A new methodology for assessing the global dynamic response of large shell structures under impact loading

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A new methodology for assessing the global dynamic

response of large shell structures under impact loading

Christophe Rouzaud, Fabrice Gatuingt, Olivier Dorival, Hervé Guillaume,

Louis Kovalevsky

To cite this version:

Christophe Rouzaud, Fabrice Gatuingt, Olivier Dorival, Hervé Guillaume, Louis Kovalevsky. A new

methodology for assessing the global dynamic response of large shell structures under impact loading.

Engineering Computations, Emerald, 2015, 32 (8), pp.2343-2382. �10.1108/EC-06-2014-0124�.

�hal-01240476�

1,2,3 1 4,5 2 6 1 2 3 4 5 6 m m m

m1 m2 k1 k2

m1x¨1(t) + k1[x1(t)− x2(t)] = 0

m2x¨2(t)− k1[x1(t)− x2(t)] + k2x2(t) = 0

x1(t)� x2(t)

m1x¨1(t) + k1x1(t) = 0 m2x¨2(t) + k2x2(t) = F (t) F (t) F (t) = Rcr(xcr) + mc(xcr) �_dx cr dt �2 mc xcr dxcr dt Rcr 0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00 0 0,2 0,4 0,6 0,8 1 N or med mas s / u nit leng th

Normed distance from front of plane

Rcr mc mc

Gd = ¨x =−�_L Rcr(xcr)n (xcr)nmc(xcr)dxcr

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 Impac t f or ce / Ma x. impa ct f or ce

f ω Ω Ωi i Γ x y z eα eβ e3 r(α, β) B R ∂Ωi n t kr kt ηr ηt U u w σ K N M h ρ η K CP E D ν � Sad Uh σh Un Cn X P ϕ θ C R Etot Ed T n l λ c c c

n

f = _2πω hi ρi U_i U_i(x, y, z) = u_i(x, y) + wi(x, y) e_3i+ zθi θi(x, y) =−grad wi(x, y)− B_iui(x, y) u_i wi B_i αi βi ri(αi, βi) Ωi ri(αi, βi) ∂r_i ∂αi = Aieαi ∂r_i ∂βi = Bieβi e_αi eβ i e3i e3i = eαi∧ eβi αi βi αi�→ ri(α_�i, β0i) βi�→ ri(α0i, βi) e_αi, eβ_i, e_3i � B_i=   − 1 Rαi 0 0 0 ₋ 1 Rβi 0 0 0 0   � e_{α i},eβ i,e3 i � Rαi Rβi Ω n Ωi Γ Ωi ∂wΩi ∂uΩi ∂w,nΩi ∂KΩi ∂NΩi ∂MΩi Ωi Ωj

(ui, wi, Ki, Ni,M_i)

ui−ktbi(1+iη1 tbi)Ni= uid ∂uΩi

wi−_ktbi(1+iη1 _tbi)Ki= wid ∂wΩi

wi,n+krbi(1+iη1 rbi)n �

i .M_i.ni= wi,nd ∂w,nΩi

u�i .ni=− αiju�_j.nj+ (1 + αij) wj−_ktij(1+iη1 _tij)(−N�i .ni− αijN�j.nj+ (1 + αij) Kj) Γij

wi= −αijwj+ (1 + αij) u�j.nj−ktij(1+iη1 tij)(−Ki− αijKj+ (1 + αij) N �

j.nj) Γij

u�i .ti=−βijuj�.tj−ktij(1+iη1 tij)

�

−N�i .ti− βijN�j.tj

�

Γij

wi,n = βijwj,n−_krij(1+iη1 _rij)

�

n�i M_ini− βijn�j.M_j.nj

�

Γij

αij= n�i .nj βij = t�i .tj

krbi ktbi ηrbi ηtbi

Ωi

krij ktij ηrij ηtij

Γij Ωi Ωj Ωi Ni− B_i � div M_i�=−ρiω2hiui Ωi div�div _M i � + T r�N i.Bi � =_−ρiω2hiwi Ωi Ni= N_i.ni− B_i.M_i.ni = Nid ∂NΩi Ki= n�i .div M_i+ � t�i .M_i.ni � ,t= Kid ∂KΩi n�i .M_i.ni=Mid ∂MΩi �� t�i .M_i.ni �� ∂Ωi = 0 Γij N�i .ni= αijN_j�.nj− (1 + αij) Kj Γij Ki = αijKj− (1 + αij) N�j.nj Γij N�i .ti= βijN�j.tj Γij n�i .M_i.ni=−βijn�j.M_j.nj Γij �n i=1(n�i .M_i.ni)ti= 0 �n i=1(N�i .ni− Ki) = 0

M_i=hi3 12K CPi :X (wi) Ωi Ni = hiK CPi : γ (ui) Ωi K CPi = (1 + iηi)K0i ρi ηi X γ K CPi = (1 + iηi)     Eαi 1−ναiνβi ναiEαi 1−ναiνβi 0 νβiEβi 1−ναiνβi Eβi 1−ναiνβi 0 0 0 √_E αiEβi 2(1+√ναiνβi)     � e_{α i},eβ i,e3 i � X (ui) = � (θi)− � B_i. �(ui+ wie_3i) � γ (ui) = �(ui+ wie_3i)

� Eα,βi να,βi e_αi

eβ_i hi st (Ui, σ_i) = (ui, wi, Ki, Ni,M_i)∈ Sad,i A   � � � � � � s1 . . . sn , � � � � � � δs1 . . . δsn   = L   � � � � � � δs1 . . . δsn   A   � � � � � � s1 . . . sn , � � � � � � δs1 . . . δsn   =Re � −iω �_�n i=1 � ∂uΩi δσ_i.ni.U∗idS + n � i=1 � ∂FΩi σ_i.ni.δU∗idS + � Γij    n−1 n �n i=1 � δσ_i.ni � .(Ui)∗+n1 � i�=j � δσ_i.ni � .�Uj �∗ +_n1�n_i=1�σ_i.ni � .(δUi)∗−n1 � i�=j � σ_i.ni � .�δUj �∗   dS      L   � � � � � � δs1 . . . δsn   = Re � −iω � _n � i=1 � ∂uΩi δσ_i.ni.U∗iddS + n � i=1 � ∂FΩi Fid.δU∗idS �� ∂uΩi Ωi ∂FΩi Ωi Γij Γij Re S0 ad,i Ωi f_d,i = 0 i = 1, . . . , n Sad,i0 ≡ Sad,i i = 1, . . . , n

Sad,i KCP i ηi > ∀ i = 1, . . . , n ∀ j �= i        kriηri≥ 0 ktiηti≥ 0 krijηrij ≥ 0 ktijηtij≥ 0 (U_ih, σh i) Ui(Xi, Pi) = � Pi�CiUni(Pi) .e P_i.X_{i Ω} i σ_i(Xi, Pi) = � P_i�CiCn_i(Pi) .e P_i.X_{i Ω} i X_i U_ni Cn i n Pi Ci � Ui, σ_i � Sad,i Pi

wi Ωi hi3 12div � div � K CPi :X (wi) �� + hiT r �� K CPi : γ (ui) � .B_i � =−ρiω2hiwi Ωi

Pi (PT_i.P_i)4₌ 12(1− ναiνβi)ρiω2 (1 + iηi)Eα,βih2i (PT_i.P_i)2 −12(1− ν_h2αiνβi) i (PT_i.R.B i.R.Pi) 2 R = � 0 ₋₁ 1 0 � RT =_−R B_i (PTi.Pi)4= 12(1_{− ν}αiνβi)ρiω2 (1 + iηi)Eα,βih2i (PTi.Pi)2 u_1i w0i= wi wi Pi R.Pi u1i u_1i=APi+BR.Pi

A = w0i1−νEα/βiαiνβi PT i.K CPi .B i.Pi (PT i.Pi)2

B = w0i1+√ν2Eα/βiαiνβi PT i.K CPi .B i.R.Pi (PT i.Pi)2 u_0i= 0 wi γ (ui) = �(ui + wie_3i) Pi    wh interior,i � Xi, Pint,i � = Wh interior,i � Pint,i � .ePint,i.Xi Ω i wh edge,i � Xi, Pedg,i � = Wh edge,i � Pedg,i � .ePedg,i.Xi _Ω i whcorner,i � Xi, Pcor,i � = Wcorner,ih � Pcor,i � .ePcor,i.Xi _Ω i

Pint,i(ϕint,i) =−ip_int,i(ϕint,i)

� cosϕint,i sinϕint,i � � e_{α i},eβ i �=−i �

pα,int,i(ϕint,i).cosϕint,i

pβ,int,i(ϕint,i).sinϕint,i

� � e_{α i},eβ i � pα/β,int,i(ϕ)

pα/β,int,i4(ϕint,i) = pplate,α/β,i4−

�

(pshell,α,i.sinϕint,i)2+ (pshell,β,i.cosϕint,i)2

�2 pplate,α/β,i= �_12ρ iω2(1−ναiνβi) (1+iηi)Eα/βih2i �1 4 pshell,α/β,i= � 12ρiω2(1−ναiνβi) R2 α/βih2i �1 4

p_int,i(ϕint,i) Cint,i

Cint,i

p_int,i ϕint,i= 0 45 90

p_int,i(ϕint,i)

Pedg,i(ϕedg,i) = Pt,edg,i(ϕedg,i)t + Pn,edg,i(ϕedg,i)n

Re(Pn,edg,i)� Im(Pn,edg,i)

t n

Pn,edg,i(ϕedg,i) = (1 + iηi) 1 4��1 + sin2(ϕ_edg,i)− iηi 4. 1 √ 1+sin2_(ϕ edg,i) � � pplate,α,i pplate,β,i � � e_{α i},eβ i �

Pt,edg,i(ϕedg,i) = i(1 + iηi) 1 4sinϕ_edg,i � pplate,α,i pplate,β,i � � e_{α i},eβ i �

Re(Pcor,i) � Im(Pcor,i)

Pcor,i(ϕcor,i) = p_cor,i

� cosϕcor,i sinϕcor,i � � e_{α i},eβ i �= �

pplate,α,i.cosϕcor,i

pplate,β,i.sinϕcor,i

� � e_{α i},eβ i � ui Ωi ui hiK CPi : γ (ui)−hi 3 12Bi.div � K CPi :X (wi) � =−ρiω2hiui Ωi (uh_{, N}h i) ui(Xi, Pi) = � Pi�Ciuni(Pi) .e Pi.Xi Ω i Ni(Xi, Pi) = � Pi�CiNni(Pi) .e P_i.X_{i Ω} i X_i P_i Ci u_ni N_ni n Xi Pi (ui, Ni) Sad,i          Ni− B_i � div _Mi�=_−ρiω2hiui Ωi Mi= hi3 12K CPi :_{X (w}i) Ωi Ni= hiK CPi : γ (ui) Ωi 0 u_0i= ui N_0i= Ni u_0pres,i Ppres,i upres,i � Xi, Ppres,i � = u_0pressure,i�Ppres,i � .ePpres,i.Xi Ω i u_0pressure,i�Ppres,i � = u0pres,i � cosθpres,i sinθpres,i � � e_{α i},eβ i �

Ppres,i(θpres,i) = ip_pres,i(θpres,i) = i(pα,pres,i.cos(θpres,i)e_αi+ pβ,pres,i.sin(θpres,i)eβ i)

p2α/β,pres,i= ρiω2(1− ναiνβi) Eα/βi(1 + iηi) p_pres,i(θpres,i) Cpres,i Cpres,i Cpres,i u_0shea,i Pshea,i e3i ushea,i � Xi, Pshea,i � = u_0shear,i�Pshea,i � .ePshea,i.Xi _Ω i u_0shear,i�Pshea,i � = u0shea,i � −sinθshea,i cosθshea,i � � e_{α i},eβ i �

Pshea,i(θshea,i) = ip_shea,i(θshea,i) = i(pα,shea,i.cos(θshea,i)e_αi+ pβ,shea,i.sin(θshea,i)eβ_i)

p2_α/β,shea,i= 2ρiω

2_{(1 + √ν} αiνβi)

Eα/βi(1 + iηi)

Cshea,i Etot = Ed+ T Etot Ed T Uhi (Pi) ϕint/edg/cor,i θpres/shea,i Cint/edg/cor/pres/shea,i Cint/edg/cor/pres/shea,i whi (xi) = � ϕint,i�Cint,i

Winterior,ih (ϕint,i)ePint,i(ϕint,i).xidϕint,i

+ �

ϕedg,i�Cedg,i

Wedge,ih (ϕedg,i)ePedg,i(ϕedg,i).xidϕedg,i

+ �

ϕcor,i�Ccor,i

Wh

corner,i(ϕcor,i)ePcor,i(ϕcor,i).xidϕcor,i

                            

Pint,i(ϕint,i) =−i

�

pα,int,i(ϕint,i).cosϕint,i

pβ,int,i(ϕint,i).sinϕint,i

� � e_{α i},eβ i � Pedg,i(ϕedg,i) =      (1 + iηi) 1 4��1 + sin2(ϕ_edg,i)− iηi 4. 1 √ 1+sin2_(ϕ edg,i) � � pplate,α,i pplate,β,i � � e_{α i},eβ i �.t +i(1 + iηi) 1 4sinϕ_edg,i � pplate,α,i pplate,β,i � � e_{α i},eβ i �.n      Pcor,i(ϕcor,i) = �

pplate,α,i.cosϕcor,i

pplate,β,i.sinϕcor,i

�

� e_{α i},eβ

i �

uh(xi) =

�

θpres,i�Cpres,i

u0h_pressure,i(θpres,i) .ePpres,i(θpres,i).xidθpres,i

+ �

θshea,i�Cshea,i

u0h_shear,i(θshea,i) .ePshea,i(θshea,i).xidθshea,i

                  

u0h_pressure,i(θpres,i) = u0pres,i

� cosθpres,i sinθpres,i � � e_{α i},eβ i �

Ppres,i(θpres,i) = i(pα,pres,i.cos(θpres,i)e_αi+ pβ,pres,i.sin(θpres,i)eβ i)

u0h_shear,i(θshea,i) = u0shea,i

� −sinθshea,i cosθshea,i � � e_{α i},eβ i �

Pshea,i(θshea,i) = i(pα,shea,i.cos(θshea,i)e_αi+ pβ,shea,i.sin(θshea,i)eβ i) Sad,i p_i Sad,i Sad,ih Uh(Pi(ϕi)) ni lα/βi Ωi ni= 2lα/βi λα/βi = ωlα/βi πc α/βi = lα/βi √_ω π 4 � ρihi Dα/βi

λα/βi e_αi eβ_i ω c α/βi = √ω4 �_D α/βi ρihi ρi hi Dα/βi Dα/βi = Eα/βih3i 12(1−ναiνβi) ni c α/βi = � _E α/βi ρi(1−ναiνβi) c α/βi = � _√ EαiEβi 2ρi(1+√ναiνβi) ϕmn wanalytical(x, y) = ∞ � m=1 ∞ � n=1 amnϕmn(x, y)

amn=

F sin₍mπxF_{Lx )}sin�nπyF_Ly � LxLy 4 ρh(wmn2 −w2) ϕmn= sin � mπx Lx � sin�nπy_L y � whanalytical(x, y) = M � m=1 N � n=1 amnϕmn(x, y) M N M � Lx π 4 � 12ω2_ρ(1−ν2₎ Eh2 N � Ly π 4 � 12ω2_ρ(1−ν2₎ Eh2 k3_h2 _k h

wF,inf inite(x, y) = −iF

8 Eh3 12(1−ν2₎ � 12ω2_ρ(1−ν2₎ Eh2 � J0( 4 � 12ω2_{ρ(1− ν}2₎ Eh2 r) − iY0( 4 � 12ω2_{ρ(1− ν}2₎ Eh2 r) −2i_πK0( 4 � 12ω2_{ρ(1− ν}2₎ Eh2 r) � r xF J0 Y0 K0 ≈

                          ͳͲ݉   ʹͲ݉   ͳͲ݉   ͵݉   ͵݉   Load ͳͲͲܯܰ  

≈ ͲǤͲ   ͲǤ͸Ͷ   ͳǤʹͺ   ͳǤͻʹ   ʹǤͷ͸   ͵ǤʹͲ   ͵ǤͺͶ   ͶǤͶͺ   ൈ ͳͲିଵ   ͲǤͲ   ͲǤʹͳ   ͲǤͶ͵   ͲǤ͸Ͷ   ͲǤͺ͸   ͳǤͲ͹   ͳǤʹ͸   ͳǤͷͲ   ൈ ͳͲିଵ  

ͲǤͲ   ͲǤ͸Ͷ   ͳǤʹͺ   ͳǤͻʹ   ʹǤͷ͸   ͵ǤʹͲ   ͵ǤͺͶ   ͶǤͶͺ   ൈ ͳͲିଵ   ͲǤͲ   ͲǤʹͳ   ͲǤͶ͵   ͲǤ͸Ͷ   ͲǤͺ͸   ͳǤͲ͹   ͳǤʹ͸   ͳǤͷͲ   ൈ ͳͲିଵ   P 2

P 1

wP 1(t) =100 sin (2π10t + 10)− 200 sin (2π20t + 20)

+ 300 sin (2π30t + 30)− 400 sin(2π40t + 40)

wP 1

rd

wP 2 P 2 P 2 -­‐200 -­‐150 -­‐100 -­‐50 0 50 100 150 0 50 100 150 200 250 300 Disp la cem ent w P2 (mm) Time (ms)

Time loading function w

P2

(t)

Frequency VTCR approach (6000 DOFs)

Temporal FEM approach (25 elements per wavelength)

A   � � � � � � s1 . . . sn , � � � � � � δs1 . . . δsn   = Re                                                                                      −iω                                            �n i=1 � ∂_w,nΩi � δniM_ini � .�wi,n+_kri(1+iη1 _ri)niM_ini �∗ dl −�ni=1 � ∂wΩiδKi. � wi−_kti(1+iη1 _ti)Ki �∗ dl₋�n_i=1�_∂uΩiδNi. � ui−kti(1+iη1 ti)Ni �∗ dl +�n_i=1�_∂ MΩi � niM_ini � .δw∗i,ndl− �n i=1 � ∂KΩiKi.δw ∗ idl− �n i=1 � ∂NΩiNi.δu ∗ idl −�ni=1 � ∂Ωi �� (tiM_ini).δw∗i �� +n−1_n �n_i=1�_Γij�δniM_ini �

.�wi,n+krij(1+iη1 rij)

� niM_ini ��∗ dl −1 n � i�=j � Γij � δniM_ini �

.�βijwj,n−_krij(1+iη1 _rij)

� −βijnjM_jnj ��∗ dl +_n1�n_i=1�_Γij�δwi,n �∗ .�niM_ini � dl + 1_n�_i�=j�_Γij�δwj,n �∗ .�βijniM_ini � dl −n−1n �n i=1 � Γij(δKi) . �

wi+_ktij(1+iη1 _tij)(−Ki)

�∗ dl −1 n � i�=j � Γij(δKi) . �

αijwj− (1 + αij) ujnj+ktij(1+iη1 tij)(−αijKj+ (1 + αij) Njnj)

�∗ dl −1 n �n i=1 � Γij(δwi) ∗_{. (K} i) dl−1_n�i�=j � Γij(δwi) ∗_.� −αijKj+ (1 + αij)Njnj � dl −n−1n �n i=1 � Γij(δNini) . �

uini+ktij(1+iη1 tij)(−Nini)

�∗ dl −1 n � i�=j � Γij(δNini) . �

αijujnj− (1 + αij) wj+_ktij(1+iη1 _tij)(−αijNjnj+ (1 + αij) Kj)

�∗ dl −1 n �n i=1 � Γij(δuini) ∗_{. (N} ini) dl−1n � i�=j � Γij(δuini) ∗_.� −αijNjnj+ (1 + αij)Kj�dl −n−1n �n i=1 � Γij(δNiti) . �

uiti+ktij(1+iη1 tij)(−Niti)

�∗ dl −1 n � i�=j � Γij(δNiti) . �

βijujtj+ktij(1+iη1 tij)

� −βijNjtj ��∗ dl −1 n �n i=1 � Γij(δuiti) ∗_{. (N} iti) dl−1n � i�=j � Γij(δuiti) ∗_.� −βijNjtj � dl                                                                                                                                 L   � � � � � � δs1 . . . δsn   = Re            −iω       �n i=1 � ∂_w,nΩi � δniM_ini � .w∗ i,nddl− �n i=1 � ∂wΩiδKi.w ∗ iddl −�ni=1 � ∂uΩiδNi.u ∗ iddl +�n_i=1�_∂ MΩiMid.δw ∗ i,ndl−� n i=1 � ∂KΩiKid.δw ∗ idl −�ni=1 � ∂NΩiNid.δu ∗ idl                  Sad,i