Sample listings
Listing 1.1.3-1
*HEADING
ELBOW31 elements with 6 ovalization modes
*restart,write
************************
*** PARAMETER DEFINITION
************************
*PARAMETER
# length a of the straight sections a = 10.0
# radius of the curved section (bend) bend_radius = 4.0
# outer radius of the pipe outer_pipe_radius = 0.5
# wall thickness of the pipe wall_thick = 0.08
# number of elements along the straight sections
# of the pipe
num_elem_s = 25
# number of elements around the bend num_elem_c = 25
# displacement at the end of the pipe disp = 1.0
# Young's modulus
young = 28.1E6 # Poisson's ratio
poisson = 0.0
# number of integration points through the
# thickness
nip_thru_thick = 5
# number of integration points around the pipe nip_around_pipe = 20
# number of ovalization modes numoval = 6
#
# total number of elements along the length of
# the pipe
#
num_elem = num_elem_c + 2*num_elem_s
#
###############################################
#
Listing 1.1.3-1
*HEADING
ELBOW31 elements with 6 ovalization modes
*restart,write
************************
*** PARAMETER DEFINITION
************************
*PARAMETER
# length a of the straight sections a = 10.0
# radius of the curved section (bend) bend_radius = 4.0
# outer radius of the pipe outer_pipe_radius = 0.5
# wall thickness of the pipe wall_thick = 0.08
# number of elements along the straight sections
# of the pipe
num_elem_s = 25
# number of elements around the bend num_elem_c = 25
# displacement at the end of the pipe disp = 1.0
# Young's modulus
young = 28.1E6 # Poisson's ratio
poisson = 0.0
# number of integration points through the
# thickness
nip_thru_thick = 5
# number of integration points around the pipe nip_around_pipe = 20
# number of ovalization modes numoval = 6
#
# total number of elements along the length of
# the pipe
#
num_elem = num_elem_c + 2*num_elem_s
#
###############################################
#
# geometrical properties of the problem:
x1_plus_100 = x1 + 100.0 y4_plus_100 = y4 + 100.0
#
# END OF PARAMETER SECTION
#
#*******************
#** Node definitions
#*******************
*NODE
<ndummy>,<a>,<a>,0.
# geometrical properties of the problem:
# points along pipe c/l
x1_plus_100 = x1 + 100.0 y4_plus_100 = y4 + 100.0
#
# END OF PARAMETER SECTION
#
#*******************
#** Node definitions
#*******************
*NODE
<ndummy>,<a>,<a>,0.
*NODE
** Element data definition
**************************
** Element data definition
**************************
*BEAM SECTION,SECTION=elbow,ELSET=straight2,
Apply displacements at the ends of the structure
*STATIC
Apply displacements at the ends of the structure
*STATIC
Listing 1.1.3-2
1.1.4 Indentation of an elastomeric foam specimen with a hemispherical punch
Products: ABAQUS/Standard ABAQUS/Explicit ABAQUS/Design
In this example we consider a cylindrical specimen of an elastomeric foam, indented by a rough, rigid, hemispherical punch. Examples of elastomeric foam materials are cellular polymers such as cushions, padding, and packaging materials. This problem illustrates a typical application of elastomeric foam materials when used in energy absorption devices. The same geometry as the crushable foam model of
``Indentation of a crushable foam specimen with a hemispherical punch, '' Section 1.1.7, is used but with a slightly different mesh. Design sensitivity analysis is carried out for a shape design parameter and a material design parameter to illustrate the usage of design sensitivity analysis for a problem involving contact.
Geometry and model
The axisymmetric model (135 linear 4-node elements) analyzed is shown in Figure 1.1.4-1. The mesh refinement is biased toward the center of the foam specimen where the largest deformation is expected.
The foam specimen has a radius of 600 mm and a thickness of 300 mm. The punch has a radius of 200 mm. The bottom nodes of the mesh are fixed, while the outer boundary is free to move.
In ABAQUS/Standard a contact pair is defined between the punch, which is modeled by a rough spherical rigid surface, and a slave surface composed of the faces of the axisymmetric elements in the contact region. A point mass of 200 kg representing the weight of the punch is attached to the rigid body reference node. In ABAQUS/Explicit the punch is modeled as either an analytical rigid surface or a rigid surface defined with RAX2 elements. In ABAQUS/Explicit the friction coefficient between the punch and the foam is 0.8.
Material
The elastomeric foam material is defined through the *HYPERFOAM option using experimental test data. The uniaxial compression and simple shear data whose stress-strain curves are shown in Figure 1.1.4-2 are defined with the *UNIAXIAL TEST DATA and *SIMPLE SHEAR TEST DATA options.
Other available test data options are *BIAXIAL TEST DATA, *PLANAR TEST DATA and
*VOLUMETRIC TEST DATA. The test data are defined in terms of nominal stress and nominal strain values. ABAQUS performs a nonlinear least-squares fit of the test data to determine the hyperfoam coefficients ¹i; ®i; and ¯i.
Details of the formulation and usage of the hyperfoam model are given in ``Elastomeric foam behavior,'' Section 10.5.2 of the ABAQUS/Standard User's Manual and Section 9.3.2 of the
ABAQUS/Explicit User's Manual; ``Hyperelastic material behavior,'' Section 4.6.1 of the ABAQUS Theory Manual; and ``Fitting of hyperelastic and hyperfoam constants,'' Section 4.6.2 of the ABAQUS Theory Manual. ``Fitting of elastomeric foam test data,'' Section 3.1.5 of the ABAQUS Benchmarks
Listing 1.1.3-2
1.1.4 Indentation of an elastomeric foam specimen with a hemispherical punch
Products: ABAQUS/Standard ABAQUS/Explicit ABAQUS/Design
In this example we consider a cylindrical specimen of an elastomeric foam, indented by a rough, rigid, hemispherical punch. Examples of elastomeric foam materials are cellular polymers such as cushions, padding, and packaging materials. This problem illustrates a typical application of elastomeric foam materials when used in energy absorption devices. The same geometry as the crushable foam model of
``Indentation of a crushable foam specimen with a hemispherical punch, '' Section 1.1.7, is used but with a slightly different mesh. Design sensitivity analysis is carried out for a shape design parameter and a material design parameter to illustrate the usage of design sensitivity analysis for a problem involving contact.
Geometry and model
The axisymmetric model (135 linear 4-node elements) analyzed is shown in Figure 1.1.4-1. The mesh refinement is biased toward the center of the foam specimen where the largest deformation is expected.
The foam specimen has a radius of 600 mm and a thickness of 300 mm. The punch has a radius of 200 mm. The bottom nodes of the mesh are fixed, while the outer boundary is free to move.
In ABAQUS/Standard a contact pair is defined between the punch, which is modeled by a rough spherical rigid surface, and a slave surface composed of the faces of the axisymmetric elements in the contact region. A point mass of 200 kg representing the weight of the punch is attached to the rigid body reference node. In ABAQUS/Explicit the punch is modeled as either an analytical rigid surface or a rigid surface defined with RAX2 elements. In ABAQUS/Explicit the friction coefficient between the punch and the foam is 0.8.
Material
The elastomeric foam material is defined through the *HYPERFOAM option using experimental test data. The uniaxial compression and simple shear data whose stress-strain curves are shown in Figure 1.1.4-2 are defined with the *UNIAXIAL TEST DATA and *SIMPLE SHEAR TEST DATA options.
Other available test data options are *BIAXIAL TEST DATA, *PLANAR TEST DATA and
*VOLUMETRIC TEST DATA. The test data are defined in terms of nominal stress and nominal strain values. ABAQUS performs a nonlinear least-squares fit of the test data to determine the hyperfoam coefficients ¹i; ®i; and ¯i.
Details of the formulation and usage of the hyperfoam model are given in ``Elastomeric foam behavior,'' Section 10.5.2 of the ABAQUS/Standard User's Manual and Section 9.3.2 of the
ABAQUS/Explicit User's Manual; ``Hyperelastic material behavior,'' Section 4.6.1 of the ABAQUS Theory Manual; and ``Fitting of hyperelastic and hyperfoam constants,'' Section 4.6.2 of the ABAQUS Theory Manual. ``Fitting of elastomeric foam test data,'' Section 3.1.5 of the ABAQUS Benchmarks
Manual, illustrates the fitting of elastomeric foam test data to derive the hyperfoam coefficients.
For the material used in this example, ¯i is zero, since the effective Poisson's ratio, º, is zero, as specified through the POISSON parameter. The order of the series expansion is chosen to be N = 2 since this fits the test data with sufficient accuracy. N = 2 also provides a more stable model than the N = 3 case.
The viscoelastic properties in ABAQUS/Standard are specified in terms of a relaxation curve (shown in Figure 1.1.4-3) of the normalized modulus M(t)=M0, where M(t) is the shear or bulk modulus as a function of time and M0 is the instantaneous modulus, as determined from the hyperfoam model. This requires the use of the TIME=RELAXATION TEST DATA parameter of the *VISCOELASTIC option. The relaxation data are specified through the *SHEAR TEST DATA option but actually apply to both shear and bulk moduli when used in conjunction with the hyperfoam model.
ABAQUS/Standard performs a nonlinear least-squares fit of the relaxation data to a Prony series to determine the coefficients, gPi , and the relaxation periods, ¿i. A maximum order of NMAX=2 for fitting the Prony series is used. If creep data are available, the TIME=CREEP TEST DATA parameter is set to specify normalized creep compliance data. The ABAQUS/Explicit analysis is purely elastic.
A rectangular material orientation is defined for the foam specimen, so stress and strain are reported in material axes that rotate with the element deformation. This is especially useful when looking at the stress and strain values in the region of the foam in contact with the punch in the direction normal to the punch (direction "22").
The rough surface of the punch is modeled by specifying a friction coefficient of 0.80 for the contact surface interaction through the *FRICTION option under the *SURFACE INTERACTION definition.
Because of the unsymmetric nature of the friction material model, set UNSYMM=YES on the *STEP option in ABAQUS/Standard.