Logarithmic formulation, BEAM56H1
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 logarithmic stress tensor $\mathbf{T}$ of a hyperelastic material with a logarithmic description of strain and linear constitutive relations is given by $$\mathbf{T}=2\mu\ln\mathbf{U}+\Lambda\ln J\mathbf{I},$$ where $\ln\mathbf{U}$ is the logarithmic strain tensor, $J=\det\mathbf{F}$ is the determinant of the deformation gradient $\mathbf{F}$ and $\mu,$ $\Lambda$ are Lamé constants such that $$\mu=\frac{E}{2(1+\nu)},\quad \Lambda=\frac{E\nu}{(1+\nu)(1−2\nu)}.$$ The relation between the Cauchy stress tensor and the logarithmic stress tensor is $$\mathbf{T}=J\mathbf{R}^\text{T}\mathbf{\sigma R}.$$ Considering one dimensional stress field, we have the only nonzero component of the Cauchy stress tensor $$\sigma_x=l_x\frac{^0A}{^tA}=\frac{l_x}{\lambda_y\lambda_z},$$ where $^0A$ and $^tA$ are the original and current size of the cross section of the bar, respectively, and $\lambda_y=\lambda_z$ are the principal stretches in the transverse direction. During the tension the rotation of the principal axes does not occur and $\mathbf{R = I}.$ Thus $$T_x=J\sigma_x=l_x\lambda_x,$$ because $J$ can be also expressed as $J=\lambda_x\lambda_y\lambda_z.$ Expressing the first component of the logarithmic stress tensor from the constitutive equations we obtain the equation describing the one-dimensional stress field in the form $$l_x\lambda_x=2\mu\ln\lambda_x+\Lambda\ln(\lambda_x\lambda_y\lambda_z).$$ For given load $l_x=0.4\text{ MPa}$ and given material parameters $\mu=422{,}500\text{ Pa},$ $\Lambda=99{,}695{,}979.5\text{ Pa},$ corresponding to $E=1.265717\text{ MPa}$ and $\nu=0.49789,$ we obtain the solution $$\begin{array}{lll} \lambda_x=1.724691 & & u_x=0.724691\text{ m} \\ \lambda_y=0.76232 & & u_y=-0.004753372\text{ m} \\ \lambda_z=0.76232 & & u_z=-0.002376686\text{ m} \end{array}$$
The Cauchy stress in the bar is $\sigma_x=0.6883\text{ MPa}.$
The computation is executed with the following commands:
- beam56h1.bat
rmd3 beam56h1.i1 rpd3 beam56h1.i2 srh3 beam56h1.i3 fefs beam56h1.i4 hpp3 beam56h1.iP hpls beam56h1.iL str3 beam56h1.i5
Input
- beam56h1.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
- beam56h1.i2
; KREST IP 1 ; E α ν MP 1 T 1 V 1265717 0 0.49789 SV 1 T 9 V 4e5 0 0 0 0 ; lx = 0.4 MPa 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
- beam56h1.i3
; KREST IP 1 EN EN
- beam56h1.i4
; KREST IP 1 EN EN
- beam56h1.iP
; KREST NLC NCYC KMOD KCRP KLARG ... IP 1 1 0 0 0 3 4*0 ; load case sequence 2 EN EN
- beam56h1.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
- beam56h1.i5
; KLC 0 KOUT ILC ... KPROB KGRAF IP 1 0 1 0 6*0 2 2 EN EN
Output
- beam56h1.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 -0.4753372E-02 0.0000000E+00 3 0.0000000E+00 -0.4753372E-02 -0.2376686E-02 4 0.0000000E+00 0.0000000E+00 -0.2376686E-02 5 0.1811728E+00 0.0000000E+00 0.0000000E+00 6 0.1811728E+00 -0.4753372E-02 0.0000000E+00 7 0.1811728E+00 -0.4753372E-02 -0.2376686E-02 8 0.1811728E+00 0.0000000E+00 -0.2376686E-02 9 0.3623455E+00 0.0000000E+00 0.0000000E+00 10 0.3623455E+00 -0.4753372E-02 0.0000000E+00 11 0.3623455E+00 -0.4753372E-02 -0.2376686E-02 12 0.3623455E+00 0.0000000E+00 -0.2376686E-02 13 0.5435183E+00 0.0000000E+00 0.0000000E+00 14 0.5435183E+00 -0.4753372E-02 0.0000000E+00 15 0.5435183E+00 -0.4753372E-02 -0.2376686E-02 16 0.5435183E+00 0.0000000E+00 -0.2376686E-02 17 0.7246910E+00 0.0000000E+00 0.0000000E+00 18 0.7246910E+00 -0.4753372E-02 0.0000000E+00 19 0.7246910E+00 -0.4753372E-02 -0.2376686E-02 20 0.7246910E+00 0.0000000E+00 -0.2376686E-02 21 0.0000000E+00 -0.2376686E-02 0.0000000E+00 22 0.0000000E+00 -0.4753372E-02 -0.1188343E-02 23 0.0000000E+00 -0.2376686E-02 -0.2376686E-02 24 0.0000000E+00 0.0000000E+00 -0.1188343E-02 25 0.9058638E-01 0.0000000E+00 0.0000000E+00 26 0.9058637E-01 -0.4753372E-02 0.0000000E+00 27 0.9058637E-01 -0.4753372E-02 -0.2376686E-02 28 0.9058638E-01 0.0000000E+00 -0.2376686E-02 29 0.1811728E+00 -0.2376686E-02 0.0000000E+00 30 0.1811728E+00 -0.4753372E-02 -0.1188343E-02 31 0.1811728E+00 -0.2376686E-02 -0.2376686E-02 32 0.1811728E+00 0.0000000E+00 -0.1188343E-02 33 0.2717591E+00 0.0000000E+00 0.0000000E+00 34 0.2717591E+00 -0.4753372E-02 0.0000000E+00 35 0.2717591E+00 -0.4753372E-02 -0.2376686E-02 36 0.2717591E+00 0.0000000E+00 -0.2376686E-02 37 0.3623455E+00 -0.2376686E-02 0.0000000E+00 38 0.3623455E+00 -0.4753372E-02 -0.1188343E-02 39 0.3623455E+00 -0.2376686E-02 -0.2376686E-02 40 0.3623455E+00 0.0000000E+00 -0.1188343E-02 41 0.4529319E+00 0.0000000E+00 0.0000000E+00 42 0.4529319E+00 -0.4753372E-02 0.0000000E+00 43 0.4529319E+00 -0.4753372E-02 -0.2376686E-02 44 0.4529319E+00 0.0000000E+00 -0.2376686E-02 45 0.5435183E+00 -0.2376686E-02 0.0000000E+00 46 0.5435183E+00 -0.4753372E-02 -0.1188343E-02 47 0.5435183E+00 -0.2376686E-02 -0.2376686E-02 48 0.5435183E+00 0.0000000E+00 -0.1188343E-02 49 0.6341046E+00 0.0000000E+00 0.0000000E+00 50 0.6341046E+00 -0.4753372E-02 0.0000000E+00 51 0.6341046E+00 -0.4753372E-02 -0.2376686E-02 52 0.6341046E+00 0.0000000E+00 -0.2376686E-02 53 0.7246910E+00 -0.2376686E-02 0.0000000E+00 54 0.7246910E+00 -0.4753372E-02 -0.1188343E-02 DISPLACEMENTS NODE U V W ALPHA BETA GAMMA [m] [m] [m] [rad] [rad] [rad] 55 0.7246910E+00 -0.2376686E-02 -0.2376686E-02 56 0.7246910E+00 0.0000000E+00 -0.1188343E-02 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 0.7 0.0 0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 2 0.2818E-01 0.1000E-01 0.1127E-02 0.7 0.0 0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 3 0.2818E-01 0.1775E-01 0.1127E-02 0.7 0.0 0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 4 0.1250E+00 0.2254E-02 0.1127E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 5 0.1250E+00 0.1000E-01 0.1127E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 6 0.1250E+00 0.1775E-01 0.1127E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 7 0.2218E+00 0.2254E-02 0.1127E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 8 0.2218E+00 0.1000E-01 0.1127E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 9 0.2218E+00 0.1775E-01 0.1127E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 10 0.2818E-01 0.2254E-02 0.5000E-02 0.7 0.0 0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 11 0.2818E-01 0.1000E-01 0.5000E-02 0.7 0.0 0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 12 0.2818E-01 0.1775E-01 0.5000E-02 0.7 0.0 0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 13 0.1250E+00 0.2254E-02 0.5000E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 14 0.1250E+00 0.1000E-01 0.5000E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 15 0.1250E+00 0.1775E-01 0.5000E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 16 0.2218E+00 0.2254E-02 0.5000E-02 0.7 0.0 0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 17 0.2218E+00 0.1000E-01 0.5000E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 18 0.2218E+00 0.1775E-01 0.5000E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 19 0.2818E-01 0.2254E-02 0.8873E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 20 0.2818E-01 0.1000E-01 0.8873E-02 0.7 -0.0 -0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 21 0.2818E-01 0.1775E-01 0.8873E-02 0.7 -0.0 -0.0 -0.0 -0.0 0.0 0.7 0.000E+00 0.00 22 0.1250E+00 0.2254E-02 0.8873E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 23 0.1250E+00 0.1000E-01 0.8873E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 24 0.1250E+00 0.1775E-01 0.8873E-02 0.7 -0.0 -0.0 -0.0 0.0 0.0 0.7 0.000E+00 0.00 25 0.2218E+00 0.2254E-02 0.8873E-02 0.7 0.0 0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 26 0.2218E+00 0.1000E-01 0.8873E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 27 0.2218E+00 0.1775E-01 0.8873E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 2 1 0.2782E+00 0.2254E-02 0.1127E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 2 0.2782E+00 0.1000E-01 0.1127E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 3 0.2782E+00 0.1775E-01 0.1127E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 4 0.3750E+00 0.2254E-02 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 5 0.3750E+00 0.1000E-01 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 6 0.3750E+00 0.1775E-01 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 7 0.4718E+00 0.2254E-02 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 8 0.4718E+00 0.1000E-01 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 9 0.4718E+00 0.1775E-01 0.1127E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 10 0.2782E+00 0.2254E-02 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 11 0.2782E+00 0.1000E-01 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 12 0.2782E+00 0.1775E-01 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 13 0.3750E+00 0.2254E-02 0.5000E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 14 0.3750E+00 0.1000E-01 0.5000E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 15 0.3750E+00 0.1775E-01 0.5000E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 16 0.4718E+00 0.2254E-02 0.5000E-02 0.7 0.0 0.0 0.0 0.0 0.0 0.7 0.000E+00 0.00 17 0.4718E+00 0.1000E-01 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 18 0.4718E+00 0.1775E-01 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 19 0.2782E+00 0.2254E-02 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 20 0.2782E+00 0.1000E-01 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 21 0.2782E+00 0.1775E-01 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 -0.0 0.7 0.000E+00 0.00 22 0.3750E+00 0.2254E-02 0.8873E-02 0.7 -0.0 -0.0 0.0 0.0 0.0 0.7 0.000E+00 0.00 23 0.3750E+00 0.1000E-01 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 24 0.3750E+00 0.1775E-01 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 25 0.4718E+00 0.2254E-02 0.8873E-02 0.7 -0.0 -0.0 0.0 0.0 0.0 0.7 0.000E+00 0.00 26 0.4718E+00 0.1000E-01 0.8873E-02 0.7 -0.0 -0.0 0.0 0.0 0.0 0.7 0.000E+00 0.00 27 0.4718E+00 0.1775E-01 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 3 1 0.5282E+00 0.2254E-02 0.1127E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 2 0.5282E+00 0.1000E-01 0.1127E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 3 0.5282E+00 0.1775E-01 0.1127E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 4 0.6250E+00 0.2254E-02 0.1127E-02 0.7 -0.0 -0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 5 0.6250E+00 0.1000E-01 0.1127E-02 0.7 -0.0 -0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 6 0.6250E+00 0.1775E-01 0.1127E-02 0.7 -0.0 -0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 7 0.7218E+00 0.2254E-02 0.1127E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 8 0.7218E+00 0.1000E-01 0.1127E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 9 0.7218E+00 0.1775E-01 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 10 0.5282E+00 0.2254E-02 0.5000E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 11 0.5282E+00 0.1000E-01 0.5000E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 12 0.5282E+00 0.1775E-01 0.5000E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 13 0.6250E+00 0.2254E-02 0.5000E-02 0.7 -0.0 -0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 14 0.6250E+00 0.1000E-01 0.5000E-02 0.7 -0.0 -0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 15 0.6250E+00 0.1775E-01 0.5000E-02 0.7 -0.0 -0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 16 0.7218E+00 0.2254E-02 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 17 0.7218E+00 0.1000E-01 0.5000E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 18 0.7218E+00 0.1775E-01 0.5000E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 19 0.5282E+00 0.2254E-02 0.8873E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 20 0.5282E+00 0.1000E-01 0.8873E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 21 0.5282E+00 0.1775E-01 0.8873E-02 0.7 0.0 0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 22 0.6250E+00 0.2254E-02 0.8873E-02 0.7 0.0 0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 23 0.6250E+00 0.1000E-01 0.8873E-02 0.7 -0.0 -0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 24 0.6250E+00 0.1775E-01 0.8873E-02 0.7 -0.0 -0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 25 0.7218E+00 0.2254E-02 0.8873E-02 0.7 -0.0 -0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 26 0.7218E+00 0.1000E-01 0.8873E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 27 0.7218E+00 0.1775E-01 0.8873E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 4 1 0.7782E+00 0.2254E-02 0.1127E-02 0.7 0.0 0.0 0.0 0.0 0.0 0.7 0.000E+00 0.00 2 0.7782E+00 0.1000E-01 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 3 0.7782E+00 0.1775E-01 0.1127E-02 0.7 0.0 0.0 0.0 -0.0 0.0 0.7 0.000E+00 0.00 4 0.8750E+00 0.2254E-02 0.1127E-02 0.7 -0.0 -0.0 -0.0 0.0 -0.0 0.7 0.000E+00 0.00 5 0.8750E+00 0.1000E-01 0.1127E-02 0.7 -0.0 -0.0 -0.0 -0.0 -0.0 0.7 0.000E+00 0.00 6 0.8750E+00 0.1775E-01 0.1127E-02 0.7 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