 ```````````````````````````````````````
 
 In order to add a new non-local material we first have to add the local
 constitutive law in Akantu (see above). We can then add the non-local version
 of the constitutive law by adding the two files (``material_XXX_non_local.hh``
 and ``material_XXX_non_local.cc``) where ``XXX`` is the name of the
 corresponding local material. The new law must inherit from the two classes,
 non-local parent class, such as the ``MaterialNonLocal`` class, and from the
 local version of the constitutive law, *i.e.* ``MaterialXXX``. It is therefore
 necessary to include the interface of those classes in the header file of your
 custom material and indicate the inheritance in the declaration of the class::
 
    /* ---------------------------------------------------------------------- */
    #include "material_non_local.hh" // the non-local parent
    #include "material_XXX.hh"
    /* ---------------------------------------------------------------------- */
 
    #ifndef __AKANTU_MATERIAL_XXX_HH__
    #define __AKANTU_MATERIAL_XXX_HH__
 
    namespace akantu {
 
    class MaterialXXXNonLocal : public MaterialXXX,
                                public MaterialNonLocal {
 
    /// declare here the interface of your material
 
    };
 
 As members of the class we only need to add the internal fields to store the
 non-local quantities, which are obtained from the averaging process::
 
    /* -------------------------------------------------------------------------- */
    /* Class members                                                              */
    /* -------------------------------------------------------------------------- */
    protected:
      InternalField<Real> grad_u_nl;
 
 The following four functions need to be implemented in the non-local material::
 
      /// initialization of the material
      void initMaterial();
      /// loop over all element and invoke stress computation
      virtual void computeNonLocalStresses(GhostType ghost_type);
      /// compute stresses after local quantities have been averaged
      virtual void computeNonLocalStress(ElementType el_type, GhostType ghost_type)
      /// compute all local quantities
      void computeStress(ElementType el_type, GhostType ghost_type);
 
 In the intialization of the non-local material we need to register the local
 quantity for the averaging process. In our example the internal field
 *grad_u_nl* is the non-local counterpart of the gradient of the displacement
 field (*grad_u_nl*)::
 
      void MaterialXXXNonLocal::initMaterial() {
        MaterialXXX::initMaterial();
        MaterialNonLocal::initMaterial();
        /// register the non-local variable in the manager
        this->model->getNonLocalManager().registerNonLocalVariable(
          this->grad_u.getName(),
          this->grad_u_nl.getName(),
          spatial_dimension * spatial_dimension);
      }
 
 The function to register the non-local variable takes as parameters the name of
 the local internal field, the name of the non-local counterpart and the number
 of components of the field we want to average. In the *computeStress* we now
 need to compute all the quantities we want to average. We can then write a loop
 for the stress computation in the function *computeNonLocalStresses* and then
 provide the constitutive law on each integration point in the function
 *computeNonLocalStress*.
diff --git a/doc/dev-doc/manual/solidmechanicsmodel.rst b/doc/dev-doc/manual/solidmechanicsmodel.rst
index 12ae4a720..374e930f3 100644
--- a/doc/dev-doc/manual/solidmechanicsmodel.rst
+++ b/doc/dev-doc/manual/solidmechanicsmodel.rst
@@ -1,880 +1,880 @@
 .. _sect-smm:
 
 Solid Mechanics Model
 =====================
 
 The solid mechanics model is a specific implementation of the :cpp:class:`Model
 <akantu::Model>` interface dedicated to handle the equations of motion or
 equations of equilibrium. The model is created for a given mesh. It will create
 its own :cpp:class:`FEEngine <akantu::FEEngine>` object to compute the
 interpolation, gradient, integration and assembly operations. A
 :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>` object can simply
 be created like this::
 
   SolidMechanicsModel model(mesh);
 
 where ``mesh`` is the mesh for which the equations are to be
 solved. A second parameter called ``spatial_dimension`` can be
 added after ``mesh`` if the spatial dimension of the problem is
 different than that of the mesh.
 
 This model contains at least the following six ``Arrays``:
 
 :cpp:func:`blocked_dofs <akantu::SolidMechanicsModel::getBlockedDOFs>`:
     contains a Boolean value for each degree of freedom specifying whether that
     degree is blocked or not. A Dirichlet boundary condition can be prescribed
     by setting the **blocked_dofs** value of a degree of freedom to
     ``true``. A Neumann boundary condition can be applied by setting the
     **blocked_dofs** value of a degree of freedom to ``false``. The
     **displacement**, **velocity** and **acceleration** are
     computed for all degrees of freedom for which the **blocked_dofs**
     value is set to ``false``. For the remaining degrees of freedom, the imposed
     values (zero by default after initialization) are kept.
 
 :cpp:func:`displacement <akantu::SolidMechanicsModel::getDisplacement>`:
     contains the displacements of all degrees of freedom. It can be either a
     computed displacement for free degrees of freedom or an imposed displacement
     in case of blocked ones (:math:`\vec{u}` in the following).
 
 :cpp:func:`velocity <akantu::SolidMechanicsModel::getVelocity>`:
     contains the velocities of all degrees of freedom. As **displacement**,
     it contains computed or imposed velocities depending on the nature of the
     degrees of freedom (:math:`\dot{\vec{u}}` in the following).
 
 :cpp:func:`acceleration <akantu::SolidMechanicsModel::getAcceleration>`:
     contains the accelerations of all degrees of freedom. As **displacement**,
     it contains computed or imposed accelerations depending on the nature of the
     degrees of freedom (:math:`\ddot{\vec{u}}` in the following).
 
 :cpp:func:`external_force <akantu::SolidMechanicsModel::getExternalForce>`:
     contains the external forces applied on the nodes
     (:math:`\vec{f}_{\st{ext}}` in the following).
 
 :cpp:func:`internal_force <akantu::SolidMechanicsModel::getInternalForce>`:
     contains the internal forces on the nodes (:math:`\vec{f}_{\mathrm{int}}` in
     the following).
 
 
 Some examples to help to understand how to use this model will be
 presented in the next sections.
 
 
 Model Setup
 -----------
 
 
 Setting Initial Conditions
 ``````````````````````````
 
 For a unique solution of the equations of motion, initial
 displacements and velocities for all degrees of freedom must be
 specified:
 
 .. math::
   \vec{u}(t=0) & = \vec{u}_0\\
   \dot{\vec u}(t=0) & = \vec{v}_0
 
 The solid mechanics model can be initialized as
 follows::
 
   model.initFull()
 
 This function initializes the internal arrays and sets them to zero. Initial
 displacements and velocities that are not equal to zero can be prescribed by
 running a loop over the total number of nodes. Here, the initial displacement in
 :math:`x`-direction and the initial velocity in :math:`y`-direction for all
 nodes is set to :math:`0.1` and :math:`1`, respectively::
 
     auto & disp = model.getDisplacement();
     auto & velo = model.getVelocity();
 
     for (UInt node = 0; node < mesh.getNbNodes(); ++node) {
         disp(node, 0) = 0.1;
         velo(node, 1) = 1.;
     }
 
 .. _sect-smm-boundary:
 
 Setting Boundary Conditions
 ```````````````````````````
 
 This section explains how to impose Dirichlet or Neumann boundary
 conditions. A Dirichlet boundary condition specifies the values that
 the displacement needs to take for every point :math:`x` at the boundary
 (:math:`\Gamma_u`) of the problem domain (:numref:`fig-smm-boundaries`):
 
 .. math::
   \vec{u} = \bar{\vec u} \quad \forall \vec{x}\in \Gamma_{u}
 
 
 A Neumann boundary condition imposes the value of the gradient of the
 solution at the boundary :math:`\Gamma_t` of the problem domain
 (:numref:`fig-smm-boundaries`):
 
 .. math::
     \vec{t} = \mat{\sigma} \vec{n} = \bar{\vec t} \quad
     \forall \vec{x}\in \Gamma_{t}
 
 .. _fig-smm-boundaries:
 .. figure:: figures/problem_domain.svg
             :align: center
 
             Problem domain :math:`\Omega` with boundary in three dimensions. The
             Dirchelet and the Neumann regions of the boundary are denoted with
             :math:`\Gamma_u` and :math:`\Gamma_t`, respecitvely.
 
 Different ways of imposing these boundary conditions exist. A basic
 way is to loop over nodes or elements at the boundary and apply local
 values. A more advanced method consists of using the notion of the
 boundary of the mesh. In the following both ways are presented.
 
 Starting with the basic approach, as mentioned, the Dirichlet boundary
 conditions can be applied by looping over the nodes and assigning the
 required values. :numref:`fig-smm-dirichlet_bc` shows a beam with a
 fixed support on the left side. On the right end of the beam, a load
 is applied. At the fixed support, the displacement has a given
 value. For this example, the displacements in both the :math:`x` and the
 :math:`y`-direction are set to zero. Implementing this displacement boundary
 condition is similar to the implementation of initial displacement
 conditions described above. However, in order to impose a displacement
 boundary condition for all time steps, the corresponding nodes need to
 be marked as boundary nodes using the function ``blocked``. While,
 in order to impose a load on the right side, the nodes are not marked.
 The detail codes are shown as follows
 
 .. code-block:: c++
 
    auto & blocked = model.getBlockedDOFs();
    const auto & pos = mesh.getNodes();
 
    UInt nb_nodes = mesh.getNbNodes();
 
    for (UInt node = 0; node < nb_nodes; ++node) {
      if(Math::are_float_equal(pos(node, _x), 0)) {
        blocked(node, _x) = true; // block dof in x-direction
        blocked(node, _y) = true; // block dof in y-direction
        disp(node, _x) = 0.; // fixed displacement in x-direction
        disp(node, _y) = 0.; // fixed displacement in y-direction
      } else if (Math::are_float_equal(pos(node, _y), 0)) {
        blocked(node, _x) = false; // unblock dof in x-direction
        forces(node, _x) = 10.;    // force in x-direction
      }
    }
 
 
 .. _fig-smm-dirichlet_bc:
 .. figure:: figures/dirichlet.svg
             :align: center
 
             Beam with fixed support and load.
 
 
 For the more advanced approach, one needs the notion of a boundary in
 the mesh. Therefore, the boundary should be created before boundary
 condition functors can be applied. Generally the boundary can be
 specified from the mesh file or the geometry.  For the first case, the
 function ``createGroupsFromMeshData`` is called.  This function
 can read any types of mesh data which are provided in the mesh
 file. If the mesh file is created with Gmsh, the function takes one
 input strings which is either ``tag_0``, ``tag_1`` or
 ``physical_names``. The first two tags are assigned by Gmsh to
 each element which shows the physical group that they belong to. In
 Gmsh, it is also possible to consider strings for different groups of
 elements. These elements can be separated by giving a string
 ``physical_names`` to the function
 ``createGroupsFromMeshData``
 
 .. code-block:: c++
 
    mesh.createGroupsFromMeshData<std::string>("physical_names").
 
 Boundary conditions support can also be created from the geometry by calling
 ``createBoundaryGroupFromGeometry``. This function gathers all the elements on
 the boundary of the geometry.
 
 To apply the required boundary conditions, the function :cpp:func:`applyBC
 <akantu::BoundaryCondition::applyBC>` needs to be called on a
 :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>`. This function
 gets a Dirichlet or Neumann functor and a string which specifies the desired
 boundary on which the boundary conditions is to be applied. The functors specify
 the type of conditions to apply. Three built-in functors for Dirichlet exist:
 :cpp:class:`FlagOnly <akantu::BC::Dirichlet::FlagOnly>`, :cpp:class:`FixedValue
 <akantu::BC::Dirichlet::FixedValue>` and :cpp:class:`IncrementValue
 <akantu::BC::Dirichlet::IncrementValue>`. The functor ``FlagOnly`` is used if a
 point is fixed in a given direction. Therefore, the input parameter to this
 functor is only the fixed direction. The ``FixedValue`` functor is used when a
 displacement value is applied in a fixed direction. The ``IncrementValue``
 applies an increment to the displacement in a given direction. The following
 code shows the utilization of three functors for the top, bottom and side
 surface of the mesh which were already defined in the Gmsh
 
 .. code-block:: c++
 
    model.applyBC(BC::Dirichlet::FixedValue(13.0, _y), "Top");
    model.applyBC(BC::Dirichlet::FlagOnly(_x), "Bottom");
    model.applyBC(BC::Dirichlet::IncrementValue(13.0, _x), "Side");
 
 To apply a Neumann boundary condition, the applied traction or stress should be
 specified before. In case of specifying the traction on the surface, the functor
 :cpp:class:`FromTraction <akantu::BC::Neumann::FromTraction>` of Neumann
 boundary conditions is called. Otherwise, the functor :cpp:class:`FromStress
 <akantu::BC::Neumann::FromStress>` should be called which gets the stress tensor
 as an input parameter
 
 .. code-block:: c++
 
    Vector<Real> surface_traction{0., 0., 1.};
    auto surface_stress(3, 3) = Matrix<Real>::eye(3);
 
    model.applyBC(BC::Neumann::FromTraction(surface_traction), "Bottom");
    model.applyBC(BC::Neumann::FromStress(surface_stress), "Top");
 
 If the boundary conditions need to be removed during the simulation, a
 functor is called from the Neumann boundary condition to free those
 boundary conditions from the desired boundary
 
 .. code-block:: c++
 
     model.applyBC(BC::Neumann::FreeBoundary(), "Side");
 
 User specified functors can also be implemented.  A full example for
 setting both initial and boundary conditions can be found in
 ``examples/boundary_conditions.cc``.  The problem solved
 in this example is shown in :numref:`fig-smm-bc_and_ic`. It consists
 of a plate that is fixed with movable supports on the left and bottom
 side. On the right side, a traction, which increases linearly with the
 number of time steps, is applied. The initial displacement and
 velocity in :math:`x`-direction at all free nodes is zero and two
 respectively.
 
 .. _fig-smm-bc_and_ic:
 .. figure:: figures/bc_and_ic_example.svg
             :align: center
             :width: 75%
 
             Plate on movable supports.
 
 ..
    \begin{figure}[!htb]
      \centering
      \includegraphics[scale=0.8]{figures/bc_and_ic_example}
      \caption{Plate on movable supports.\label{fig-smm-bc_and_ic}}
    \end{figure}
 
 As it is mentioned in Section \ref{sect:common:groups}, node and
 element groups can be used to assign the boundary conditions. A
 generic example is given below with a Dirichlet boundary condition::
 
      // create a node group
      NodeGroup & node_group = mesh.createNodeGroup("nodes_fix");
 
      /* fill the node group with the nodes you want */
 
      // create an element group using the existing node group
      mesh.createElementGroupFromNodeGroup("el_fix",
                                           "nodes_fix",
                                           spatial_dimension-1);
 
      // boundary condition can be applied using the element group name
      model.applyBC(BC::Dirichlet::FixedValue(0.0, _x), "el_fix");
 
 Material Selector
 `````````````````
 
 If the user wants to assign different materials to different
 finite elements groups in ``Akantu``, a material selector has to be
 used. By default, ``Akantu`` assigns the first valid material in the
 material file to all elements present in the model (regular continuum
 materials are assigned to the regular elements and cohesive materials
 are assigned to cohesive elements or element facets).
 
 To assign different materials to specific elements, mesh data information such
 as tag information or specified physical names can be used.
 :cpp:class:`MeshDataMaterialSelector <akantu::MeshDataMaterialSelector>` class
 uses this information to assign different materials. With the proper physical
 name or tag name and index, different materials can be assigned as demonstrated
 in the examples below::
 
      auto mat_selector =
         std::make_shared<MeshDataMaterialSelector<std::string>>("physical_names",
                                                                 model);
      model.setMaterialSelector(mat_selector);
 
 In this example the physical names specified in a GMSH geometry file will by
 used to match the material names in the input file.
 
 Another example would be to use the first (``tag_0``) or the second
 (``tag_1``) tag associated to each elements in the mesh::
 
      auto mat_selector = std::make_shared<MeshDataMaterialSelector<UInt>>(
          "tag_1", model, first_index);
      model.setMaterialSelector(*mat_selector);
 
 where ``first_index`` (default is 1) is the value of ``tag_1`` that will
 be associated to the first material in the material input file. The following
 values of the tag will be associated with the following materials.
 
 There are four different material selectors pre-defined in ``Akantu``.
 :cpp:class:`MaterialSelector <akantu::MaterialSelector>` and
 :cpp:class:`DefaultMaterialSelector <akantu::DefaultMaterialSelector>` is used
 to assign a material to regular elements by default. For the regular elements,
 as in the example above, :cpp:class:`MeshDataMaterialSelector
 <akantu::MeshDataMaterialSelector>` can be used to assign different materials to
 different elements.
 
 Apart from the ``Akantu``'s default material selectors, users can always
 develop their own classes in the main code to tackle various
 multi-material assignment situations.
 
 For cohesive material, ``Akantu`` has a pre-defined material selector to assign
 the first cohesive material by default to the cohesive elements which is called
 :cpp:class:`DefaultMaterialCohesiveSelector
 <akantu::DefaultMaterialCohesiveSelector>` and it inherits its properties from
 :cpp:class:`DefaultMaterialSelector <akantu::DefaultMaterialSelector>`. Multiple
 cohesive materials can be assigned using mesh data information (for more
 details, see :ref:`sect-smm-intrinsic-insertion`).
 
 
 Insertion of Cohesive Elements
 ``````````````````````````````
 Cohesive elements are currently compatible only with static simulation
 and dynamic simulation with an explicit time integration scheme (see
 section :ref:`ssect-smm-expl-time-integration`). They do not have to be
 inserted when the mesh is generated (intrinsic) but can be added
 during the simulation (extrinsic). At any time during the simulation,
 it is possible to access the following energies with the relative
 function:
 
 .. code-block:: c++
 
   Real Ed = model.getEnergy("dissipated");
   Real Er = model.getEnergy("reversible");
   Real Ec = model.getEnergy("contact");
 
 A new model have to be call in a very similar way that the solid
 mechanics model:
 
 .. code-block:: c++
 
   SolidMechanicsModelCohesive model(mesh);
   model.initFull(_analysis_method = _explicit_lumped_mass,
                  _is_extrinsic = true);
 
 Cohesive element insertion can be either realized at the beginning of
 the simulation or it can be carried out dynamically during the
 simulation. The first approach is called ``intrinsic``, the second
 one ``extrinsic``. When an element is present from the beginning, a
 bi-linear or exponential cohesive law should be used instead of a
 linear one. A bi-linear law works exactly like a linear one except for
 an additional parameter :math:`\delta_0` separating an initial linear
 elastic part from the linear irreversible one. For additional details
 concerning cohesive laws see Section~\ref{sec:cohesive-laws}.
 
 .. _fig-smm-coh-insertion:
 .. figure:: figures/insertion.svg
             :align: center
 
             Insertion of a cohesive element.
 
 Extrinsic cohesive elements are dynamically inserted between two
 standard elements when
 
 .. math::
    \sigma_\mathrm{eff} > \sigma_\mathrm{c} \quad\text {with} \quad \sigma_\mathrm{eff} = \sqrt{\sigma_\mathrm{n} ^ 2 + \frac{\tau ^ 2} {\beta ^ 2 }}
 
 in which :math:`\sigma_\mathrm { n }
 ` is the tensile normal traction and $\tau$ the resulting tangential one(  :numref:`fig-smm-coh-insertion`).
 
 Extrinsic approach
 ''''''''''''''''''
 
 During the simulation, stress has to be checked along each facet in order to
 insert cohesive elements where the stress criterion is reached. This check is
 performed by calling the method :cpp:func:`checkCohesiveStress
 <akantu::SolidMechanicsModelCohesive::checkCohesiveStress>`, as example before
 each step resolution:
 
 .. code-block:: c++
 
    model.checkCohesiveStress();
    model.solveStep();
 
 The area where stresses are checked and cohesive elements inserted can be
 limited using the method :cpp:func:`setLimit
 <akantu::CohesiveInserter::setLimit>` on the :cpp:class:`CohesiveInserter
 <akantu::CohesiveInserter>` during initialization. As example, to limit
 insertion in the range :math:`[-1.5, 1.5]` in the $x$ direction:
 
 .. code-block:: c++
 
   auto & inserter = model.getElementInserter();
   inserter.setLimit(_x, -1.5, 1.5);
 
 Additional restrictions with respect to :math:`_y` and :math:`_z` directions can
 be added as well.
 
 .. _sect-smm-intrinsic-insertion:
 
 Intrinsic approach
 ''''''''''''''''''
 
 Intrinsic cohesive elements are inserted in the mesh with the method
 :cpp:func:`initFull <akantu::SolidMechanicsModelCohesive::initFull>`.
 Similarly, the range of insertion can be limited with :cpp:func:`setLimit
 <akantu::CohesiveInserter::setLimit>` before the :cpp:func:`initFull
 <akantu::SolidMechanicsModelCohesive::initFull>` call.
 
 In both cases extrinsic and intrinsic the insertion can be restricted to
 surfaces or element groups. To do so the list of groups should be specified in
 the input file.
 
 .. code-block::
 
   model solid_mechanics_model_cohesive [
     cohesive_inserter [
       cohesive_surfaces = [surface1, surface2, ...]
       cohesive_zones = [group1, group2, ...]
     ]
 
     material cohesive_linear [
       name = insertion
       beta = 1
       G_c = 10
       sigma_c = 1e6
     ]
   ]
 
 
 
 Static Analysis
 ---------------
 
 The :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>` class can
 handle different analysis methods, the first one being presented is the static
 case. In this case, the equation to solve is
 
 .. math::
      \mat{K} \vec{u} = \vec{f}_{\mathrm{ext}}
      :label: eqn-smm-static
 
 where :math:`\mat{K}` is the global stiffness matrix, :math:`\vec{u}` the
 displacement vector and :math:`\vec{f}_{\st{ext}}` the vector of external
 forces applied to the system.
 
 To solve such a problem, the static solver of the
 :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>` object is used.
 First, a model has to be created and initialized. To create the model, a mesh
 (which can be read from a file) is needed, as explained in
 Section~\ref{sect:common:mesh}. Once an instance of a
 :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>` is obtained, the
 easiest way to initialize it is to use the :cpp:func:`initFull
 <akantu::SolidMechanicsModel::initFull>` method by giving the
 :cpp:class:`SolidMechanicsModelOptions <akantu::SolidMechanicsModelOptions>`.
 These options specify the type of analysis to be performed and whether the
 materials should be initialized with :cpp:func:`initMaterials
 <akantu::SolidMechanicsModel::initMaterials>` or not
 
 .. code-block:: c++
 
    SolidMechanicsModel model(mesh);
    model.initFull(_analysis_method = _static);
 
 Here, a static analysis is chosen by passing the argument
 :cpp:enumerator:`_static <akantu::_static>` to the method. By default, the
 Boolean for no initialization of the materials is set to false, so that they are
 initialized during the ``initFull``. The method ``initFull`` also initializes
 all appropriate vectors to zero. Once the model is created and initialized, the
 boundary conditions can be set as explained in Section :ref:`sect-smm-boundary`.
 Boundary conditions will prescribe the external forces for some free degrees of
 freedom :math:`\vec{f}_{\st{ext}}` and displacements for some others. At this
 point of the analysis, the function
 :cpp:func:`solveStep <akantu::SolidMechanicsModel::solveStep>` can be called
 
 .. code-block:: c++
 
    auto & solver = model.getNonLinearSolver();
    solver.set("max_iterations", 1);
    solver.set("threshold", 1e-4);
    solver.set("convergence_type", SolveConvergenceCriteria::_residual);
 
    model.solveStep();
 
 This function is templated by the solving method and the convergence criterion
 and takes two arguments: the tolerance and the maximum number of iterations (100
 by default), which are :math:`10^{-4}` and :math:`1` for this example. The
 modified Newton-Raphson method is chosen to solve the system. In this method,
 the equilibrium equation (:eq:`eqn-smm-static`) is modified in order to apply a
 Newton-Raphson convergence algorithm:
 
 .. math::
      \mat{K}^{i+1}\delta\vec{u}^{i+1} &= \vec{r} \\
      &= \vec{f}_{\st{ext}} -\vec{f}_{\st{int}}\\
      &= \vec{f}_{\st{ext}} - \mat{K}^{i} \vec{u}^{i}\\
      \vec{u}^{i+1} &= \vec{u}^{i} + \delta\vec{u}^{i+1}~,
 
 where :math:`\delta\vec{u}` is the increment of displacement to be added from
 one iteration to the other, and :math:`i` is the Newton-Raphson iteration
 counter. By invoking the ``solveStep`` method in the first step, the global
 stiffness matrix :math:`\mat{K}` from (:eq:`eqn-smm-static`) is automatically
 assembled. A Newton-Raphson iteration is subsequently started, :math:`\mat{K}`
 is updated according to the displacement computed at the previous iteration and
 one loops until the forces are balanced
 (:cpp:enumerator:`SolveConvergenceCriteria::_residual
 <akantu::SolveConvergenceCriteria::_residual>`), i.e. :math:`||\vec{r}|| <`
 :cpp:member:`threshold
 <akantu::NonLinearSolverNewtonRaphson::convergence_criteria>`. One can also
 iterate until the increment of displacement is zero
 (:cpp:enumerator:`SolveConvergenceCriteria::_solution
 <akantu::SolveConvergenceCriteria::_solution>`) which also means that the
 equilibrium is found. For a linear elastic problem, the solution is obtained in
 one iteration and therefore the maximum number of iterations can be set to one.
 But for a non-linear case, one needs to iterate as long as the norm of the
 residual exceeds the tolerance threshold and therefore the maximum number of
 iterations has to be higher, e.g. :math:`100`
 
 .. code-block:: c++
 
    solver.set("max_iterations", 100);
    model.solveStep();
 
 At the end of the analysis, the final solution is stored in the
 **displacement** vector.  A full example of how to solve a static
 problem is presented in the code ``examples/static/static.cc``.
 This example is composed of a 2D plate of steel, blocked with rollers
 on the left and bottom sides as shown in :numref:`fig-smm-static`.
 The nodes from the right side of the sample are displaced by :math:`0.01\%`
 of the length of the plate.
 
 .. _fig-smm-static:
 .. figure:: figures/static.svg
             :align: center
             :width: 75%
 
             Numerical setup.
 
 The results of this analysis is depicted in
 :numref:`fig-smm-implicit:static_solution`.
 
 .. _fig-smm-implicit:static_solution:
 .. figure:: figures/static_analysis.png
             :align: center
             :width: 75%
 
             Solution of the static analysis. Left: the initial condition, right:
             the solution (deformation magnified 50 times).
 
 
 Dynamic Methods
 ---------------
 
 Different ways to solve the equations of motion are implemented in the
 solid mechanics model.  The complete equations that should be solved
 are:
 
 .. math:: \mat{M}\ddot{\vec{u}} + \mat{C}\dot{\vec{u}} + \mat{K}\vec{u} = \vec{f}_{\mathrm{ext}}
    :label: eqn-equation-motion
 
 where :math:`\mat{M}`, :math:`\mat{C}` and :math:`\mat{K}` are the mass,
 damping and stiffness matrices, respectively.
 
 In the previous section, it has already been discussed how to solve this
 equation in the static case, where :math:`\ddot{\vec{u}} = \dot{\vec{u}} = 0`.
 Here the method to solve this equation in the general case will be presented.
 For this purpose, a time discretization has to be specified. The most common
 discretization method in solid mechanics is the Newmark-:math:`\beta` method,
 which is also the default in ``Akantu``.
 
 For the Newmark-:math:`\beta` method, (:eq:`eqn-equation-motion`) becomes a
 system of three equations (see :cite:`curnier92a,hughes-83a` for more details):
 
 .. math::
    \begin{eqnarray}
      \mat{M} \ddot{\vec{u}}_{n+1} + \mat{C}\dot{\vec{u}}_{n+1} + \mat{K} \vec{u}_{n+1} &={\vec{f}_{\st{ext}}}_{\, n+1}\\
      \vec{u}_{n+1} &= \vec{u}_{n}
                       + \left(1 - \alpha\right) \Delta t \dot{\vec{u}}_{n}
                       + \alpha \Delta t \dot{\vec{u}}_{n+1}
                       + \left(\frac{1}{2} - \alpha\right) \Delta t^2 \ddot{\vec{u}}_{n}\\
      \dot{\vec{u}}_{n+1} &= \dot{\vec{u}}_{n}
                             + \left(1 - \beta\right) \Delta t \ddot{\vec{u}}_{n}
                             + \beta \Delta t \ddot{\vec{u}}_{n+1}
    \end{eqnarray}
    :label: eqn-equation-motion-discret
 
 In these new equations, :math:`\ddot{\vec{u}}_{n}`, :math:`\dot{\vec{u}}_{n}`
 and :math:`\vec{u}_{n}` are the approximations of :math:`\ddot{\vec{u}}(t_n)`,
 :math:`\dot{\vec{u}}(t_n)` and :math:`\vec{u}(t_n)`.
 Equation~(:eq:`eqn-equation-motion-discret`) is the equation of motion
 discretized in space (finite-element discretization), and the equations above
 are discretized in both space and time (Newmark discretization). The
 :math:`\alpha` and :math:`\beta` parameters determine the stability and the
 accuracy of the algorithm. Classical values for :math:`\alpha` and :math:`\beta`
 are usually :math:`\beta = 1/2` for no numerical damping and :math:`0 < \alpha <
 1/2`.
 
 .. csv-table::
    :header: ":math:`\\alpha`", "Method (:math:`\\beta = 1/2`)", "Type"
 
    ":math:`0`", "central difference", "explicit"
    ":math:`\frac{1}{6}`", "Fox-Goodwin(royal road)", "implicit"
    ":math:`\frac{1}{3}`", "Linear acceleration", "implicit"
    ":math:`\frac{1}{2}`", "Average acceleration (trapeziodal rule)", "implicit"
 
 
 The solution of this system of equations,
 (:eq:`eqn-equation-motion-discret`) is
 split into a predictor and a corrector system of equations.  Moreover,
 in the case of a non-linear equations, an iterative algorithm such as
 the Newton-Raphson method is applied. The system of equations can be
 written as:
 
 - *Predictor*:
 
   .. math:: \vec{u}_{n+1}^{0} &= \vec{u}_{n} + \Delta t \dot{\vec{u}}_{n} + \frac{\Delta t^2}{2} \ddot{\vec{u}}_{n} \\
             \dot{\vec{u}}_{n+1}^{0} &= \dot{\vec{u}}_{n} + \Delta t \ddot{\vec{u}}_{n} \\
             \ddot{\vec{u}}_{n+1}^{0} &= \ddot{\vec{u}}_{n}
 
 - *Solve*:
 
   .. math:: \left(c \mat{M} + d \mat{C} + e \mat{K}_{n+1}^i\right)
             \vec{w} &= {\vec{f}_{\st{ext}}}_{\,n+1} - {\vec{f}_{\st{int}}}_{\,n+1}^i -
             \mat{C} \dot{\vec{u}}_{n+1}^i - \mat{M} \ddot{\vec{u}}_{n+1}^i\\
             &= \vec{r}_{n+1}^i
 
 - *Corrector*:
 
   .. math:: \ddot{\vec{u}}_{n+1}^{i+1} &= \ddot{\vec{u}}_{n+1}^{i} +c \vec{w} \\
             \dot{\vec{u}}_{n+1}^{i+1} &= \dot{\vec{u}}_{n+1}^{i} + d\vec{w} \\
             \vec{u}_{n+1}^{i+1} &= \vec{u}_{n+1}^{i} + e \vec{w}
 
 where :math:`i` is the Newton-Raphson iteration counter and :math:`c`, :math:`d` and :math:`e`
 are parameters depending on the method used to solve the equations
 
 
 .. csv-table::
    :header: "", ":math:`\\vec{w}`", ":math:`e`", ":math:`d`", ":math:`c`"
 
    "in acceleration", ":math:`\delta\ddot{\vec{u}}`", ":math:`\alpha\beta\Delta t^2`", ":math:`\beta\Delta t`", ":math:`1`"
    "in velocity", ":math:`\delta\dot{\vec{u}}`", ":math:`\alpha\Delta t`", ":math:`1`", ":math:`\frac{1}{\beta\Delta t}`"
    "in displacement", ":math:`\delta\vec{u}`", ":math:`1`", ":math:`\frac{1}{\alpha\Delta t}`", ":math:`\frac{1}{\alpha\beta \Delta t^2}`"
 
 
 .. note:: If you want to use the implicit solver ``Akantu`` should be compiled at
           least with one sparse matrix solver such as `Mumps
           <http://mumps.enseeiht.fr/>`_ :cite:`mumps`.
 
 
 Implicit Time Integration
 `````````````````````````
 
 To solve a problem with an implicit time integration scheme, first a
 :cpp:class:`SolidMechanicsModel <akantu::SolidMechanicsModel>` object has to be
 created and initialized. Then the initial and boundary conditions have to be
 set. Everything is similar to the example in the static case
 (Section~\ref{sect:smm:static}), however, in this case the implicit dynamic
 scheme is selected at the initialization of the model::
 
    SolidMechanicsModel model(mesh);
    model.initFull(_analysis_method = _implicit_dynamic);
 
 Because a dynamic simulation is conducted, an integration time step
 :math:`\Delta t` has to be specified. In the case of implicit simulations,
 ``Akantu`` implements a trapezoidal rule by default.  That is to say
 :math:`\alpha = 1/2` and :math:`\beta = 1/2` which is unconditionally
 stable. Therefore the value of the time step can be chosen arbitrarily
 within reason::
 
    model.setTimeStep(time_step);
 
 Since the system has to be solved for a given amount of time steps, the
 method ``solveStep()``, (which has already been used in the static
 example in Section~\ref{sect:smm:static}), is called inside a time
 loop::
 
    /// time loop
    Real time = 0.;
 
    auto & solver = model.getNonLinearSolver();
    solver.set("max_iterations", 100);
    solver.set("threshold", 1e-12);
    solver.set("convergence_type", SolveConvergenceCriteria::_solution);
 
    for (UInt s = 1; time <max_time; ++s, time += time_step) {
        model.solveStep();
    }
 
 An example of solid mechanics with an implicit time integration scheme is
 presented in ``examples/implicit/implicit_dynamic.cc``. This example consists of
 a 3D beam of
 :math:`10\mathrm{m}\times1\mathrm{m}\times1\mathrm{m}` blocked
 on one side and is on a roller on the other side. A constant force of
 :math:`5\mathrm{kN}` is applied in its middle.
 :numref:`fig-smm-implicit-dynamic` presents the geometry of this case. The
 material used is a fictitious linear elastic material with a density of
 :math:`1000 \mathrm{kg/m}^3`, a Young's Modulus of
 :math:`120 \mathrm{MPa}` and Poisson's ratio of :math:`0.3`. These values
 were chosen to simplify the analytical solution.
 
 An approximation of the dynamic response of the middle point of the
 beam is given by:
 
 .. math::
 
     u\left(\frac{L}{2}, t\right)
     = \frac{1}{\pi^4} \left(1 - cos\left(\pi^2 t\right) +
     \frac{1}{81}\left(1 - cos\left(3^2 \pi^2 t\right)\right) +
     \frac{1}{625}\left(1 - cos\left(5^2 \pi^2 t\right)\right)\right)
 
 .. _fig-smm-implicit-dynamic:
 .. figure:: figures/implicit_dynamic.svg
             :align: center
             :width: 75%
 
             Numerical setup.
 
 ..
    \begin{figure}[!htb]
      \centering
      \includegraphics[scale=.6]{figures/implicit_dynamic}
      \caption{Numerical setup}
      \label{fig-smm-implicit:dynamic}
    \end{figure}
 
-Figure :numref:`fig-smm-implicit-dynamic_solution` presents the deformed
+:numref:`fig-smm-implicit-dynamic_solution` presents the deformed
 beam at 3 different times during the simulation: time steps 0, 1000 and
 2000.
 
 .. _fig-smm-implicit-dynamic_solution:
 .. figure:: figures/dynamic_analysis.png
             :align: center
             :width: 60%
 
             Deformed beam at three different times (displacement :math:`\times
             10`).
 
 .. _ssect-smm-expl-time-integration:
 
 Explicit Time Integration
 `````````````````````````
 
 The explicit dynamic time integration scheme is based on the
 Newmark-:math:`\beta` scheme with :math:`\alpha=0` (see equations
 \ref{eqn:equation-motion-discret}-\ref{eqn:finite-difference-2}).  In
 ``Akantu``, :math:`\beta` is defaults to :math:`\beta=1/2`, see section
 \ref{sect:smm:Dynamic_methods}.
 
 The initialization of the simulation is similar to the static and implicit
 dynamic version. The model is created from the :cpp:class:`SolidMechanicsModel
 <akantu::SolidMechanicsModel>` class. In the initialization, the explicit scheme
 is selected using the ``_explicit_lumped_mass`` constant::
 
    SolidMechanicsModel model(mesh);
    model.initFull(_analysis_method = _explicit_lumped_mass);
 
 
 .. note::
     Writing ``model.initFull()`` or ``model.initFull();`` is
     equivalent to use the ``_explicit_lumped_mass`` keyword, as this
     is the default case.
 
 The explicit time integration scheme implemented in ``Akantu`` uses a
 lumped mass matrix :math:`\mat{M}` (reducing the computational cost). This
 matrix is assembled by distributing the mass of each element onto its
 nodes. The resulting :math:`\mat{M}` is therefore a diagonal matrix stored
 in the **mass** vector of the model.
 
 The explicit integration scheme is conditionally stable. The time step
 has to be smaller than the stable time step which is obtained in
 Akantu as follows::
 
    critical_time_step = model.getStableTimeStep();
 
 The stable time  step corresponds to the time the fastest wave (the compressive
 wave) needs to travel the characteristic length of the mesh:
 
 .. math::
    \Delta t_{\st{crit}} = \frac{\Delta x}{c}
 
 where :math:`\Delta x` is a characteristic length (\eg the inradius in the case
 of linear triangle element) and :math:`c` is the celerity of the fastest wave in
 the material. It is generally the compressive wave of celerity :math:`c =
 \sqrt{\frac{2 \mu + \lambda}{\rho}}`, :math:`\mu` and :math:`\lambda` are the
 first and second Lame's coefficients and :math:`\rho` is the density. However,
 it is recommended to impose a time step that is smaller than the stable time
 step, for instance, by multiplying the stable time step by a safety factor
 smaller than one::
 
    const Real safety_time_factor = 0.8;
    Real applied_time_step = critical_time_step * safety_time_factor;
    model.setTimeStep(applied_time_step);
 
 The initial displacement and velocity fields are, by default, equal to zero if
 not given specifically by the user (see \ref{sect:smm:initial_condition}).
 
 Like in implicit dynamics, a time loop is used in which the
 displacement, velocity and acceleration fields are updated at each
 time step. The values of these fields are obtained from the
 Newmark:math:`-\beta` equations with :math:`\beta=1/2` and :math:`\alpha=0`. In ``Akantu``
 these computations at each time step are invoked by calling the
 function ``solveStep``::
 
    for (UInt s = 1; (s-1)*applied_time_step < total_time; ++s) {
      model.solveStep();
    }
 
 The method ``solveStep`` wraps the four following functions:
 
 - ``model.explicitPred()`` allows to compute the displacement
   field at :math:`t+1` and a part of the velocity field at :math:`t+1`, denoted by
   :math:`\vec{\dot{u}^{\st{p}}}_{n+1}`, which will be used later in the method
   ``model.explicitCorr()``. The equations are:
 
    .. math::
       \vec{u}_{n+1} &= \vec{u}_{n} + \Delta t
       \vec{\dot{u}}_{n} + \frac{\Delta t^2}{2} \vec{\ddot{u}}_{n}\\
       \vec{\dot{u}^{\st{p}}}_{n+1} &= \vec{\dot{u}}_{n} + \Delta t
       \vec{\ddot{u}}_{n}
 
 - ``model.updateResidual()`` and ``model.updateAcceleration()`` compute the acceleration increment
   :math:`\delta \vec{\ddot{u}}`:
 
   .. math::
      \left(\mat{M} + \frac{1}{2} \Delta t \mat{C}\right)
      \delta \vec{\ddot{u}} = \vec{f_{\st{ext}}} - \vec{f}_{\st{int}\, n+1}
      - \mat{C} \vec{\dot{u}^{\st{p}}}_{n+1} - \mat{M} \vec{\ddot{u}}_{n}
 
   The internal force :math:`\vec{f}_{\st{int}\, n+1}` is computed from the
     displacement :math:`\vec{u}_{n+1}` based on the constitutive law.
 
 - ``model.explicitCorr()`` computes the velocity and
   acceleration fields at :math:`t+1`:
 
   .. math::
      \vec{\dot{u}}_{n+1} &= \vec{\dot{u}^{\st{p}}}_{n+1} + \frac{\Delta t}{2}
      \delta \vec{\ddot{u}} \\ \vec{\ddot{u}}_{n+1} &=
      \vec{\ddot{u}}_{n} + \delta \vec{\ddot{u}}
 
 The use of an explicit time integration scheme is illustrated by the example:
 ``examples/explicit/explicit_dynamic.cc``. This example models the propagation
 of a wave in a steel beam. The beam and the applied displacement in the
 :math:`x` direction are shown in :numref:`fig-smm-explicit`.
 
 
 .. _fig-smm-explicit:
 .. figure:: figures/explicit.svg
             :align: center
             :width: 90%
 
             Numerical setup.
 
 
 The length and height of the beam are :math:`L={10}\textrm{m}` and :math:`h =
 {1}\textrm{m}`, respectively. The material is linear elastic, homogeneous and
 isotropic (density: :math:`7800\mathrm{kg/m}^3`, Young's modulus:
 :math:`210\mathrm{GPa}` and Poisson's ratio: :math:`0.3`). The imposed
 displacement follow a Gaussian function with a maximum amplitude of :math:`A =
 {0.01}\textrm{m}`. The potential, kinetic and total energies are computed. The
 safety factor is equal to :math:`0.8`.
 
 .. include:: ./constitutive-laws.rst
 
 .. include:: ./new-constitutive-laws.rst