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Copyright Dassault Systèmes | www.3ds.com
Abaqus/Explicit VUMAT for Bergstrom-Boyce Hysteresis Model
Viscoelastic rubber-like materials often exhibit hysteresis in cyclic loadings. This behavior arises from the frictional sliding of the long molecules across one another. Bergstrom and Boyce [1] observed in experiments that both filled and unfilled elastomers show significant amounts of hysteresis during cyclic loading, and both filled and unfilled elastomers are strain-rate-dependent. This document describes an Abaqus/Explicit user material model (VUMAT) that is largely analogous to the functionality provided in the Abaqus/Standard hysteresis material model. Based on the work of Bergstrom and Boyce [1], the model is meant for the simulation of large-strain, time-dependent hysteretic behavior in rubber-like elastomers.
Hysteresis Material Model
The hysteresis material model for elastomers is based on [1]. The model decomposes the mechanical behavior of the material into two parts. Using the analogy of a polymeric network, we have an equilibrium, or purely elastic, response (network A) and a time-dependent deviation from equilibrium (network B). The model is characterized by the schematic diagram in Figure 1:
Figure 1: Schematic of Bergstrom-Boyce hysteresis model.
The equilibrium network A corresponds to the state that is approached in long-time stress relaxation tests. The time-dependent network B captures the nonlinear rate-dependent deviation from the equilibrium state. The time dependence of network B is assumed to be governed by the repeated motion of molecules that have the ability to significantly change configuration, thereby relaxing the overall stress state. An isotropic hyperelastic material model defines network A. The hysteresis material model defines network B. The total stress of the model is assumed to be the sum of the stresses in the two networks. The deformation gradient
F
is assumed to act on both networks. It is decomposed into elastic (
) and inelastic (
) components in network B through the multiplicative decomposition
e cr B B
F F F
=
(1)
Copyright Dassault Systèmes | www.3ds.com Abaqus/Explicit VUMAT for Hysteresis
2
The constitutive response of network A is governed by standard isotropic hyperelasticity. The stress response of network B is solely dependent on the elastic deformation gradient component
and is governed by the same hyperelastic strain energy potential as network A. Given a deformation gradient
F
acting on both networks, the elastic deformation gradient component
in network B is obtained through an evolution equation,
1 1
e cr cr e cr B B B B B B B
S F F F F
ε σ
− −
=
(2) Where
̇
is the effective creep strain rate in network B,
S
B
is the Cauchy stress deviator tensor in network B, and
σ
B
is the effective stress in network B. The effective creep strain rate in network B is given by the expression
( )
1
C cr cr m B
A E
ε λ σ
= − +
(3) where the positive exponent
m
, generally greater than 1, characterizes the effective stress dependence of the effective creep strain rate; the exponent C, restricted to the interval [
−
1, 0], characterizes the creep strain dependence (through the creep stretch
λ
cr
) on the creep strain rate; the non-negative constant
A
maintains dimensional consistency in the equation; and the parameter
E
helps
regularize the creep strain rate in the vicinity of the undeformed state. In addition to the above material constants, the hysteresis model is characterized by a stress scaling factor,
S
, that defines the ratio of the stress carried by network B to the stress carried by network A under instantaneous loading. The constants for a typical elastomer are as indicated below:
( )
( ) ( )
1
1.65341.00.01
mm
S A s MPamC E
− −
====−=
(4)
VUMAT Hysteresis Model for Explicit Dynamics
The VUMAT material model supports the polynomial and reduced polynomial forms of the hyperelastic strain energy potential up to order 3.
Copyright Dassault Systèmes | www.3ds.com Abaqus/Explicit VUMAT for Hysteresis
3
User Interface
The input file user interface is shown below, with a third order polynomial form as an example.
*Material, name=POLY *Density
density
,
*Depvar
18,
*User Material, constants=17 **c10, c01, c20, c11, c02, c30, c21, c12
1.5, 0.5, 0., 0., 0., 0., 0., 0.
**c03, d1, d2, d3, s, A, m, C
0., 4E-07, 0., 0., 1., 0.0122474, 1.,-1.
**Eta
0.01, The first nine state variables (SDV1 to SDV9) indicate the creep stretch and can be output.
Verification
To verify the VUMAT hysteresis model, uniaxial tension and simple shear simulations were conducted and the behavior of the explicit and implicit implementations under quasi-static loading conditions was compared. The implicit simulation used the native Abaqus/Standard hysteresis constitutive model. The uniaxial tension and simple shear simulations were performed at a cycling frequency of 0.1 Hz. A single element unit cube was used. A sinusoidal extension displacement was applied to one face in the uniaxial tension model. A sinusoidal shear displacement was applied to one edge in the shear model. The corresponding reaction forces were recorded. The forces and displacements were plotted on the same graph to form hysteresis loops. The displaced shape of the uniaxial extension model and the comparison of force-displacement curves from implicit and explicit solutions are shown in Figs. 2 and 3, respectively. A similar displaced shape and a force-displacement curve comparison for the simple shear model are shown in Figs. 4 and 5, respectively. As can be seen, the results compare quite favorably for these simple modes of deformation at low cycling frequencies where inertia effects are minimal.
Copyright Dassault Systèmes | www.3ds.com Abaqus/Explicit VUMAT for Hysteresis
4
Figure 2: Displaced shape of the uniaxial tension unit cube model. Figure 3: Comparison of force-displacement loops for unit cube uniaxial tension models using Abaqus/Standard and Abaqus/Explicit.

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