II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

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By convention, a comma preceding a lowercase subscript denotes differentiation with respect to the spatial coordinates (e.g., Ui,j denotes ~lii 1 dxj). The two-dimensional isoparametric-shape functions map the unit square in ~-~ to an arbitrary quadrilateral in x- y, as shown in Figure 3.1.1. We choose to center the unit square at the origin in
m’ I 2$=2 y 4 L x I 1 2 1- 1 I I I I_ E A, I :~ IJ2”a” A21 TRI-6348-5-O Figure 3.1.1. Mode shapes for the four-node constant strain quadrilateral element. 3.1.1 Plane Strain Case In the finite element method, we replace the momentum Equation (2.3.5) with a weak form of the equation. Using the principal of virtual work, we write the weak form of the equation as ‘J v ~(Tij,j ‘Pbl e ._piii F uidV=O (3. 1.7) where &i represents an arbitrary virtual displacement field, with the same interpolation as Equation (3.1.2), which satisfies the kinematic constraints. In plane strain, the thickness of the body is considered uniform and arbitrary and therefore can be eliminated from the preceding expression. Integrating by parts and applying Gauss’ divergence theorem to Equation (3. 1.7) then gives 1 *Tij&ui,jdA+~A ~ ~ Tijnj~Uid/ -~ pbi&uidA_~A piii6uidA = O s e [ e e e e (3.1.8) The summation symbol represents the assembly of element force vectors into a global nodal force array. that the reader understands the details of this assembly; we will not discuss it further in this document. We assume 11
Page 1 and 2: SANDIA REPORT SAND90-0543 l UC-721
Page 3 and 4: SAND90-0543 Unlimited Release Print
Page 5 and 6: CONTENTS 1.0 IN~ODUCTION . . . . .
Page 7 and 8: Figures 2.1.1 2.2.1 2.2.2 3.1.1 3.7
Page 9 and 10: 1.0 INTRODUCTION SANTOS is a finite
Page 11 and 12: 2.0 GOVERNING EQUATIONS In this cha
Page 13 and 14: Using the left decomposition of Equ
Page 15 and 16: 1.5 1.0 0.5 a 0.0 i% \ +7797 / \ &
Page 17: 3.0 NUMERICAL FORMULATION In this c
Page 21 and 22: Consequently, the gradient operator
Page 23 and 24: We are now faced with a three-dimen
Page 25 and 26: We now see that the internal force
Page 27 and 28: ‘t+At = QAt % (3.3.2) For a const
Page 29 and 30: If the strain increments are insign
Page 31 and 32: The mean coordinates Z, correspond
Page 33 and 34: for arbitrary Ui]. Using Equation (
Page 35 and 36: TRI-6348-6-O Figure 3.7.1. A model
Page 37 and 38: 3.8 Convergence Measures When an it
Page 39 and 40: 4.0 CONSTITUTIVE MODELS One of the
Page 41 and 42: where y is a column vector containi
Page 43 and 44: ~=~y —=Kdkk (4.3.3) where K is th
Page 45 and 46: Q TRI-6348-7-O Figure 4.5.1. Yield
Page 47 and 48: @r=R . (4.5.16) Because S is deviat
Page 49 and 50: and (4.5.27) Combining Equations (4
Page 51 and 52: TRI-6348-1o-o Figure 4.5.4. Effect
Page 53 and 54: The subscripts n and n+ 1 refer to
Page 55 and 56: The elastic plastic material model
Page 57 and 58: (a) a. P, Compression (b) 16 a. ‘
Page 59 and 60: The mean volumetric strain is updat
Page 61 and 62: ~=$s:s . (4.6.9) J The yield condit
Page 63 and 64: in Section 4.6, use such decomposit
Page 65 and 66: not amenable to vectorization. Vect
Page 67 and 68: We make use of the following identi
Page 69 and 70:
law secondary creep model is reques
Page 71 and 72:
PROP(3) - Modulus function identifi
Page 73 and 74:
where K @ () n+l is the O;:l =3K @n
Page 75 and 76:
2 11 F=exp-1-~ 6 for q>&; . (4.12.8
Page 77 and 78:
Accuracy of the method is assured i
Page 79 and 80:
The volumetric part of the model ca
Page 81 and 82:
for the bulk response, where G(t) a
Page 83 and 84:
then G(t,@) = G(g) . (4.14.9) This
Page 85 and 86:
PROP(7) - Third Short Time Shear Mo
Page 87 and 88:
5.0 CONTACT SURFACES Many structure
Page 89 and 90:
5.2 Application Phase The operation
Page 91 and 92:
6.0 LOADS AND BOUNDARY CONDITIONS S
Page 93 and 94:
TRI-6348-43-O Figure 6.2.1. Definit
Page 95 and 96:
diagonal mass matrix routines with
Page 97 and 98:
Biffle, J. H., and M.L. Blanford. 1
Page 99 and 100:
Schreyer,H.L., R.F. Kulak, and J.M.
Page 101 and 102:
APPENDIX A: SANTOS Users Manual A-1
Page 103 and 104:
SANTOS Users Manual Listed below, i
Page 105 and 106:
tol ... tolerance value used for co
Page 107 and 108:
See the theory section for definiti
Page 109 and 110:
omega ... specified angular velocit
Page 111 and 112:
dence of the pressure scale factor
Page 113 and 114:
TWO MU(2P) BULK MODULUS (K) AO Al A
Page 115 and 116:
RN3 (exponent of workhardening and
Page 117 and 118:
APPENDIX B: User Subroutines B-1 ,.
Page 119 and 120:
‘ser Subroutines SANTOS allows th
Page 121 and 122:
600 CONTINUE END IF 1000 CONTINUE R
Page 123 and 124:
APPENDIX C: Printed Output Descript
Page 125 and 126:
Printed Output Description The SANT
Page 127 and 128:
FUNCTION DEFI NIT IONS FUNCTION ID
Page 129 and 130:
1 SANTOS, VERSION SANTOS 2.0 ,RUN O
Page 131 and 132:
APPENDIX D: Adding a New Constituti
Page 133 and 134:
Adding A New Constitutive Model to
Page 135 and 136:
APPENDIX E: Verification and Sample
Page 137 and 138:
Verification and Sample Problems Sa
Page 139 and 140:
To obtain a solution to the gravity
Page 141 and 142:
Figure E-5. Deformed Shape of the B
Page 143 and 144:
P= () 4:(1-V) 1-: pc3 +2(1-m)lllpc
Page 145 and 146:
L < 1 t i I I t I , I 1 3 3 1 1 I i
Page 147 and 148:
Side Set ID for the Rigid Surface C
Page 149 and 150:
TITLE UPSETTING OF A CYLINDRICAL BI
Page 151 and 152:
Table 1: M-D Argillaceous Salt Cree
Page 153 and 154:
4.0 3.5 3.0 1.0 0.5 0.0 I I I 1 1 1
Page 155 and 156:
WIPP UC721 - DISTRIBUTIONLIST SAND9
Page 157 and 158:
Sue B. Clark University of Georgia
Page 159 and 160:
Mr. Steven Crouch GeoLogic Research
Page 161:
[:1 . 1 .i i- ) r
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II II II II II I - Waste Isolation Pilot Plant - U.S. Department of Energy

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