A trilinear cylinder (r interpolation) of incompressible Mooney-Rivlin material is inflated to 1.5 kPa, axially stretched by 20% and twisted (torsion angle 30 degrees). An online analytic solution is available for the problem.
#Example_538 Inflation, axial extension and torsion of an isotropic, homogeneous cylinder (analytic solution is available)
fem
fem define coordinates;r;cylanalyt;example #Define 3 GLOBAL CYLINDRICAL
# ! POLAR coordinates for an UNSYMMETRIC
# ! GEOMETRY, RADIAL INTERPOLATION in r.
# ! Written to file CYLIND.ipcoor.
fem define node;r;;example #4 NODES, 3 COORDINATES, NO VERSION
# ! PROMPTING, 0 DERIVATIVES for all
# ! coordinates. Nodal coordinates (Xj)
# ! are (R,theta,Z): 1=(1,0,0);
# ! 2=(1,0,1); 3=(1.5,0,0); 4=(1.5,0,1).
fem define base;r;;example #Define 3 BASIS FUNCTIONS. Choose
# ! LAGRANGE/HERMITE for basis function 1
# ! with 3 Xi COORDINATES, a LINEAR LAGRANGE interpolant
# ! and 1 GAUSS POINT each in Xi(1) (circumferential) and
# ! Xi(2) (axial) directions but 3 in Xi(3) (radial).
# ! For basis function 2 (hydrostatic pressure), choose an
# ! AUXILIARY BASIS with 5 AUXILIARY ELEMENT PARAMETERS,
# ! PRESSURE BASIS with 3 Xi coordinates and the same
# ! number of Gauss points as basis function 1. For all 5
# ! parameters choose POLYNOMIAL DEGREE of 0 (ie constant)
# ! in Xi1 and Xi2 directions. For parameters 1;2;3;4;5
# ! choose POLYNOMIAL DEGREE 0;1;2;-1;-2 respectively
# ! (0;1;2 denote a quadratic variation in the hydrostatic
# ! pressure in Xi3 whilst -1;-2 denotes parameters for
# ! Xi3=0;Xi3=1 face pressure bcs respectively.) Basis
# ! function 3 describes surfaces and thus has 2 Xi
# ! COORDINATES with LINEAR LAGRANGE interpolant, and 3
# ! Gauss points in Xi(1) and Xi(2) directions.
fem define element;r;;example #1 ELEMENT with 3 Xj COORDINATES. BASIS
# ! FUNCTION 1 describes each coordinate.
# ! GLOBAL NODE # is 1,1,2,2,3,3,4,4.
fem define fibres;r;;example
fem define elements;r;;example fibre
fem define window;r;;example #Define window dimensions in X,Y,Z
# ! directions as (min,max): (-2,2),
# ! (-2,2), (-1,2).
fem draw lines #Make line segments visible on window.
fem define equation;r;;example #Defines equation the same as in the
# ! uniaxial cube extension problem.
fem define material;r;;example #Defines a Mooney-Rivlin material as in
# ! the uniaxial cube extension problem.
fem define initial;r;;example #BOUNDARY PRESSURE INCREMENTS will be
# ! entered and HYDROSTATIC PRESSURE will
# ! be matched across elements with adjacent Xi3 faces
# ! (none in this example). INITIAL DISPLACEMENTS are ALL
# ! ZERO. For dep var 2 (theta-direction) fix nodes 1,3 and
# ! ans set increments of 0.524 (30 deg) for nodes 2,4; for
# ! dep var 3 (Z-direction), fix nodes 1,3 and apply 0.2
# ! displacement to nodes 2,4. Pressure bcs are applied
# ! via DEPENDENT VARIABLE/EQUATION 4. For ELEMENT 1 DO NOT
# ! prescribe bcs for parameters 1,2,3; to apply a pressure
# ! of magnitude 1.5 kPa on Xi3=0 face (inside) prescribe
# ! an INCREMENT of 1.5 to AUXILIARY VARIABLE 4. To apply
# ! a pressure of magnitude 0 kPa on Xi(3)=1 face (outside
# ! cylinder) prescribe an INCREMENT of 0.0 to AUXILIARY
# ! VARIABLE 5 (note these param #'s correlate with the
# ! basis description file).
fem define solve;r;;example #Defines solution information.
fem solve step 1 #Solve the problem, (should converge
# ! in about 6 iterations).
fem draw lines def dotted #Draw deformed mesh.
fem list elem tot
fem list elem deformed total
# !Note volume is not conserved with quadratic transmural
# ! hydrostatic pressure and r interpolation (I3 not equal
# ! to 1).
fem list strain xi 3 points 3 ref #List strain wrt reference coords at
# ! 3 pts in xi3 dirn (xi1=xi2=0.5).
fem list stress xi 3 points 3 ref #List stress wrt reference coords at
# ! 3 pts in xi3 dirn (xi1=xi2=0.5).
fem list node solution
# The radial stress at the first point (ng=1) is
# at the inner face where the stress should be close
# to the applied pressure (-1.5) but is quite different.
# At the third point (ng=3), at the outer face, the
# radial stress should be near zero but is again quite
# different from what is expected.
# Contrast the results at the second point (ng=2)
# and the deformed radii with the analytic solution.
Name Modified Size
example_538.com 17-Jan-2005 5.1k cylanalyt.ipbase 10-Apr-2000 4.2k cylanalyt.ipcoor 25-Jun-2014 647 cylanalyt.ipelem 10-Apr-2000 409 cylanalyt.ipelfb 17-Jan-2005 233 cylanalyt.ipequa 02-May-2004 2.1k cylanalyt.ipfibr 17-Jan-2005 734 cylanalyt.ipinit 05-Dec-2002 2.2k cylanalyt.ipmate 05-Dec-2002 2.3k cylanalyt.ipnode 10-Apr-2000 1.1k cylanalyt.ipsolv 25-Jun-2014 2.3k cylanalyt.ipsolv.old 13-Apr-2007 2.2k cylanalyt.ipwind 10-Apr-2000 262 newtest_output.com 10-Apr-2000 50
Name Modified Size
examples_5_53_538.tar.gz 08-Aug-2014 8.5k
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