Benchmark solution ===================== Calculation method used for the reference solution -------------------------------------------------------- In document [:ref:`1 <1>`], a model for the diffusion of hydrogen atoms in steels is proposed. It considers two types of concentration of hydrogen atoms: * :math:`{C}_{L}` is the concentration in the crystal lattice (Lattice) * :math:`{C}_{T}` is the concentration in the "pitfalls" or gaps (Traps) Without repeating here the whole approach followed by the authors of [:ref:`1 <1>`], the proposed formulation for the :math:`{C}_{L}` diffusion equation is: :math:`\frac{{C}_{L}+{C}_{T}(1\mathrm{-}{\theta }_{T})}{{C}_{L}}\frac{\mathrm{\partial }{C}_{L}}{\mathrm{\partial }t}\mathrm{-}\mathrm{\nabla }\mathrm{\cdot }({D}_{L}\mathrm{\nabla }{C}_{L})+\mathrm{\nabla }\mathrm{\cdot }(\frac{{D}_{L}{C}_{L}{V}_{H}}{RT}\mathrm{\nabla }{\sigma }_{H})+{\theta }_{T}\frac{d{N}_{T}}{d{\varepsilon }_{\mathit{eq}}^{p}}\frac{d{\varepsilon }_{\mathit{eq}}^{p}}{\mathit{dt}}\mathrm{=}0` EQ1 It is therefore observed that the diffusion equation takes into account the local gradient of the stress trace (hydrostatic stress :math:`{\sigma }_{H}\mathrm{=}1\mathrm{/}3\mathit{tr}(\sigma )`) and the equivalent Von Mises plastic deformation. The relationships defining the different quantities are: :math:`{\theta }_{L}=\frac{{C}_{L}}{{N}_{L}}` is the occupancy rate of sites in the crystal lattice, with :math:`{N}_{T}` the number of sites per unit volume. :math:`{N}_{L}` is a constant estimated at :math:`{N}_{L}\mathrm{=}\mathrm{5,1}{10}^{29}{m}^{\text{-3}}` for iron :math:`\alpha` in [:ref:`1 <1>`]. :math:`{\theta }_{T}=\frac{{C}_{T}}{{N}_{T}}` is the occupancy rate of the trap sites, with :math:`{N}_{T}` the density of the traps, i.e. the number of sites corresponding to traps per unit volume. Unlike :math:`{N}_{L}` which is a constant, :math:`{N}_{T}` is a function of plastic deformation according to the expression: :math:`{\mathrm{log}}_{10}({N}_{T})={a}_{1}-{a}_{2}\mathrm{exp}(-{a}_{3}{\varepsilon }_{\mathrm{eq}}^{p})`, with: :math:`{a}_{1}\mathrm{=}23.26,{a}_{2}\mathrm{=}\mathrm{2,33},{a}_{3}\mathrm{=}\mathrm{-}\mathrm{5,5}` [:ref:`1 <1>`]. :math:`{D}_{L}=\mathrm{1,27}{10}^{\text{-8}}{m}^{2}/s` :math:`{V}_{H}=2{10}^{\text{-6}}{m}^{3}` for iron :math:`\alpha` at room temperature, :math:`R=\mathrm{8,3144}J/\mathrm{mol}/K` is the ideal gas constant, and :math:`T` the temperature in :math:`°K`. It remains to define :math:`{C}_{T}` according to :math:`{C}_{L}`: according to [:ref:`1 <1>`] at equilibrium, which is the case for CSC: :math:`{C}_{T}=\frac{{N}_{T}}{1+\frac{1}{{K}_{T}{\theta }_{L}}}`, with :math:`{K}_{T}\mathrm{=}\mathrm{exp}(\mathrm{-}\Delta {E}_{T}\mathrm{/}RT)\mathrm{=}\mathrm{4,97}{10}^{10}` at room temperature, following [:ref:`1 <1>`], taking :math:`\Delta {E}_{T}=-60\mathrm{KJ}/\mathrm{mol}`. In a manner similar to non-linear thermal, the variational formulation is then written: Let :math:`\Omega` be an open of :math:`{R}^{3}`, of border :math:`\Gamma ={\Gamma }_{1}\cup {\Gamma }_{2}`. We need to solve the equation [:ref:`eq. 1 `] in :math:`{C}_{L}` over :math:`\Omega \mathrm{\times }\mathrm{]}\mathrm{0,}t\mathrm{[}` with the boundary conditions: .. _RefEquation 3: :math:`\{\begin{array}{ccc}{C}_{L}={C}_{L}^{d}& \text{sur}& {\Gamma }_{1}\\ J\mathrm{.}n=\phi & \text{sur}& {\Gamma }_{2}\end{array}` eq 3 with :math:`J=-{D}_{L}\nabla {C}_{L}+\frac{{D}_{L}{V}_{H}}{RT}{C}_{L}\nabla {\sigma }_{H}` and with initial conditions :math:`{C}_{L}(t=0)` (and :math:`{C}_{T}(t=0)`). Let :math:`v` be a sufficiently regular function cancelling itself out of :math:`{\Gamma }_{1}`, the variational formulation of the problem is written as: .. _RefEquation 4: :math:`\begin{array}{c}\underset{\Omega }{\mathrm{\int }}{D}^{\text{*}}({C}_{L})\frac{{\mathit{dC}}_{L}}{\text{dt}}vd\Omega +\underset{\Omega }{\mathrm{\int }}{D}_{L}\mathrm{\nabla }{C}_{L}\mathrm{\cdot }\mathrm{\nabla }vd\Omega \mathrm{-}\underset{\Omega }{\mathrm{\int }}\mathrm{\nabla }v\mathrm{\cdot }(\frac{{D}_{L}{C}_{L}{V}_{H}}{RT}\mathrm{\nabla }{\sigma }_{H})d\Omega \mathrm{=}\\ \underset{\Omega }{\mathrm{\int }}{r}_{\text{vol}}vd\Omega +\underset{{\Gamma }_{2}}{\mathrm{\int }}\phi vd{\Gamma }_{2}\end{array}` eq 4 with: :math:`{D}^{\text{*}}({C}_{L})=\frac{{C}_{L}+{C}_{T}(1-{\theta }_{T})}{{C}_{L}}` :math:`\phi \mathrm{=}({D}_{L}\mathrm{\nabla }{C}_{L}\mathrm{-}\frac{{D}_{L}{V}_{H}}{RT}{C}_{L}\mathrm{\nabla }{\sigma }_{H})\mathrm{.}n` and :math:`{r}_{\mathrm{vol}}=-{\theta }_{T}\frac{d{N}_{T}}{d{\varepsilon }_{\mathrm{eq}}^{p}}\frac{d{\varepsilon }_{\mathrm{eq}}^{p}}{\mathrm{dt}}=0` The numerical resolution by finite elements is therefore similar to that of non-linear thermics, and is based on a :math:`\theta` -method, modulo two particularities: * the source term :math:`{r}_{\mathrm{vol}}` is non-linear, as in drying modeling [:ref:`R7.01.12 `], and will be explicitly integrated; * the term :math:`\underset{\Omega }{\int }\nabla v\cdot (\frac{{D}_{L}{C}_{L}{V}_{H}}{RT}\nabla {\sigma }_{H})d\Omega` is not symmetric, and will also be carried over to the second member, as in [:ref:`1 <1>`] and integrated explicitly. This explicit discretization of these two terms is not binding: in fact, the resolution of equation [:ref:`eq. 4 `] is carried out at each time step, chained with mechanical resolution. On the other hand, the term :math:`\underset{\Omega }{\int }\nabla v\cdot (\frac{{D}_{L}{C}_{L}{V}_{H}}{RT}\nabla {\sigma }_{H})d\Omega` requires applying a temperature gradient :math:`\nabla {T}_{\mathrm{ini}}` that is assumed to be uniform in the element. The second elementary member calculated is: :math:`{\int }_{\Omega }\nabla {T}_{\mathrm{ini}}K\nabla vd\Omega` where :math:`K` is the thermal conductivity tensor. Knowing the initial conditions :math:`{C}_{L}^{0}={C}_{L}(t=0)` and :math:`{C}_{T}^{0}={C}_{T}(t=0)`, :math:`{C}_{\mathrm{tot}}^{0}={C}_{L}^{0}+{C}_{T}^{0}` is calculated. and the unknown (concentration) is dimensioned; the variational formulation is then: :math:`\begin{array}{c}\underset{\Omega }{\mathrm{\int }}{D}^{\text{*}}\frac{{\mathit{dc}}_{L}}{\text{dt}}vd\Omega +\underset{\Omega }{\mathrm{\int }}{D}_{L}\mathrm{\nabla }{c}_{L}\mathrm{\cdot }\mathrm{\nabla }vd\Omega \mathrm{=}\underset{\Omega }{\mathrm{\int }}\mathrm{\nabla }v\mathrm{\cdot }(\frac{{D}_{L}{V}_{H}}{RT}{c}_{L}\mathrm{\nabla }{\sigma }_{H})d\Omega +\underset{\Omega }{\mathrm{\int }}{\stackrel{ˉ}{r}}_{\text{vol}}vd\Omega +\underset{{\Gamma }_{2}}{\mathrm{\int }}\stackrel{ˉ}{\phi }vd{\Gamma }_{2}\end{array}` with: :math:`{c}_{L}=\frac{{C}_{L}}{{C}_{\mathrm{tot}}^{0}}` :math:`{\stackrel{ˉ}{r}}_{\mathit{vol}}\mathrm{=}\frac{\mathrm{-}{\theta }_{T}}{{C}_{\mathit{tot}}^{0}}\frac{d{N}_{T}}{d{\varepsilon }_{\mathit{eq}}^{p}}\frac{d{\varepsilon }_{\mathit{eq}}^{p}}{\mathit{dt}}` and :math:`\stackrel{ˉ}{\phi }\mathrm{=}(\mathrm{-}{D}_{L}\mathrm{\nabla }{c}_{L}+\frac{{D}_{L}{V}_{H}}{RT}{c}_{L}\mathrm{\nabla }{\sigma }_{H})\mathrm{.}n` The heat capacity coefficient :math:`{D}^{\text{*}}` is a field represented by a control variable (NEUT1). In an isolated volume element (:math:`\nabla {C}_{L}=0` on the border) and loaded so that the state of stresses and strains is uniform, the total concentration :math:`{C}_{\mathrm{tot}}={C}_{l}+{C}_{t}` is constant. The analytical solution described in [:ref:`1 <1>`] is obtained directly in the form of the quadratic equation: :math:`{C}_{T}^{2}-B{C}_{T}+{N}_{T}{C}_{\mathrm{tot}}=0`, with :math:`B={N}_{L}/{K}_{T}+{C}_{\mathrm{tot}}+{N}_{T}`: one of the two solutions would lead to negative values of :math:`{C}_{L}`, so the solution is. :math:`{C}_{T}=1/2(B-\sqrt{{B}^{2}-4{N}_{T}{C}_{\mathrm{tot}}})` The representation of :math:`{C}_{T}({\varepsilon }_{\mathrm{eq}}^{p})` and :math:`{C}_{L}({\varepsilon }_{\mathrm{eq}}^{p})` obtained analytically and numerically in [:ref:`1 <1>`] is: Benchmark results ---------------------- .. image:: images/10000000000014F700000BC0CE3B0073CD2004C9.png :width: 6.889in :height: 3.8602in .. _RefImage_10000000000014F700000BC0CE3B0073CD2004C9.png: Uncertainty about the solution --------------------------- Uncertainty less than :math:`\text{1 \%}`. Bibliographical references --------------------------- 1. "Hydrogen transport near a blunting crack tip" A.H.M. Krom, R.W.J.Koers, A.Bakker in "Journal of the Mechanics and Physics of Solids" 45 (1999) 971-992