B4 model
========

Theory
------

The present model is based on the B4 model proposed by Bazant et al. (2015) generalized to 3D.
The constitutive law is given in total form as

.. math::

   \sig = \Cel : \epsE  = \Cel : \left( \eps - \epsVE - \epsF - \epsDC -\epsSHR \right)

relating the nominal stress tensor :math:`\sig`
to the elastic strain tensor :math:`\epsE`
in terms of the fourth order stiffness tensor :math:`\Cel`.
Where
:math:`\eps` is the total strain,
:math:`\epsVE` is the viscoelastic strain,
:math:`\epsF` is the viscous flow strain (with the sum of :math:`\epsVE` and :math:`\epsF` representing the basic creep strain),
:math:`\epsDC` is the strain due to drying creep and
:math:`\epsSHR` is the total shrinkage strain.
A potential temperature dependence is not taken into account.

The implementation of the evolution of the visoelastic strain is using a solidifying Kelvin chain according to Bazant and Jirasek (2018). The drying creep strain is computed using an aging Kelvin chain.

The evolution of the instantaneous elastic strain and the basic creep strain are represented by the Solidification Theory generalized to three-dimensional stress states.
The evolution of the instantaneous elastic strain :math:`\epsE` is expressed as

.. math::

   \epsERate(t) = q_1 \DelNu : \sigRate(t),

in which :math:`q_1` denotes the age-independent asymptotic compliance and :math:`\DelNu` the unit compliance tensor.
The evolution of the viscoelastic strain :math:`\epsVE` is defined as

.. math::

   \epsVERate(t) = \frac{1}{v(t)} \int_0^t \PhiRate(t-t') \DelNu : d\sig(t'),

in terms of the solidified volume function :math:`v(t)` and the compliance rate of the hydrated constituent :math:`\PhiRate(t-t')`.
The solidified volume function

.. math::

   v(t) =  \left[ \left( \frac{\lambda_0}{t} \right) ^m + \frac{q_3}{q_2} \right] ^{-1}

represents the time-dependent aging of the material. Where :math:`\lambda_0` and :math:`m` are material parameters with default values :math:`\lambda_0 = 1 \; \mathrm{d}` and :math:`m = 0.5`.
The anti-derivative of the compliance rate :math:`\PhiRate(t-t')` is defined as

.. math::

   \Phi(t-t') = q_2 \ln\left( 1 + \left( \frac{t-t'}{\lambda_0} \right) ^n \right),

in which :math:`n` is the exponent of the log-power law, with the default value :math:`n = 0.1`.
The evolution of the viscous strain :math:`\epsF` is governed by

.. math::

   \epsFRate(t) = q_4 \frac{1}{t} \DelNu : \sig(t).

In the above equations, :math:`q_2`, :math:`q_3` and :math:`q_4` denote the compliance parameters.
The evolution of the drying strain :math:`\epsDC` is described as

.. math::

   \epsDCRate (t) = \frac{1}{\exp(p_{\mathrm{5H}}/2)} \int_0^t \PhiRate_\mathrm{d}(t, t', t_0) \DelNu : d\sig(t'),

where :math:`t_0` denotes the start of drying and :math:`p_{\mathrm{5H}}` is a cement type dependent parameter, which is set to :math:`p_{\mathrm{5H}} = 8` for slow and regular hardening cement. The anti-derivative of the drying creep compliance rate :math:`\PhiRate_d` is definded as

.. math::

   \Phi_d(t,t',t_0) = q_5 \sqrt{ \left\langle \exp\left( b \tanh \sqrt{\xi-\xi_0} \right) - \exp \left( b \tanh \sqrt{-\xi_0} \right) \right\rangle}

in which :math:`q_5` is the drying creep parameter, :math:`\xi = (t-t')/\tau_{\mathrm{shr,d}}` and :math:`\xi_0 = (t_0-t')/\tau_{\mathrm{shr,d}}` are dimensionless times, and :math:`b= p_{\mathrm{5H}}(1-h_{\mathrm{env}})`, in which :math:`h_{\mathrm{env}}` is the ambient relative humidity.

The total shrinkage strain :math:`\epsSHR` is assumed as the sum of the autogenous shrinkage strain :math:`\eps^\mathrm{shr,au}` and the drying shrinkage strain :math:`\eps^\mathrm{shr,d}`. The evolution of the autogenous shrinkage strain :math:`\eps^\mathrm{shr,au}` is described as

.. math::

   \eps^\mathrm{shr,au}(t) = \boldsymbol{\mathrm{I}} \eps^\mathrm{shr,au}_\infty \left[ 1+ \left( \frac{\tau_\mathrm{shr,au}}{t} \right) ^\alpha \right]^{-4.5}

in which :math:`\boldsymbol{\mathrm{I}}` is the second order identity tensor, :math:`\eps^\mathrm{shr,au}_\infty`, :math:`\tau_\mathrm{shr,au}` and :math:`\alpha` are the ultimate autogenous shrinkage strain, the autogenous shrinkage halftime, and a parameter depending on the concrete composition and on the cement type, respectively.
The evoluztion of the drying shrinkage strain :math:`\eps^\mathrm{shr,d}` is defined as

.. math::

   \eps^\mathrm{shr,d}(t) = \boldsymbol{\mathrm{I}} \eps^\mathrm{shr,d}_\infty k_\mathrm{h} \tanh \sqrt{\frac{\left \langle t-t_0 \right \rangle }{\tau_{shr,d}}}

in which :math:`\eps^\mathrm{shr,d}_\infty` denotes the ultimate drying shrinkage strain at zero humidity and :math:`\tau_{shr,d}` the drying shrinkage halftime. The paramter :math:`k_{\mathrm{h}}` is given as

.. math::

   k_{\mathrm{h}}= \begin{cases}1-h_{\mathrm{env}}^3 & \text { if } h_{\mathrm{env}} \leq 0.98, \\ 12.94\left(1-h_{\mathrm{env}}\right)-0.2 & \text { otherwise, }\end{cases}

where currently only the first variant is implemented.


Implementation
--------------

.. doxygenclass:: Marmot::Materials::B4
   :allow-dot-graphs: