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• Measured Variability Of Southern Yellow Pine - Manual for LS-DYNA Wood Material Model 143

Measured Variability Of Southern Yellow Pine - Manual for LS-DYNA Wood Material Model 143

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508 Compliance Captions for Equations

Equation 1. This equation reads Stress component subscript 1 equals C subscript 11, C subscript 12, C subscript 13, 0, 0, 0 equals strain component subscript 1. Stress component subscript 2 equals C subscript 12, C subscript 22, C subscript 23, 0, 0, 0 equals strain component subscript 2. Stress component subscript 3 equals C subscript 13, C subscript 23, C subscript 33, 0, 0, 0 equals strain component subscript 3.  Stress component subscript 4 equals 0, 0, 0, 2C subscript 44, 0, 0 equals strain component subscript 4.  Stress component subscript 5 equals 0, 0, 0, 0, 2c subscript 55, 0 equals strain component subscript 5.  Stress component subscript 6 equals 0, 0, 0, 0, 0, 2C subscript 66 equals strain component subscript 6.

Equation 2. This equation reads C subscript 11 equals times the product of E subscript 11 and the quantity 1 minus Impact Velocity subscript 23  times  Impact Velocity subscript 32, all over delta.

Equation 3. This equation reads C subscript 22 equals E subscript 22 times the quantity 1 minus Impact Velocity subscript 31 times Impact Velocity subscript 13, all over delta.

Equation 4. This equation reads C subscript 33 equals E subscript 33 times the quantity 1 minus Impact Velocity subscript 12 times Impact Velocity subscript 21, all divided by delta.

Equation 5. This equation reads C subscript 12 equals  the quantity of the sum of Impact Velocity subscript 21 plus the product of  Impact Velocity subscript 31 times Impact Velocity subscript 23, end quantity, times E subscript 11, all divided by delta.

Equation 6. This equation reads C subscript 13 equals the quantity of the sum of Impact Velocity subscript 31 plus the product of Impact Velocity subscript 21times Impact Velocity subscript 32, end quantity, times E subscript 11, all divided by delta.

Equation 7. This equation reads C subscript23 equals  the quantity of the sum of Impact Velocity subscript 32 plus the product of Impact Velocity subscript 12 times Impact Velocity subscript 31, end quantity, times E subscript 22, all divided by delta.

Equation 8. This equation reads C subscript 44 equals G subscript 12.

Equation 9. This equation reads C subscript 55 equals G subscript 23.

Equation 10. This equation reads C subscript 66 equals G subscript 13.

Equation 11. This equation reads delta equals 1 minus Impact Velocity subscript 12 times Impact Velocity subscript 21 minus Impact Velocity subscript 23 times Impact Velocity subscript 32 minus Impact Velocity subscript 31 times Impact Velocity subscript 13 minus 2 times Impact Velocity subscript 21 times Impact Velocity subscript 32 times Impact Velocity subscript 13.

Equation 12. This equation reads Poisson's Ratios over Normal Moduli subscript I equals Poisson's Ratios over Normal Moduli subscript j for I, when j equals 1,2, 3.

Equation 13. This equation reads Parallel yield surface function equals stress component subscript 11 superscript 2 over Parallel wood strength superscript 2 plus the quotient of the quantity of Stress component subscript 12 superscript 2 plus stress component subscript 13 superscript 2 end quantity, all over Parallel shear strength superscript 2 minus 1.   Parallel wood strength equals parallel wood strength tension for stress component subscript 11 is greater than zero and parallel wood strength compression for stress component subscript 11 is less than zero.

Equation 14. This equation reads Perpendicular yield surface function equals the quotient of the quantity of Stress component subscript 22 plus stress component subscript 33 end quantity, superscript 2, all over perpendicular wood strength superscript 2 plus the quotient of the numerator quantity of Stress component subscript 23 superscript 2 minus stress component subscript 22 times stress component subscript 33 end quantity, all over perpendicular shear strength superscript 2 minus 1.    Perpendicular wood strength equals perpendicular wood strength tension for stress component subscript 22 plus stress component subscript 33 is greater than zero and perpendicular wood strength compression for stress component subscript 22 plus stress component subscript 33 is less than zero.

Equation 15. This equation reads Parallel yield surface as a function of stress invariant subscript 1, stress invariant subscript 4, equals stress invariant subscript 1 superscript 2 over parallel wood strength superscript 2 plus stress invariant subscript 4 over parallel shear strength superscript 2 minus 1.

Equation 16. This equation reads Parallel plastic consistency parameters equal negative parallel trial elastic yield surface function all over the denominator of lowercase delta of parallel F over lowercase delta I subscript 1, evaluated at N, times of lowercase delta I subscript 1 over lowercase delta parallel lambda, evaluated at N, all plus lowercase delta of parallel F over lowercase delta I subscript 4, evaluated at N, times lowercase delta of I subscript 4 over lowercase delta parallel lambda, evaluated at N.

Equation 17. This equation reads lowercase delta parallel F over lowercase delta I subscript 1 equals 2 stress invariant subscript 1 over general parallel wood strength superscript 2.

Equation 18. This equation reads lowercase delta parallel F over lowercase delta I subscript 4 equals 1 over parallel shear stress superscript 2.

Equation 19. This equation reads lowercase delta I subscript 1 over lowercase delta parallel lambda equals negative 2 times C subscript 11 times stress invariant subscript 1 over general parallel wood strength superscript 2.

Equation 20. This equation reads lowercase delta I subscript 4 over lowercase delta parallel lambda equals negative 4 shear moduli of an orthotropic material subscript 12 stress invariant subscript 4 over parallel shear strength superscript 2.

Equation 21. This equation reads Perpendicular yield surface as a function of isotropic stress invariant subscript 2, isotropic stress invariant subscript 3, equals isotropic stress invariant subscript 2 superscript 2 over perpendicular wood strength superscript 2 plus isotropic stress invariant subscript 3 over perpendicular shear strength superscript 2 minus 1.

Equation 22. This equation reads the perpendicular plasticity consistency equals the quotient of the numerator of negative F perpendicular superscript star, all over the denominator the sum of the product of the quotient of lowercase delta perpendicular F over lowercase delta I subscript 2, evaluated at N, times the quotient of lowercase delta I subscript 2 over lowercase delta perpendicular lambda evaluated at N, plus the product of the quotient of lowercase delta perpendicular F over lowercase delta I subscript 3, evaluated at N, times the quotient of lowercase delta I subscript 3 over lowercase delta perpendicular lambda, evaluated at N.

Equation 23. This equation reads lowercase delta perpendicular F over delta I subscript 2 equals 2 stress invariant subscript 2 over general perpendicular wood strength subscript 2.

Equation 24. This equation reads lowercase delta perpendicular F over lowercase delta I subscript 3 equals 1 over perpendicular shear strength superscript 2.

Equation 25. This equation reads the quotient of lowercase delta I subscript 2 over lowercase delta perpendicular lambda equals  the quantity C subscript 22 plus C subscript 23 end quantity, times stress invariant subscript 2, times the quantity of negative 4 over general perpendicular wood strength superscript 2 plus 1 over perpendicular shear strength superscript end quantity.

Equation 26. This equation reads lowercase delta I subscript 3 over lowercase delta perpendicular lambda equals 2 times the quantity of C subscript 22 plus C subscript 23 end quantity, times the stress invariant subscript 2 superscript 2 over general perpendicular wood strength superscript 2, all minus C subscript 22 times stress invariant subscript 2 superscript 2, all over perpendicular shear strength superscript 2, all minus 4 shear moduli of an orthotropic material subscript 23 times stress invariant subscript 3, all over perpendicular shear strength superscript 2.

Equation 27. This equation reads inviscid stress tensor superscript N plus 1 equals trial elastic stress tensor superscript N plus 1 minus the product of material stiffness tensor plasticity consistency parameter times delta lambda, times lowercase delta F over lowercase delta sigma subscript KL, evaluated at N.

Equation 28. This equation reads Trial elastic stress tensor superscript N plus1 equals viscid with damage stress tensor superscript N plus material stiffness times delta epsilon subscript KL.

Equation 29. This equation reads lowercase delta parallel F over orthotropic stress component subscript 11 equals 2 times orthotropic stress component subscript 11 over general parallel wood strength superscript 2.

Equation 30. This equation reads lowercase delta perpendicular F over lowercase delta sigma subscript 22 equals 2 stress invariant subscript 2 over general perpendicular wood strength superscript 2 minus orthotropic stress component subscript 33 over perpendicular shear strength superscript 2.

Equation 31. This equation reads lowercase delta perpendicular F over lowercase delta sigma subscript 33 equals 2 times the stress invariant subscript 2 over general perpendicular wood strength superscript 2, all minus orthotropic stress component subscript 22 over perpendicular shear strength superscript 2.

Equation 32. This equation reads lowercase delta parallel F over lowercase sigma subscript 12 equals 2 times orthotropic stress component subscript 12 over parallel shear strength superscript 2.

Equation 33. This equation reads lowercase delta parallel F over lowercase sigma subscript 13 equals 2 times orthotropic stress component subscript 13 over parallel shear strength superscript 2.

Equation 34. This equation reads lowercase delta perpendicular F over lowercase sigma subscript 23  equals 2 times orthotropic stress component subscript 23 over perpendicular shear strength superscript 2.

Equation 35. This figure shows two separate equations.  The first equation reads Parallel hardening rate parameter equals 400 over compression quality factor superscript 2 Parallel.  The second equation reads Perpendicular hardening rate parameter equals 100 over compression quality factor superscript 2 Perpendicular.

Equation 36. This equation reads delta alpha subscript 11 equals Parallel hardening rate times parallel hardening model translation limit function times the quantity of orthotropic stress component subscript 11 minus alpha subscript 11 end quantity, times parallel scalar effective strain rate, times time-step increment.

Equation 37. This equation reads Parallel scalar effective strain-rate increment equals the square root of the quantity of strain-rate increments parallel to the grain superscript 2 plus 2 times the strain-rate increments parallel to the grain superscript 2 plus 2 times strain-rate increments parallel to the grain superscript 2, end quantity.

Equation 38. This equation reads Parallel hardening model transitional limit function equals 1 minus alpha subscript 11 over parallel hardening initiation parameter orthotropic stress component superscript F equals zero.

Equation 39. This equation reads Orthotropic stress component subscript 11 superscript F equals compression perpendicular wood strength times the square root of the quantity 1 minus isotropic material stress invariant subscript 4 over parallel shear strength superscript 2 end quantity.

Equation 40. This figure shows two separate equations.  The first equation reads Delta alpha subscript 22 equals perpendicular hardening-rate parameter times perpendicular hardening model translational limit function times the quantity of orthotropic stress component subscript 22 minus alpha subscript 22 end quantity, times perpendicular scalar effective strain-rate increments times time-step increment.  The second equation reads Delta alpha subscript 33 equals perpendicular hardening-rate parameter times perpendicular hardening model translational limit function times the quantity of orthotropic stress component subscript 33 minus alpha subscript 33 end quantity, times perpendicular scalar effective strain-rate increments times time-step increment.

Equation 41. This equation read Perpendicular scalar effective strain-rate increment equals the square root of the quantity of strain-rate increment perpendicular to the grain superscript 2 plus strain-rate increment perpendicular to the grain superscript 2 plus 2 strain-rate increment perpendicular to the grain superscript 2, end quantity.

Equation 42. This equation reads Perpendicular hardening model translational limit function equals 1 minus the quotient of the numerator quantity of alpha subscript 22 plus alpha subscript 33, all over perpendicular hardening initiation parameter times orthotropic stress component subscript 22 superscript F.

Equation 43. This equation reads Orthotropic stress component subscript 22 superscript F equals compression perpendicular wood strength times the square root of the quantity of 1 minus the stress invariant of a transversely isotropic material subscript 3 over the perpendicular shear strength superscript 2, end quantity.

Equation 44. This equation reads Parallel yield surface function equals orthotropic stress component subscript 11 superscript 2 over parallel wood strength compression superscript 2 times the quantity of 1 minus parallel hardening initiation parameter end quantity, times superscript 2 plus the quotient of the numerator quantity orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 end quantity, all over perpendicular shear strength superscript 2 minus 1, when orthotropic stress component subscript 11 is less than zero.

Equation 45. This equation reads Perpendicular yield surface function equals the squared quantity of orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 end quantity, all over perpendicular wood strength superscript 2 times the quantity 1 minus perpendicular hardening initiation parameter end quantity superscript 2, all plus the numerator quantity of orthotropic stress component subscript 23 superscript 2 minus orthotropic stress component subscript 22 times orthotropic stress component subscript 33 end quantity, over perpendicular shear strength minus one, when orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is less than zero.

Equation 46. This equation reads Backstress tensor for hardening model superscript N plus 1 equals backstress tensor superscript N plus delta alpha backstress tensor.

Equation 47. This equation reads Inviscid with backstress tensor superscript N plus 1 equals inviscid stress tensor superscript N plus 1 plus backstress tensor for hardening model superscript N plus 1.

Equation 48. This equation read Viscid with damage stress tensor equals the quantity of 1 minus D end quantity, times viscid stress tensor.

Equation 49. This equation reads D as a function of instantaneous parallel strain energy type term for damage accumulation equals parallel maximum damage allowed, over parallel softening parameter times the quantity of the numerator 1 plus parallel softening parameter, all over 1 plus parallel softening parameter subscript E superscript minus A times instantaneous parallel strain energy minus initial parallel strain energy type for damage initiation minus 1 end quantity.

Equation 50. This equation reads D as a function of instantaneous parallel strain energy type term for damage accumulation equals perpendicular maximum damage allowed over perpendicular softening parameter times the quantity of the numerator 1 plus perpendicular softening parameter, all over 1 plus perpendicular softening parameter subscript E superscript minus C times instantaneous perpendicular strain energy type term for damage accumulation minus initial perpendicular strain energy type value for damage initiation, all minus 1, end quantity.

Equation 51. This equation reads tau equals the square root of the quantity of trial elastic stress sensor subscript 11 times orthotropic stress component subscript 11, plus trial elastic stress sensor subscript 22 times orthotropic stress component subscript 22, plus trial elastic stress sensor subscript 33 times orthotropic stress component subscript 33, plus 2 times the quantity trial elastic stress sensor subscript 12 times orthotropic stress component subscript 12, plus trial elastic stress sensor subscript 13 times orthotropic stress component subscript 13, plus trial elastic stress sensor subscript 23 times orthotropic stress component subscript 23 end quantity, end quantity.

Equation 52. This equation reads Instantaneous parallel strain energy type value for damage accumulation equals the piecewise function of the square root of  trial elastic stress sensor subscript 11 times orthotropic stress component subscript 11, plus 2 times the quantity trial elastic stress sensor subscript 12 times orthotropic stress component subscript 12 plus trial elastic stress sensor subscript 13 times orthotropic stress component subscript 13end quantity, for values of orthotropic stress component subscript 11 is greater or equal to zero. The second half of the piecewise function reads the square root of 2 times the quantity of trial elastic stress sensor subscript 12 times orthotropic stress component subscript 12,  plus trial elastic stress sensor subscript 13 times orthotropic stress component subscript 13 end quantity, for values of orthotropic stress component subscript 11 is less than zero.

Equation 53. This equation reads Instantaneous perpendicular strain energy type term for damage accumulation equals the piecewise function of the square root of the equation of trial elastic stress sensor subscript 22 times orthotropic stress component subscript 22, plus trial elastic stress sensor subscript 33 times orthotropic stress component subscript 33, plus 2 times the trial elastic stress sensor subscript 23 times orthotropic stress component subscript 23, for values of orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is greater than or equal to zero. The second part of the piecewise function reads the square root of 2 times trial elastic stress sensor subscript 23 times orthotropic stress component subscript 23 end quantity, for values of orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is less than zero.

Equation 54. This equation reads D subscript m equals Max D as a function of instantaneous parallel strain energy type term for damage accumulation, and D as a function of instantaneous perpendicular strain energy type term for damage accumulation.

Equation 55. This equation reads D parallel equals D as a function of instantaneous parallel strain energy type term for damage accumulation.

Equation 56. This equation reads orthotropic stress component subscript 11 equals 1 minus D parallel times viscid stress tensor subscript 11.

Equation 57. This equation reads orthotropic stress component subscript 22 equals 1 minus D subscript M times viscid stress tensor subscript 22.

Equation 58. This equation reads orthotropic stress component subscript 33 equals 1 minus D subscript M times viscid stress tensor subscript 33.

Equation 59. This equation reads orthotropic stress component subscript 12 equals 1 minus D parallel times viscid stress tensor subscript 12.

Equation 60. This equation reads orthotropic stress component subscript 13 equals 1 minus D parallel times viscid stress tensor subscript 13.

Equation 61. The equation reads orthotropic stress component subscript 23 equals 1 minus D subscript M viscid stress tensor subscript 23.

Equation 62. This equation reads Shear fracture energy equals initial parallel strain energy type value for damage initiation Element length  times the quantity of 1 plus parallel softening parameter over A parallel softening parameter end quantity, times the log of 1 plus parallel softening parameter.

Equation 63. This equation reads Shear fracture energy equals the integral from X not of the function 1 minus D, times parallel wood strength tension superscript DX.

Equation 64. This equation reads A equals Initial parallel strain energy type value for damage initiation Element length times the quantity of the numerator of 1 plus parallel softening parameter, all over parallel softening parameter tension fracture energy end quantity, times the log of 1 plus parallel softening parameter.

Equation 65. This equation reads Tension fracture energy equals the piecewise function of tension fracture energy times orthotropic stress component subscript 11 superscript 2 over parallel wood strength tension superscript 2, plus shear parallel fracture energy times the quantity orthotropic stress component subscript 12 superscript 2, plus orthotropic subscript 13 superscript 2 over parallel shear strength superscript 2, for values of orthotropic stress component subscript 11 is greater than or equal to zero. The other part of the piecewise function reads shear parallel fracture energy times parallel shear strength superscript 2 over orthotropic stress component subscript 12 superscript 2 plus orthotropic subscript 13 superscript 2, for values of orthotropic stress component subscript 11 is less than zero.

Equation 66. This equation reads Shear fracture energy equals initial parallel strain energy type value for damage initiation Element length times the quantity of 1 plus perpendicular softening parameter over C perpendicular softening parameter end quantity, times the log of 1 plus perpendicular softening parameter.

Equation 67. This equation reads C equals initial parallel strain energy type value for damage initiation Element length times  the quantity of 1 plus perpendicular softening parameter over perpendicular softening parameter Shear fracture energy end quantity, times the log of 1 plus perpendicular softening parameter.

Equation 68. This equation reads Shear fracture energy equals the piecewise function of shear perpendicular fracture energy times the quantity of orthotropic stress component subscript 22, plus orthotropic stress component subscript 33 end quantity superscript 2, all over perpendicular wood strength tension end quantity plus shear perpendicular fracture energy times the quantity orthotropic stress component subscript 23 superscript 2 minus orthotropic stress component subscript 22 times orthotropic stress component subscript 33, all over perpendicular shear strength superscript 2 end quantity  for the values of orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is greater than or equal to zero. The other part of the piecewise function reads shear perpendicular fracture energy times the quantity of perpendicular shear strength superscript 2 over orthotropic stress component subscript 23 superscript 2 minus orthotropic stress component subscript 22 times orthotropic stress component subscript 33 end quantity, for values of orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is less than zero.

Equation 69. This figure shows two lines of equation.  The first line reads Tension fracture energy equals C subscript I tension fracture intensity superscript 2.  The second line reads Shear fracture energy equals C subscript II shear fracture intensity superscript 2.

Equation 70. This figure shows two lines of equation.  The first line reads C subscript l equals the quantity of shear strength subscript 11 times shear strength subscript 22 over 2 end quantity, superscript one half , times the quantity of shear strength subscript 22 over shear strength subscript 11, plus 2 times shear strength subscript 12 plus shear strength subscript 66 over 2 shear strength subscript 11end quantity superscript one half.  The second equation reads C subscript ll equals shear strength subscript 11 over the square root of of 2 times the quantity of shear strength subscript 22 over shear strength subscript 11, plus 2 times shear strength subscript 12 plus shear strength subscript 66 over 2 shear strength subscript 11end quantity, superscript one half.

Equation 71. This equation reads Shear strength subscript 11 equals 1 over normal moduli of an orthotropic material subscript 11, Shear strength subscript 22 equals 1 over normal moduli of an orthotropic material subscript 22, Shear strength subscript 12 equals negative V subscript 11 over normal moduli of an orthotropic material subscript 22, Shear strength subscript 66 equals 1 over shear moduli of an orthotropic material subscript 12.

Equation 72. This figure shows two separate equations.  The first equation reads Parallel tension fracture energy equals 106 perpendicular tension fracture energy tension per shear quality factor.  The second equation reads Parallel shear fracture energy equals 106 perpendicular shear fracture energy tension per shear quality factor.

Equation 73. This equation reads Stress enhancement factor at zero degrees equals 1 plus wood solid phase impact velocity superscript 2 over 150 times the quantity 1 minus alpha times the density of wood over wood solid phase density end quantity.

Equation 74. This equation reads Stress enhancement factor at ninety degrees equals 1 plus wood solid phase impact velocity superscript 2 over 70 density of wood over wood solid phase density  times the quantity of 1 minus alpha times the density of wood over wood solid phase density end quantity.

Equation 75. This equation reads Stress enhancement factor at zero degrees equals 1 plus 19 times impact velocity over 1000 superscript 2.

Equation 76. This equation reads Stress enhancement factor at ninety degrees equals 1 plus the product of 116 times parenthesis the quotient of impact velocity divided by 1000 parenthesis superscript 2.

Equation 77. This equation reads parallel wood strength tension superscript DYNAMIC equals parallel wood strength tension plus the product of normal moduli subscript L parallel superscript scalar effective strain rate times parallel tension/shear rate-effect fluidity parameter.

Equation 78. This equation reads parallel wood strength compression superscript DYNAMIC equals parallel wood strength compression plus the product of normal moduli subscript L superscript parallel scalar effective strain rate times parallel compression rate-effect fluidity parameter.

Equation 79. This equation reads parallel shear strength superscript DYNAMIC equals parallel shear strength plus the product of shear moduli subscript LT superscript parallel scalar effective strain rate times parallel tension/shear rate-effect fluidity parameter.

Equation 80. This equation reads Perpendicular wood strength tension superscript DYNAMIC equals perpendicular wood strength tension plus the product of normal moduli subscript T superscript perpendicular scalar effective strain rate times perpendicular tension/shear rate-effect fluidity parameter.

Equation 81. This equation reads Perpendicular wood strength compression superscript DYNAMIC equals perpendicular wood strength compression plus the product of normal moduli subscript T superscript perpendicular scalar effective strain rate times perpendicular compression rate-effect fluidity parameter.

Equation 82. This equation reads perpendicular shear strength superscript DYNAMIC equals perpendicular shear strength plus the product of shear moduli subscript TR superscript perpendicular scalar effective strain rate times perpendicular tension/shear rate-effect fluidity parameter.

Equation 83. This equation reads parallel tension/shear rate-effect fluidity parameter equals the quotient of parallel tension/shear rate-effect fluidity parameter divided by parallel scalar effective strain rate superscript rate-effect power parameter.

Equation 84. This equation reads Parallel compression rate-effect fluidity parameter equals the quotient of parallel compression rate-effect fluidity parameter divided by parallel scalar effective strain rate superscript rate-effect power parameter.

Equation 85. This equation reads Perpendicular tension/shear rate-effect fluidity parameter equals the quotient of perpendicular tension/shear rate-effect fluidity parameter divided by perpendicular scalar effective strain rate superscript perpendicular rate-effect power parameter.

Equation 86. This equation reads Perpendicular compression rate-effect fluidity parameter equals the quotient of perpendicular compression rate-effect fluidity parameter divided by perpendicular scalar effective strain rate superscript perpendicular rate-power parameter.

Equation 87. This equation reads General rate-effect fluidity parameter equals the quotient of general rate-effect fluidity parameter divided by scalar effective strain rate superscript rate-effect power parameter.

Equation 88. This equation reads General viscoplastic interpolation parameter equals the quotient of 1 divided by the summation of 1 plus the quotient time-step increment divided by general rate-effect fluidity parameter.

Equation 89. This equation reads Backstress viscid stress tensor equals the product of parenthesis 1 minus general viscoplastic interpolation parameter parenthesis times inviscid stress tensor plus general viscoplastic interpolation parameter trial elastic stress tensor.

Equation 90. This equation reads as General rate-effect fluidity parameter approaches 0, backstress viscid stress tensor equals inviscid stress tensor and is labeled Inviscid Solution following the equation.

Equation 91. The equation reads as General rate-effect fluidity parameter approaches infinity, backstress viscid stress tensor equals trial elastic stress tensor and is labeled Elastic Solution following the equation.

Equation 92. This equation reads Shear moduli of an orthotropic material subscript 12 equals 619 plus the product parenthesis the quotient of the difference of normal moduli subscript 11 minus 6000 divided by 12000 parenthesis times parenthesis 835 minus 619 parenthesis.

Equation 93. This equation reads Shear moduli of an orthotropic material subscript 23 equals the quotient of normal moduli subscript 22 divided by the product of 2 times parenthesis1 plus Impact velocity subscript 23 parenthesis.

Equation 94. This equation reads Douglas fir strength as a function of moisture content equals the product of Douglas fir strength at 20-percent moisture content times parenthesis the quotient of the parameter Capital P as a function of moisture content divided by the parameter Capital P at 20-percent moisture content subscript pine.

Equation 95. This equation reads Scale moduli factor as a function of temperature equals the product of the mean of lowercase A times parenthesis temperature minus 20 parenthesis superscript 2 plus the product of the mean of lowercase B times parenthesis temperature minus 20 parenthesis plus 1.

Equation 96. This equation reads the mean of lowercase A equals the product of lowercase A subscript 1 times moisture content superscript 2 plus the product of lowercase A subscript 2 times moisture content plus lowercase A subscript 3.

Equation 97. This equation reads the mean of lowercase B equals the product of lowercase B subscript 1 times moisture content superscript 2 plus the product of lowercase B subscript 2 times moisture content plus lowercase B subscript 3.

Equation 98. This equation reads strength factor as a function of temperature equals the product of 2 times parenthesis the difference of the scale moduli factor as a function of temperature minus 1 parenthesis plus 1.

Equation 99. This equation reads Parallel tension fracture energy equals the product of parenthesis 0.1 plus the quotient of temperature divided by 22.2223 parenthesis times parallel tension fracture energy superscript RoomTemp.

Equation 100. This equation reads Perpendicular tension fracture energy equals the product of parenthesis 0.1 plus the quotient of temperature divided by 22.2223 parenthesis times perpendicular tension fracture energy superscript RoomTemp.

Equation 101. This equation reads Stress component subscript 1 equals C subscript 11, C subscript 12, C subscript 13, 0, 0, 0 equals strain component subscript 1. Stress component subscript 2 equals C subscript 12, C subscript 22, C subscript 23, 0, 0, 0 equals strain component subscript 2. Stress component subscript 3 equals C subscript 13, C subscript 23, C subscript 33, 0, 0, 0 equals strain component subscript 3.  Stress component subscript 4 equals 0, 0, 0, 2C subscript 44, 0, 0 equals strain component subscript 4.  Stress component subscript 5 equals 0, 0, 0, 0, 2c subscript 55, 0 equals strain component subscript 5.  Stress component subscript 6 equals 0, 0, 0, 0, 0, 2C subscript 66 equals strain component subscript 6.

Equation 102. This equation reads C subscript 11 equals the product of Capital E subscript 11 times parenthesis 1 minus the product of IMPACT VELOCITY subscript 23 times IMPACT VELOCITY subscript 32 parenthesis divided by  Delta.

Equation 103. This equation reads C subscript 22 equals the product of Capital E subscript 22 times parenthesis 1 minus the product of IMPACT VELOCITY subscript 31 times IMPACT VELOCITY subscript 13 parenthesis divided by Delta.

Equation 104. This equation reads Capital C subscript 33 equals E subscript 33 parenthesis 1 minus the product of IMPACT VELOCITY subscript 12 times IMPACT VELOCITY subscript 21 parenthesis divided by Delta.

Equation 105. This equation reads Capital C subscript 12 equals the product of parenthesis IMPACT VELOCITY subscript 21 plus the product of IMPACT VELOCITY subscript 31 times IMPACT VELOCITY subscript 23 parenthesis times E subscript 11 divided by Delta.

Equation 106. This equation reads Capital C subscript13 equals the product of parenthesis IMPACT VELOCITY subscript 31 plus the product of IMPACT VELOCITY subscript 21 times IMPACT VELOCITY subscript 32 parenthesis times E subscript 11 divided by Delta.

Equation 107. This equation reads Capital C subscript 23 equals the product of parenthesis IMPACT VELOCITY subscript 32 plus the product of IMPACT VELOCITY subscript 12 times IMPACT VELOCITY subscript 31 parenthesis times E subscript 22 divided by Delta.

Equation 108. This equation reads Capital C subscript 44 equals Capital G subscript 12.

Equation 109. This equation reads Capital C subscript 55 equals Capital G subscript 13.

Equation 110. This equation reads Capital C subscript 66 equals Capital G subscript 23.

Equation 111. This equation reads Delta equals 1 minus the product of IMPACT VELOCITY subscript 12 times IMPACT VELOCITY subscript 21 minus the product of IMPACT VELOCITY subscript 23 times IMPACT VELOCITY subscript 32 minus the product of IMPACT VELOCITY subscript 31 times IMPACT VELOCITY subscript 13 minus the product of 2 times IMPACT VELOCITY subscript 21 times IMPACT VELOCITY subscript 32 times IMPACT VELOCITY subscript 13.

Equation 112. This equation reads Poisson's ratio over normal moduli subscript I equals Poisson's ratio over normal moduli subscript j for i, j equals 1, 2, 3.

Equation 113. This equation reads Parallel yield surface function equals orthotropic stress component subscript 11 superscript 2 over general wood strength superscript 2 plus parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis over parallel shear strength superscript 2 minus 1. General parallel wood strength equals tension parallel wood strength when orthotropic stress component subscript 11 is greater that zero and general parallel wood strength equals compression parallel wood strength when orthotropic stress component subscript 11 is less than zero.

Equation 114. This equation reads Perpendicular yield surface function equals parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis superscript 2 over general perpendicular wood strength superscript 2 plus parenthesis orthotropic stress component subscript 23 superscript 2 minus orthotropic stress component subscript 22 times orthotropic stress component subscript 33 parenthesis over perpendicular shear strength superscript 2 minus 1.  General perpendicular wood strength equals perpendicular tension wood strength when orthotropic stress component subscript 22 plus orthotropic stress material subscript 33 is greater than zero and general perpendicular wood strength equals perpendicular compression wood strength when orthotropic stress component subscript 22 plus orthotropic stress material subscript 33 is less than zero.

Equation 115. This equation reads D subscript M equals Max d instantaneous parallel strain energy type term for damage accumulation, d instantaneous perpendicular strain energy type term for damage accumulation.

Equation 116. This equation reads d parallel equals d instantaneous parallel strain energy type term for damage accumulation.

Equation 117. This equation reads orthotropic stress component subscript 11 equals the product of parenthesis 1 minus d parallel parenthesis times viscid stress tensor subscript 11.

Equation 118. This equation reads orthotropic stress component subscript 22 equals the product of parenthesis 1 minus d subscript m parenthesis times viscid stress tensor subscript 22.

Equation 119. This equation reads orthotropic stress component subscript 33 equals the product of  parenthesis 1 minus d subscript m parenthesis times viscid stress tensor subscript 33.

Equation 120. This equation reads orthotropic stress component subscript 12 equals the product of parenthesis 1 minus d parallel parenthesis times viscid stress tensor subscript 12.

Equation 121. This equation reads orthotropic stress component subscript 13 equals the product of parenthesis 1 minus d parallel parenthesis times viscid stress tensor subscript 13.

Equation 122. This equation reads orthotropic stress component subscript 23 equals the product of parenthesis 1 minus d subscript m parenthesis times viscid stress tensor subscript 23.

Equation 123. This set of equations is labeled Parallel.  The first equation in this first set reads Parallel wood strength tension superscript DYNAMIC equals parallel wood strength tension plus the product of normal moduli subscript Capital L parallel scalar effective strain rate to the power parenthesis 1 minus parallel rate-effect power parameter parenthesis times parallel tension/shear rate-effect fluidity parameter. 

The second reads Parallel wood strength compression superscript DYNAMIC equals parallel wood strength compression plus the product of normal moduli subscript Capital L parallel scalar effective strain rate to the power parenthesis 1 minus parallel rate-effect power parameter parenthesis times parallel compression rate-effect fluidity parameter. 

The third equation reads Parallel shear strength superscript DYNAMIC equals parallel shear strength plus the product of shear moduli subscript Capital LT parallel scalar effective strain rate to the power parenthesis 1 minus parallel rate-effect power parameter parenthesis times parallel tension/shear rate-effect fluidity parameter.

Equation 124. This set of equations is labeled Perpendicular.  The first equation in this set reads Perpendicular wood strength tension superscript DYNAMIC equals perpendicular wood strength tension plus the product of the normal moduli subscript T perpendicular scalar effective strain rate to the power parenthesis 1 minus perpendicular rate-effect power parameter parenthesis times perpendicular tension/shear rate-effect fluidity parameter.

The second reads Perpendicular wood strength compression superscript DYNAMIC equals perpendicular wood strength compression plus the product of normal moduli subscript T perpendicular scalar effective strain rate to the power parenthesis 1 minus perpendicular rate-effect power parameter parenthesis times perpendicular compression rate-effect fluidity parameter. 

The third equation reads Perpendicular shear strength superscript DYNAMIC equals perpendicular shear strength plus the product of shear moduli subscript TR perpendicular scalar effective strain rate the power of parenthesis1 minus perpendicular rate-effect power parameter parenthesis times perpendicular tension/shear rate-effect fluidity parameter.

Equation 125. This equation reads Orthotropic stress component subscript 11 is equal to or greater than parallel tension wood strength when orthotropic stress component subscript 11 is greater than zero.

Equation 126. This equation reads the absolute value of Orthotropic stress component subscript 11 is greater than or equal to parallel compression wood strength when orthotropic stress component subscript 11 is less than zero.

Equation 127. This equation reads Orthotropic stress component subscript 22 is greater than or equal to perpendicular tension wood strength when orthotropic stress component subscript 22 is greater than zero.

Equation 128. This equation reads the absolute value of Orthotropic stress component subscript 22 is greater than or equal to perpendicular compression wood strength when orthotropic stress component subscript 22 is less than zero.

Equation 129. This equation reads Orthotropic stress component subscript 33 is equal to or greater than Capital Z subscript Capital T when orthotropic stress component subscript 33 is less than zero.

Equation 130. This equation reads the absolute value of Orthotropic stress component subscript 33 is equal to or greater than Capital Z subscript Capital C when orthotropic stress component subscript 33 is less than zero.

Equation 131. This equation reads the absolute value of Orthotropic stress component subscript 12 is greater than or equal to shear strength subscript Capital XY.

Equation 132. This equation reads the absolute value of orthotropic stress component subscript 13 is greater than or equal to shear strength subscript Capital XZ.

Equation 133. This equation reads the absolute value of orthotropic stress component subscript 23 is greater than or equal to shear strength subscript capital YZ.

Equation 134. This equation reads the product of Capital F subscript 1 times orthotropic stress component subscript 11 plus the product of Capital F subscript 2 times parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis plus the product of Capital F subscript 11 times orthotropic stress component subscript 11 superscript 2 plus the product of Capital F times subscript 22 times parenthesis orthotropic stress component subscript 22 superscript 2 plus orthotropic stress component subscript 33 superscript 2 plus 2 times orthotropic stress component subscript 23 superscript 2 parenthesis plus the product of Capital F subscript 66 times parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis plus the product of 2 times Capital F subscript 12 times parenthesis the product of orthotropic stress component subscript 11 times orthotropic stress component subscript 22 plus the product of orthotropic stress component subscript 11 times orthotropic stress component subscript 33 parenthesis plus the product of 2 times Capital F subscript 23 times parenthesis orthotropic stress component subscript 23 minus the product of orthotropic stress component subscript 22 times orthotropic stress component subscript 33 parenthesis is great than or equal to 1.

Equation 135. This equation reads F subscript 1 equals the quotient of 1 divided by parallel wood strength tension minus the quotient 1 divided by parallel wood strength compression.

Equation 136. This equation reads F subscript 2 equals the quotient 1 divided by perpendicular wood strength tension minus the quotient 1 divided by perpendicular wood strength compression.

Equation 137. This equation reads F subscript 11 equals the quotient of 1 divided by the product of  parallel wood strength tension times parallel wood strength compression.

Equation 138. This equation reads F subscript 22 equals the quotient of 1 divided by the product of  perpendicular wood strength tension times perpendicular wood strength compression.

Equation 139. This equation reads F subscript 66 equals the quotient of 1 divided by parallel shear strength superscript 2.

Equation 140. This equation reads the product of 2 times F subscript 23 equals the quotient 1 divided by  perpendicular shear strength superscript 2 minus the quotient of 2 divided by the product of perpendicular wood strength tension times perpendicular wood strength compression.

Equation 141. This equation reads the product of 2 times Capital F subscript 12 equals the quotient of 1 divided by stress component superscript 2 minus the product of the quotient 1 divided by stress component times parenthesis Capital F subscript 1 plus Capital F subscript 2 parenthesis minus parenthesis Capital F subscript 11 plus Capital F subscript 22 plus F subscript 66 parenthesis.

Equation 142. This equation reads the product of Capital A times parenthesis orthotropic stress component subscript 11 minus orthotropic stress component subscript 22 parenthesis superscript 2 plus the product of Capital B times parenthesis orthotropic stress component subscript 22 minus orthotropic stress component subscript 33 parenthesis superscript 2 plus the product of Capital C times parenthesis orthotropic stress component subscript 33 minus orthotropic stress component subscript 11 parenthesis superscript 2 plus the product of Capital D times orthotropic stress component subscript 12 superscript 2 plus the product of Capital E times orthotropic normal moduli subscript 23 superscript 2 plus the product of Capital F times orthotropic stress component subscript 13 superscript 2 equals 1.

Equation 143. This equation reads the product of Capital A times parenthesis orthotropic stress component subscript 11 minus orthotropic stress component subscript 22 parenthesis superscript 2 plus the product of Capital B times parenthesis orthotropic stress component subscript 22 minus orthotropic stress component subscript 33 parenthesis superscript 2 plus the product of C times parenthesis orthotropic stress component subscript 33 minus orthotropic stress component subscript 11 parenthesis superscript 2 plus the product of Capital D times orthotropic stress component subscript 12 superscript 2 plus the product of Capital E times orthotropic normal moduli subscript 23 superscript 2 plus the product of Capital F times orthotropic stress component subscript 13 superscript 2 plus the product of Capital G times orthotropic stress component subscript 11 plus Capital H orthotropic stress component subscript 22 plus the product of Trial elastic stress invariant times orthotropic stress component subscript 33 is greater than or equal to 1.

Equation 144. This equation reads Capital A equals the quotient of 1 divided by the product of 2 times parallel wood strength tension times parallel wood strength compression plus the quotient of 1 divided by 2 times perpendicular wood strength tension times perpendicular wood strength compression minus the quotient 1 divided by 2 times Z tension times Z compression.

Equation 145. This equation reads Capital B equals the quotient of negative 1 divided by the product of 2 times parallel wood strength tension times parallel wood strength compression plus the quotient of 1 divided by the product of 2 times perpendicular wood strength tension times perpendicular wood strength compression plus the quotient of 1 divided by 2 times Capital Z tension times Capital Z compression.

Equation 146. This equation reads Capital C equals the quotient of 1 divided by the product of 2 times parallel wood strength tension times parallel wood strength compression minus the quotient 1 divided by the product of 2 times perpendicular wood strength tension times perpendicular wood strength compression plus the quotient of 1 divided by the product of 2 times Z tension times Z compression.

Equation 147. This equation reads Capital D equals the quotient 1 divided by the product of shear strength times x times y superscript 2.

Equation 148. This equation reads Capital E equals the quotient of 1 divided by shear strength times x times z superscript 2.

Equation 149. This equation reads Capital F equals the quotient of 1 divided by shear strength times y times z superscript 2.

Equation 150. This equation reads Capital G equals the quotient of 1 divided by parallel wood strength tension minus the quotient 1 divided by parallel wood strength compression.

Equation 151. This equation reads Capital H equals the quotient of 1 divided by perpendicular wood strength tension minus the quotient 1 divided by perpendicular wood strength compression.

Equation 152. This equation reads Capital I equals 1 divided by Capital Z tension minus the quotient 1 divided by Capital Z compression.

Equation 153. This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general parallel wood strength superscript 2 minus the quotient of the product of orthotropic stress component subscript 11 times orthotropic stress component subscript 22 divided by general parallel wood strength superscript 2 plus the quotient of orthotropic stress component subscript 22 superscript 2 divided by general perpendicular wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength superscript 2 is greater than or equal to 1.

Equation 154. This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general parallel wood strength superscript 2 minus the quotient of the product of orthotropic stress component subscript 11 times orthotropic stress component subscript 22 divided by general parallel/perpendicular wood strength plus the quotient of orthotropic stress component subscript 22 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength superscript 2 is greater than or equal to 1.

Equation 155. This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength superscript 2 is greater than or equal to 1.

Equation 156. This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength subscript XY superscript 2 plus the quotient of orthotropic stress component subscript 13 superscript 2 divided by shear strength subscript XZ superscript 2 is greater than or equal to 1.

Equation 157. This equation reads the quotient of Orthotropic stress component subscript 22 superscript 2 divided by general perpendicular wood strength superscript 2 plus the quotient of orthotropic stress component subscript 12 superscript 2 divided by shear strength subscript XY superscript 2 plus the quotient of orthotropic stress component subscript 23 superscript 2 divided by shear strength subscript YZ superscript 2 is greater than or equal to 1.

Equation 158. This equation reads the quotient of Orthotropic stress component subscript 33 superscript 2 divided by Capital Z superscript 2 plus the quotient of orthotropic stress component subscript 13 superscript 2 divided by shear strength subscript X lowercase Z superscript 2 plus the quotient of orthotropic stress component subscript 23 superscript 2 divided by shear strength subscript Y lowercase Z superscript 2 is greater than or equal to 1.

Equation 159. This equation reads Capital I subscript 1 equals orthotropic stress component subscript 11.

Equation 160. This equation reads Capital I subscript 2 equals orthotropic stress component subscript 22 plus orthotropic stress component subscript 33.

Equation 161. This equation reads Capital I subscript 3 equals orthotropic stress component subscript 33 superscript 2 minus the product of orthotropic stress component subscript 22 times orthotropic stress component subscript 33.

Equation 162. This equation reads Capital I subscript 4 equals orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2.

Equation 163. This equation reads Capital I subscript 5 equals the product of 2 times orthotropic stress component subscript 12 times orthotropic stress component subscript 23 times orthotropic stress component subscript 13 minus the product of orthotropic stress component subscript 22 times orthotropic stress component subscript 13 superscript 2 minus the product of orthotropic stress component subscript 33 times orthotropic stress component subscript 12 superscript 2.

Equation 164. This equation reads the product of Capital A subscript 1 times Capital I subscript 1 plus the product of parallel softening parameter subscript 1 times Capital I subscript 1 superscript 2 plus the product of Capital A subscript 2 times Capital I subscript 2 plus the product of parallel softening parameter subscript 2 times Capital I subscript 2 superscript 2 plus the product of Capital C subscript 12 times Capital I subscript 1 times Capital I subscript 2 plus the product of Capital A subscript 3 times Capital I subscript 3 plus the product of Capital A subscript 4 times Capital I subscript 4 is greater than or equal to 1.

Equation 165. This equation reads the product of Capital A subscript F times orthotropic stress component subscript 11 plus the product of parallel softening parameter subscript F times orthotropic stress component subscript 11 superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 equals 1.

Equation 166. This equation reads the product of Capital A subscript M times Parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis plus the product of parallel softening parameter subscript M times parentheses orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 23 superscript 2 minus orthotropic stress component subscript 22 times orthotropic stress component subscript 33 parenthesis divided by perpendicular shear strength superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 equals 1.

Equation 167. This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by parallel wood strength tension superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 is greater than or equal to 1 when orthotropic stress component subscript 11 is greater than zero.

Equation 168. This equation reads Orthotropic stress component subscript 11 is greater than or equal to parallel wood strength compression when orthotropic stress component subscript 11 is less than zero.

Equation 169. This equation reads the quotient of parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis superscript 2 divided by perpendicular wood strength tension superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 23 superscript 2 minus the product of orthotropic stress component subscript 22 times orthotropic stress component subscript 33 parenthesis divided by perpendicular shear strength superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 is greater than or equal to 1 when orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is greater than zero.

Equation 170. This equation reads the quotient of parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33parenthesis superscript 2 divided by the product of 4 times parallel shear strength superscript 2 plus the product of bracket parenthesis the quotient of perpendicular wood strength compression divided by the product of 2 times perpendicular shear strength parenthesis superscript 2 minus 1 bracket times the quotient parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis superscript 2 divided by perpendicular wood strength compression superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 23 superscript 2 minus the product of orthotropic stress component subscript 22 times orthotropic stress component 33 parenthesis divided by perpendicular shear strength superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 is greater than or equal to 1 when orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 is less than 0.

Equation 171. This equation reads the quotient of Orthotropic stress component subscript 11 superscript 2 divided by general wood strength superscript 2 plus the quotient of parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis divided by parallel shear strength superscript 2 is greater than or equal to 1. General parallel wood strength equals tension parallel wood strength if orthotropic stress component subscript 11 is greater that zero and equals compression parallel wood strength if orthotropic stress component subscript 11 is less than zero.

Equation 172. This equation reads the quotient of parenthesis orthotropic stress component subscript 22 plus orthotropic stress component subscript 33 parenthesis superscript 2 divided by general perpendicular wood strength superscript 2 plus parenthesis orthotropic stress component subscript 23 superscript 2 minus orthotropic stress component subscript 22 times orthotropic stress component subscript 33 parenthesis over perpendicular shear strength superscript 2 plus parenthesis orthotropic stress component subscript 12 superscript 2 plus orthotropic stress component subscript 13 superscript 2 parenthesis over perpendicular shear strength superscript 2 is greater than or equal to 1.  General perpendicular wood strength equals perpendicular tension wood strength if orthotropic stress component subscript 22 plus orthotropic stress material subscript 33 is greater than zero and equals perpendicular compression wood strength of the orthotropic stress component subscript 22 plus orthotropic stress material subscript 33 is less than zero}.

Equation 173. This equation reads Stress component subscript ULT equals the product of general parallel wood strength times the quotient of Capital F subscript lowercase phi divided by the sum of sine superscript 2 Capital theta plus F subscript lowercase phi cosine superscript 2 lowercase theta.

Equation 174. This equation reads F subscript lowercase phi equals F subscript transverse plus the product of lowercase phi times the quotient of parenthesis F subscript radial minus F subscript transverse parenthesis divided by 90 plus the product of Fracture intensity times parenthesis negative sine of the product of 2 Capital Theta parenthesis times the quotient of parenthesis  F subscript radial plus F subscript transverse parenthesis divided by 2.

Equation 175. This equation reads Stress component subscript ULT equals the quotient of the product of general parallel wood strength times perpendicular wood strength divided by the sum of general parallel wood strength sine superscript 2 Capital Theta plus general perpendicular wood strength cosine superscript 2 Capital Theta.

Equation 176. This equation reads bracket stress component subscript Capital T, stress component subscript Capital R, stress component subscript Capital TR bracket equals stress component subscript lowercase ULT bracket lowercase M superscript 2, lowercase N superscript 2, negative lowercase MN bracket.

Equation 177. This equation reads Strain increment equals strain increment superscript e plus strain increment superscript p.

Equation 178. This equation reads viscid with damage stress tensor superscript the sum of n plus 1 equals viscid with damage stress tensor superscript n plus the product of material stiffness tensor times parenthesis strain increment subscript KL minus delta strain increment subscript kl superscript p parenthesis.

Equation 179. This equation reads trial elastic stress tensor superscript n plus 1 equals viscid with damage stress tensor superscript n plus the product of material stiffness tensor times strain increment  subscript KL.

Equation 180. This equation reads Strain increment superscript p equals the product of plasticity consistency parameter divided by delta viscid with damage stress tensor evaluated at n.

Equation 181. This equation reads delta yield surface function equals the difference of yield surface function superscript the sum of n plus 1 minus yield surface function superscript n which in turn equals zero.

Equation 182. This equation reads parallel yield surface function parenthesis Trial elastic stress invariant subscript 1, Trial elastic stress invariant subscript 4 parenthesis equals the quotient of Trial elastic stress invariant subscript 1 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of Trial elastic stress invariant subscript 4 divided by parallel shear strength superscript 2 minus 1.

Equation 183. This equation reads the product of the quotient of delta parallel yield surface function divided by delta Trial elastic stress invariant subscript 1 evaluated at lowercase N times delta trial elastic stress invariant subscript 1 plus the product of the quotient delta parallel yield surface function divided by delta trial elastic stress invariant subscript 4 subscript evaluated at n times delta I subscript 4 equals zero.

Equation 184. This equation reads delta Trial elastic stress invariant subscript 1 equals trial elastic increment subscript 1 plus the product of the quotient delta Trial elastic stress invariant subscript 1 divided by parallel plasticity consistency parameter evaluated at n times parallel plasticity consistency parameter.

Equation 185. This equation reads Trial elastic stress invariant subscript 4 equals trial elastic stress invariant subscript 4 plus the product of the quotient of delta Trial elastic stress invariant subscript 4 divided by parallel consistency parameter evaluated at n times the parallel consistency parameter.

Equation 186. This equation reads parallel plasticity consistency parameter equals the quotient of parenthesis the product of the quotient of delta parallel trial elastic yield surface function divided by delta Trial elastic stress invariant subscript 1 evaluated at n times trial elastic stress invariant subscript 1 minus the product of the quotient of delta parallel trial elastic yield surface function divided by delta Trial elastic stress invariant subscript 4 evaluated at n times trial elastic stress invariant subscript 4 parenthesis divided by parenthesis the product of the quotient delta parallel yield surface function divided by delta Trial elastic stress invariant subscript 1 evaluated at n times delta trial elastic stress invariant subscript 1 divided by delta parallel consistency parameter evaluated at n plus the product of the quotient delta parallel yield surface function divided by delta Trial elastic stress invariant subscript 4 evaluated at n times delta trial elastic stress invariant subscript 4 divided by delta parallel consistency parameter evaluated at n parenthesis.

Equation 187. This equation reads the parallel consistency parameter equals the quotient of negative parallel trial elastic yield surface function divided by parenthesis the product of the quotient delta parallel yield surface function divided by delta Trial elastic stress invariant subscript 1 evaluated at n times delta trial elastic stress invariant subscript 1 divided by delta parallel consistency parameter evaluated at n plus the product of the quotient delta parallel yield surface function divided by delta Trial elastic stress invariant subscript 4 evaluated at n times delta trial elastic stress invariant subscript 4 divided by delta parallel consistency parameter evaluated at n parenthesis.

Equation 188. This equation reads perpendicular yield surface as a function of Trial elastic stress invariant subscript 2, Trial elastic stress invariant subscript 3  equals the quotient of Trial elastic stress invariant subscript 2 superscript 2 divided by perpendicular wood strength superscript 2 plus the quotient of Trial elastic stress invariant subscript 3 divided by perpendicular shear strength superscript 2 minus 1.

Equation 189. This equation reads the quotient of delta perpendicular yield surface function divided by delta Trial elastic stress invariant subscript 2 evaluated at n times delta Trial elastic stress invariant subscript 2 plus the quotient delta perpendicular yield surface function divided by delta Trial elastic stress invariant subscript 3 evaluated at n times Trial elastic stress invariant subscript 3 equals 0.

Equation 190. This equation reads delta Trial elastic stress invariant subscript 2 equals trial elastic increment subscript 2 plus the product of the quotient delta Trial elastic stress invariant subscript 2 divided by perpendicular plasticity consistency parameter evaluated at n times perpendicular plasticity consistency parameter.

Equation 191. This equation reads delta Trial elastic stress invariant subscript 3 equals trial elastic increment subscript 3 plus the product of the quotient delta Trial elastic stress invariant subscript 3 divided by perpendicular plasticity consistency parameter evaluated at n times perpendicular plasticity consistency parameter.

Equation 192. This equation reads Perpendicular plasticity consistency parameter equals the quotient of negative perpendicular trial elastic yield surface function divided by parenthesis the product of the quotient delta perpendicular yield surface function divided by delta Trial elastic stress invariant subscript 2 evaluated at n times delta trial elastic stress invariant subscript 2 divided by delta perpendicular consistency parameter evaluated at n plus the product of the quotient delta perpendicular yield surface function divided by delta Trial elastic stress invariant subscript 3 evaluated at n times delta trial elastic stress invariant subscript 3 divided by delta perpendicular consistency parameter evaluated at n parenthesis.

Equation 193. This equation reads Viscid with damage stress tensor superscript the sum of  n plus 1 equals trial elastic stress tensor superscript the sum of n plus 1.

Equation 194. This equation reads Viscid with damage stress tensor superscript the sum of n plus 1 equals the difference of trial elastic stress tensor superscript the sum of n plus 1 minus the product of material stiffness tensor times plasticity consistency parameter times the quotient of yield surface function divided by stress component subscript lowercase KL evaluated at n.

Equation 195. This equation reads the initial yield strength in compression subscript 11 equals the ultimate strength in compression subscript 11 superscript Capital F times parenthesis1 minus parallel hardening initiation parameter parenthesis.

Equation 196. This equation reads ultimate strength in compression subscript 11 superscript Capital F equals the product of parallel wood strength compression times the square-root of parenthesis 1 minus the quotient Trial elastic stress invariant subscript 4 divided by parallel shear strength superscript 2 parenthesis.

Equation 197. This equation reads Orthotropic stress component subscript 11 equals initial yield strength in compression subscript 11 plus backstress tensor subscript 11.

Equation 198. This equation reads ultimate strength in compression subscript 11 superscript Capital F equals yield strength in compression subscript 11 plus backstress tensor subscript 11 superscript max.

Equation 199. This equation reads the difference of 1 minus backstress subscript 11 superscript max divided by the product of  parallel hardening initiation parameter times strength in compression subscript 11 superscript Capital F equals zero.

Equation 200. This equation reads Parallel hardening model translational limit function equals 1 minus the quotient of backstress subscript 11 divided by the product of parallel hardening initiation parameter times strength in compression subscript 11 superscript Capital F.

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