UANISOHYPER_INV

User subroutine to define anisotropic hyperelastic material behavior using the invariant formulation.

can be used to define the strain energy potential of anisotropic hyperelastic materials as a function of an irreducible set of scalar invariants;
is called at all material calculation points of elements for which the material definition contains user-defined anisotropic hyperelastic behavior with an invariant-based formulation (Anisotropic hyperelastic behavior);
can include material behavior dependent on field variables or state variables;
requires that the values of the derivatives of the strain energy density function of the anisotropic hyperelastic material be defined with respect to the scalar invariants; and
is called twice per material point in each iteration.

The following topics are discussed:

Enumeration of invariants
Storage of arrays of derivatives of the energy function
Special considerations for various element types
User subroutine interface
Variables to be defined
Variables passed in for information
Example: Anisotropic hyperelastic model of Kaliske and Schmidt
Additional reference

Enumeration of invariants

To facilitate coding and provide easy access to the array of invariants passed to user subroutine UANISOHYPER_INV, an enumerated representation of each invariant is introduced. Any scalar invariant can, therefore, be represented uniquely by an enumerated invariant, $I_{n}^{*}$ , where the subscript n denotes the order of the invariant according to the enumeration scheme in the following table:

Invariant	Enumeration, $n$
${\bar{I}}_{1}$	$1$
${\bar{I}}_{2}$	$2$
$J$	$3$
${\bar{I}}_{4 (α β)}$	$4 + 2 (α - 1) + β (β - 1); α \leq β$
${\bar{I}}_{5 (α β)}$	$5 + 2 (α - 1) + β (β - 1); α \leq β$

For example, in the case of three families of fibers there are a total of 15 invariants: ${\bar{I}}_{1}$ , ${\bar{I}}_{2}$ , $J$ , six invariants of type ${\bar{I}}_{4 (α β)}$ , and six invariants of type ${\bar{I}}_{5 (α β)}$ , with $α, β = 1, 2, 3 (α \leq β)$ . The following correspondence exists between each of these invariants and their enumerated counterpart:

Enumerated invariant	Invariant
$I_{1}^{*}$	${\bar{I}}_{1}$
$I_{2}^{*}$	${\bar{I}}_{2}$
$I_{3}^{*}$	$J$
$I_{4}^{*}$	${\bar{I}}_{4 (11)}$
$I_{5}^{*}$	${\bar{I}}_{5 (11)}$
$I_{6}^{*}$	${\bar{I}}_{4 (12)}$
$I_{7}^{*}$	${\bar{I}}_{5 (12)}$
$I_{8}^{*}$	${\bar{I}}_{4 (22)}$
$I_{9}^{*}$	${\bar{I}}_{5 (22)}$
$I_{10}^{*}$	${\bar{I}}_{4 (13)}$
$I_{11}^{*}$	${\bar{I}}_{5 (13)}$
$I_{12}^{*}$	${\bar{I}}_{4 (23)}$
$I_{13}^{*}$	${\bar{I}}_{5 (23)}$
$I_{14}^{*}$	${\bar{I}}_{4 (33)}$
$I_{15}^{*}$	${\bar{I}}_{5 (33)}$

A similar scheme is used for the array ZETA of terms $ζ_{α β} = A_{α} \cdot A_{β}$ . Each term can be represented uniquely by an enumerated counterpart $ζ_{m}^{*}$ , as shown below:

Dot product	Enumeration, $m$
$ζ_{α β}$	$α + \frac{1}{2} (β - 2) (β - 1); α < β$

As an example, for the case of three families of fibers there are three $ζ_{α β}$ terms: $ζ_{12}$ , $ζ_{13}$ , and $ζ_{23}$ . These are stored in the ZETA array as $(ζ_{1}^{*}, ζ_{2}^{*}, ζ_{3}^{*})$ .

Storage of arrays of derivatives of the energy function

The components of the array UI1 of first derivatives of the strain energy potential with respect to the scalar invariants, $\partial U / \partial I_{i}^{*}$ , are stored using the enumeration scheme discussed above for the scalar invariants.

The elements of the array UI2 of second derivatives of the strain energy function, $\partial^{2} U / \partial I_{i}^{*} \partial I_{j}^{*}$ , are laid out in memory using triangular storage: if $k$ denotes the component in this array corresponding to the term $\partial^{2} U / \partial I_{i}^{*} \partial I_{j}^{*}$ , then $k = i + j \times (j - 1) / 2; (i \leq j)$ . For example, the term $\partial^{2} U / \partial I_{2}^{*} \partial I_{5}^{*}$ is stored in component $k = 2 + (5 \times 4) / 2 = 12$ in the UI2 array.

Special considerations for various element types

There are several special considerations that need to be noted.

Shells that calculate transverse shear energy

When UANISOHYPER_INV is used to define the material response of shell elements that calculate transverse shear energy, Abaqus/Standard 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 for guidelines on choosing this stiffness.

Elements with hourglassing modes

When UANISOHYPER_INV is used to define the material response of elements with hourglassing modes, you must define the hourglass stiffness for hourglass control based on the total stiffness approach. The hourglass stiffness is not required for enhanced hourglass control, but you can define a scaling factor for the stiffness associated with the drill degree of freedom (rotation about the surface normal). See Section controls.

User subroutine interface

      SUBROUTINE UANISOHYPER_INV (AINV, UA, ZETA, NFIBERS, NINV,
     1     UI1, UI2, UI3, TEMP, NOEL, CMNAME, INCMPFLAG, IHYBFLAG,
     2     NUMSTATEV, STATEV, NUMFIELDV, FIELDV, FIELDVINC,
     3     NUMPROPS, PROPS)
C
	     INCLUDE 'ABA_PARAM.INC' 
C
      CHARACTER*80 CMNAME
      DIMENSION AINV(NINV), UA(2), 
     2     ZETA(NFIBERS*(NFIBERS-1)/2)), UI1(NINV),
     3     UI2(NINV*(NINV+1)/2), UI3(NINV*(NINV+1)/2),
     4     STATEV(NUMSTATEV), FIELDV(NUMFIELDV),
     5     FIELDVINC(NUMFIELDV), PROPS(NUMPROPS)


       user coding to define UA,UI1,UI2,UI3,STATEV


      RETURN
      END

Variables to be defined

UA(1): U, strain energy density function. For a compressible material at least one derivative involving J should be nonzero. For an incompressible material all derivatives involving J are ignored.
UA(2): ${\tilde{U}}_{d e v}$ , the deviatoric part of the strain energy density of the primary material response. This quantity is needed only if the current material definition also includes Mullins effect (see Mullins effect).
UI1(NINV): Array of derivatives of strain energy potential with respect to the scalar invariants, $\partial U / \partial I_{i}^{*}$ , ordered using the enumeration scheme discussed above.
UI2(NINV*(NINV+1)/2): Array of second derivatives of strain energy potential with respect to the scalar invariants (using triangular storage), $\partial^{2} U / \partial I_{i}^{*} \partial I_{j}^{*}$ .
UI3(NINV*(NINV+1)/2): Array of derivatives with respect to J of the second derivatives of the strain energy potential (using triangular storage), $\partial^{3} U / \partial I_{i}^{*} \partial I_{j}^{*} \partial J$ . This quantity is needed only for compressible materials with a hybrid formulation (when INCMPFLAG = 0 and IHYBFLAG = 1).
STATEV: Array containing the user-defined solution-dependent state variables at this point. These are supplied as values at the start of the increment or as values updated by other user subroutines (see About user subroutines and utilities) and must be returned as values at the end of the increment.

Variables passed in for information

NFIBERS: Number of families of fibers defined for this material.
NINV: Number of scalar invariants.
TEMP: Current temperature at this point.
NOEL: Element number.
CMNAME: User-specified material name, left justified.
INCMPFLAG: Incompressibility flag defined to be 1 if the material is specified as incompressible or 0 if the material is specified as compressible.
IHYBFLAG: Hybrid formulation flag defined to be 1 for hybrid elements; 0 otherwise.
NUMSTATEV: User-defined number of solution-dependent state variables associated with this material (see Allocating space).
NUMFIELDV: Number of field variables.
FIELDV: Array of interpolated values of predefined field variables at this material point at the end of the increment based on the values read in at the nodes (initial values at the beginning of the analysis and current values during the analysis).
FIELDVINC: Array of increments of predefined field variables at this material point for this increment, including any values updated by user subroutine USDFLD.
NUMPROPS: Number of material properties entered for this user-defined hyperelastic material.
PROPS: Array of material properties entered for this user-defined hyperelastic material.
AINV(NINV): Array of scalar invariants, $I_{i}^{*}$ , at each material point at the end of the increment. The invariants are ordered using the enumeration scheme discussed above.
ZETA(NFIBERS*(NFIBERS-1)/2)): Array of dot product between the directions of different families of fiber in the reference configuration, $ζ_{α β} = A_{α} \cdot A_{β}$ . The array contains the enumerated values $ζ_{m}^{*}$ using the scheme discussed above.

Example: Anisotropic hyperelastic model of Kaliske and Schmidt

As an example of the coding of user subroutine UANISOHYPER_INV, consider the model proposed by Kaliske and Schmidt (2005) for nonlinear anisotropic elasticity with two families of fibers. The strain energy function is given by a polynomial series expansion in the form

\begin{matrix} U = & \frac{1}{D} {(J - 1)}^{2} + \sum_{i = 1}^{3} a_{i} {({\bar{I}}_{1} - 3)}^{i} + \sum_{j = 1}^{3} b_{j} {({\bar{I}}_{2} - 3)}^{j} + \sum_{k = 2}^{6} c_{k} {({\bar{I}}_{4 (11)} - 1)}^{k} + \sum_{l = 2}^{6} d_{l} {({\bar{I}}_{5 (11)} - 1)}^{l} \\ + \sum_{m = 2}^{6} e_{m} {({\bar{I}}_{4 (22)} - 1)}^{m} + \sum_{n = 2}^{6} f_{n} {({\bar{I}}_{5 (22)} - 1)}^{n} + \sum_{p = 2}^{6} g_{p} {(ζ_{12} {\bar{I}}_{4 (12)} - ζ_{12}^{2})}^{p} . \end{matrix}

The code in user subroutine UANISOHYPER_INV must return the derivatives of the strain energy function with respect to the scalar invariants, which are readily computed from the above expression. In this example auxiliary functions are used to facilitate enumeration of pseudo-invariants of type ${\bar{I}}_{4 (α β)}$ and ${\bar{I}}_{5 (α β)}$ , as well as for indexing into the array of second derivatives using symmetric storage. The user subroutine would be coded as follows:

      subroutine uanisohyper_inv (aInv, ua, zeta, nFibers, nInv,
     *                            ui1, ui2, ui3, temp, noel,
     *                            cmname, incmpFlag, ihybFlag,
     *                            numStatev, statev,
     *                            numFieldv, fieldv, fieldvInc,
     *                            numProps, props)
C
      include 'aba_param.inc'
C
      character *80 cmname
      dimension aInv(nInv), ua(2), zeta(nFibers*(nFibers-1)/2)
      dimension ui1(nInv), ui2(nInv*(nInv+1)/2)
      dimension ui3(nInv*(nInv+1)/2), statev(numStatev)
      dimension fieldv(numFieldv), fieldvInc(numFieldv)
      dimension props(numProps)
C
      parameter ( zero  = 0.d0,
     *            one   = 1.d0,
     *            two   = 2.d0,
     *            three = 3.d0,
     *            four  = 4.d0,
     *            five  = 5.d0,
     *            six   = 6.d0 )
C
C Kaliske-Schmidtt energy function (3D)
C
C     Read material properties
      d=props(1)
      dInv = one / d
      a1=props(2)
      a2=props(3)
      a3=props(4)
      b1=props(5)
      b2=props(6)
      b3=props(7)
      c2=props(8)
      c3=props(9)
      c4=props(10)
      c5=props(11)
      c6=props(12)
      d2=props(13)
      d3=props(14)
      d4=props(15)
      d5=props(16)
      d6=props(17)
      e2=props(18)
      e3=props(19)
      e4=props(20)
      e5=props(21)
      e6=props(22)
      f2=props(23)
      f3=props(24)
      f4=props(25)
      f5=props(26)
      f6=props(27)
      g2=props(28)
      g3=props(29)
      g4=props(30)
      g5=props(31)
      g6=props(32)
C
C     Compute Udev and 1st and 2nd derivatives w.r.t invariants
C     - I1
      bi1 = aInv(1)
      term = bi1-three
      ua(2)  = a1*term + a2*term**2 + a3*term**3
      ui1(1) = a1 + two*a2*term + three*a3*term**2
      ui2(indx(1,1)) = two*a2 + three*two*a3*term
C     - I2       
      bi2 = aInv(2)
      term = bi2-three
      ua(2)  = ua(2) + b1*term + b2*term**2 + b3*term**3
      ui1(2) = b1 + two*b2*term + three*b3*term**2
      ui2(indx(2,2)) = two*b2 + three*two*b3*term
C     - I3 (=J)
      bi3 = aInv(3)
      term = bi3-one
      ui1(3) = two*dInv*term
      ui2(indx(3,3)) = two*dInv
C     - I4(11)
      nI411 = indxInv4(1,1)
      bi411 = aInv(nI411)
      term = bi411-one
      ua(2) = ua(2)
     *      + c2*term**2 + c3*term**3 + c4*term**4
     *      + c5*term**5 + c6*term**6
      ui1(nI411) =
     *          two*c2*term
     *        + three*c3*term**2
     *        + four*c4*term**3
     *        + five*c5*term**4
     *        + six*c6*term**5
      ui2(indx(nI411,nI411)) =
     *          two*c2
     *        + three*two*c3*term
     *        + four*three*c4*term**2
     *        + five*four*c5*term**3
     *        + six*five*c6*term**4
C     - I5(11)
      nI511 = indxInv5(1,1)
      bi511 = aInv(nI511)
      term = bi511-one
      ua(2) = ua(2)
     *      + d2*term**2 + d3*term**3 + d4*term**4
     *      + d5*term**5 + d6*term**6
      ui1(nI511) =
     *          two*d2*term
     *        + three*d3*term**2
     *        + four*d4*term**3
     *        + five*d5*term**4
     *        + six*d6*term**5
      ui2(indx(nI511,nI511)) =
     *          two*d2
     *        + three*two*d3*term
     *        + four*three*d4*term**2
     *        + five*four*d5*term**3
     *        + six*five*d6*term**4
C     - I4(22)
      nI422 = indxInv4(2,2)
      bi422 = aInv(nI422)
      term = bi422-one
      ua(2) = ua(2)
     *      + e2*term**2 + e3*term**3 + e4*term**4
     *        + e5*term**5 + e6*term**6
      ui1(nI422) =
     *          two*e2*term
     *        + three*e3*term**2
     *        + four*e4*term**3
     *        + five*e5*term**4
     *        + six*e6*term**5
      ui2(indx(nI422,nI422)) =
     *          two*e2
     *        + three*two*e3*term
     *        + four*three*e4*term**2
     *        + five*four*e5*term**3
     *        + six*five*e6*term**4
C     - I5(22)
      nI522 = indxInv5(2,2)
      bi522 = aInv(nI522)
      term = bi522-one
      ua(2) = ua(2)
     *      + f2*term**2 + f3*term**3 + f4*term**4
     *        + f5*term**5 + f6*term**6
      ui1(nI522) =
     *          two*f2*term
     *        + three*f3*term**2
     *        + four*f4*term**3
     *        + five*f5*term**4
     *        + six*f6*term**5
      ui2(indx(nI522,nI522)) =
     *          two*f2
     *        + three*two*f3*term
     *        + four*three*f4*term**2
     *        + five*four*f5*term**3
     *        + six*five*f6*term**4
C     - I4(12)
      nI412 = indxInv4(1,2)
      bi412 = aInv(nI412)
      term = zeta(1)*(bi412-zeta(1))
      ua(2) = ua(2)
     *        + g2*term**2 + g3*term**3
     *        + g4*term**4 + g5*term**5
     *        + g6*term**6
      ui1(nI412) = zeta(1) * (
     *          two*g2*term
     *        + three*g3*term**2
     *        + four*g4*term**3
     *        + five*g5*term**4
     *        + six*g6*term**5 )
      ui2(indx(nI412,nI412)) = zeta(1)**2 * (
     *          two*g2
     *        + three*two*g3*term
     *        + four*three*g4*term**2
     *        + five*four*g5*term**3
     *        + six*five*g6*term**4 )
C
C Add volumetric energy
C
      term = aInv(3) - one
      ua(1) = ua(2) + dInv*term*term
C
      return
      end
C
C Maps index from Square to Triangular storage of symmetric 
C matrix 
C
      integer function indx( i, j )
C
      include 'aba_param.inc'
C
      ii = min(i,j)
      jj = max(i,j)
C
      indx = ii + jj*(jj-1)/2
C
      return
      end
C
C
C Generate enumeration of Anisotropic Pseudo Invariants of 
C type 4 
C
      integer function indxInv4( i, j )
C     
      include 'aba_param.inc'
C
      ii = min(i,j)
      jj = max(i,j)
C     
      indxInv4 = 4 + jj*(jj-1) + 2*(ii-1)
C
      return
      end
C
C
C Generate enumeration of Anisotropic Pseudo Invariants of 
C type 5 
C
      integer function indxInv5( i, j )
C
      include 'aba_param.inc'
C
      ii = min(i,j)
      jj = max(i,j)
C     
      indxInv5 = 5 + jj*(jj-1) + 2*(ii-1)
C
      return
      end

Additional reference

Kaliske, M., and J. Schmidt, “Formulation of Finite Nonlinear Anisotropic Elasticity,” CADFEM GmbH Infoplaner 2/2005, vol. 2, pp. 22–23, 2005.