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stifd T
stifp = [ P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 P 9 ]


D 1 P1 0 0 0 0
D 2 P2 P3 P5 0 T
kprof = [ 1 2 4 7 9 ]
D 3 P4 P6 0
K=
D 4 P7 P8
symmetric D 5 P9
D6


FIGURE 8.4 Pro¬le storage pointers.


the well-known Cuthill“McKee algorithm. This results in a nonsequential
allocation of degrees of freedom to nodes that, although transparent to the
user, are stored in the matrix ldgof(1:ndime,1:npoin).


8.6 CONSTITUTIVE EQUATION SUMMARY

For the purpose of facilitating the understanding of the implementation of
the various constitutive equations, the boxes below summarize the required
constitutive and kinematic equations for each material type. These equa-
tions are presented here in an indicial form to concur with the code.

Material 8.1: Three-dimensional or plane strain compressible
neo-Hookean


µ »
σij = (bij ’ δij ) + (ln J)δij (5.29)
J J
c ijkl = » δij δkl + 2µ δik δjl (5.37)
» µ ’ » ln J
» = ; µ= (5.38)
J J


Material 8.2: Not de¬ned
12 COMPUTER IMPLEMENTATION




Material 8.3: Three-dimensional or plane strain hyperelasticity
in principal directions


2µ »
σ±± = ln »± + ln J (5.90)
J J
3
σij = σ±± T±i T±j ; (T±i = n± · ei ) (5.77)
±=1
3 3
» + 2(µ ’ σ±± )δ±β
c ijkl = T±i T±j Tβk Tβl + 2µ±β T±i Tβj T±k Tβl
J
±,β=1 ±,β=1
±=β
(5.86, 91)
σ±± »2 ’ σββ »2 µ
±
β
µ±β = ; if »± = »β or µ±β = ’ σ±± if »± = »β
»2 ’ »2 J
± β
(5.86“7, 91)



Material 8.4: Plane stress hyperelasticity in principal directions




γ = (5.112)
» + 2µ
¯
» = γ» (5.112)
= jγ ;
J (J = dv/dV ; j = da/dA) (5.113)
¯
2µ »
σ±± = ln »± + γ ln j (5.114)
jγ j
2
σij = σ±± T±i T±j ; (T±i = n± · ei ) (5.77)
±=1
2 ¯
» + 2(µ ’ σ±± )δ±β
c ijkl = T±i T±j Tβk Tβl + 2µ12 T1i T2j T1k T2l
(5.86, 115)

±,β=1
σ11 »2 ’ σ22 »2 µ
2 1
µ12 = ; if »1 = »2 or µ12 = ’ σ11 if »1 = »2
»2 2 jγ
’ »2
1
(5.86“7, 115)
HJ
h = (Exercise 3.3)
j
13
8.6 CONSTITUTIVE EQUATION SUMMARY




Material 8.5: Three-dimensional or plane strain nearly incom-
pressible neo-Hookean


v (e)
¯
J = (e) (6.50a)
V
¯
p = κ(J ’ 1) (6.52)
v (e)
κ
¯ = κ (e) (6.61)
V
= µJ ’5/3 (bij ’ 1 Ibδij )
σij (5.51)
3
σij = σij + pδij (4.49a)
1
’ 1 bij δkl ’ 3 δij bkl + 1 Ibδij δkl
1
= 2µJ ’5/3
ˆ 3 Ibδik δjl (5.55a)
c ijkl 3 9
c p,ijkl = p(δij δkl ’ 2δik δjl ) (6.55b)


Material 8.6: Plane stress incompressible neo-Hookean (exer-
cise 5.1)


J =1; (J = dv/dV, j = da/dA)
= µbij ’ j 2 δij ; j 2 = det b
σij
2—2

c ijkl = » δij δkl + 2µ δik δjl

» =
j2
µ
µ =2
j
H
h =
j


Material 8.7: Nearly incompressible in principal directions



¯
J = v (e) /V (e) (6.50a)
¯
κ ln J
p= ¯ (5.99)
J
14 COMPUTER IMPLEMENTATION




Material 8.8:


¯ dp κ
κ
¯ =J ¯ = ¯’p (6.59)
dJ J
2µ 2µ
σ±± =’ ln J + ln »± (5.103)
3J J
3
σij = σ±± T±i T±j ; (T±i = n± · ei ) (5.77)
±=1

σij = σij + pδij (4.49a)
c p,ijkl = p(δij δkl ’ 2δik δjl ) (5.55b)
3 3
2 1
ˆ = (µ ’ σ±± )δ±β ’ 3 µ T±i T±j Tβk Tβl + 2µ±β T±i Tβj T±k Tβl
c ijkl
J
±,β=1 ±,β=1
±=β
(5.106“7)
σ±± »2 ’ σββ »2 µ
±
β
µ±β = ; if »± = »β or µ±β = ’ σ±± if »± = »β
»2 ’ »2 J
± β
(5.106, 87, 107)
15
8.6 CONSTITUTIVE EQUATION SUMMARY




Material 8.9: Plane stress incompressible in principal directions


¯
» ’ ∞; γ = 0; » = 2µ (5.112)
J = 1; (J = dv/dV ; j = da/dA) (5.113)
¯
σ±± = 2µ ln »± + » ln j (5.114)
2
σij = σ±± T±i T±j ; (T±i = n± · ei ) (5.77)
±=1
2
¯
c ijkl = » + 2(µ ’ σ±± )δ±β T±i T±j Tβk Tβl + 2µ12 T1i T2j T1k T2l
±,β=1
(5.86, 115)
σ11 »2 ’ σ22 »2
2 1
µ12 = ; if »1 = »2 or µ±β = µ ’ σ11 if »1 = »2
2 ’ »2
»1 2
(5.86“7, 115)
H
h = (Exercise 3.3)
j
16 COMPUTER IMPLEMENTATION




BOX 8.4: FLagSHyP Structure

¬‚agshyp ..............master routine
welcome ...........types welcome message and reads file names
elinfo ............reads element type
lin2db ........evaluates the 2-noded line element data
qua3db ........evaluates the 3-noded line element data
tria3db .......evaluates the 3-noded triangle data
tria6db .......evaluates the 6-noded triangle data
tetr4db .......evaluates the 4-noded tetrahedron data
tetr10db ......evaluates the 10-noded tetrahedron data
quad4db .......evaluates the 4-noded quadrilateral data
hexa8db .......evaluates the 8-noded hexahedron data
innodes ...........reads nodal coordinates and boundary codes
inelems ...........reads element connectivities and material types
nodecon ...........evaluates node to node connectivities
degfrm ............numbers degrees of freedom with profile minimization
profile ...........determines the profile column heights and addressing
matprop ...........reads material properties
inloads ...........reads loads and prescribed displacements
incontr ...........reads solution control parameters
initno ............initializes nodal data such as coordinates

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