ANSYS_NeoHook_usermat

A large deformation Neo–Hookean user material in ANSYS. ANSYS, Inc. 275 Technology Drive, Canonsburg, PA 15317 December 2008 Abstract The development and implementation of a 3–dimensional, large deformation, hyperelastic user material routine for ANSYS is presented. The constitutive model development follows the well–known Neo–Hookean hyperelastic formulation which yields the stress and material tangent in the spatial configuration. The model is recast in the corotated frame required by ANSYS. An algorithm for determination of the corotated frame rotation via the polar decomposition of the deformation gradient is also presented along with the conversion of the stress, material tangent and change of basis operations to Voigt notation. The Fortran code listing for the user material routine and a sample input file simulating the simple shearing of a block are given.

1

Neo–Hookean hyperelastic material

The strain energy potential for a nearly incompressible hyperelastic material is
¯ + φ (J) , W ≡Φ C

(1-1)

¯ = F¯ T F¯ is the isochoric right Cauchy–Green deformation tensor, F¯ = J −1/3 F is the isochoric where C deformation gradient, F is the deformation gradient and J = det (F ). The 2nd Piola–Kirchhoff stress is S=2

∂W ∂Φ ∂φ =2 + JC −1 , ∂C ∂C ∂J

(1-2)

and material stiffness tensor is C=2

∂2Φ ∂ ∂S =4 +2 ∂C ∂C∂C ∂C



∂φ ∂J



⊗ JC −1 + 2


∂φ ∂ JC −1 . ∂J ∂C

(1-3)

For a Neo–Hookean material

¯ = µ I¯C − 3 , Φ C 2 K 2 φ (J) = (J − 1) , 2

(1-4)


¯ , and µ and K are the Neo–Hookean shear and bulk moduli. The 1st order derivatives where I¯C = trace C in Eqs. (1-2) and (1-3) are   ∂Φ µ IC −1 = J −2/3 I − C , ∂C 2 3 ∂φ = K (J − 1) , ∂J

∂ JC −1 = J C −1 ⊗ C −1 − 2I , (1-5) ∂C where IC = trace (C), I is the 2nd order identity tensor, and I is a 4th order tensor with components −1 −1 Iijkl = Cik Cjl . The 2nd order derivatives in Eqs. (1-2) and (1-3) are   ∂2Φ µ J −2/3 IC −1 = −C −1 ⊗ I + C ⊗ C −1 − I ⊗ C −1 + IC I , ∂C∂C 2 3 3   ∂ ∂φ J (1-6) = KC −1 . ∂C ∂J 2

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DECEMBER 2008

Stress

Substituting Eq. (1-5) into (1-2), the 2nd Piola–Kirchhoff stress is then   IC −1 −2/3 S = µJ I− C + KJ (J − 1) C −1 , 3

(1-7)

and the Cauchy stress is 1 1 τ = F SF T J J    IC 1 µJ −2/3 b − I + KJ (J − 1) I , = J 3   ¯ IC 1 b− = µ ¯ I + K (J − 1) I, J 3

σ=

(1-8)

where τ is the Kirchhoff stress, b = F F T is the left Cauchy–Green deformation tensor and ¯ b = F¯ F¯ T .

1.2

Tangent stiffness

Substituting Eqs. (1-5) and (1-6) into (1-3), the material tangent stiffness is C=

  IC −1 2 −2/3 µJ −C −1 ⊗ I − I ⊗ C −1 + IC I + C ⊗ C −1 + 3 3 J 2 KC −1 ⊗ C −1 + KJ (J − 1) C −1 ⊗ C −1 − 2I




(1-9)

or, with some simplification,   2 −2/3 IC −1 −1 −1 −1 C = µJ −C ⊗ I − I ⊗ C + IC I + C ⊗C + 3 3   KJ (2J − 1) C −1 ⊗ C −1 − 2 (J − 1) I . (1-10) Using the Piola transform, the spatial tangent is   I¯C 2 −1 ¯ ¯ ¯ −b ⊗ I − I ⊗ b + IC i + I ⊗ I + K [(2J − 1) I ⊗ I − 2 (J − 1) i] . c = µJ 3 3

(1-11)

where i is a 4th order identity tensor with components iijkl = Iik Ijl .

2

Corotational frame

Large deformation in ANSYS is formulated in a corotated frame given by the rotation, R, from the polar decomposition of the deformation gradient F = RU , (2-1) where U is the stretch tensor. The rotation of the corotated frame is then R = F U −1 .

(2-2)

The stress and tangent stiffness tensors are returned to ANSYS in the corotated frame.

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Polar decomposition

Given the deformation gradient, the frame rotation from Eq. (2-2) is determined from the closed–form solution of the inverse stretch, 1 [C − IU U + IIU I] , (2-3) U −1 = IIIU where IU , IIU , and IIIU are the principal invariants of U . The stretch tensor is given in closed form by U=


 1 −C 2 + IU2 − IIU C + IU IIIU I . IU IIU − IIIU

(2-4)

Noting the eigenvalues of U are the square root of the eigenvalues of C, the principal invariants of U in terms of the eigenvalues of C are p p p IU = λ1 + λ2 + λ3 , p p p IIU = λ1 λ2 + λ2 λ3 + λ1 λ3 , p IIIU = λ1 λ2 λ3 , (2-5) where λ1 , λ2 and λ3 are the eigenvalues of C given by the roots of the characteristic polynomial, P (λ) = −λ3 + IC λ2 − IIC λ + IIIC ,

(2-6)

IC = trace (C) ,

1 2 IC − trace C 2 , IIC = 2 IIIC = det (C) = J 2 .

(2-7)

where the principal invariants are

C is a symmetric, positive–definite, rank 3 matrix. Equation (2-6) then has 3 positive real roots given by    q β + (i − 1) 2π 1 λi = IC + 2 IC2 − 3IIC cos , i = 1, 2, 3 3 3 2I 3 − 9IC IIC + 27IIIC β = arccos C q . (2-8) 3 2 (IC2 − 3IIC )

2.2

Stress

The constitutive model defined by the Neo–Hookean hyperelastic material in §1 gives the stress in the spatial configuration by direct evaluation of Eq. (1-8). No rate form is used and no integration is required. The Cauchy stress in the corotated frame is returned to ANSYS.

2.3

Tangent stiffness

The corotational rate relative to the corotated frame is the Green–Naghdi rate. However, the constitutive response of the Neo–Hookean material from §1 is in terms of the Lie derivative, or Truesdell rate, of the Kirchhoff stress Lτ = Jcd, (2-9) where c is the spatial tangent given by Eq. (1-11), and ◦

Lτ = τ˙ − lτ − τ lT = J σ, l = F˙ F −1 = d + w,
1 l + lT , 2
1 w= l − lT , 2 d=

©2008 ANSYS, Inc.

(2-10)

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◦

where σ is the Truesdell rate of the Cauchy stress and the superposed dot indicates the time derivative in the reference configuration. A convenient approximation of the Green–Naghdi rate is the Jaumann rate, ∇

σ= σ˙ + σw − wσ.

(2-11)

Using the Jaumann rate, the constitutive response in the corotated frame is given by ∇

∇

σ= c d,

(2-12)

∇

where c is the Jaumann tangent stiffness tensor and is implicitly defined by combining Eqs. (2-9) to (2-11), giving ∇

σ= cd + dσ + σd.

(2-13)

The Jaumann tangent stiffness in the corotated frame is the material stiffness returned to ANSYS.

3

Voigt notation

Voigt notation is a reduced order representation of tensors and is the form required for the stress and tangent stiffness tensors in ANSYS. In practice, Voigt notation is given by converting pairs of indices to a single index via {11 22 33 12 23 31} = {11 22 33 21 32 13} ⇒ {1 2 3 4 5 6} and redefining the rate of deformation tensor as     d1  d11               d2  d22             d3 d33 d⇒ ≡ . (3-1) d4  2d12              d5    2d23           d6 2d31 Then, in component form, ∇ ∇ σ α = c αβ

dβ ,

α, β = 1, 2, . . . , 6,

(3-2)

where,   ∇   σ  11       ∇     σ 22        ∇  ∇ σ 33 σα = ∇ ,    σ 12       ∇     σ 23        ∇  σ 31  ∇ ∇ c c ∇1111 ∇1122 c  2211 c 2222 ∇  c 3311 ∇ ∇ c 3322 c αβ =  ∇ ∇  c 1211 c 1222 ∇ ∇ c  2311 c 2322 ∇ ∇ c 3111 c 3122

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

∇

c 1133 c 2233

c 3333

c 1233 c 2333

c 3133

c 1112

c 2212

c 3312 c 1212

c 2312

c 3112

c 1123

c 2223

c 3323 c 1223

c 2323

c 3123

c 1131



 c 2231    ∇ c 3331  . ∇  c 1231   ∇ c 2331   ∇ c 3131

(3-3)

Minor symmetry with respect to the index pairs is required for this conversion. The 4th order identity i and the Jaumann tangent stiffness lack minor symmetry. They are redefined to ensure minor symmetry as 1 (Iik Ijl + Iil Ijk ) , 2 1 ≡ cijkl + (Iik σjl + σik Ijl + Iil σjk + σil Ijk ) . 2

iijkl ≡ ∇

c ijkl

©2008 ANSYS, Inc.

(3-4)

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Converting to Voigt notation, I ⇒ {1 1 1 0 0 0} ,  1 0 0 0 0 1 0 0  0 0 1 0 i⇒ 0 0 0 1 2  0 0 0 0 0 0 0 0  1 1 1 0 1 1 1 0  1 1 1 0 I ⊗I ⇒ 0 0 0 0  0 0 0 0 0 0 0 0

3.1

 0 0  0 , 0  0

0 0 0 0 1 2

0

1 2

0 0 0 0 0 0

 0 0  0 . 0  0 0

(3-5)

 − 1   − 1    −1 . 0    0     0

(3-6)

Stress

The Cauchy stress in Voigt notation is then    I¯C ¯ J     b1 − ¯3       ¯b2 − IC  J      3   µ ¯b3 − I¯C  J 3 +K σ⇒ ¯   J b    4            ¯b5     ¯b6

3.2

Tangent stiffness

The terms ¯ b ⊗ I and I ⊗ ¯ b in Voigt notation are  ¯b1 ¯b1  ¯b1 ¯ b⊗I ⇒ 0  0 0  ¯b1 ¯b2  ¯b3 I ⊗¯ b⇒ ¯b4  ¯b5 ¯b6

©2008 ANSYS, Inc.

¯b2 ¯b2 ¯b2 0 0 0 ¯b1 ¯b2 ¯b3 ¯b4 ¯b5 ¯b6

¯b3 ¯b3 ¯b3 0 0 0 ¯b1 ¯b2 ¯b3 ¯b4 ¯b5 ¯b6

¯b4 ¯b4 ¯b4 0 0 0 0 0 0 0 0 0

¯b5 ¯b5 ¯b5 0 0 0 0 0 0 0 0 0

 ¯b6 ¯b6   ¯b6  , 0  0 0 

0 0  0 . 0  0 0

(3-7)

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The spatial tangent in Voigt notation is then −2¯b1 + 43 I¯C −¯b − ¯b + I¯C 2  1 3 I¯C ¯ ¯ 2 −1  − b − b  1 3+ 3 c ⇒ µJ   3 −¯b4   −¯b5 −¯b6 

¯ −¯b2 − ¯b1 + I3C −2¯b2 + 34 I¯C ¯ −¯b2 − ¯b3 + I3C −¯b4 −¯b5 −¯b6 

¯ −¯b3 − ¯b1 + I3C ¯ −¯b3 − ¯b2 + I3C 4 −2¯b3 + 3 I¯C −¯b4 −¯b5 −¯b6

−¯b4 −¯b4 −¯b4 I¯C 2

−¯b5 −¯b5 −¯b5 0

0 0

0

I¯C 2

1 2J − 1 2J − 1 2J − 1 1 2J − 1  2J − 1 2J − 1 1 K  0 0 0   0 0 0 0 0 0

 −¯b6 −¯b6    −¯b6  + 0   0  I¯C 2

0 0 0 −J + 1 0 0

0 0 0 0 −J + 1 0

 0 0   0   . (3-8) 0   0  −J + 1

The stress terms in the Jaumann tangent stiffness are 1 1 1 (Iik σjl + σik Ijl + Iil σjk + σil Ijk ) = (Iik σjl + σik Ijl ) + (Iil σjk + σil Ijk ) 2 2  2  2σ1 0 0 σ4 0 σ6 2σ1 0 0 σ4  0  0  2σ 0 σ σ 0 2σ 0 σ 2 4 5 2 4     1 0 1 0 0 2σ 0 σ σ 0 2σ 0 3 5 6 3 +  ⇒   σ4 0 σ2 + σ1 0 0  σ4 0 0 2  σ4  2  σ4  0  0 σ5 σ5 0 σ3 + σ2 0  σ5 σ5 σ6 σ6 0 σ6 0 0 σ1 + σ3 σ6 0 σ6 σ5   2σ1 0 0 σ4 0 σ6  0 2σ 0 σ σ 0  2 4 5    0 0 2σ3 0 σ5 σ6   = σ6 σ5  . σ2 +σ1  σ4 σ4 0 2 2 2   σ3 +σ2 σ4  σ6  0 σ5 σ5 2 2 2 σ5 σ4 σ1 +σ3 σ6 0 σ6 2 2 2

0 σ5 σ5 σ6 0 σ4

 σ6 0  σ6   σ5   σ4  0

(3-9)

After some simplification, ¯b1 + I¯C 3 −¯b − ¯b + 2  1  ¯ 2 ∇ −b − ¯b + c⇒ µJ −1  1 ¯b43  3 2   −¯b5 

¯ b6 2

©2008 ANSYS, Inc.

I¯C 3 I¯C 3

¯ −¯b2 − ¯b1 + I3C ¯ I −¯b2 + 3C ¯ ¯ −b2 − ¯b3 + I3C ¯ b4 2 ¯ b5 2

−¯b6

−¯b3 − ¯b1 + −¯b3 − ¯b2 + ¯b3 + I¯C 3 −¯b4 ¯ b5 2 ¯ b6 2

I¯C 3 I¯C 3

¯ b4 2 ¯ b4 2

¯

−b4 3 ¯ ¯ ( b 4 1 + b2 ) 3¯ 4 b6 3¯ 4 b5

−¯b5 ¯ b5 2 ¯ b5 2 3¯ 4 b6

3 ¯ ¯ 4 (b2 + b3 ) 3¯ 4 b4

¯ b6 2

−¯b6 ¯ b6 2 3¯ 4 b5 3¯ 4 b4

     +   

3 ¯ ¯ 4 (b1 + b3 )   2J − 1 2J − 1 2J − 1 0 0 0 2J − 1 2J − 1 2J − 1 0 0 0   2J − 1 2J − 1 2J − 1 0 0 0 . K  0 0 0 0 0 0    0 0 0 0 0 0 0 0 0 0 0 0 (3-10)

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Change of basis

For an arbitrary change of basis given by the orthogonal tensor T , the components of the 2nd order stress tensor and the 4th order tangent stiffness tensors transform as 0 σij = Tim σmn Tjn , ∇0

∇

c ijkl = Tim Tjn c mnop Tko Tlp .

(3-11)

And in Voigt notation, σα0 = Qαξ σξ , ∇0

∇

c αβ = Qαξ c ξη Qβη .

(3-12)

Comparing Eqs. (3-11) and (3-12), the change of basis tensor Q is given by   [P1 ] 2 [P2 ] Q [T ] = , [P3 ] [P4 ] where

 T11 T21 [P3 ] = T21 T31 T31 T11

 2 T11 2 [P1 ] = T21 2 T31 T12 T22 T22 T32 T32 T12

2 T12 2 T22 2 T32 

T13 T23 T23 T33  , T33 T13

 2 T13 2  T23 , 2 T33

 T11 T12 [P2 ] = T21 T22 T31 T32

 T11 T22 + T12 T21 [P4 ] = T21 T32 + T22 T31 T31 T12 + T32 T11

T12 T13 T22 T23 T32 T33

(3-13)

 T11 T13 T21 T23  , T31 T33

T12 T23 + T13 T22 T22 T33 + T23 T32 T32 T13 + T33 T12

 T11 T23 + T13 T21 T21 T33 + T23 T31  . T31 T13 + T33 T11 (3-14)

For a change of basis from the spatial frame to the corotated frame, T = RT .

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APPENDIX A

Implementation

Shown below in §A.2 is the usermat3d() subroutine that implements the Neo–Hookean Cauchy stress and tangent stiffness calculations of Eqs. (1-8) and (3-10). Two auxiliary subroutines to perform the polar decomposition and form the 6–dimensional change of basis matrix are also given. The coding is done in fixed source form modern Fortran and has favored clarity over computational efficiency. The implementation is 3-dimensional and, as presented above, a displacement only formulation. It requires the Neo-Hookean Young’s Modulus and Poisson’s ratio as the first and second material properties respectively and has no user state variables. The stored energy used by ANSYS as an output variable is calculated as Eq. (1-1) with (1-4) and the dissipated energy is zero. The transverse shear stiffness terms are arbitrarily set to the (6, 6) component of the material tangent stiffness in the corotated frame. A sample simulation input file and solution output are given below in §A.1. It simulates a cycle of simple shear of a block that has the initial dimensions 1 × 1 × 1. The block is discretized with a 4 × 4 × 4 array of SOLID185, 8–node, trilinear, hexahedral elements. The Neo–Hookean Young’s Modulus and Poisson’s ratio are 3 × 107 and 0.3 respectively. The simulation uses a fixed 10 substeps for the loading and unloading steps and it writes the time, effective stress and effective strain, for an element near the center of the block, to a text file named stressstrain usr.log.

A.1 A.1.1

Sample simulation Input

!*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* ! Sample input file to demonstrate the large deformation Neo-Hookean ! user material implementation. Simulates a simple shear cycle of a ! block with initial dimensions 1x1x1. !*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-*-* !--------------------------------------------------------! Preprocessing !--------------------------------------------------------/PREP7 ! Geometry and mesh Lx=1 Ly=1 Lz=1 BLOCK,0,Lx,0,Ly,0,Lz ET,1,SOLID185 ESIZE,Lx/4. MAT,1 VMESH,1 ! Material ! User material parameters E = 30E6 ! Young’s Modulus nu = 0.3 ! Poisson’s ratio TB,USER,1,1,2, TBDATA,1,E,nu ! Neo-hookean hyperelastic parameters !G = E/2/(1+nu) ! shear modulus ©2008 ANSYS, Inc.

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!K = E/3/(1-2*nu) ! bulk modulus !d = 2/K ! incompressibility parameter !TB,HYPER,1,1,,NEO !TBDATA,1,G,d !--------------------------------------------------------! Solution !--------------------------------------------------------/SOL ANTYPE,STATIC NLGEOM,ON NSUBST,10,10,10 OUTRES,ALL,ALL ! Fix surfaces @ y=0 and z=0 NSEL,S,LOC,Y,0. D,All,Ux,0. D,All,Uy,0. NSEL,S,LOC,Z,0. D,All,Uz,0. !Simple shear NSEL,S,LOC,Y,Ly D,All,Ux,Lx D,All,Uy,0. TIME,1.0 ALLSEL,ALL SOLVE !Return to initial configuration NSEL,S,LOC,Y,Ly D,All,Ux,0. D,All,Uy,0. TIME,2.0 ALLSEL,ALL SOLVE !--------------------------------------------------------! postprocessing !--------------------------------------------------------/POST26 NN=node(Lx/2,Ly/2,Lz/2) EN=enearn(NN) ESOL,5,EN,NN,EPEL,EQV,EE ESOL,6,EN,NN,S,EQV,SE NSUB=20 *dim,timeo,array,NSUB *dim,stress2,array,NSUB *dim,strain2,array,NSUB VGET,timeo(1),1 VGET,strain2(1),5 VGET,stress2(1),6 *cfopen,stressstrain_usr,log *vwrite ©2008 ANSYS, Inc.

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(5x,’Time’,10x,’Stress eqv.’,5x,’Strain eqv.’) *vwrite,timeo(1),stress2(1),strain2(1) (2X,E14.6,2X,E14.8,2X,E14.8) *cfclos A.1.2

Output

Time 0.100000E+00 0.200000E+00 0.300000E+00 0.400000E+00 0.500000E+00 0.600000E+00 0.700000E+00 0.800000E+00 0.900000E+00 0.100000E+01 0.110000E+01 0.120000E+01 0.130000E+01 0.140000E+01 0.150000E+01 0.160000E+01 0.170000E+01 0.180000E+01 0.190000E+01 0.200000E+01

A.2

Stress eqv. 0.18511346E+07 0.36665547E+07 0.54384537E+07 0.71593019E+07 0.88222967E+07 0.10422123E+08 0.11955693E+08 0.13422645E+08 0.14825551E+08 0.16169843E+08 0.14825540E+08 0.13422581E+08 0.11955591E+08 0.10422004E+08 0.88221827E+07 0.71592219E+07 0.54384351E+07 0.36666162E+07 0.18512794E+07 0.46409597E-08

Strain eqv. 0.53645216E-01 0.10602121E+00 0.15643180E+00 0.20437866E+00 0.24957578E+00 0.29193486E+00 0.33153319E+00 0.36857480E+00 0.40335214E+00 0.43621175E+00 0.40324373E+00 0.36836731E+00 0.33123410E+00 0.29155654E+00 0.24913378E+00 0.20389315E+00 0.15592898E+00 0.10553450E+00 0.53215926E-01 0.66716967E-03

Code listing

*deck,usermat3d parallel user jmgerken subroutine usermat3d(matId, elemId,kDomIntPt, kLayer, kSectPt, & ldstep,isubst,keycut, & nDirect,nShear,ncomp,nStatev,nProp, & Time,dTime,Temp,dTemp, & stress,ustatev,dsdePl,sedEl,sedPl,epseq, & Strain,dStrain, epsPl, prop, coords, & var0, F0, F1, & tsstif, epsZZ, & var1, var2, var3, var4, var5, & var6, var7, var8) !---------------------------------------------------------------------!- user material constitutive routine !- 3D Neo-Hookean hyperelastic material !---------------------------------------------------------------------IMPLICIT NONE INTEGER,PARAMETER :: RK=KIND(1.D0) ! real kind ! external procedures EXTERNAL polarRU ! polar decomposition subroutine EXTERNAL trans_matrx_6 ! 6D voigt rotation matrix subroutine ! arguments INTEGER,INTENT(IN) :: matID ! material ID number INTEGER,INTENT(IN) :: elemID ! element ID number

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LARGE DEFORMATION USER MATERIAL IN ANSYS

DECEMBER 2008

INTEGER,INTENT(IN) :: kDomIntPt ! integration point INTEGER,INTENT(IN) :: kLayer ! layer number INTEGER,INTENT(IN) :: kSectPt ! section number INTEGER,INTENT(IN) :: ldstep ! load step number INTEGER,INTENT(IN) :: isubst ! substep number INTEGER,INTENT(OUT) :: keycut ! time step reduction INTEGER,INTENT(IN) :: ncomp ! no. stress components INTEGER,INTENT(IN) :: nDirect ! no. direct stress components INTEGER,INTENT(IN) :: nShear ! no. shear stress components INTEGER,INTENT(IN) :: nStatev ! size of state variable array INTEGER,INTENT(IN) :: nProp ! size of property array REAL(RK),INTENT(IN) :: Time ! time at begining of increment REAL(RK),INTENT(IN) :: dTime ! time increment REAL(RK),INTENT(IN) :: Temp ! temperature at begining of increment REAL(RK),INTENT(IN) :: dTemp ! temperature increment REAL(RK),INTENT(INOUT) :: sedEl ! stored (elastic) energy REAL(RK),INTENT(INOUT) :: sedPl ! dissipated (plastic) energy REAL(RK),INTENT(INOUT) :: epseq ! equivalent plastic strain REAL(RK),INTENT(INOUT) :: epsZZ ! plane stress thickness strain REAL(RK),INTENT(INOUT) :: stress(ncomp) !stress REAL(RK),INTENT(INOUT) :: ustatev(nStatev) !state variables REAL(RK),INTENT(OUT) :: dsdePl(ncomp,ncomp) !tangent stiffness REAL(RK),INTENT(IN) :: Strain(ncomp) !strain at begining of inc REAL(RK),INTENT(IN) :: dStrain(ncomp) !strain increment REAL(RK),INTENT(IN) :: epsPl(ncomp) !work space REAL(RK),INTENT(IN) :: prop(nProp) ! mat props, user input REAL(RK),INTENT(IN) :: coords(3) ! coordinates of material point REAL(RK),INTENT(IN) :: F1(3,3) ! deformation gradient at t+dt REAL(RK),INTENT(IN) :: F0(3,3) ! deformation gradient at t REAL(RK),INTENT(OUT) :: tsstif(2) ! transverse shear stiffness REAL(RK),INTENT(IN) :: var0(*),var1,var2,var3,var4 !place holder REAL(RK),INTENT(IN) :: var5,var6,var7,var8 !place holder ! local REAL(RK) :: R(3,3), Q(6,6) !rotations REAL(RK) :: U(3,3) !stretch REAL(RK) :: Bbar(6) ! vector form isochoric left cauchy green REAL(RK) :: tauiso(6),tauvol ! isochoric and volumetric stress REAL(RK) :: s(6) ! Cauchy stress REAL(RK) :: M(6,6) ! material jacobian REAL(RK) :: G,K ! shear & bulk modulus REAL(RK) :: I1bar3,detF,detF23 ! deformation invariants REAL(RK) :: C1,C2 ! constants !---------------------------------------------------------------------! deformation gradient determinant detF = F1(1,1) * (F1(2,2)*F1(3,3) - F1(2,3)*F1(3,2)) & + F1(1,2) * (F1(2,3)*F1(3,1) - F1(2,1)*F1(3,3)) & + F1(1,3) * (F1(2,1)*F1(3,2) - F1(2,2)*F1(3,1)) if(detF