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Publication Number: FHWA-HRT-08-073
Date: September 2009

Development of A Multiaxial Viscoelastoplastic Continuum Damage Model for Asphalt Mixtures

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508 Captions

Equation 36. Definition of general transverse isotropic stiffness terms. The first two diagonal terms in the general transverse isotropic matrix, Z subscript 11 and Z subscript 22, are given by stiffness on the isotropy plane, E, multiplied by the Poisson's ratio factor, uppercase curled upsilon, multiplied by one minus the product of Poisson's ratio between axis of symmetry and isotropy plane, lowercase nu subscript 3,132, and Poisson's ratio between isotropy plane and axis of symmetry, lowercase nu subscript 1,323. The third diagonal term in the general transverse isotropic matrix, Z subscript 33, is given by the stiffness on the symmetric axis, E subscript 3, multiplied by the Poisson's ratio factor, uppercase curled upsilon, multiplied by 1 minus the Poisson's ratio on the isotropic plane, lowercase nu subscript 12. The stiffness term in column two row one, Z subscript 12, is equal to the stiffness on the transverse isotropic plane, E, multiplied by the Poisson's ratio factor, uppercase curled upsilon, multiplied by the sum of the Poisson's ratio on the isotropic plane, lowercase nu subscript 12, and the product of Poisson's ratio between axis of symmetry and isotropy plane, lowercase nu subscript 3,132, and Poisson's ratio between isotropy plane and axis of symmetry, lowercase nu subscript 1,323. The stiffness term column three rows one and two, Z subscript 13 and Z subscript 23, is equal to the stiffness on the transverse isotropic plane, E, multiplied by the Poisson's ratio factor, uppercase curled upsilon, multiplied by the sum of the Poisson's ratio between the axis of symmetry and the isotropic plane, lowercase nu subscript 3,132, and the product of Poisson's ratio between axis of symmetry and isotropy plane, lowercase nu subscript 3,132, and Poisson's ratio on the isotropy plane, lowercase nu subscript 12. Or, these terms equal the stiffness on the transverse isotropic plane, E, multiplied by the Poisson's ratio factor, uppercase curled upsilon, multiplied by the sum of the Poisson's ratio between the transverse isotropy plane and the axis of symmetry lowercase nu subscript 1,323, and the product of Poisson's ratio between the transverse isotropy plane and the axis of symmetry, lowercase nu subscript 1,323, and Poisson's ratio on the isotropy plane, lowercase nu subscript 12. The stiffness terms in the fourth and fifth diagonal terms, Z subscript 44 and Z subscript 55, are equal to the shear modulus between the transverse plan and the axis of, G subscript 13 or G subscript 23. The stiffness term in the sixth diagonal position, Z subscript 66, is equal to the shear modulus between isotropic planes, G subscript 12. The Poisson's ratio factor, uppercase curled upsilon, is given by the reciprocal of 1 minus the Poisson's ratio on the isotropy plane, lowercase nu subscript 12, squared, minus 2 multiplied by the product of Poisson's ratio between axis of symmetry and isotropy plane, lowercase nu subscript 3,132, and Poisson's ratio between isotropy plane and axis of symmetry, lowercase nu subscript 1,323 minus 2 times multiplied by the product of Poisson's ratio between axis of symmetry and isotropy plane, lowercase nu subscript 3,132, Poisson's ratio between isotropy plane and axis of symmetry, lowercase nu subscript 1,323, and Poisson's ratio on the isotropy plane, lowercase nu subscript 12.

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Equation 37. Definition of stiffness on isotropy plane from generalized stiffness terms. The stiffness on the isotropy plane, E, is equal to the first stiffness term, Z subscript 11, squared multiplied by the third diagonal term, Z subscript 33, plus 2 multiplied by the product of the third column first row term, Z subscript 13, squared and the second column first row term, Z subscript 12, minus 2 multiplied by the product of the third column first row term, Z subscript 13, squared and the first diagonal term, Z subscript 11, minus the product of the third diagonal term, Z subscript 33, and the second column first row term, Z subscript 12, squared, divided by the difference of the product of the first diagonal term, Z subscript 11, and the third diagonal term, Z subscript 33, and the third column first row term, Z subscript 13, squared. The stiffness along the axis of symmetry, E subscript 3, is equal to the first stiffness term, Z subscript 11, squared multiplied by the third diagonal term, Z subscript 33, plus 2 multiplied by the product of the third column first row term, Z subscript 13, squared and the second column first row term, Z subscript 12, minus 2 multiplied by the product of the third column first row term, Z subscript 13, squared and the first diagonal term, Z subscript 11, minus the product of the third diagonal term, Z subscript 33, and the second column first row term, Z subscript 12, squared, divided by the difference of the first diagonal term, Z subscript 11, squared and the second column first row term, Z subscript 12, squared. The Poisson's ratio on the isotropy plane, lowercase nu subscript 12, is equal to the product of the second column first row stiffness term, Z subscript 12, and the third diagonal term, Z subscript 33, minus the third column first row term, Z subscript 13, squared; divided by the difference of the product of the first diagonal term, Z subscript 11, and the third diagonal term, Z subscript 33, and the third column first row term, Z subscript 13, squared. The Poisson's ratio between the axis of symmetry and the isotropy plane, lowercase nu subscript 3132, is equal to the product of the first stiffness term, Z subscript 11, and the third column first row term, Z subscript 13, minus the product of the third column first row term, Z subscript 13, and the second column first row term, Z subscript 12, divided by the difference of the first diagonal term, Z subscript 11, squared and the second column first row term, Z subscript 12, squared. The Poisson's ratio between the isotropy plane and axis of symmetry, lowercase nu subscript 1323, is equal to the product of the first stiffness term, Z subscript 11, and the third column first row term, Z subscript 13, minus the product of the third column first row term, Z subscript 13, and the second column first row term, Z subscript 12, divided by the difference of the product of the first diagonal term, Z subscript 11, and the third diagonal term, Z subscript 33, and the third column first row term, Z subscript 13, squared. The shear modulus between the transverse plan and the axis of symmetry, G subscript 13 or G subscript 23, is equal to the fourth, or equivalently fifth diagonal stiffness term, Z subscript 44 or Z subscript 55. The shear modulus between the isotropic planes, G subscript 12, is equal to the sixth diagonal stiffness term, Z subscript 66.

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Equation 38. Compliance matrix for transversely isotropic material in terms of engineering parameters. The compliance matrix for transversely isotropic material, strong S, is equal to 6 by 6 matrix, where the matrix is symmetric about the diagonal with the first two diagonal elements equal to the reciprocal of the stiffness on the isotropy plane, E, the first two elements in the third column are also the same and equal the negative of the Poisson's ratio between the axis of symmetry, lowercase nu subscript 3,132, divided by the stiffness along the axis of symmetry, E subscript 3, but differ from the value in the second column first row which is equal to the negative of the Poisson's ratio on the isotropic planes, lowercase nu subscript 12, divided by stiffness on the isotropic plane, E, the first two elements in the third row are the same and equal the negative of Poisson's ratio between the isotropic plane and the axis of symmetry, nu subscript 1,323, divided by the stiffness on the isotropic plane, E, the element in the third column and third row is equal to the reciprocal of the stiffness along the axis of symmetry, E subscript 3, the fourth and fifth diagonal elements are equal to the reciprocal of the shear modulus between the transverse plane and the axis of symmetry, G subscript 23, the sixth diagonal element is equal to the reciprocal of the shear modulus on the isotropic plane, G subscript 12, and all other elements are 0.

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Equation 39. Strain energy density function from generalized stiffness terms. The strain energy, W, is equal to one half multiplied by the first diagonal term, Z subscript 11, multiplied by the strain along the direction of the first diagonal term, epsilon subscript 1, squared plus the first diagonal term, Z subscript 11, multiplied by the strain along the second diagonal term direction, epsilon subscript 2, squared, plus the third diagonal term, Z subscript 33, multiplied by the strain along the third diagonal direction, epsilon subscript 3, squared, plus 2 multiplied by the product of the second column first row term, Z subscript 12, the strain along the direction of the first diagonal term, epsilon subscript 1, and the strain along the direction of the second diagonal term, epsilon subscript 2, plus 2 multiplied by the product of the third column first row term, Z subscript 13, the strain along the first diagonal term direction, epsilon subscript 1, and the strain along the third diagonal term direction, epsilon subscript 3, plus 2 multiplied by the product of the second column third row term, Z subscript 23, the strain along the second diagonal direction, epsilon subscript 2, and the strain along the third diagonal direction, epsilon subscript 3.

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Equation 42. Definition of the primary loading direction deviatoric strain. The primary loading direction deviatoric strain, e subscript 3, is equal to primary direction strain, epsilon subscript 3, minus the dilation, e subscript v, divided by 3. The strain difference term, e subscript 2, is equal to the difference between the strains on the transverse plane, epsilon subscript 2 minus epsilon subscript 1. The first diagonal general stiffness term, Z subscript 11, is equal to the second diagonal stiffness term, Z subscript 22, and equal to the first Schapery term, A subscript 11, plus the second Schapery stiffness term, A subscript 22, divided by 9, minus 2 multiplied by the third Schapery stiffness term, A subscript 12, divided by 3, plus the fourth Schapery stiffness term, A subscript 66. The third diagonal general stiffness term, Z subscript 33, equals the first Schapery term, A subscript 11, plus 4 multiplied by the second Schapery stiffness term, A subscript 22, divided by 9, plus 4 multiplied by the third Schapery stiffness term, A subscript 12, divided by 3. The second column first row general stiffness term, Z subscript 12, is equal to the first Schapery term, A subscript 11, plus the second Schapery stiffness term, A subscript 22, divided by 9, minus 2 multiplied by the third Schapery stiffness term, A subscript 12, divided by 3, minus the fourth Schapery stiffness term, A subscript 66. The third column first row general stiffness term, Z subscript 13, is equal to the second column third row general stiffness term, Z subscript 23, and equals the first Schapery term, A subscript 11, minus 2 multiplied by the second Schapery stiffness term, A subscript 22, divided by 9, plus the third Schapery stiffness term, A subscript 12, divided by 3

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Equation 45. Definition of stiffness along axis of symmetry from material integrity terms. The stiffness along the axis of symmetry, E subscript 3, is equal to the first material integrity term, C subscript 11. The stiffness on the isotropy plane, E, is equal to product of 4, initial shear modulus, G subscript 0, and the first material integrity term, C subscript 11, divided by the first material integrity term, C subscript 11, plus the product of the initial shear modulus, G subscript 0, and 1 plus the second material integrity term, C subscript 12, squared minus 2 multiplied by the second material integrity parameter, C subscript 12, minus the product of the first and third material integrity parameters, C subscript 11 and C subscript 22. The Poisson's ratio between the axis of symmetry and the isotropy plane, lowercase nu subscript 3132, is equal to one half multiplied by one minus the second material integrity term, C subscript 12. The Poisson's ratio between the isotropy plane and axis of symmetry, lowercase nu subscript 1323, is equal to is equal to product of 2, initial shear modulus, G subscript 0, and the second material integrity parameter, C subscript 12, minus one; divided by the first material integrity term, C subscript 11, plus the product of the initial shear modulus, G subscript 0, and 1 plus the second material integrity term, C subscript 12, squared minus 2 multiplied by the second material integrity parameter, C subscript 12, minus the product of the first and third material integrity parameters, C subscript 11 and C subscript 22. The Poisson's ratio on the isotropy plane, lowercase nu subscript 12, is equal the first material integrity term, C subscript 11, minus the product of the initial shear modulus, G subscript 0, and 1 plus the second material integrity term, C subscript 12, squared minus 2 multiplied by the second material integrity parameter, C subscript 12, minus the product of the first and third material integrity parameters, C subscript 11 and C subscript 22; divided by the first material integrity term, C subscript 11, plus the product of the initial shear modulus, G subscript 0, and 1 plus the second material integrity term, C subscript 12, squared minus 2 multiplied by the second material integrity parameter, C subscript 12, minus the product of the first and third material integrity parameters, C subscript 11 and C subscript 22.

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Equation 46. Compliance matrix for transversely isotropic material in terms of engineering parameters. The compliance matrix for transversely isotropic material, strong S, is equal to 6 by 6 matrix, where the matrix is symmetric about the diagonal with the first two diagonal elements equal to the first material integrity term, C subscript 11, plus the product of the initial shear modulus, G subscript 0, and 1 plus the second material integrity term, C subscript 12, squared minus 2 multiplied by the second material integrity parameter, C subscript 12, minus the product of the first and third material integrity parameters, C subscript 11 and C subscript 22, divided by the product of 4, initial shear modulus, G subscript 0, and the first material integrity term, C subscript 11, the first two elements in the third column are also the same and equal the negative of 1 minus the second material integrity term, C subscript 12, divided by 2 times the first material integrity parameter, C subscript 11, but differ from the value in the second column first row which is equal to the negative of the first material integrity term, C subscript 11, minus the product of the initial shear modulus, G subscript 0, and 1 plus the second material integrity term, C subscript 12, squared minus 2 multiplied by the second material integrity parameter, C subscript 12, minus the product of the first and third material integrity parameters, C subscript 11 and C subscript 22; divided by the product of 4, initial shear modulus, G subscript 0, and the first material integrity term, C subscript 11, the first two elements in the third row are the same and equal the negative of 1 minus the second material integrity term, C subscript 12; divided by 2 times the first material integrity parameter, C subscript 11, the element in the third column and third row is equal to the reciprocal of the first material integrity term, C subscript 11, the remaining diagonal elements are equal to the reciprocal of the initial shear modulus, G subscript 0.

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Figure 69. Illustration. Mathematical equivalence of the formulation used by Lee, Daniel, and Kim. This figure shows a single haversine pseudo strain, epsilon superscript R, pulse. The x axis shows time, and the y axis shows the pseudo strain. The time from the beginning of the loading pulse to the end of the loading pulse is labeled as pulse time, t subscript p. The value of pseudostrain at the peak of the loading pulse is labeled, epsilon subscript m superscript R. Also shown in this graph is a square shaded area centered in the haversine pulse. This shaded area represents the previous formulation calculation and has a width equal to 1 divided by adjustment factor M multiplied by pulse time, t subscript p. The complete solution for this loading form is labeled and is damage, S, equals one half multiplied by pseudo strain at the peak, epsilon subscript m superscript R squared multiplied by the rate of change of material integrity with respect to time, dC divided by dt, raised to the damage evolution rate, alpha, divided by one plus the damage evolution rate; multiplied by pulse time, t subscript p, divided by adjustment factor M. The complete solution for the haversine pulse is shown as damage, S, is equal to the peak pseudo strain value, epsilon subscript m superscript R, divided by; the peak pseudo strain value, epsilon subscript m superscript R, minus the permanent pseudo strain, epsilon subscript s superscript R; multiplied by the integral of one half multiplied by the time dependent effective pseudostrain, epsilon subscript e superscript R as a function of time, squared multiplied by the rate of change of material integrity with respect to time, dC divided by dt, raised to the damage evolution rate, alpha, divided by one plus the damage evolution rate.

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Equation 127. Discrete form of damage growth equation used for numerical solution in rigorous formulation. The damage at the current time step, S subscript i, equals damage at the previous step, S subscript i minus 1, plus one half multiplied by the peak pseudo strain value at the current time step, epsilon subscript m superscript R subscript i, squared and multiplied the form adjustment factor, Q, multiplied by the pseudo stiffness at the previous time step, C subscript i minus 1, minus the pseudo stiffness at the current time step, C subscript i, raised to the damage evolution rate, alpha, divided by 1 plus the damage evolution rate, alpha, multiplied by the change in time between step i and i plus 1, uppercase delta multiplied by time, t, divided by time adjustment factor, M, raised to 1 divided by the damage evolution rate, alpha, plus 1, which equals damage at the previous step, S subscript i minus 1, plus one half multiplied by the peak pseudo strain value at the current time step, epsilon subscript m superscript R subscript i, squared and multiplied the form adjustment factor, Q, multiplied by the change in pseudo stiffness between peaks, uppercase delta of C subscript between peaks, raised to the damage evolution rate, alpha, divided by 1 plus the damage evolution rate, alpha; multiplied by the change in time between step i and i plus 1, uppercase delta multiplied by time, t, divided by time adjustment factor, M, raised to 1 divided by the damage evolution rate, alpha.

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