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VUMAT
1.2.17 VUMAT: User subroutine to define material behavior.
Product: Abaqus/Explicit
WARNING: The use of this user subroutine generally requires considerable
expertise. You are cautioned that the implementation of any realistic constitutive
model requires extensive development and testing. Initial testing on a single-element
model with prescribed traction loading is strongly recommended.
The component ordering of the symmetric and nonsymmetric tensors for
the three-dimensional case using C3D8R elements is different from the ordering
specified in “Three-dimensional solid element library,” Section 25.1.4 of the Abaqus
Analysis User’s Manual, and the ordering used in Abaqus/Standard.
References
• “User-defined mechanical material behavior,” Section 23.8.1 of the Abaqus Analysis User’s Manual
• *USER MATERIAL
Overview
User subroutine VUMAT:
• is used to define the mechanical constitutive behavior of a material;
• will be called for blocks of material calculation points for which the material is defined in a user
subroutine (“Material data definition,” Section 18.1.2 of the Abaqus Analysis User’s Manual);
• can use and update solution-dependent state variables;
• can use any field variables that are passed in; and
• can be used in an adiabatic analysis, provided you define both the inelastic heat fraction and the
specific heat for the appropriate material definitions and you store the temperatures and integrate
them as user-defined state variables.
Component ordering in tensors
The component ordering depends upon whether the tensor is symmetric or nonsymmetric.
Symmetric tensors
For symmetric tensors such as the stress and strain tensors, there are ndir+nshr components, and the
component order is given as a natural permutation of the indices of the tensor. The direct components are
first and then the indirect components, beginning with the 12-component. For example, a stress tensor
contains ndir direct stress components and nshr shear stress components, which are passed in as
1.2.17–1
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VUMAT
Component 2-D Case 3-D Case
1
2
3
4
5
6
The shear strain components in user subroutine VUMAT are stored as tensor components and not as
engineering components; this is different from user subroutine UMAT in Abaqus/Standard, which uses
engineering components.
Nonsymmetric tensors
For nonsymmetric tensors there are ndir+2*nshr components, and the component order is given as
a natural permutation of the indices of the tensor. The direct components are first and then the indirect
components, beginning with the 12-component. For example, the deformation gradient is passed as
Component 2-D Case 3-D Case
1
2
3
4
5
6
7
8
9
Initial calculations and checks
In the data check phase of the analysis Abaqus/Explicit calls user subroutineVUMATwith a set of fictitious
strains and a totalTime and stepTime both equal to 0.0. This is done as a check on your constitutive
relation and to calculate the equivalent initial material properties, based upon which the initial elastic
wave speeds are computed.
1.2.17–2
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VUMAT
Defining local orientations
All stresses, strains, stretches, and state variables are in the orientation of the local material axes. These
local material axes form a basis system in which stress and strain components are stored. This represents
a corotational coordinate system in which the basis system rotates with the material. If a user-specified
coordinate system (“Orientations,” Section 2.2.5 of the Abaqus Analysis User’s Manual) is used, it
defines the local material axes in the undeformed configuration.
Special considerations for various element types
The use of user subroutine VUMAT requires special consideration for various element types.
Shell and plane stress elements
You must define the stresses and internal state variables. In the case of shell or plane stress elements,
NDIR=3 and NSHR=1; you must define strainInc(*,3), the thickness strain increment. The
internal energies can be defined if desired. If they are not defined, the energy balance provided by
Abaqus/Explicit will not be meaningful.
Shell elements
When VUMAT is used to define the material response of shell elements, Abaqus/Explicit cannot calculate
a default value for the transverse shear stiffness of the element. Hence, you must define the element’s
transverse shear stiffness. See “Shell section behavior,” Section 26.6.4 of the Abaqus Analysis User’s
Manual, for guidelines on choosing this stiffness.
Beam elements
For beam elements the stretch tensor and the deformation gradient tensor are not available. For
beams in space you must define the thickness strains, strainInc(*,2) and strainInc(*,3).strainInc(*,4) is the shear strain associated with twist. Thickness stresses, stressNew(*,2)and stressNew(*,3), are assumed to be zero, and any values you assign are ignored.
Pipe elements
For pipe elements the stretch tensor and the deformation gradient tensor are not available. The axial
strain, strainInc(*,1), and the shear strain, strainInc(*,4), associated with twist are
provided along with the hoop stress, stressNew(*,2). The hoop stress is predefined based on your
pipe internal and external pressure load definitions (PE, PI, HPE, HPI, PENU, and PINU), and it should
not be modified here. The thickness stress, stressNew(*,3), is assumed to be zero and any value
you assign is ignored. You must define the axial stress, stressNew(*,1), and the shear stress,
stressNew(*,4). You must also define hoop strain, strainInc(*,2), and the pipe thickness
strain, strainInc(*,3).
1.2.17–3
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VUMAT
Deformation gradient
The polar decomposition of the deformation gradient is written as , where and
are the right and left symmetric stretch tensors, respectively. The constitutive model is defined in a
corotational coordinate system in which the basis system rotates with the material. All stress and strain
tensor quantities are defined with respect to the corotational basis system. The right stretch tensor, , is
used. The relative spin tensor represents the spin (the antisymmetric part of the velocity gradient)
defined with respect to the corotational basis system.
Special considerations for hyperelasticity
Hyperelastic constitutive models in VUMAT should be defined in a corotational coordinate system in
which the basis system rotates with the material. This is most effectively accomplished by formulating
the hyperelastic constitutive model in terms of the stretch tensor, , instead of in terms of the
deformation gradient, . Using the deformation gradient can present some difficulties because
the deformation gradient includes the rotation tensor and the resulting stresses would need to be rotated
back to the corotational basis.
Objective stress rates
The Green-Naghdi stress rate is used when the mechanical behavior of the material is defined using user
subroutine VUMAT. The stress rate obtained with user subroutine VUMAT may differ from that obtained
with a built-in Abaqus material model. For example, most material models used with solid (continuum)
elements in Abaqus/Explicit employ the Jaumann stress rate. This difference in the formulation will
cause significant differences in the results only if finite rotation of a material point is accompanied by
finite shear. For a discussion of the objective stress rates used in Abaqus, see “Stress rates,” Section 1.5.3
of the Abaqus Theory Manual.
Material point deletion
Material points that satisfy a user-defined failure criterion can be deleted from the model (see “User-
defined mechanical material behavior,” Section 23.8.1 of the Abaqus Analysis User’s Manual). You
must specify the state variable number controlling the element deletion flag when you allocate space
for the solution-dependent state variables, as explained in “User-defined mechanical material behavior,”
Section 23.8.1 of the Abaqus Analysis User’s Manual. The deletion state variable should be set to a value
of one or zero in VUMAT. A value of one indicates that the material point is active, while a value of zero
indicates that Abaqus/Explicit should delete the material point from the model by setting the stresses to
zero. The structure of the block of material points passed to user subroutine VUMAT remains unchanged
during the analysis; deleted material points are not removed from the block. Abaqus/Explicit will pass
zero stresses and strain increments for all deleted material points. Once a material point has been flagged
as deleted, it cannot be reactivated.
1.2.17–4
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VUMAT
User subroutine interface
subroutine vumat(C Read only (unmodifiable)variables -
1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal,2 stepTime, totalTime, dt, cmname, coordMp, charLength,3 props, density, strainInc, relSpinInc,4 tempOld, stretchOld, defgradOld, fieldOld,5 stressOld, stateOld, enerInternOld, enerInelasOld,6 tempNew, stretchNew, defgradNew, fieldNew,
C Write only (modifiable) variables -7 stressNew, stateNew, enerInternNew, enerInelasNew )
Cinclude 'vaba_param.inc'
Cdimension props(nprops), density(nblock), coordMp(nblock,*),
1 charLength(nblock), strainInc(nblock,ndir+nshr),2 relSpinInc(nblock,nshr), tempOld(nblock),3 stretchOld(nblock,ndir+nshr),4 defgradOld(nblock,ndir+nshr+nshr),5 fieldOld(nblock,nfieldv), stressOld(nblock,ndir+nshr),6 stateOld(nblock,nstatev), enerInternOld(nblock),7 enerInelasOld(nblock), tempNew(nblock),8 stretchNew(nblock,ndir+nshr),8 defgradNew(nblock,ndir+nshr+nshr),9 fieldNew(nblock,nfieldv),1 stressNew(nblock,ndir+nshr), stateNew(nblock,nstatev),2 enerInternNew(nblock), enerInelasNew(nblock),
Ccharacter*80 cmname
C
do 100 km = 1,nblockuser coding
100 continue
returnend
1.2.17–5
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VUMAT
Variables to be defined
stressNew (nblock, ndir+nshr)Stress tensor at each material point at the end of the increment.
stateNew (nblock, nstatev)State variables at each material point at the end of the increment. You define the size of this array by
allocating space for it (see “User subroutines: overview,” Section 15.1.1 of the Abaqus Analysis User’s
Manual, for more information).
Variables that can be updated
enerInternNew (nblock)Internal energy per unit mass at each material point at the end of the increment.
enerInelasNew (nblock)Dissipated inelastic energy per unit mass at each material point at the end of the increment.
Variables passed in for information
nblockNumber of material points to be processed in this call to VUMAT.
ndirNumber of direct components in a symmetric tensor.
nshrNumber of indirect components in a symmetric tensor.
nstatevNumber of user-defined state variables that are associated with this material type (you define this
as described in “Allocating space” in “User subroutines: overview,” Section 15.1.1 of the Abaqus
Analysis User’s Manual).
nfieldvNumber of user-defined external field variables.
npropsUser-specified number of user-defined material properties.
lannealFlag indicating whether the routine is being called during an annealing process. lanneal=0 indicatesthat the routine is being called during a normal mechanics increment. lanneal=1 indicates that this isan annealing process and you should re-initialize the internal state variables, stateNew, if necessary.Abaqus/Explicit will automatically set the stresses, stretches, and state to a value of zero during the
annealing process.
1.2.17–6
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VUMAT
stepTimeValue of time since the step began.
totalTimeValue of total time. The time at the beginning of the step is given by totalTime - stepTime.
dtTime increment size.
cmnameUser-specified material name, left justified. It is passed in as an uppercase character string. Some
internal material models are given names starting with the “ABQ_” character string. To avoid conflict,
you should not use “ABQ_” as the leading string for cmname.
coordMp(nblock,*)Material point coordinates. It is the midplane material point for shell elements and the centroid for
beam and pipe elements.
charLength(nblock)Characteristic element length. This is a typical length of a line across an element for a first-order
element; it is half of the same typical length for a second-order element. For beams, pipes, and trusses,
it is a characteristic length along the element axis. For membranes and shells it is a characteristic length
in the reference surface. For axisymmetric elements it is a characteristic length in the r–z plane only.
For cohesive elements it is equal to the constitutive thickness.
props(nprops)User-supplied material properties.
density(nblock)Current density at the material points in the midstep configuration. This value may be inaccurate in
problems where the volumetric strain increment is very small. If an accurate value of the density is
required in such cases, the analysis should be run in double precision. This value of the density is not
affected by mass scaling.
strainInc (nblock, ndir+nshr)Strain increment tensor at each material point.
relSpinInc (nblock, nshr)Incremental relative rotation vector at each material point defined in the corotational system. Defined
as , where is the antisymmetric part of the velocity gradient, , and .
Stored in 3-D as and in 2-D as .
tempOld(nblock)Temperatures at each material point at the beginning of the increment.
1.2.17–7
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VUMAT
stretchOld (nblock, ndir+nshr)Stretch tensor, , at each material point at the beginning of the increment defined from the polar
decomposition of the deformation gradient by .
defgradOld (nblock,ndir+2*nshr)Deformation gradient tensor at each material point at the beginning of the increment. Stored in 3-D as
( , , , , , , , , ) and in 2-D as ( , , , , ).
fieldOld (nblock, nfieldv)Values of the user-defined field variables at each material point at the beginning of the increment.
stressOld (nblock, ndir+nshr)Stress tensor at each material point at the beginning of the increment.
stateOld (nblock, nstatev)State variables at each material point at the beginning of the increment.
enerInternOld (nblock)Internal energy per unit mass at each material point at the beginning of the increment.
enerInelasOld (nblock)Dissipated inelastic energy per unit mass at each material point at the beginning of the increment.
tempNew(nblock)Temperatures at each material point at the end of the increment.
stretchNew (nblock, ndir+nshr)Stretch tensor, , at each material point at the end of the increment defined from the polar
decomposition of the deformation gradient by .
defgradNew (nblock,ndir+2*nshr)Deformation gradient tensor at each material point at the end of the increment. Stored in 3-D as ( ,
, , , , , , , ) and in 2-D as ( , , , , ).
fieldNew (nblock, nfieldv)Values of the user-defined field variables at each material point at the end of the increment.
Example: Using more than one user-defined material model
To use more than one user-defined material model, the variable cmname can be tested for different
material names inside user subroutine VUMAT, as illustrated below:if (cmname(1:4) .eq. 'MAT1') then
call VUMAT_MAT1(argument_list)else if (cmname(1:4) .eq. 'MAT2') then
call VUMAT_MAT2(argument_list)end if
1.2.17–8
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VUMAT
VUMAT_MAT1 and VUMAT_MAT2 are the actual user material subroutines containing the constitutive
material models for each material MAT1 and MAT2, respectively. Subroutine VUMAT merely acts as a
directory here. The argument list can be the same as that used in subroutine VUMAT. The material names
must be in uppercase characters since cmname is passed in as an uppercase character string.
Example: Elastic/plastic material with kinematic hardening
As a simple example of the coding of subroutine VUMAT, consider the generalized plane strain case for
an elastic/plastic material with kinematic hardening. The basic assumptions and definitions of the model
are as follows.
Let be the current value of the stress, and define to be the deviatoric part of the stress. The
center of the yield surface in deviatoric stress space is given by the tensor , which has initial values of
zero. The stress difference, , is the stress measured from the center of the yield surface and is given by
The von Mises yield surface is defined as
where is the uniaxial equivalent yield stress. The von Mises yield surface is a cylinder in deviatoric
stress space with a radius of
For the kinematic hardening model, R is a constant. The normal to the Mises yield surface can be written
as
We decompose the strain rate into an elastic and plastic part using an additive decomposition:
The plastic part of the strain rate is given by a normality condition
where the scalar multiplier must be determined. A scalar measure of equivalent plastic strain rate is
defined by
1.2.17–9
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VUMAT
The stress rate is assumed to be purely due to the elastic part of the strain rate and is expressed in terms
of Hooke’s law by
where and are the Lamés constants for the material.
The evolution law for is given as
where H is the slope of the uniaxial yield stress versus plastic strain curve.
During active plastic loading the stress must remain on the yield surface, so that
The equivalent plastic strain rate is related to by
The kinematic hardening constitutive model is integrated in a rate form as follows. A trial elastic
stress is computed as
where the subscripts and refer to the beginning and end of the increment, respectively. If the
trial stress does not exceed the yield stress, the new stress is set equal to the trial stress. If the yield
stress is exceeded, plasticity occurs in the increment. We then write the incremental analogs of the rate
equations as
where
From the definition of the normal to the yield surface at the end of the increment, ,
1.2.17–10
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VUMAT
This can be expanded using the incremental equations as
Taking the tensor product of this equation with , using the yield condition at the end of the increment,
and solving for :
The value for is used in the incremental equations to determine , , and .
This algorithm is often referred to as an elastic predictor, radial return algorithm because the
correction to the trial stress under the active plastic loading condition returns the stress state to the yield
surface along the direction defined by the vector from the center of the yield surface to the elastic trial
stress. The subroutine would be coded as follows:
subroutine vumat(C Read only -
1 nblock, ndir, nshr, nstatev, nfieldv, nprops, lanneal,2 stepTime, totalTime, dt, cmname, coordMp, charLength,3 props, density, strainInc, relSpinInc,4 tempOld, stretchOld, defgradOld, fieldOld,3 stressOld, stateOld, enerInternOld, enerInelasOld,6 tempNew, stretchNew, defgradNew, fieldNew,
C Write only -5 stressNew, stateNew, enerInternNew, enerInelasNew )
Cinclude 'vaba_param.inc'
CC J2 Mises Plasticity with kinematic hardening for planeC strain case.C Elastic predictor, radial corrector algorithm.CC The state variables are stored as:C STATE(*,1) = back stress component 11C STATE(*,2) = back stress component 22C STATE(*,3) = back stress component 33C STATE(*,4) = back stress component 12C STATE(*,5) = equivalent plastic strain
1.2.17–11
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VUMAT
CCC All arrays dimensioned by (*) are not used in this algorithm
dimension props(nprops), density(nblock),1 coordMp(nblock,*),2 charLength(*), strainInc(nblock,ndir+nshr),3 relSpinInc(*), tempOld(*),4 stretchOld(*), defgradOld(*),5 fieldOld(*), stressOld(nblock,ndir+nshr),6 stateOld(nblock,nstatev), enerInternOld(nblock),7 enerInelasOld(nblock), tempNew(*),8 stretchNew(*), defgradNew(*), fieldNew(*),9 stressNew(nblock,ndir+nshr), stateNew(nblock,nstatev),1 enerInternNew(nblock), enerInelasNew(nblock)
Ccharacter*80 cmname
Cparameter( zero = 0., one = 1., two = 2., three = 3.,
1 third = one/three, half = .5, twoThirds = two/three,2 threeHalfs = 1.5 )
Ce = props(1)xnu = props(2)yield = props(3)hard = props(4)
Ctwomu = e / ( one + xnu )thremu = threeHalfs * twomusixmu = three * twomualamda = twomu * ( e - twomu ) / ( sixmu - two * e )term = one / ( twomu * ( one + hard/thremu ) )con1 = sqrt( twoThirds )
Cdo 100 i = 1,nblock
CC Trial stress
trace = strainInc(i,1) + strainInc(i,2) + strainInc(i,3)sig1 = stressOld(i,1) + alamda*trace + twomu*strainInc(i,1)sig2 = stressOld(i,2) + alamda*trace + twomu*strainInc(i,2)sig3 = stressOld(i,3) + alamda*trace + twomu*strainInc(i,3)sig4 = stressOld(i,4) + twomu*strainInc(i,4)
C
1.2.17–12
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C Trial stress measured from the back stresss1 = sig1 - stateOld(i,1)s2 = sig2 - stateOld(i,2)s3 = sig3 - stateOld(i,3)s4 = sig4 - stateOld(i,4)
CC Deviatoric part of trial stress measured from the back stress
smean = third * ( s1 + s2 + s3 )ds1 = s1 - smeands2 = s2 - smeands3 = s3 - smean
CC Magnitude of the deviatoric trial stress difference
dsmag = sqrt( ds1**2 + ds2**2 + ds3**2 + 2.*s4**2 )CC Check for yield by determining the factor for plasticity,C zero for elastic, one for yield
radius = con1 * yieldfacyld = zeroif( dsmag - radius .ge. zero ) facyld = one
CC Add a protective addition factor to prevent a divide by zeroC when dsmag is zero. If dsmag is zero, we will not have exceededC the yield stress and facyld will be zero.
dsmag = dsmag + ( one - facyld )CC Calculated increment in gamma (this explicitly includes theC time step)
diff = dsmag - radiusdgamma = facyld * term * diff
CC Update equivalent plastic strain
deqps = con1 * dgammastateNew(i,5) = stateOld(i,5) + deqps
CC Divide dgamma by dsmag so that the deviatoric stresses areC explicitly converted to tensors of unit magnitude in theC following calculations
dgamma = dgamma / dsmagCC Update back stress
factor = hard * dgamma * twoThirds
1.2.17–13
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VUMAT
stateNew(i,1) = stateOld(i,1) + factor * ds1stateNew(i,2) = stateOld(i,2) + factor * ds2stateNew(i,3) = stateOld(i,3) + factor * ds3stateNew(i,4) = stateOld(i,4) + factor * s4
CC Update the stress
factor = twomu * dgammastressNew(i,1) = sig1 - factor * ds1stressNew(i,2) = sig2 - factor * ds2stressNew(i,3) = sig3 - factor * ds3stressNew(i,4) = sig4 - factor * s4
CC Update the specific internal energy -
stressPower = half * (1 ( stressOld(i,1)+stressNew(i,1) )*strainInc(i,1)1 + ( stressOld(i,2)+stressNew(i,2) )*strainInc(i,2)1 + ( stressOld(i,3)+stressNew(i,3) )*strainInc(i,3)1 + two*( stressOld(i,4)+stressNew(i,4) )*strainInc(i,4) )
CenerInternNew(i) = enerInternOld(i)
1 + stressPower / density(i)CC Update the dissipated inelastic specific energy -
plasticWorkInc = dgamma * half * (1 ( stressOld(i,1)+stressNew(i,1) )*ds11 + ( stressOld(i,2)+stressNew(i,2) )*ds21 + ( stressOld(i,3)+stressNew(i,3) )*ds31 + two*( stressOld(i,4)+stressNew(i,4) )*s4 )
enerInelasNew(i) = enerInelasOld(i)1 + plasticWorkInc / density(i)
100 continueC
returnend
1.2.17–14
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