Implementation of VUMAT

Implementation of VUMAT Wenhai Wang Advisor : Dr. Antonios Zavaliangos October 15th 2003

UMAT & VUMAT UMAT 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 Jacobian matrix.

Where Does VUMAT “Fit” in ABAQUS? ABAQUS

X i (t ), Vi (t ), Fi (t )

σ i (t )

Solving equations of equilibrium

Fi (t + ∆t )

σ i (t + ∆t )

VUMAT Solving equations of mechanics

X i (t + ∆t )

∆ε i

Why Is VUMAT Important? ABAQUS

Stress at t time & strain increment σ t , ∆ετ

SUBROUTINE

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

σ (t + dt ) = σ * P(t + dt ) = P*

YIELD CONDITION

σ* σ * , P*

hydrostatic Pressure & equivalent stress

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

VUMAT

•ABAQUS constitutive library is extensive BUT; 9Some models are missing; 9Some models are not flexible enough; Example: Gurson Model *POROUS METAL PLASTICITY, RELATIVE DENSITY=0.95 1. , 1. , 1.

•We can develop our models.

Debug and Run a VUMAT Input File:

Subroutine (VUMAT):

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

Structure of VUMAT Elasticity σ (t + dt ) = σ (t ) + Lijkl (∆ε el )

Strain decomposition

= σ (t ) + Lijkl (∆ε − ∆ε pl ) = σ (t ) + Lijkl ∆ε − Lijkl ∆ε pl

Flow rule

∂Φ 3 σ kl 1 ∂Φ δ kl ) + = σ (t + dt ) − Lijkl λ ( 3 ∂P ∂σ 2 σ '

*

Stress: Strain increment:

σ (t ) ∆ε f (t )

State variables:

...

σ (t + dt ) f (t + dt ) ...

An 1-D Example of UMAT u (t )

σ

u (t ); σ (t ) know ABAQUS:

L

1.Calculate u (t + dt ) from Boundary Conditions

?

σ (t )

2.Strain increment ∆ε =

ε

u

ε (t )

ε (t + dt ) = ε (t ) + ∆ε

t

u (t + dt ) − u (t ) L

VUMAT: ∆ε = ∆ε el + ∆ε pl ∆ε =

σ (t + dt ) − σ (t )

If λ = 0 If λ ≠ 0

E

∂Φ +λ ∂σ 11

, step is elastic; , step is plastic;

An 1-D Example of UMAT (cont.) STEPI: Assume step is elastic ∆ε =

σ (t + dt ) − σ (t ) E

+λ

∂Φ ∂σ 11

If σ (t + dt ) > σ Y , If σ (t + dt ) < σ Y ,

σ (t + dt ) = σ (t ) + E∆ε Plastic. Go to STEP II; Elastic. End and return to ABAQUS;

STEPII: Only if plastic ∆ε =

σ (t + dt ) − σ (t ) E

∂Φ +λ ∂σ 11

σ (t + dt ) = σ (t ) + E∆ε − λ

∂Φ ∂σ 11

Plastic means σ (t + dt ) = σ Y σ (t ) + E∆ε − σ Y λ= ∂Φ / ∂σ 11 Problem is complex for 3D and complex models

Idea of The Algorithm

(σ ) ' (t + dt ) =

1

∂Φ 1 ⎞ ⎛ + 1 3 G λ ⎜ ⎟ ∂σ σ ⎠ ⎝ ∂Φ P(t + dt ) = P * (t + dt ) − Kλ ∂P

(σ *)' (t + dt )

Yield Condtions ELLIPSE MODEL

Φ (σ , p, D) = A( D)σ 2 + B( D) p 2 − 1 = 0 GURSON MODEL σ2 3q p Φ(σ , p, D) = + 2q1 (1 − D) cosh( 2 ) − (1 + q12 (1 − D) 2 ) = 0 σY 2 σY

CRITICAL STATE MODEL/CAM CLAY MODEL Φ (σ , p, f ) = A( f )(

p σ 2 ) + B ( f )( − pcr ( f )) 2 − 1 = 0 σY σY

DRUCKER-PRAGER Fs = σ − p tan β − d = 0 Fc = A( p − p a ) 2 + B( Rσ ) 2 − 1 = 0

Associated and Non-associated

Non-associated

Associated

Outline of A Typical VUMAT 1. Trial stress:

σ * (t + dt ) = σ (t ) + L * ∆ε

2. Check if Φ * ≤ 0

if yes, the new stress is equal to the trial stress:

σ (t + dt ) = σ * (t + dt ) if no, call VUMAT_SUBROUTINE

3. VUMAT Subroutine: calculate the

λ

by using Newton-Raphson method

4. Update the new stress (σ ) ' (t + dt ) =

1

(σ *)' (t + dt )

∂Φ 1 ⎞ ⎛ ⎟ ⎜1 + 3Gλ ∂σ σ ⎠ ⎝ ∂Φ P (t + dt ) = P * (t + dt ) − Kλ ∂P

5. Return to ABAQUS

Difficulties & Disadvantages ¾Convergence of newton-raphson method; ¾Time step increment selection; ¾Time consuming.

Single Element Compress 4

3

2

1

1

Initial geometry:

2

Final geometry:

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

S22

ABAQUS VUMAT

VVF (Porosity)

ABAQUS VUMAT

Cylinder Compression Initial geometry:

Final geometry:

CPU Time: ABAQUS : VUMAT = 00:08:15 : 01:25:04 If 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 Points S22 at element15:

S22 at element 1915:

ABAQUS VUMAT

VVF at Selected Points VVF at element 30:

VVF at element 801:

ABAQUS VUMAT

Rolling ABAQUS:

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

RF2 ABAQUS VUMAT

Future Work ¾Implementation of the Drucker-Prager Model for Explicit; ¾Calibrate from the experimental data and derive the models; ¾Develop other models for powder compaction.