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 - 2016

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

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.

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

(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

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

(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

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

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

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 = 0.68947MPa, C20 = -0.006894MPa, C30 = 0.06894MPa 11/20

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

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

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

BVP1: COMPARISON OF IMPLICIT INTEGRATION RESULTS

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.

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

THANK YOU QUESTIONS