4. Digital integration#
The separation into two parts in the formulation: damage — sliding, allows us to treat each of them separately. Thus, the integration of the damage portion is achieved explicitly by defining the two threshold surfaces. On the other hand, the « sliding » part is solved implicitly by a conventional method, namely the « return-mapping » algorithm proposed by Ortiz & Simo, Cf [bib4], which will ensure convergence effectively.
4.1. Calculation of the « crack friction » part with an implicit integration method#
The effects on the bond associated with the phenomenon of crack friction can be calculated in the framework of pseudoplastic behavior with non-linear kinematic work hardening. For the implementation with the proposed integration method, we will perform a linearization of the threshold function around the current values of the associated internal variables. In iteration \((i+1)\), the threshold surface is written as:
\({\varphi }_{f}\mathrm{=}{\varphi }_{f}^{(i)}+\frac{\mathrm{\partial }{\varphi }_{f}^{(i)}}{\mathrm{\partial }{\sigma }_{T}^{f}}\mathrm{:}({\sigma }_{T}^{{f}^{(i+1)}}\mathrm{-}{\sigma }_{T}^{{f}^{(i)}})+\frac{\mathrm{\partial }{\varphi }_{f}^{(i)}}{\mathrm{\partial }X}\mathrm{:}({X}^{(i+1)}\mathrm{-}{X}^{(i)})\mathrm{\approx }0\) eq 4.1-1
According to the equations [éq 3.1-7], [éq 3.1-8], and [eq 3.6-5], we have:
\(\stackrel{\text{.}}{X}\mathrm{=}\gamma \mathrm{\cdot }\stackrel{\text{.}}{\alpha }\mathrm{=}\mathrm{-}\gamma \mathrm{\cdot }\stackrel{\text{.}}{{\lambda }_{f}}\mathrm{\cdot }\frac{\mathrm{\partial }{\varphi }_{f}^{p}}{\mathrm{\partial }X}\) eq 4.1-2
\(\stackrel{\text{.}}{{\sigma }_{T}^{f}}\mathrm{=}\mathrm{-}G\mathrm{\cdot }{D}_{T}\mathrm{\cdot }\stackrel{\text{.}}{{\varepsilon }_{T}^{f}}\mathrm{=}\mathrm{-}G\mathrm{\cdot }{D}_{T}\mathrm{\cdot }\stackrel{\text{.}}{{\lambda }_{f}}\mathrm{\cdot }\frac{\mathrm{\partial }{\varphi }_{f}^{p}}{\mathrm{\partial }{\sigma }_{T}^{f}}\) eq 4.1-3
That can be discretized in the following way:
\(\Delta X\mathrm{=}{X}^{(i+1)}\mathrm{-}{X}^{(i)}\mathrm{=}\gamma \mathrm{\cdot }\Delta \alpha \mathrm{=}\mathrm{-}\gamma \mathrm{\cdot }\Delta {\lambda }_{f}\mathrm{\cdot }\frac{\mathrm{\partial }{\varphi }_{f}^{p}}{\mathrm{\partial }X}\) eq 4.1-4
\(\Delta {\sigma }_{T}^{f}\mathrm{=}{\sigma }_{T}^{{f}^{(i+1)}}\mathrm{-}{\sigma }_{T}^{{f}^{(i)}}\mathrm{=}\mathrm{-}G\mathrm{\cdot }{D}_{T}\mathrm{\cdot }\Delta {\varepsilon }_{T}^{f}\mathrm{=}\mathrm{-}G\mathrm{\cdot }{D}_{T}\mathrm{\cdot }\Delta {\lambda }_{f}\mathrm{\cdot }\frac{\mathrm{\partial }{\varphi }_{f}^{p}}{\mathrm{\partial }{\sigma }_{T}^{f}}\) eq 4.1-5
By combining these expressions with the threshold surface expression and by writing that is equal to \({\varphi }_{f}\) to zero, we can deduce the \(\Delta {\lambda }_{f}\) multiplier increment at each iteration \(i\):
\(\Delta {\lambda }_{f}\mathrm{=}\frac{{\varphi }_{{f}^{(i)}}}{{\frac{\mathrm{\partial }{\varphi }_{f}}{\mathrm{\partial }{\sigma }_{T}^{f}}}^{(i)}\mathrm{\cdot }G\mathrm{\cdot }{D}_{T}\mathrm{\cdot }{\frac{\mathrm{\partial }{\varphi }_{f}^{p}}{\mathrm{\partial }{\sigma }_{T}^{f}}}^{(i)}+{\frac{\mathrm{\partial }{\varphi }_{f}}{\mathrm{\partial }X}}^{(i)}\mathrm{\cdot }\gamma \mathrm{\cdot }{\frac{\mathrm{\partial }{\varphi }_{f}^{p}}{\mathrm{\partial }X}}^{(i)}}\) eq 4.1-6
After obtaining the value of \({\mathrm{\Delta \lambda }}_{f}\), it can be substituted into equations [éq 4.1-4] and [éq 4.1-5] in order to update the thermodynamic forces \({\sigma }_{T}^{f}\) and \(\mathrm{\Khi }\). Iterations should continue until the consistency condition is met.
4.2. The resolution algorithm#
In general, the aim is to verify the balance of the structure at each moment, in an incremental form. As explained above, for damage, a simple scalar equation makes it possible to obtain the corresponding value, which makes it possible to avoid recourse to iterative methods. On the other hand, an iterative method is applied to integrate the friction part of the cracks. So, the algorithm is as follows:
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; \({\alpha }_{n+1}^{(0)}={\alpha }_{n}\) |
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if OUI, end of cycle; if NON, start of iterations |
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OUI: |
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NON: |
\(\Delta {\lambda }_{f}=\frac{{({\varphi }_{f})}_{n+1}^{(i)}}{{(\partial {\varphi }_{f}/\partial {\sigma }_{T}^{f})}_{n+1}^{(i)}G\text{.}{D}_{T}{(\partial {\varphi }_{f}^{p}/\partial {\sigma }_{T}^{f})}_{n+1}^{(i)}+{(\partial {\varphi }_{f}/\partial X)}_{n+1}^{(i)}\text{.}\gamma \text{.}{(\partial {\varphi }_{f}^{p}/\partial X)}_{n+1}^{(i)}}\) |
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\({\sigma }_{n+1}^{(i+1)}={\sigma }_{n+1}^{(i)}-G\text{.}{D}_{T}\text{.}\Delta {\lambda }_{f}\text{.}{(\partial {\varphi }_{f}^{p}/\partial {\sigma }_{T}^{f})}^{(i)}-\gamma \text{.}\Delta {\lambda }_{f}\text{.}{(\partial {\varphi }_{f}^{p}/\partial X)}^{(i)}\) |
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\({\alpha }_{n+1}^{(i+1)}={\alpha }_{n+1}^{(i)}+\Delta {\lambda }_{f}\text{.}{(\partial {\phi }_{f}^{p}/\partial X)}^{(i)}\) |
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\({({\phi }_{f})}_{n+1}^{(i+1)}\le \mathrm{TOL}∣{({\phi }_{f})}_{n+1}^{(0)}∣?\) |
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if OUI, end of the cycle; if NON, continue the iterations in (iv) |
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OUI: |
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NON: |
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4.3. Internal variables of the model#
Here we show the internal variables stored at each Gauss point in the model implementation: