Copyright Dassault Systèmes | www.3ds.com Abaqus/Explicit Honeycomb Material Model 1

Abaqus/Explicit Honeycomb Material Model

Introduction A constitutive model for honeycomb materials is provided in the form of a built-in user material model for Abaqus/Explicit. One application of this model is in the simulation of deformable barriers in automobile crash applications. Before a honeycomb-type material is fully compacted, it will exhibit orthotropic behavior. The material will be stiffest in the direction aligned with the honeycomb cells, and less stiff in the lateral directions. In the remainder of this document, the material direction aligned with the cell axes will be referred to as the “strong” direction. The formulation of this model is based on the assumption that the stress-strain response in each direction is uncoupled from that of the other directions, i.e., the deformation in one direction will only produce stress in that direction. Currently only elements with three dimensional stress states are supported. When specifying material orientations, the strong direction is assumed to be the local x material direction; the two lateral directions align with the local y and z material directions respectively. If no material orientation is specified, the honeycomb material is assumed to be initially aligned with the global directions (i.e., the “strong” direction is aligned with the global X-axis). The effect of damage initiation and evolution is also included in this model. The mechanics of the model will be discussed together with its user interface. An example input file is included. No user subroutine code is needed if the honeycomb model is the only user material included in the analysis. Uniaxial tensile and compressive behavior The uniaxial tensile and compressive behavior in the strong ( x ) direction is depicted in Figure 1. The behavior in the other two directions is similar in nature, but with lower stiffness, yielding strain and yield stress magnitudes; therefore only the response in the x direction will be considered. The discussion below is also applicable to the y and z directions (replace the subscript xx in the material constants with yy and zz respectively). Under tensile loading the behavior is linear elastic, with a Young’s modulus of 0

xxE , up to the initial tensile yield stress Txxσ .

Beyond initial tensile yielding, the material response is represented as one-dimensional rate-independent perfect plasticity. Under compressive loading, the behavior is also linear elastic, with a Young’s modulus of 0

xxE , up to the compressive yield

strain yxxε . Beyond initial compressive yielding, the material behavior is represented as one-dimensional rate-independent

plasticity with isotropic hardening and a tangent stiffness of 1xxE . Upon reaching the compaction strain cε (this value is

assumed to be the same in all directions), the behavior is elastic with a tangent stiffness of 2xxE . An example loading,

unloading, and reloading path is shown in Figure 1 as a-b-c-d-d’-c’-e-f.

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1xxE

0xxE

2xxE

Txxσ

yxxε

cε

xxσ

xxεa

b cxxσ

c, c’

d, d’

e

f

0xxE

Figure 1. Uniaxial tensile and compressive behavior of the honeycomb material model.

Shear behavior The shear behavior in the yx − plane is depicted in Figure 2. The behavior in the zy − and zx − planes is similar in nature but with different stiffness and yielding strain magnitudes; therefore only the shear response in yx − plane will be discussed. All the discussions below are applicable to behavior in the zy − and zx − planes; simply replace the subscript xy in the material constants with yz and xz respectively. The shear behavior is linear elastic with the initial shear modulus 0

xyG in the region between the compressive and tensile

shear yield strains ),( yxy

yxy εε− . Beyond yielding, the material behavior is represented as one-dimensional rate-independent

plasticity with isotropic hardening, with tangent shear modulus of 1xyG . An example loading, unloading and reloading path is

shown in Figure 2 as a-b-c-d-d’-c’-e.

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11 2 xyxy GE =

00 2 xyxy GE =yxyε−

00 2 xyxy GE =

yxyε

11 2 xyxy GE =

xyε

xyσ

a

b c

d (d’)

(c’) e

00 2 xyxy GE =

Figure 2. Shear behavior of the honeycomb material model.

Material damage The damage initiation and evolution for the normal and shear behavior are assumed to be isotropic. Two damage variables,

normald and sheard , govern the damaged response in the normal and shear directions,

iinormalii d σσ )1( −= zyxi ,,= , no summation on i

ijshearij d σσ )1( −= zyxji ,,, = , ji ≠ (1)

where σ is the effective stress tensor computed in the current increment. Note that σ is the stress that exists in the material when no damage is present. Damage initiation Damage initiates in shear when the equivalent plastic shear strain reaches the specified initiation value eq

iε . The equivalent

plastic shear strain eqshearε is defined as

222 )()()( p

xzpyz

pxy

pvol

eqshear εεεεαε +++>−<−= (2)

where

{ })(,0max pzz

pyy

pxx

pvol εεεε ++−>=−< .

The first term on the right hand side of Equation (2) introduces the effect of the volumetric plastic strain on the shear damage. Specifically, it can be seen that the equivalent plastic shear strain is reduced by the scaled plastic volumetric strain; subsequently, the onset of shear damage is delayed in the presence of compressive plastic strains.

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Damage initiates in the normal directions when the equivalent plastic normal strain eqnormalε reaches the specified initiation

value eqtensε . The equivalent plastic normal strain is defined as

222 ><+><+><= pzz

pyy

pxx

eqnormal εεεε (3)

where ),0max( p

iip

ii εε >=< , no summation on i . Damage evolution The shear and normal damage variables evolve during the solution as bilinear functions of the equivalent plastic shear and equivalent plastic normal strains, respectively. Curves A and B in Figure 3 display the behavior of sheard and normald , respectively.

A

eqiε eq

2εeq3ε

2d

3d

)( eqnormal

eqshear εε

)( normalshear dd

eqtensε

B

Figure 3. Damage evolution laws.

The evolution law of the shear damage variable is determined by the shear damage initiation strain eq

iε , intermediate shear

damage equivalent plastic strain eq2ε , maximum shear damage equivalent plastic strain eq

3ε , intermediate damage variable

2d , and the maximum damage variable 3d , as depicted in Figure 3. The evolution law of the tensile damage variable has the same shape as that of the shear damage variable, only shifted from the shear damage curve by eq

ieqtens εε − , as depicted in Figure 3.

An additional parameter d

volε can be used to deactivate the evolution of shear damage once a threshold value of volumetric plastic strain is reached. That is, when the value of the negative volumetric strain

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)( pzz

pyy

pxx

pvol εεεε ++−=−

is smaller (more compressive) than the value of dvolε , the shear damage is not further evolved.

User Interface When using the honeycomb model, the material name must begin with ABQ_HONEYCOMB, e.g. ABQ_HONEYCOMB_1. Thirty five material constants must be specified, as shown in Table 1, and 17 solution dependent state variables are used, as shown in Table 2.

Number Description 1 0

xxE Initial Young’s modulus in x direction 2 y

xxε Yield strain for compression in x direction 3 1

xxE Tangent stiffness in the compressive plastic region in the x direction 4 2

xxE Tangent stiffness in the elastic region beyond full compaction for compression in xdirection

5 0xyG Initial shear modulus in yx − plane

6 yxyε Shear yield strain in yx − plane

7 1xyG Tangent modulus in shear plastic region in yx − plane

8 Txxσ Initial yield stress in tension in x direction; when 0=T

xxσ , the behavior will be elastic in tension

9 0yyE Initial Young’s modulus in y direction

10 yyyε Yield strain for compression in y direction

11 1yyE Tangent stiffness in the compressive plastic region in the y direction

12 2yyE Tangent stiffness in elastic region beyond full compaction for compression in y

direction 13 0

yzG Initial shear modulus in zy − plane

14 yyzε Shear yield strain in zy − plane

15 1yzG Tangent modulus in shear plastic region in zy − plane

16 Tyyσ Initial yield stress for tension in y direction; when 0=T

yyσ , the behavior will be elastic in tension

17 0zzE Initial Young’s modulus in z direction

18 yzzε Yield strain for compression in z direction

19 1zzE Tangent stiffness in the compressive plastic region in the z direction

20 2zzE Tangent stiffness in elastic region beyond full compaction for compression in z

direction 21 0

zxG Initial shear modulus in xz − plane 22 y

zxε Shear yield strain in xz − plane

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23 1zxG Tangent modulus in shear plastic region in xz − plane

24 Tzzσ Initial yield stress for tension in z direction; when 0=T

zzσ , the behavior will be elastic in tension

25 cε Compaction strain in all directions 26

2d Intermediate damage value 27

3d Maximum damage value 28 eq

iε Shear damage initiation strain 29 eq

2ε Intermediate shear damage equivalent plastic strain 30 eq

3ε Maximum shear damage equivalent plastic strain 31 Not currently used 32 Not currently used 33 α Scale factor for the volumetric effect in shear damage 34 d

volε Deactivation volumetric strain for the shear damage; no deactivation will be used when

0=dvolε

35 eqtensε Damage initiation strain in tension

Table 1. Material constants.

Number Description 1-6 Plastic strains 7-9 Equivalent plastic strain in tension

10-12 Equivalent plastic strain in compression 13-15 Equivalent plastic strain in shear

16 Shear damage variable 17 Tensile damage variable

Table 2. Solution dependent state variables.

To include the material model from ABAQUS/CAE complete these steps in the Property Module:

1. Select Material → Create... 2. In the Edit Material dialog box specify the name such that it begins with ABQ_HONEYCOMB. 3. In the Edit Material dialog box select General → User Material 4. Set the User material type as Mechanical. 5. In the Data portion of the Edit Material dialog box, enter the user material constants in the order shown in Table 1. 6. Select General → Depvar and specify 17 solution-dependent state variables.

For a detailed example of the keyword usage of the honeycomb model, please refer to the attached verification input file honey_comb_uni_x.zip