Derivation of shape derivative for the nanoelasticity problem

APPENDIX A

Derivation of shape derivative for the nanoelasticity problem Firstly, the total potential energy objective function is considered. The objective function and its constraints are as follows, Minimize



J () =



u.b d  + 

u.t d  N

subject to: a(u, δ u, ) + as (u, δ u, ) = −ls (u, ) + l(u, ) (i.e.) 

 (δ u) : Cbulk : (u) d +

(P(δ u)P) : τ s d+     s (P(δ u)P) : C : (P(u)P) d = u.b d + u.t d. 



N

a˙ (u, w, ) + a˙ s (u, w, ) + ˙ls (w, ) 

=

 

(u ) : Cbulk : (w)d + (u) : Cbulk : (w )d +



(u) : Cbulk : (w)Vn d +   P(w )P : τ s d + (∇s (P(w)P : τ s ) · n 





+ κ(P(w)P : τ s ))Vn d +  (P(u )P) : Cs : (P(w)P)d+  (P(u)P) : Cs : (P(w )P)d+   (∇s (P(u)P) : Cs : P(w)P) · n Vn d+

(A.1)



Extended Finite Element and Meshfree Methods Copyright © 2020 Elsevier Inc. https://doi.org/10.1016/B978-0-12-814106-9.00019-X All rights reserved.

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Extended Finite Element and Meshfree Methods

 

(P(u)P) : Cs : ∇s (P(w)P)) · n Vn d+



κ(P(u)P : Cs : P(w)P) Vn d, 

˙l(w, ) =









w .td

N

(A.2)

(∇(w.t).n + κ w.t) Vn d



N

u .b d  + 

w.b Vn d +



+

J˙ =



w .bd + 







u . b Vn d  + 

(∇(u.t).n + κ u.t) Vn d.

(A.3)

N

The Lagrangian of the objective functional is, L = J + l(w, ) − a(u, w, ) − as (u, w, ) − ls (w, ).

(A.4)

The material derivative of the Lagrangian is defined as, L˙ = J˙ + ˙l(w, ) − a˙ (u, w, ) − a˙ s (u, w, ) − ˙ls (w, ).

(A.5)

All the terms that contain w in the material derivative of Lagrangian are collected and the sum of these terms is set to zero, to get the weak form of the state equation,  

w .b d +



w .t d =

N

 

+

(u) : Cbulk : (w ) d





P(w )P : τ s d



+ 

(P(u)P : Cs : P(w )P)

(A.6)