Implementation of VUMAT

Wenhai Wang

Advisor : Dr. Antonios Zavaliangos

October 15th 2003

UMAT & VUMATUMAT and VUMAT are the user subroutines for the definition of USER based constitutive models

UMAT ABAQUS/Standard--implicit time integration, must provide "material stiffness matrix" for use in forming the Jacobian of the nonlinear equilibrium equations;

VUMAT ABAQUS/Explicit--Explicit time integration, not necessary forming the Jacobianmatrix.

Where Does VUMAT “Fit” in ABAQUS?

ABAQUS

Solving equations of equilibrium

)(),(),( tFtVtX iii )( ttX i ∆+

iε∆)( tti ∆+σ

VUMAT

Solving equations of mechanics

)(tiσ

)( ttFi ∆+

•ABAQUS constitutive library is extensive BUT;

Some models are missing;Some models are not flexible

enough;Example: Gurson Model*POROUS METAL PLASTICITY, RELATIVE DENSITY=0.951. , 1. , 1.

•We can develop our models.

Why Is VUMAT Important?

ABAQUS SUBROUTINEτεσ ∆,t

Stress at t time &strain increment

YIELDCONDITION

Combine the flow rule,Update the stress at t+dt time

*σ

hydrostatic Pressure & equivalent stress

** ,Pσ

trial stress:update the stress at t+dt time using trial stress

*

*

)()(PdttP

dtt=+

=+ σσ

VUMAT

ABAQUS SUBROUTINEτεσ ∆,t

Stress at t time &strain increment

YIELDCONDITION

Combine the flow rule,Update the stress at t+dt time

*σ

hydrostatic Pressure & equivalent stress

** ,Pσ

trial stress:update the stress at t+dt time using trial stress

*

*

)()(PdttP

dtt=+

=+ σσ

VUMAT

ABAQUS SUBROUTINEτεσ ∆,t

Stress at t time &strain increment

ABAQUS SUBROUTINEτεσ ∆,t

Stress at t time &strain increment

ABAQUS SUBROUTINEτεσ ∆,t

Stress at t time &strain increment

YIELDCONDITION

Combine the flow rule,Update the stress at t+dt time

*σ

hydrostatic Pressure & equivalent stress

** ,Pσ

trial stress:update the stress at t+dt time using trial stress

*

*

)()(PdttP

dtt=+

=+ σσ

VUMAT

YIELDCONDITION

Combine the flow rule,Update the stress at t+dt time

*σ

hydrostatic Pressure & equivalent stress

** ,Pσ

trial stress:update the stress at t+dt time using trial stress

*

*

)()(PdttP

dtt=+

=+ σσ YIELDCONDITION

Combine the flow rule,Update the stress at t+dt time

*σ

hydrostatic Pressure & equivalent stress

** ,Pσ

trial stress:update the stress at t+dt time using trial stress

*

*

)()(PdttP

dtt=+

=+ σσ*

*

)()(PdttP

dtt=+

=+ σσ

VUMAT

Debug and Run a VUMATInput File: Subroutine (VUMAT):

C:\>abaqus job=test input=dpnodev user=vumat datacheck

C:\>abaqus job=test input=dpnodev user=vumat

ABAQUS Command:

Structure of VUMAT

)31

23()(

)(

)()(

)()()(

'*

klkl

ijkl

plijklijkl

plijkl

elijkl

PLdtt

LLt

Lt

Ltdtt

δσσ

σλσ

εεσ

εεσ

εσσ

∂Φ∂

+∂Φ∂

−+=

∆−∆+=

∆−∆+=

∆+=+

εσ∆

)(t

...)(tf

Elasticity Strain decomposition

Stress:

Strain increment:

State variables:

)( dtt +σ

...)( dttf +

Flow rule

An 1-D Example of UMAT)(tu

L

)(tσ

)(tε εεε ∆+=+ )()( tdtt

?

σ

ε

u

t

)();( ttu σ know

ABAQUS:1.Calculate from Boundary Conditions)( dttu +

2.Strain incrementL

tudttu )()( −+=∆ε

VUMAT:plel εεε ∆+∆=∆

11

)()(σ

λσσε∂Φ∂

+−+

=∆E

tdtt

If , step is elastic;If , step is plastic;0

0≠=

λλ

An 1-D Example of UMAT (cont.)

If , Plastic. Go to STEP II;If , Elastic. End and return to ABAQUS;Y

Y

dttdtt

σσσσ

<+>+

)()(

11

)()(σ

λσσε∂Φ∂

+−+

=∆E

tdtt

STEPI: Assume step is elastic

11

)()(σ

λσσε∂Φ∂

+−+

=∆E

tdtt εσσ ∆+=+ Etdtt )()(

STEPII: Only if plastic

11

)()(σ

λεσσ∂Φ∂

−∆+=+ Etdtt

Plastic means Ydtt σσ =+ )(

11/)(

σσεσλ

∂Φ∂−∆+

= YEt

Problem is complex for 3D and complex models

Idea of The Algorithm

)(*)(131

1)()( '' dttG

dtt +⎟⎠⎞

⎜⎝⎛

∂Φ∂+

=+ σ

σσλ

σ

PKdttPdttP

∂Φ∂

−+=+ λ)()( *

Yield CondtionsELLIPSE MODEL

GURSON MODEL

DRUCKER-PRAGER

01)()(),,( 22 =−+=Φ pDBDADp σσ

0))1(1()23cosh()1(2),,( 22

12

1

2

=−+−−+=Φ DqpqDqDpYY σσ

σσ

01)()(

0tan22 =−+−=

=−−=

σ

βσ

RBppAF

dpF

ac

s

01))()(())((),,( 22 =−−+=Φ fpcrpfBfAfpYY σσ

σσ

CRITICAL STATE MODEL/CAM CLAY MODEL

Associated and Non-associated

Non-associated Associated

Outline of A Typical VUMAT1. Trial stress: εσσ ∆+=+ *)()(* Ltdtt

2. Check if 0* ≤Φ if yes, the new stress is equal to the trial stress:

)()( * dttdtt +=+ σσif no, call VUMAT_SUBROUTINE

3. VUMAT Subroutine:

calculate the λ by using Newton-Raphson method

4. Update the new stress

)(*)(131

1)()( '' dttG

dtt +⎟⎠⎞

⎜⎝⎛

∂Φ∂

+=+ σ

σσλ

σ

PKdttPdttP

∂Φ∂

−+=+ λ)()( *

5. Return to ABAQUS

Difficulties & Disadvantages

Convergence of newton-raphson method;

Time step increment selection;

Time consuming.

Single Element Compress

1 2

4 3

2

1

Initial geometry: Final geometry:

CPU Time:ABAQUS : VUMAT = 00:00:04 : 00:00:10

S22

ABAQUSVUMAT

VVF (Porosity)

ABAQUSVUMAT

Cylinder CompressionInitial geometry: Final geometry:

CPU Time:ABAQUS : VUMAT = 00:08:15 : 01:25:04If Purely elastic problem, ABAQUS : VUMAT = 00:04:25 : 00:05:01

Most of the CPU time on VUMAT subroutine !

S22

ABAQUS:

VUMAT:

VVF

ABAQUS:

VUMAT:

S22 at Selected PointsS22 at element15: S22 at element 1915:

ABAQUSVUMAT

VVF at Selected PointsVVF at element 30: VVF at element 801:

ABAQUSVUMAT

Rolling

ABAQUS:

VUMAT:

CPU Time:ABAQUS : VUMAT = 00:58:04 : 06:04:23

RF2ABAQUSVUMAT

Future WorkImplementation of the Drucker-Prager Model for Explicit;

Calibrate from the experimental data and derive the models;

Develop other models for powder compaction.