Figure 1.1.4-14 shows the variation of the normalized sensitivity of CPRESS with respect to the design dr dL = 0 dz dL = r 600
are given under the *PARAMETER SHAPE VARIATION option.
Results and discussion
This problem tests the hyperfoam material model in ABAQUS but does not provide independent verification of the model. The results for all analyses are discussed in the following paragraphs.
ABAQUS/Standard: Case 1
Deformation and contour plots for oriented S22 stress and E22 strain are shown for the viscoelastic foam in Figure 1.1.4-4. Even though the foam has been subjected to large strains, only moderate distortions occur because of the zero Poisson's ratio. The maximum (logarithmic) total strain is of the order of -1.8, which is equivalent to a stretch of ¸ = e-1.8= 0.17, or a nominal compressive strain of
83% indicating severe compression of the foam.
In the viscoelastic case the stresses relax during loading and, consequently, lead to a softer response than in the purely elastic case, as shown in Figure 1.1.4-5. The force-displacement responses are shown in Figure 1.1.4-6. The purely elastic material is reversible, while the viscoelastic material shows hysteresis.
ABAQUS/Standard: Case 2
Various displaced configurations during the Case 2 analysis are shown in Figure 1.1.4-7.
Displacement, velocity, and acceleration histories for the punch are shown in Figure 1.1.4-8. The displacement is shown to reach a steady value at the stress relaxation stage, followed by a severe drop due to the impulsive dynamic load. This is followed by a rebound and then finally by a rapid decay of the subsequent oscillations due to the strong damping provided by the viscoelasticity of the foam.
ABAQUS/Explicit
Figure 1.1.4-9 shows a plot of the initial configurations. Figure 1.1.4-10 shows a contour plot of stress in the y-direction. Figure 1.1.4-11 shows a contour plot of logarithmic strain in the y-direction at 0.06 seconds when the maximum indentation is reached. The elements underneath the punch are seen to be subject to large strains. The history of the punch reaction force (reference node 1000) is shown in Figure 1.1.4-12. A plot of the punch displacement versus the punch reaction force is shown in Figure 1.1.4-13. Because of the purely elastic material behavior, there is no hysteresis and the punch reaction force in the unloading stage follows the same curve as during loading.
ABAQUS/Design
variables L and ¹1 on the contact surface. The sensitivities have been normalized by multiplying with
the value of the design parameter and dividing by the maximum value of CPRESS. Figure 1.1.4-15 and Figure 1.1.4-16 show the contours of sensitivity of the displacement in the z-direction to the design paramaters L and ¹1, respectively. Figure 1.1.4-17 and Figure 1.1.4-18 show the contours of
sensitivity of S22 to the design parameters L and ¹1, respectively. To provide an independent
assessment of the results provided by ABAQUS, sensitivities were computed using the overall finite difference (OFD) technique. The central difference method with a perturbation size of 0.1% of the value of the design parameter was used to obtain the OFD results. Table 1.1.4-1 shows that the sensitivities computed using ABAQUS compare well with the overall finite difference results.
Input files
indentfoamhemipunch_case1.inp
Case 1 of the ABAQUS/Standard example using test data for both elastic and viscoelastic properties of the foam, which is statically deformed in two *VISCO steps.
indentfoamhemipunch_case2.inp
Case 2 in which the ABAQUS/Standard analysis is performed in three steps subjecting the punch to both static and dynamic loading.
hyperfoam_anl.inp
ABAQUS/Explicit analysis using an analytical rigid surface.
hyperfoam.inp
ABAQUS/Explicit model using a faceted rigid surface.
indentfoamhemipunch_dsa.inp
Design sensitivity analysis.
Table
Table 1.1.4-1 Comparison of normalized sensitivities computed using ABAQUS and overall finite difference.
Normalized sensitivity ABAQUS OFD
¹1 S22max ³ dS22 d¹1 ´ max 0.5024 0.5018 L S22max ¡dS22 dL ¢ max -0.1134 -0.1107 ¹1 umax 2 ³ du2 d¹1 ´ max -0.0075 -0.0075 L umax 2 ¡du2 dL ¢ max 0.2918 0.2922 ¹1 CPRESSmax ³ dCPRESS d¹1 ´ max 0.5578 0.5582 L CPRESS max ¡dCPRESS dL ¢ max 0.8012 0.8038
Figures
variables L and ¹1 on the contact surface. The sensitivities have been normalized by multiplying with
the value of the design parameter and dividing by the maximum value of CPRESS. Figure 1.1.4-15 and Figure 1.1.4-16 show the contours of sensitivity of the displacement in the z-direction to the design paramaters L and ¹1, respectively. Figure 1.1.4-17 and Figure 1.1.4-18 show the contours of
sensitivity of S22 to the design parameters L and ¹1, respectively. To provide an independent
assessment of the results provided by ABAQUS, sensitivities were computed using the overall finite difference (OFD) technique. The central difference method with a perturbation size of 0.1% of the value of the design parameter was used to obtain the OFD results. Table 1.1.4-1 shows that the sensitivities computed using ABAQUS compare well with the overall finite difference results.
Input files
indentfoamhemipunch_case1.inp
Case 1 of the ABAQUS/Standard example using test data for both elastic and viscoelastic properties of the foam, which is statically deformed in two *VISCO steps.
indentfoamhemipunch_case2.inp
Case 2 in which the ABAQUS/Standard analysis is performed in three steps subjecting the punch to both static and dynamic loading.
hyperfoam_anl.inp
ABAQUS/Explicit analysis using an analytical rigid surface.
hyperfoam.inp
ABAQUS/Explicit model using a faceted rigid surface.
indentfoamhemipunch_dsa.inp
Design sensitivity analysis.
Table
Table 1.1.4-1 Comparison of normalized sensitivities computed using ABAQUS and overall finite difference.
Normalized sensitivity ABAQUS OFD
¹1 S22max ³ dS22 d¹1 ´ max 0.5024 0.5018 L S22max ¡dS22 dL ¢ max -0.1134 -0.1107 ¹1 umax 2 ³ du2 d¹1 ´ max -0.0075 -0.0075 L umax 2 ¡du2 dL ¢ max 0.2918 0.2922 ¹1 CPRESSmax ³ dCPRESS d¹1 ´ max 0.5578 0.5582 L CPRESS max ¡dCPRESS dL ¢ max 0.8012 0.8038
Figures
Figure 1.1.4-1 Model for foam indentation by a spherical punch.
Figure 1.1.4-2 Elastomeric foam stress-strain curves.
Figure 1.1.4-3 Elastic modulus relaxation curve.
Figure 1.1.4-1 Model for foam indentation by a spherical punch.
Figure 1.1.4-2 Elastomeric foam stress-strain curves.
Figure 1.1.4-4 Deformation and contour plots of viscoelastic foam, ABAQUS/Standard.
Figure 1.1.4-5 Punch reaction force history: static and viscoelastic cases, ABAQUS/Standard.
Figure 1.1.4-6 Punch reaction force versus displacement response: static and viscoelastic cases, (loading-unloading curves); ABAQUS/Standard.
Figure 1.1.4-7 Deformation plots for the visco and dynamic steps, ABAQUS/Standard.
Figure 1.1.4-6 Punch reaction force versus displacement response: static and viscoelastic cases, (loading-unloading curves); ABAQUS/Standard.
Figure 1.1.4-8 Displacement, velocity, and acceleration histories of the punch; ABAQUS/Standard.
Figure 1.1.4-9 Initial (undeformed) configuration, ABAQUS/Explicit.
Figure 1.1.4-10 Stress S22 contour plot at 0.06 s, ABAQUS/Explicit.
Figure 1.1.4-11 Logarithmic strain in the z-direction at 0.06 s, ABAQUS/Explicit.
Figure 1.1.4-10 Stress S22 contour plot at 0.06 s, ABAQUS/Explicit.
Figure 1.1.4-12 Punch reaction force, ABAQUS/Explicit.
Figure 1.1.4-13 Punch reaction force versus indentation, ABAQUS/Explicit.
Figure 1.1.4-14 Normalized sensitivities of contact pressure at the end of the analysis.
Figure 1.1.4-12 Punch reaction force, ABAQUS/Explicit.
Figure 1.1.4-13 Punch reaction force versus indentation, ABAQUS/Explicit.
Figure 1.1.4-15 Sensitivities at the end of the analysis for displacement in the z-direction with respect to L.
Figure 1.1.4-16 Sensitivities at the end of the analysis for displacement in the z-direction with respect to ¹1.
Figure 1.1.4-15 Sensitivities at the end of the analysis for displacement in the z-direction with respect to L.
Figure 1.1.4-16 Sensitivities at the end of the analysis for displacement in the z-direction with respect to ¹1.
Figure 1.1.4-17 Sensitivities at the end of the analysis for stress S22 with respect to L.
Figure 1.1.4-18 Sensitivities at the end of the analysis for stress S22 with respect to ¹1.
Figure 1.1.4-17 Sensitivities at the end of the analysis for stress S22 with respect to L.