Material implementation in ABAQUS

||Institute for Building Materials

Mechanics of Building MaterialsABAQUS UMAT Implementation

F. Wittel


Overview on ABAQUS structures

How subroutines interact with ABAQUS

Calling subroutines inside a model

Writing of subroutines

UMAT and VUMAT

UMAT in some detail

Examples of UMATs for plasticity

Material implementation in ABAQUS


User Subroutines:

• Extension of the functionality and applicability of ABAQUS beyond thecurrent implementation.

• Offers a powerful, flexible analysis tool by adaptation to individual needs.

• Are typically written in Fortran77 code, that needs to be included into amodel for the calculation.

• Can call utility routines to simplify coding.

• ABAQUS User Subroutine Reference Manual describes (55) differentuser subroutines for Abaqus/Standard, (21) Abaqus/Explicit subroutines,and all available utility routines (18) in detail.

Material implementation in ABAQUS: structures


Popular user subroutines for ABAQUS/Standard (for material behavior)

CREEP: Definition of time dependent, visco-plastic material behavior. Deformations with decomposition into deviatoric(creep) and volumetric (swelling) part.

DLOAD: Definition of non-uniformly, distributed mechanical loads in form of surface or volumetric loads.

UEL: Definition of own finite elements, that are not contained in the element library.

URDFIL: Input of data from result files (*.fil) at the end of the increment.

USDFLD: Definition of filed properties directly at the integration point of elements. Those can depend on stresses andstrains.

DISP: Definition of boundary conditions.

ORIENT: Definition of material orientations.

UMAT: Definition of own, complex constitutive material models, that are not part of the material library of ABAQUS.



Popular user subroutines for ABAQUS/Explicit for material behavior

VDISP: Definition of displacement boundary conditions.

VDLOAD: Definition of non-uniform loads.

VUEL: Element definition.

VUFIELD: Definition of field variables.

VUHARD: Definition of yield surfaces and hardening rules for isotropic plasticity or combined hardening models.

VUSDFLD: Definition of new field variables at the material point.

VUMAT: Definition of own, complex constitutive material models, not included in the material library of ABAQUS.

Utility routines: Obtaining values of ABAQUS environment variables, Job name, path name, output path, parallelization information, part information, information at material points and node averages, node information, element information, stress-/strain invariants and rotation tensor. Output to message file, and more…



Material implementation in ABAQUS: Interaction with ABAQUS

Analysis start

Definition of initial conditions

Initiation of step

Initiation of increment

Initiation of iteration

Definition of global stiffness matrix [K]

Definition of loads {F}

UEXTERNALDB

CREEP, FRIC, UEL,UEXPAN, UGENS,UMAT, USDFLD

DLOADFILM,

HETVAL,UWAVE

HARDINISDVINI

UPOREPSIGINIVOIDRI


Did it converge?

End of step?

Go to iteration startsolve[K]{U}={F}

Write outputGo to next step

Go to next increment start

Yes

No

YesNo



Analysis start

Initial conditions

Step initiation

Increment initiation

Iteration start

System stiffness matrix [K]

Load definition{F}

Increment start

Calculation of field properties at integration points from node values

Iteration start

calculate

calculate ,

defineload

Px

CREEP

UEXPAN

FRIC

UGENS

,c r s w t h

/

/N E

UEL

UMATUSDFLD

FILM

HETVAL

/d h d /r



Before you can start you have to…

Install Microsoft Visual Studio

Install a FORTRAN compiler

Set environment variable

Change starting properties

start of a calculation by giving the name of the user subroutine with the User parameter in the ABAQUS command shell.Input file usage: Abaqus job=job-name User={Source-file | Object-file}

Abaqus job=Analysis User=Sample _subroutine.for

Material implementation in ABAQUS: Using subroutines in ABAQUS


Great care while programming Conventions and guidelines have to be followed!

INCLUDE call for FORTRAN compiler and linker.• Always the first statement after the argument list:

• ABAQUS/STANDARD: 'ABA_PARAM.INC‘ /EXPLICIT: 'VABA_PARAM.INC‘• Has to be stated in all main and sub routines.

'INCLUDE ABA_PARAM.INC' is the first statement after the argument list.

Include call in sub routines



Naming conventions:

For name of COMMON blocks or called subroutines the initial letter K is reserved (Example: SUBROUTINE KS_Eff(D_f,Eff_val)).

Overwriting of variables:

User subroutines are not allowed to overwrite other parts of ABAQUS. Passed in variables are:

already declared variables.

Variables that need to be declares.

Variables passed in for information purposes (do not overwrite).

Output of variables (debugging):

Into the message file (*.msg) handler 7 or output file (*.dat) handler 6. handler 15-18 and > 100 can also be used for reading/writing into files.

Write statement to standard out goes into the *.log file in the cwd.



Golden rules:

Develop user routines always with the smallest meaningful model and single elements inits simplest form (no contact ..).

Adding and modifying models requires tests before and after modification..

If possible test subroutines in a way that only node DOFs are specified. In a next steptests of combinations e.g. with volumetric forces and node values can be made.

Attention when assigning the number of solution dependent variables (SDV) with the*Depvar keyword. It 20 are needed but only 18 SDVs specified the consequences areunpredictable.



Interface for defining own, complicated constitutive material laws that are not part of the material library of ABAQUS-

ABAQUS/Standard UMAT ABAQUS/Explicit VUMAT

What you need for UMATs …

Explicit (total) stress

Stress rate (only in co-rotational framework)

Definition of time-, temperature- or field property dependence (if defined)

Definition of solution dependent state variables (SDV if defined)

Transformation of rate dependent constitutive models into incremental form with integration method (ForwardEuler (explicit); Backward Euler (implicit); Mid-point method (Semi implicit)

Incremental statement for internal SDV

Conventions of FORTRAN have to be followed correct variable declaration and initiation

Memory allocation for SDV with *DEPVAR statement

Calculation of a consistent Jacobian matrix (UMAT)

Material implementation in ABAQUS: UMAT and VUMAT


Consistent Jacobian matrix (DDSDDE):

For small deformations or large deformations with small volumetric change (plasticity) the Jacobian is:

CAUCHY stress increment; strain increment.

Stain increments are approximated for small strains as logarithmic strain.

Jacobian can be non-symmetric, depending on the constitutive model or integration scheme.

For complicated constitutive models the Jacobian is only approximated loss of quadratic convergence.

D D S D D E J



Verification of implementations…

… on the simplest case: one or a few elements with input.

… on calculations with prescribed displacements to test the integration algorithm for stress- and state variables..

… for representative cases: Uni-axial, uniaxial under an angle, uniaxial with rotation, shear.

… on calculations with prescribed load to test the accuracy of the Jacobian.

… by comparing with analytical solutions or material models from ABAQUS.



UMAT Variables stresses (STRESS), strains (STRAN), and solution dependent state variables (SDVs) (STATEV) at the start of the increment. Strain increment (DSTRAN), rotation increment (DROT(3,3)), deformation gradient at the increment start (DFGRD0(3,3)) and (DFGRD1(3,3))

Increment end. Total time (TIME(1)) and incremental time (DTIME), temperature (TEMP and DTEMP), and user defined field properties (PREDEF and

DPRED). Material constants (PROPS), material point positions (COORDS) and char. element length (CELENT). Element number (NOEL), integration point (NPT) and composite layer number for shell and layered bodies (LAYER). Present step (KSTEP) and increment number (KINC).

Has to be updated by the UMAT: Stress (STRESS), SDVs (STATEV), and Jacobian (DDSDDE). Specific elastic strain energy (SSE), plastic dissipation (SPD) , creep dissipation (SCD). Predicted new (reduced) time increment (PNEWDT).

Utility Routines: SINV returns the fist and second invariant of a tensor. SPRINC returns principal values of a tensor. SPRIND returns principal orientations and values of a tensor. ROTSIG rotates a tensor with a rotation matrix. XIT terminates the calculation and closed all files.

Material implementation in ABAQUS: UMAT variables


Units: ABAQUS has no fixed unit system except for rotations (rad). Units have to be self- consitent.

Stresses and strains are stored in vectorial form:

• Plane stress state (PS):

• Plane strain state (PE)/ axisymmetric:

• 3D elements:

Shear strain: ABAQUS always uses engineering strains.

Deformation gradient: The deformation gradient is always stored as a 3x3 matrix.

1 1 2 2 1 2, ,

1 1 2 2 3 3 1 2, , ,

1 1 2 2 3 3 1 2 1 3 2 3, , , , ,

2i j i j j i i j

i jF

Material implementation in ABAQUS: UMAT variables


ABAQUS CAE

Part Module: Define part Property module:

Define material. With General>Depvar number of SDVs isdefined

General>User define material parameter forUser Subroutine.

Material implementation in ABAQUS: UMAT usage


Job module: Job definition with correct path to user subroutine.

Material implementation in ABAQUS: UMAT usage


The Jaumann incremental form is integrated into the scheme of corotational formulation:

, : Lames constants

Material implementation in ABAQUS: UMAT for kinematic hardening

Constitutive relations

Elasticity:

In Jaumann (Corotational) rate form:


:y Yield stress

:h Material property

, i j i jS : Deviatoric stress and strain tensor (back stress)

Constitutive Relation Plasticity:

Loading function with von MISES stress (J2)

Equivalent plastic strain rate

Plastic flow rule

Prager-Ziegler linear kinematic hardening in rate form:



Back stress tensor at increment start0 :i j

Integration procedure:

1. Elastic predictor: equivalent stress with purely elastic deformation is calculated:

2. If the equivalent predictor stress > yield stress plastic flow sets in.The backward Euler integration scheme is used for integration of the equations.

3. Equivalent plastic strain increment:



1 ( )e l e l e l e l p ln n n 1 1

e l e ln n σ C ε

Cel : Elastic stiffness matrix

Update of back stress, stress and strain tensor:

Direction of the plastic strain increment (normality condition)

Back stress tensor increment Plastic strain increment

Stress at increment end



Definition of a consistent Jacobian

Detailed documentation in the ABAQUS User’s manual

D D S D D E J

3(1 2 )Ek k: bulk modulus

Jacobian matrix



User Subroutine UMAT header and argument list:



Definition and memory allocation for local variables:



Definition of LAMEs constants and of the stiffness matrix (=Jacobian for elastic deformation):



Calling and rotating strain tensors (elastic and plastic) and of back stress tensors for large displacementsand deformations.

Renaming initial values for stress and strains (el. + pl.). Calculation of predictor stress and elastic strain.



Calculation of equivalent von MISES stress based on predictor stress. Comparison of equivalent von MISES stress with yield surface.



Calculation of deviatoric part of the predictor stress. Calculation of direction of the plastic strain increment with normality condition. Calculation of equivalent plastic strain increment with consistency condition ( equivalent stress

= yield stress).



Update of back stress tensor, elastic and plastic strain tensors and stress tensor for incremnt end. (Note *2for shear stain components since engineering definition is used in ABAQUS).

Calculation of plastic work or dissipation:

11 2 ( )( )p ln nS P D



Calculation of the consistent Jacobian (material tangent stiffness matrix):



Update of SDVs with values at the increment end. Output request write to *.log file(optional for debugging)



The Jaumann incremental form is integrated into the scheme of corotational formulation:

, : Lames constants

Material implementation in ABAQUS: UMAT for isotropic hardening

Constitutive relations

Elasticity:

In Jaumann (Corotational) rate form:

no difference to kinematic hardening


Loading function with vonMISES stress (J2):

equivalent plastic strain in rate form:

Yield rate:

:y Yield stress

:i jS Deviatoric stress

constitutive relation plasticity:



Hardening rule( ) :p ly

p l

( ) 0p lp l

ff d

Yield function:f

Integration procedure:

1. Elastic predictor: equivalent stress calculated with purely elastic deformation:

2. If the equivalent predictor stress > yield stress plastic flow.The backward Euler integration scheme is used for integration the equation.

3. In general non-linear equation in e.g. solved via Newton method



Algorithms for solving non-linear equations

1

( )

( )n

n nn

f xx x

f x

Newton Raphson method:

Newton Raphson method multi dimensional:1

1 ( ( ) ) ( )n n n nx x J x f x

1 1 1

1 2

2 2 1

1 2 1

1

1 1

( ) : ( )

n

n n

n

f f fx x x

f f fx x xi

nj

f ffx x x

fJ x f x

x

1: ( ( ) ) ( )n n nx J x f x

( ) ( )n n nJ x x f x

1n n nx x x

Jacobian-matrix:

LES:


1

( )

( )n

n nn

f xx x

f x

Newton Raphson method: Quasi Newton method:Modifizierte Newton Raphson method:

⁺ Iterative solution with identical tangent (from predictor step) in each increment.

⁺ Accurate results.⁺ Low computational cost within one

iteration.

⁻ Slow convergence.⁻ high number of iterations needed.

⁺ Very accurate results.⁺ Close to the solution quadratic

convergence.⁺ Only few iterations needed.

⁻ Iterative solution with adapted tangent.⁻ Large computational cost within one

iteration.

1

( )

( )n

n no

f xx x

f x

⁺ Inverse of the Hesse matrix is only approximated.

⁺ low computational cost⁺ Good convergence.

⁻ Approximation of the Hesse matrix can be costly.

⁻ Diverse approaches for the approximation.

1

( )

( )n

n nn

f xx x

B x


Arc-length method

⁺ Usable for calculating snap-through and snap-back problems. Also for singular tangent stiffness matrices in buckling / post-buckling⁺ Fast convergence for implicit solution and autonomous control.⁺ Can deal with local and global extrema without stopping.

⁻ More complicated solution, since the equation system becomes unsymmetrical band structure gets lost.⁻ Unusable for contact problems, visco-elastic material and bifurcation problems.

Extension of the classical Newton-Raphson method.Usage of arc-length for the approximation progress.Introduction of a loading parameter as pre-factor (Load step control)Displacements and loads are iterated simultaneously.

Algorithms for solving non-linear equations


1 ( )e l e l e l e l p ln n n 1 1

e l e ln n σ C ε

Cel : Elastic stiffness matrix

1( ) ( ) ( ( ) )p l k p l k p ld

Update of stress and strain tensor

Direction of the plastic strain increment (normality condition):

Equivalent plastic strain increment:

Plastic strain increment:

Stress at increment end:



D D S D D E J

3(1 2 )Ek k: bulk modulus

Definition if the consistent Jacobian

Detailed documentation in ABAQUS User’s manual

Jacobian



User Subroutine UMAT header and argument list:



Definition and memory allocation for local variables:



Definition of LAMEs constants and the stiffness matrix (=Jacobian for elastic deformations)



Materialimplementierungen in ABAQUS

Calling and rotating strain tensors (elastic and plastic) for large displacements and deformations. Calculating predictor stress and elastic strain Calculation of equivalent vonMises stress from predictor stress


Calculation of yield stress that correspond to the present equivalent plastic strain via the UHARD subroutine. Comparison of equivalent vonMISES stress with the present yield stress to check for yielding. Calculation of the flow direction.



UHARD: User subroutine for defining yield surfaces and hardening parameters for isotropic or combined hardeningmodels.

User Subroutine UHARD header and arguments

(1)

( 2 )

( 3 )

y

p l

y

p l

y

H a rd h

H a rd

H a rd



Definition of present yield values as function of the equivalent plastic strain and isotropic hardening parameters.



Calculation of equivalent plastic strain increments and extension of consecutive yield surfaces with the iterative NEWTONMethod and UHARD.

Plastic material behavior is defined as multi-linear curve by pairs of material strength and corresponding equivalent plasticstrain.

( ) 0plpl

ff d



11 2 ( )( )p ln nS P D S y S y

Update of the stress tensors, elastic and plastic strain tensor and equivalent plastic strain at incrementend.

Calculation of plastic work or dissipation:



Calculation of the consistent Jacobian (material tangent stiffness matrix)



Update of SDVs with values at the increment end. Output request output to *.log file (optional for debugging purposes)

Radial return mapping



Thank you for your attention.

09.09.2013 53

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Material implementation in ABAQUS