Table of Contents
Ogden material model, BEAM56H3
Problem
Mesh
| element | type | nodes |
|---|---|---|
| 1 | 56 | 1 2 3 4 5 6 7 8 21 22 23 24 25 26 27 28 29 30 31 32 |
| 2 | 56 | 5 6 7 8 9 10 11 12 29 30 31 32 33 34 35 36 37 38 39 40 |
| 3 | 56 | 9 10 11 12 13 14 15 16 37 38 39 40 41 42 43 44 45 46 47 48 |
| 4 | 56 | 13 45 15 16 17 18 19 20 45 46 47 48 49 50 51 52 53 54 55 56 |
Boundary conditions
$u=0$ at all nodes in the $yz$ plane
$v=0$ at all nodes in the $zx$ plane
$w=0$ at all nodes in the $xy$ plane
Computation
The Ogden material model for compressible hyperelastic materials is defined by the strain energy function given $$\psi=\psi_v+\psi_d,\quad \psi_v=\frac{1}{2}\kappa(J-1)^2,\quad \psi_d=\sum^\infty_{r=1}\frac{\mu_r}{\alpha_r}\left(\bar\lambda_1^{\alpha_r}+\bar\lambda_2^{\alpha_r}+\bar\lambda_3^{\alpha_r}-3\right),$$ where $\alpha_r,$ $\mu_r$ are pairs of material parameters of the Ogden material model and $\kappa$ is bulk modulus. $J=\det\mathbf{F}$ is the determinant of the deformation gradient $\mathbf{F}.$ $\bar\lambda_A=J^{-2/3}\lambda_A$ are modified principal stretches.
Even if the material model contains only parameters $\alpha_r,$ $\mu_r$ and $\kappa,$ it is convenient to define the shear modulus as $2\mu=\sum^\infty_{r=1}\mu_r\alpha_r$ and then establish the tangent modulus of elasticity in the non-deformed state and the Poisson ratio as $$E=\frac{9\kappa\mu}{3\kappa+\mu},\quad \nu=\frac{3\kappa-2\mu}{2(3\kappa+\mu)}.$$ In our example we have $\mu=422{,}500\text{ Pa},$ $E=1{,}265{,}717\text{ Pa}$ and $\nu=0.49789.$ The principal stresses of the Cauchy stress tensor corresponding to the strain energy function given above are $$\sigma_A=\sigma_{dA}+\sigma_{vA},$$ where $$\sigma_{dA}=J^{-4/3}\lambda_A\frac{\partial\psi_d}{\partial\bar\lambda_A}-\frac{1}{3}J^{-4/3}\sum^3_{B=1}\frac{\partial\psi_d}{\partial\bar\lambda_B}\lambda_B$$ with $$\frac{\partial\psi_d}{\partial\bar\lambda_A}=\sum^\infty_{r=1}\mu_r\bar\lambda_A^{\alpha_r-1}.$$ The principal stretches of the volumetric part of the Cauchy stress tensor are $$\sigma_{vA}=\kappa(J−1).$$ It can be proved that for given load $l_x=0.4\text{ MPa}$ and given material parameters $\alpha_1=1.3,$ $\mu_1=0.63\text{ MPa},$ $\alpha_2=5,$ $\mu_2=0.0012\text{ MPa},$ $\alpha_3=−2,$ $\mu_3=−0.01\text{ MPa}$ and $\kappa=10^8\text{ Pa},$ the following principal stretches and displacements fit the above written equations $$\begin{array}{lll} \lambda_x=1.4984 & & u_x=0.4984\text{ m} \\ \lambda_y=0.8177 & & u_y=-0.0036455\text{ m} \\ \lambda_z=0.8177 & & u_z=-0.001823\text{ m} \end{array}$$ The Cauchy stress in the bar is $\sigma_x=0.5982\text{ MPa}.$
The computation is executed with the following commands:
- beam56h3.bat
rmd3 beam56h3.i1 rpd3 beam56h3.i2 srh3 beam56h3.i3 fefs beam56h3.i4 hpp3 beam56h3.iP hpls beam56h3.iL str3 beam56h3.i5
Input
- beam56h3.i1
; NELEM NNOD ITED IP 4 56 56 ; CRIT SCALE RP 1.01 1 XY N 1:20 X 4*0 4*0.25 4*0.5 4*0.75 4*1 Y 5*(0 0.02 0.02 0) Z 5*(0 0 0.01 0.01) EL E 1:4 N =A 1:8 =A =B 21:32 =B =4A =8B =8A =16B =12A =24B EN EN
- beam56h3.i2
; KREST IP 1 IV 1 T 7 V 0 ; E α ν α1 α2 α3 μ2 μ3 MP 1 T 1 I 1 V 1265717 0 0.49789 7*0 1.3 5.0 -2.0 1200 -10000 SV 1 T 9 V 4e5 0 0 0 0 AS 1 /M 1 /B 0 C 1 N 1:4 21:24 /B 0 C 3 N 1 2 5 6 9 10 13 14 17 18 21 29 37 45 53 25 26 33 34 41 42 49 50 /B 0 C 2 N 1 4 5 8 9 12 13 16 17 20 24 32 40 48 56 25 28 33 36 41 44 49 52 AS 2 /S 1 E 4 S6 EN EN
- beam56h3.i3
; KREST IP 1 EN EN
- beam56h3.i4
; KREST IP 1 EN EN
- beam56h3.iP
; KREST NLC NCYC KMOD KCRP KLARG ... IP 1 1 0 0 0 6 4*0 ; load case sequence 2 EN EN
- beam56h3.iL
; KMET KOUT NSUBI NITER NINT KTPR IP 0 1 50 10000 0 0 ; UTOL RTOL XTOL PENAL RP 0 1e-4 1e-4 0 EN EN
- beam56h3.i5
; KLC 0 KOUT ILC ... KPROB KGRAF IP 1 0 1 0 6*0 2 2 EN EN
Output
- beam56h3.o5
; KLC 0 KOUT ILC ... KPROB KGRAF IP 1 0 1 0 6*0 2 2 EN EN 1TH LOAD CASE DISPLACEMENTS NODE U V W ALPHA BETA GAMMA [m] [m] [m] [rad] [rad] [rad] 1 0.0000000E+00 0.0000000E+00 0.0000000E+00 2 0.0000000E+00 -0.3645074E-02 0.0000000E+00 3 0.0000000E+00 -0.3645074E-02 -0.1822537E-02 4 0.0000000E+00 0.0000000E+00 -0.1822537E-02 5 0.1246002E+00 0.0000000E+00 0.0000000E+00 6 0.1246002E+00 -0.3645073E-02 0.0000000E+00 7 0.1246002E+00 -0.3645073E-02 -0.1822536E-02 8 0.1246002E+00 0.0000000E+00 -0.1822536E-02 9 0.2492005E+00 0.0000000E+00 0.0000000E+00 10 0.2492005E+00 -0.3645073E-02 0.0000000E+00 11 0.2492005E+00 -0.3645073E-02 -0.1822537E-02 12 0.2492005E+00 0.0000000E+00 -0.1822537E-02 13 0.3738007E+00 0.0000000E+00 0.0000000E+00 14 0.3738007E+00 -0.3645074E-02 0.0000000E+00 15 0.3738007E+00 -0.3645074E-02 -0.1822537E-02 16 0.3738007E+00 0.0000000E+00 -0.1822537E-02 17 0.4984010E+00 0.0000000E+00 0.0000000E+00 18 0.4984010E+00 -0.3645073E-02 0.0000000E+00 19 0.4984010E+00 -0.3645073E-02 -0.1822537E-02 20 0.4984010E+00 0.0000000E+00 -0.1822537E-02 21 0.0000000E+00 -0.1822537E-02 0.0000000E+00 22 0.0000000E+00 -0.3645074E-02 -0.9112684E-03 23 0.0000000E+00 -0.1822537E-02 -0.1822537E-02 24 0.0000000E+00 0.0000000E+00 -0.9112684E-03 25 0.6230013E-01 0.0000000E+00 0.0000000E+00 26 0.6230013E-01 -0.3645073E-02 0.0000000E+00 27 0.6230013E-01 -0.3645073E-02 -0.1822537E-02 28 0.6230013E-01 0.0000000E+00 -0.1822537E-02 29 0.1246002E+00 -0.1822536E-02 0.0000000E+00 30 0.1246002E+00 -0.3645073E-02 -0.9112682E-03 31 0.1246002E+00 -0.1822536E-02 -0.1822536E-02 32 0.1246002E+00 0.0000000E+00 -0.9112682E-03 33 0.1869004E+00 0.0000000E+00 0.0000000E+00 34 0.1869004E+00 -0.3645073E-02 0.0000000E+00 35 0.1869004E+00 -0.3645073E-02 -0.1822536E-02 36 0.1869004E+00 0.0000000E+00 -0.1822536E-02 37 0.2492005E+00 -0.1822537E-02 0.0000000E+00 38 0.2492005E+00 -0.3645073E-02 -0.9112683E-03 39 0.2492005E+00 -0.1822537E-02 -0.1822537E-02 40 0.2492005E+00 0.0000000E+00 -0.9112683E-03 41 0.3115006E+00 0.0000000E+00 0.0000000E+00 42 0.3115006E+00 -0.3645073E-02 0.0000000E+00 43 0.3115006E+00 -0.3645073E-02 -0.1822537E-02 44 0.3115006E+00 0.0000000E+00 -0.1822537E-02 45 0.3738007E+00 -0.1822537E-02 0.0000000E+00 46 0.3738007E+00 -0.3645074E-02 -0.9112684E-03 47 0.3738007E+00 -0.1822537E-02 -0.1822537E-02 48 0.3738007E+00 0.0000000E+00 -0.9112684E-03 49 0.4361009E+00 0.0000000E+00 0.0000000E+00 50 0.4361009E+00 -0.3645073E-02 0.0000000E+00 51 0.4361009E+00 -0.3645073E-02 -0.1822537E-02 52 0.4361009E+00 0.0000000E+00 -0.1822537E-02 53 0.4984010E+00 -0.1822537E-02 0.0000000E+00 54 0.4984010E+00 -0.3645073E-02 -0.9112683E-03 DISPLACEMENTS NODE U V W ALPHA BETA GAMMA [m] [m] [m] [rad] [rad] [rad] 55 0.4984010E+00 -0.1822537E-02 -0.1822537E-02 56 0.4984010E+00 0.0000000E+00 -0.9112683E-03 STRESSES AT GAUSSPOINTS IE IGP XG YG ZG SIG-X SIG-Y SIG-Z SIG-XY SIG-YZ SIG-XZ SIG-HMH EPSP DMG [m] [m] [m] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [-] [%] 1 1 0.2818E-01 0.2254E-02 0.1127E-02 0.6 0.0 0.0 -0.0 0.0 0.0 0.6 0.000E+00 0.00 2 0.2818E-01 0.1000E-01 0.1127E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 3 0.2818E-01 0.1775E-01 0.1127E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 4 0.1250E+00 0.2254E-02 0.1127E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 5 0.1250E+00 0.1000E-01 0.1127E-02 0.6 0.0 0.0 0.0 -0.0 -0.0 0.6 0.000E+00 0.00 6 0.1250E+00 0.1775E-01 0.1127E-02 0.6 0.0 0.0 0.0 -0.0 -0.0 0.6 0.000E+00 0.00 7 0.2218E+00 0.2254E-02 0.1127E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 8 0.2218E+00 0.1000E-01 0.1127E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 9 0.2218E+00 0.1775E-01 0.1127E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 10 0.2818E-01 0.2254E-02 0.5000E-02 0.6 0.0 0.0 -0.0 0.0 0.0 0.6 0.000E+00 0.00 11 0.2818E-01 0.1000E-01 0.5000E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 12 0.2818E-01 0.1775E-01 0.5000E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 13 0.1250E+00 0.2254E-02 0.5000E-02 0.6 0.0 0.0 -0.0 -0.0 0.0 0.6 0.000E+00 0.00 14 0.1250E+00 0.1000E-01 0.5000E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 15 0.1250E+00 0.1775E-01 0.5000E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 16 0.2218E+00 0.2254E-02 0.5000E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 17 0.2218E+00 0.1000E-01 0.5000E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 18 0.2218E+00 0.1775E-01 0.5000E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 19 0.2818E-01 0.2254E-02 0.8873E-02 0.6 0.0 0.0 -0.0 0.0 0.0 0.6 0.000E+00 0.00 20 0.2818E-01 0.1000E-01 0.8873E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 21 0.2818E-01 0.1775E-01 0.8873E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 22 0.1250E+00 0.2254E-02 0.8873E-02 0.6 0.0 0.0 -0.0 -0.0 0.0 0.6 0.000E+00 0.00 23 0.1250E+00 0.1000E-01 0.8873E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 24 0.1250E+00 0.1775E-01 0.8873E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 25 0.2218E+00 0.2254E-02 0.8873E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 26 0.2218E+00 0.1000E-01 0.8873E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 27 0.2218E+00 0.1775E-01 0.8873E-02 0.6 -0.0 -0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 2 1 0.2782E+00 0.2254E-02 0.1127E-02 0.6 0.0 0.0 0.0 -0.0 0.0 0.6 0.000E+00 0.00 2 0.2782E+00 0.1000E-01 0.1127E-02 0.6 0.0 0.0 -0.0 0.0 0.0 0.6 0.000E+00 0.00 3 0.2782E+00 0.1775E-01 0.1127E-02 0.6 0.0 0.0 -0.0 0.0 0.0 0.6 0.000E+00 0.00 4 0.3750E+00 0.2254E-02 0.1127E-02 0.6 -0.0 -0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 5 0.3750E+00 0.1000E-01 0.1127E-02 0.6 -0.0 -0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 6 0.3750E+00 0.1775E-01 0.1127E-02 0.6 -0.0 -0.0 -0.0 0.0 0.0 0.6 0.000E+00 0.00 7 0.4718E+00 0.2254E-02 0.1127E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 8 0.4718E+00 0.1000E-01 0.1127E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 9 0.4718E+00 0.1775E-01 0.1127E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 10 0.2782E+00 0.2254E-02 0.5000E-02 0.6 0.0 0.0 0.0 -0.0 -0.0 0.6 0.000E+00 0.00 11 0.2782E+00 0.1000E-01 0.5000E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 12 0.2782E+00 0.1775E-01 0.5000E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 13 0.3750E+00 0.2254E-02 0.5000E-02 0.6 -0.0 -0.0 0.0 0.0 -0.0 0.6 0.000E+00 0.00 14 0.3750E+00 0.1000E-01 0.5000E-02 0.6 -0.0 -0.0 0.0 0.0 -0.0 0.6 0.000E+00 0.00 15 0.3750E+00 0.1775E-01 0.5000E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 16 0.4718E+00 0.2254E-02 0.5000E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 17 0.4718E+00 0.1000E-01 0.5000E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 18 0.4718E+00 0.1775E-01 0.5000E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 19 0.2782E+00 0.2254E-02 0.8873E-02 0.6 0.0 0.0 0.0 -0.0 -0.0 0.6 0.000E+00 0.00 20 0.2782E+00 0.1000E-01 0.8873E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 21 0.2782E+00 0.1775E-01 0.8873E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 22 0.3750E+00 0.2254E-02 0.8873E-02 0.6 -0.0 -0.0 0.0 0.0 -0.0 0.6 0.000E+00 0.00 23 0.3750E+00 0.1000E-01 0.8873E-02 0.6 -0.0 -0.0 0.0 0.0 -0.0 0.6 0.000E+00 0.00 24 0.3750E+00 0.1775E-01 0.8873E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 25 0.4718E+00 0.2254E-02 0.8873E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 26 0.4718E+00 0.1000E-01 0.8873E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 27 0.4718E+00 0.1775E-01 0.8873E-02 0.6 0.0 0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 3 1 0.5282E+00 0.2254E-02 0.1127E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 2 0.5282E+00 0.1000E-01 0.1127E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 3 0.5282E+00 0.1775E-01 0.1127E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 4 0.6250E+00 0.2254E-02 0.1127E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 5 0.6250E+00 0.1000E-01 0.1127E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 6 0.6250E+00 0.1775E-01 0.1127E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 7 0.7218E+00 0.2254E-02 0.1127E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 8 0.7218E+00 0.1000E-01 0.1127E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 9 0.7218E+00 0.1775E-01 0.1127E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 10 0.5282E+00 0.2254E-02 0.5000E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 11 0.5282E+00 0.1000E-01 0.5000E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 12 0.5282E+00 0.1775E-01 0.5000E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 13 0.6250E+00 0.2254E-02 0.5000E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 14 0.6250E+00 0.1000E-01 0.5000E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 15 0.6250E+00 0.1775E-01 0.5000E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 16 0.7218E+00 0.2254E-02 0.5000E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 17 0.7218E+00 0.1000E-01 0.5000E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 18 0.7218E+00 0.1775E-01 0.5000E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 19 0.5282E+00 0.2254E-02 0.8873E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 20 0.5282E+00 0.1000E-01 0.8873E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 21 0.5282E+00 0.1775E-01 0.8873E-02 0.6 -0.0 -0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 22 0.6250E+00 0.2254E-02 0.8873E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 23 0.6250E+00 0.1000E-01 0.8873E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 24 0.6250E+00 0.1775E-01 0.8873E-02 0.6 0.0 0.0 -0.0 -0.0 -0.0 0.6 0.000E+00 0.00 25 0.7218E+00 0.2254E-02 0.8873E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 26 0.7218E+00 0.1000E-01 0.8873E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 27 0.7218E+00 0.1775E-01 0.8873E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 4 1 0.7782E+00 0.2254E-02 0.1127E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 2 0.7782E+00 0.1000E-01 0.1127E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 3 0.7782E+00 0.1775E-01 0.1127E-02 0.6 0.0 0.0 0.0 0.0 0.0 0.6 0.000E+00 0.00 4 0.8750E+00 0.2254E-02 0.1127E-02 0.6 -0.0 -0.0 -0.0 0.0 -0.0 0.6 0.000E+00 0.00 5 0.8750E+00 0.1000E-01 0.1127E-02 0.6 -0.0 -0.0 0.0 0.0 -0.0 0.6 0.000E+00 0.00 6 0.8750E+00 0.1775E-01 0.1127E-02 0.6 -0.0 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