KOKO\Mimi
314 lines
9.3 KiB
Markdown
314 lines
9.3 KiB
Markdown
<!-- source-page: 151 -->
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```txt
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dE12Dj = term1 * E12
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dE13Dj = term1 * E13
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dE23Dj = term1 * E23
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term2 = - third * detuInv
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d2E11DjDj = term2 * dE11Dj
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d2E22DjDj = term2 * dE22Dj
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d2E33DjDj = term2 * dE33Dj
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d2E12DjDj = term2 * dE12Dj
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d2E13DjDj = term2 * dE13Dj
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d2E23DjDj = term2 * dE23Dj
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*
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dUdE11 = d2UdE11dE11 * E11
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* + d2UdE11dE22 * E22
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* + d2UdE11dE33 * E33
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dUdE22 = d2UdE22dE11 * E11
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* + d2UdE22dE22 * E22
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* + d2UdE22dE33 * E33
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dUdE33 = d2UdE33dE11 * E11
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* + d2UdE33dE22 * E22
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* + d2UdE33dE33 * E33
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dUdE12 = two * d2UdE12dE12 * E12
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dUdE13 = two * d2UdE13dE13 * E13
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dUdE23 = two * d2UdE23dE23 * E23
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*
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U = half * ( E11*dUdE11 + E22*dUdE22 + E33*dUdE33 )
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* + E12*dUdE12 + E13*dUdE13 + E23*dUdE23
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*
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ua(2) = U
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ua(1) = ua(2)
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*
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du1(1) = xpow * dUdE11
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du1(2) = xpow * dUdE22
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du1(3) = xpow * dUdE33
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du1(4) = xpow * dUdE12
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du1(5) = xpow * dUdE13
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du1(6) = xpow * dUdE23
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du1(7) = dUdE11*dE11Dj + dUdE22*dE22Dj + dUdE33*dE33Dj
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* + two * ( dUdE12*dE12Dj
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* + dUdE13*dE13Dj
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* + dUdE23*dE23Dj )
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*
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xpow2 = xpow * xpow
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```
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<!-- source-page: 152 -->
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\*
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```txt
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du2 (indx(1,1)) = xpow2 * d2UdE11dE11
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du2 (indx(1,2)) = xpow2 * d2UdE11dE22
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du2 (indx(2,2)) = xpow2 * d2UdE22dE22
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du2 (indx(1,3)) = xpow2 * d2UdE11dE33
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du2 (indx(2,3)) = xpow2 * d2UdE22dE33
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du2 (indx(3,3)) = xpow2 * d2UdE33dE33
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du2 (indx(1,4)) = zero
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du2 (indx(2,4)) = zero
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du2 (indx(3,4)) = zero
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du2 (indx(4,4)) = xpow2 * d2UdE12dE12
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du2 (indx(1,5)) = zero
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du2 (indx(2,5)) = zero
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du2 (indx(3,5)) = zero
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du2 (indx(4,5)) = zero
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du2 (indx(5,5)) = xpow2 * d2UdE13dE13
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du2 (indx(1,6)) = zero
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du2 (indx(2,6)) = zero
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du2 (indx(3,6)) = zero
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du2 (indx(4,6)) = zero
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du2 (indx(5,6)) = zero
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du2 (indx(6,6)) = xpow2 * d2UdE23dE23
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```
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\*
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```txt
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du2(indx(1,7)) = xpow * (term1 * dUdE11
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* + d2UdE11dE11 * dE11Dj
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* + d2UdE11dE22 * dE22Dj
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* + d2UdE11dE33 * dE33Dj)
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du2(indx(2,7)) = xpow * (term1 * dUdE22
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* + d2UdE22dE11 * dE11Dj
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* + d2UdE22dE22 * dE22Dj
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* + d2UdE22dE33 * dE33Dj)
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du2(indx(3,7)) = xpow * (term1 * dUdE33
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* + d2UdE33dE11 * dE11Dj
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* + d2UdE33dE22 * dE22Dj
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* + d2UdE33dE33 * dE33Dj)
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du2(indx(4,7)) = xpow * (term1 * dUdE12
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* + two * d2UdE12dE12 * dE12Dj)
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du2(indx(5,7)) = xpow * (term1 * dUdE13
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* + two * d2UdE13dE13 * dE23Dj)
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du2(indx(6,7)) = xpow * (term1 * dUdE23
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* + two * d2UdE23dE23 * dE13Dj)
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du2(indx(7,7)) = dUdE11*d2E11DjDj
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```
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<!-- source-page: 153 -->
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```txt
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* +dUdE22*d2E22DjDj
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* +dUdE33*d2E33DjDj
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* + two*( dUdE12*d2E12DjDj
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* +dUdE13*d2E13DjDj
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* +dUdE23*d2E23DjDj)
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* + d2UdE11dE11 * dE11Dj * dE11Dj
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* + d2UdE22dE22 * dE22Dj * dE22Dj
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* + d2UdE33dE33 * dE33Dj * dE33Dj
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* + two * ( d2UdE11dE22 * dE11Dj * dE22Dj
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* +d2UdE11dE33 * dE11Dj * dE33Dj
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* +d2UdE22dE33 * dE22Dj * dE33Dj )
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* + four * ( d2UdE12dE12 * dE12Dj * dE12Dj
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* +d2UdE13dE13 * dE13Dj * dE13Dj
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* +d2UdE23dE23 * dE23Dj * dE23Dj )
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*
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return
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end
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*
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* Maps index from Square to Triangular storage
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* of symmetric matrix
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*
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integer function index( i, j )
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*
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include 'aba_param.inc'
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*
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ii = min(i,j)
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jj = max(i,j)
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*
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indx = ii + jj*(jj-1)/2
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*
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return
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end
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```
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<!-- source-page: 154 -->
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<!-- source-page: 155 -->
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# 1.1.22 UCORR: User subroutine to define cross-correlation properties for random response loading.
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Product: Abaqus/Standard
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# References
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• “Random response analysis,” Section 6.3.11 of the Abaqus Analysis User’s Guide
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• \*CORRELATION
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• “Random response to jet noise excitation,” Section 1.4.10 of the Abaqus Benchmarks Guide
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# Overview
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User subroutine UCORR:
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• can be used to define the coefficients for the cross-correlation matrix in a random response analysis;
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• will be called once for the combination of any two degrees of freedom with nonzero prescribed loads for each load case specified as a concentrated or distributed load or once for the combination of any two excitation directions specified as a base motion;
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• allows correlation coefficients to be defined as a function of nodal coordinates; and
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• ignores any data specified outside the user subroutine for the associated cross-correlation matrix.
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# Cross-correlation for base motion excitation
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The spatial correlation matrix for base motion excitation is defined by the coefficients $\Psi _ { i j } ^ { I J }$ in user subroutine UCORR, where $i , j$ are excitation directions and J corresponds to the Jth frequency function referenced under load case I.
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# Cross-correlation for point loads and distributed loads
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The spatial correlation matrix of the load is defined as follows. Let $F _ { ( N , i ) } ^ { I }$ be the load applied to degree of freedom i at node N in load case I, through the use of a concentrated or distributed load. Let J correspond to the Jth frequency function referenced under load case I. The spatial correlation matrix used in the random response analysis for this load case is then
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$$
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\Psi_ {(N, i) (M, j)} ^ {I J} = C _ {(N, i) (M, j)} ^ {I J} F _ {(N, i)} ^ {I} F _ {(M, j)} ^ {I},
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$$
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where $C _ { ( N , i ) ( M , j ) } ^ { I J }$ are the coefficients defined in user subroutine UCORR. Typically the load magnitude is given as 1.0; therefore, the load definition is simply selecting the nonzero terms that will appear in $\Psi _ { ( N , i ) ( M , j ) } ^ { \bar { I } , J }$ .
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<!-- source-page: 156 -->
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User subroutine interface
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```txt
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SUBROUTINE UCORR(PSD,CORRR,CORRI,KSTEP,LCASE,JNODE1,JDOF1,1 JNODE2,JDOF2,COOR1,COOR2)
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C
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INCLUDE 'ABA_PARAM.INC'
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C
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DIMENSION COOR1(3),COOR2(3)
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CHARACTER*80 PSD
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user coding to define CORRR and CORRI
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RETURN
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END
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```
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Variables to be defined
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```txt
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CORRR
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Real part of the cross-correlation scaling factor.
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CORRI
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Imaginary part of the cross-correlation scaling factor.
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```
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Variables passed in for information
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```txt
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PSD
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User-specified name for the frequency function that references this correlation, left justified.
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KSTEP
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Step number.
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LCASE
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Load case number, I.
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JNODE1
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First node involved, N (not used for base motion excitation).
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JDOF1
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Degree of freedom i at the first node (for concentrated or distributed load excitation) or global e direction i (for base motion excitation).
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JNODE2
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Second node involved, M (not used for base motion excitation).
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```
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<!-- source-page: 157 -->
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# JDOF2
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Degree of freedom $j$ at the second node (for concentrated or distributed load excitation) or global excitation direction $j$ (for base motion excitation).
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# COOR1
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An array containing the coordinates of the first node (not used for base motion excitation).
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# COOR2
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An array containing the coordinates of the second node (not used for base motion excitation).
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<!-- source-page: 158 -->
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<!-- source-page: 159 -->
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# 1.1.23 UCREEPNETWORK: User subroutine to define time-dependent behavior (creep) for models defined within the parallel rheological framework.
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# Product: Abaqus/Standard
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# References
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• “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide
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• “Nonlinear large-strain viscoelasticity with hyperelasticity,” Section 2.2.8 of the Abaqus Verification Guide
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• \*VISCOELASTIC
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# Overview
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User subroutine UCREEPNETWORK:
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• is intended to provide creep laws for nonlinear viscoelastic networks for models defined using the parallel rheological framework (see “Parallel rheological framework,” Section 22.8.2 of the Abaqus Analysis User’s Guide);
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• can use and update solution-dependent state variables; and
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• can be used in conjunction with user subroutine USDFLD to redefine any field variables before they are passed in.
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# Model description
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The user subroutine allows a creep law of the following general form to be defined:
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$$
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\dot {\bar {\varepsilon}} ^ {c r} = g ^ {c r} (\bar {\varepsilon} ^ {c r}, I _ {1} ^ {c r}, \bar {I} _ {1}, \bar {I} _ {2}, J, p, \tilde {q}, t, \theta , F V),
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$$
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where
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$$
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I _ {1} ^ {c r} = \mathbf {I}: \mathbf {C} ^ {c r},
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$$
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and
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I is the identity tensor, $\mathbf{C}^{cr}$ is the right Cauchy-Green creep strain tensor, $\dot{\bar{\varepsilon}}^{cr}$ is the equivalent creep strain rate, $\bar{\varepsilon}^{cr}$ is the equivalent creep strain, $\bar{I}_1$ is the first invariant of $\bar{\mathbf{B}}$ , $\bar{I}_2$ is the second invariant of $\bar{\mathbf{B}}$ , $J$ is the determinant of the deformation gradient, $\mathbf{F}$ ,
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<!-- source-page: 160 -->
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p is the Kirchhoff pressure, $\tilde{q}$ is the equivalent deviatoric Kirchhoff stress, $t$ is the time, $\theta$ is the temperature, and $FV$ are field variables.
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The left Cauchy-Green strain tensor, , is defined as
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$$
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\bar {\mathbf {B}} = \bar {\mathbf {F}} \bar {\mathbf {F}} ^ {T},
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$$
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where is the deformation gradient with volume change eliminated, which is computed using
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$$
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\bar {\mathbf {F}} = J ^ {- \frac {1}{3}} \mathbf {F}.
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$$
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The user subroutine must define the increment of creep equivalent strain, $\Delta \bar { \varepsilon } ^ { c r }$ , as a function of the time increment, $\Delta t ,$ and the variables used in the definition of $\boldsymbol { \cdot } \boldsymbol { g } ^ { c r }$ , as well as the derivatives of the equivalent creep strain increment with respect to those variables. If any solution-dependent state variables are included in the definition of $\boldsymbol { g } ^ { c r }$ , they must also be integrated forward in time in this user subroutine.
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User subroutine interface ```csv
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subroutine ucreepnetwork (
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C Must be updated
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* outputData,
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C Can be updated
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* statev,
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C Information (Read only)
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* nOutput,
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* nstatv,
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* networkid,
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* coords,
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* temp,
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* dtemp,
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* nfield,
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* predef,
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* dpred,
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* nprops,
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* props,
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* i_array,
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* niarray,
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* r_array,
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* nrarray,
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* c_array,
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```
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