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I am currently studying the phase-field method for fracture modeling. In an article by Miehe -"Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementation", I came across on an identity that I don't understand basically it says: $$ d = \mbox{Arg}\left\{\inf_{d \in W} I(d)\right\}. $$ This expression came from the Euler type differential equation $$ d(x) - l^2d''(x) = 0, $$ which has the solution as an exponential function $$ d(x)\ = \ \exp(-|x|/l) $$ Variational principle of this differential equation is , $$ d = \mbox{Arg}\left\{\inf_{d \in W} I(d)\right\}. $$ with $$ W = \{d\ |\ d(0)=1,\ d(\pm \infty)=0\}$$ where I(d) is a functional defined as an integral $$ I(d) = \frac{1}{2}\int [d^2 + l^2d'^2] dV, $$ and $l$ is a parameter.

Can anyone tell me what does that mean?

postmortes
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Cro Simpson2.0
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1 Answers1

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It returns the argument that makes the functional $I\equiv I(d)$ infimum such that the function $d$ belongs to the space of kinematically acceptable functions $W$, i.e., it satisfies the prescribed boundary conditions. It just explicitly emphasises that we are after the function (or argument) that makes the functional infimum and not the infimum value itself.

Zia Javanbakht
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