1. Explicit Methods in Quasi-Static Analyses of Rubber-Like Materials
Volkan Yurdabak, Şebnem Özüpek
Department of Mechanical Engineering
Boğaziçi University, Istanbul
SIMULIA Abaqus Türkiye Kullanıcılar Toplantısı

2. PROBLEM STATEMENT
When a finite element analysis is highly nonlinear, convergence problems may be encountered during static analysis using implicit time integration. F
Linear
Nonlinear
Stress
Strain

3. SOLUTION
Performing quasi-static analyses with explicit time integration which is typically used for dynamic analysis.
Static:
Dynamic:
Quasi-static:
When the load is applied slowly enough, the terms related to velocity and acceleration vanish.

4. OBJECTIVE
Quasi-static finite element analysis of a class of rubber problems using
explicit and implicit time integration with varying
Rubber compressibility
Loading velocity
Density
Degree of confinement
Tool : ABAQUS/Standart - ABAQUS/Explicit

5. (IN)COMPRESSIBILITY OF RUBBER
Compressibility of a hyperelastic/rubber-like material can be assessed using the ratio of the initial bulk modulus K0 to the initial shear modulus µ0.
In terms of Poisson’s ratio this can be defined as:
Incompressible
K0/µ0
Poisson's ratio 10
0.452 20
0.475 50
0.490
100
0.495
1,000
0.4995
Infinite
0.5
Bulk Modulus, K0
Shear Modulus, µ0

6. IMPLICIT VS EXPLICIT TIME INTEGRATION
Implicit Time Integration
Explicit Time Integration
There is equilibrium check.
Both compressible and incompressible rubbers can be analyzed.
There is no equilibrium check.
Δt = α (Le / w)
Only compressible rubbers can be analyzed. Wave speed must be defined.
α : Reduction factor
Le : Characteristic element
length
Load P
Time Δt

7. (IN)COMPRESSIBILITY OF RUBBER:Computational aspect
Explicit solvers are much more efficient than implicit ones for very large problems.
Reference for the figure: ABAQUS 6.14 Reference Manual

8. BOUNDARY VALUE PROBLEM
Plane Stress (CPS4R)
Static Analysis
with Implicit Integration:
Quasi-Static Analysis
with Explicit Integration:
Mesh convergence study
Reference Results
The effect of the compressibility
The effect of the loading rate

9. symmetry b.c. symmetry b.c.
BVP: STRESS CONCENTRATION AROUND A CIRCULAR HOLE IN A THIN PLATE
25 mm
100 mm
150 mm
Vertical displacement
symmetry b.c.
100 mm
Compressibilities: 10 50
100
25.4 mm
symmetry b.c.
100 mm
Thickness = 6 mm
Density = 1 g/cm3

10. MATERIAL MODEL Hyperelastic Models :
-Arruda-Boyce -Marlow Mooney Rivlin Neo Hookean
-Ogden -Polynomial form -Reduced polynomial -Van der Waals
-Yeoh
Yeoh strain energy density function given as :
Deviatoric strain invariants
Material constants
C10 = MPa, C20 = MPa, C30 = MPa
11/20

11. BVP1: COMPARISON OF EXPLICIT INTEGRATION RESULTS WITH IMPLICIT ONES
25 mm
At 10 m/s, KE/SE is
more than 10%

12. BVP1: COMPARISON OF EXPLICIT INTEGRATION RESULTS WITH IMPLICIT ONES
100 mm
At 10 m/s, KE/SE is
more than 10%

13. BVP1: COMPARISON OF EXPLICIT INTEGRATION RESULTS WITH IMPLICIT ONES
150 mm
At 10 m/s, KE/SE is
more than 10%

14. BVP1: COMPARISON OF IMPLICIT INTEGRATION RESULTS

15. CONCLUSIONS
The accuracy of the quasi-static analyses are affected by compressibility and loading velocity.
For the problem, the effect of compressibility on maximum von Mises stress increases with the increase in strain rate.
The velocity determines the KE/SE ratio and it must be between 0.001% and 10% for a density of 1 g/cm3 if von Mises stress is considered.

16. FUTURE WORK
The methodology used for the problem will be applied to the compression of a disc made of rubber.
Plane Stress (CPS4R)
Axisymmetry (CAX4R)
Reference for the figure: White J., R., De, S., K.,Rubber Technologist’s Handbook,
Rapra Technology Limited, 2011

17. THANK YOU
QUESTIONS