246 lines
9.0 KiB
TeX
246 lines
9.0 KiB
TeX
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The correspondence matrix method allows predicting how crystallographic directions and planes transform during twinning. This approach is crucial for analyzing defects inherited from twinning, interactions between slip dislocations and twin boundaries, incorporating parent dislocations into the twinned lattice, and dealing with twin-twin intersections and double twinning. According to Niewczas \cite{Niewczas121}, most current continuum models for plastic deformation of crystals treat twinning incorrectly by only considering rotations, missing important aspects of the lattice transformation. Correspondence matrix approach (an example given in Table \ref{tab:Shear and correspondence matrices for Mg}) is implemented in DAMASK and utilized in the discrete twinning model.
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\begin{table}[H]
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\centering
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\caption{Characteristic Shear and Correspondence matrices for different twinning modes of Mg with c/a=1.624.}
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\begin{tabular}{ccc}
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\hline
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$K_1 / \eta_1$ & Magnitude of shear $s$ & Correspondence matrix $C$ \\
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\hline
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$( \bar{1} 0 1 2 ) [1 0 \bar{1} 1] $ & 0.128917 & $\begin{bmatrix} -0.25 & 0.433 & -0.924 \\ 0.433 & -0.75 & -0.533 \\ -0.812 & -0.47 & 0 \end{bmatrix}$ \\
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$( 1 0 \bar{1} 1 ) [1 0 \bar{1} \bar{2}] $ & 0.137717 & $\begin{bmatrix} 0.125 & 0.65 & 0.693 \\ 0.65 & -0.625 & 0.4 \\ 0.812 & 0.47 & -0.5 \end{bmatrix}$ \\
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$( 1 0 \bar{2} \bar{2} ) [1 1 \bar{2} \bar{3}] $ & 0.261649 & $\begin{bmatrix} -0.67 & 0.577 & 0.411 \\ 0.577 & 0 & 0.711 \\ 0.54 & 0.937 & -0.333 \end{bmatrix}$ \\
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$( 1 1 \bar{2} 1) [\bar{1}\bar{1} 2 6] $ & 0.615764 & $\begin{bmatrix} -0.5 & 0.866 & 0.308 \\ 0.866 & 0.5 & 0.533 \\ 0.0 & 0.0 & -1.0 \end{bmatrix}$ \\
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\hline
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\end{tabular}
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\label{tab:Shear and correspondence matrices for Mg}
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\end{table}
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\begin{minted}[fontsize=\scriptsize, frame=single]{fortran}
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module math
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implicit none
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contains
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function math_axisAngleToR(axis,omega) result(math_axisAngleToR1)
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!------------------------------------------------
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!> Function to generate rotation matrix around
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!> arbitrary direction and arbitrary angle
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!------------------------------------------------
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implicit none
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real, dimension(3), intent(in) :: axis
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real, intent(in) :: omega
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real, dimension(3) :: n
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real :: norm,s,c,c1
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real, dimension(3,3), parameter :: &
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I3 = real(reshape([&
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1, 0, 0, &
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0, 1, 0, &
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0, 0, 1 &
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],shape(I3))) !< 3x3 Identity
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real, dimension(3,3) :: math_axisAngleToR1
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norm = norm2(axis)
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wellDefined: if (norm > 1.0e-8) then
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n = axis/norm ! normalize axis to be sure
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s = sin(omega)
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c = cos(omega)
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c1 = 1.0 - c
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math_axisAngleToR1(1,1) = c + c1*n(1)**2.0
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math_axisAngleToR1(1,2) = c1*n(1)*n(2) - s*n(3)
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math_axisAngleToR1(1,3) = c1*n(1)*n(3) + s*n(2)
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math_axisAngleToR1(2,1) = c1*n(1)*n(2) + s*n(3)
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math_axisAngleToR1(2,2) = c + c1*n(2)**2.0
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math_axisAngleToR1(2,3) = c1*n(2)*n(3) - s*n(1)
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math_axisAngleToR1(3,1) = c1*n(1)*n(3) - s*n(2)
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math_axisAngleToR1(3,2) = c1*n(2)*n(3) + s*n(1)
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math_axisAngleToR1(3,3) = c + c1*n(3)**2.0
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else wellDefined
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math_axisAngleToR1 = I3
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endif wellDefined
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end function
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end module math
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program corresponcence_matrix
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use math
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implicit none
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integer, dimension(4) :: &
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active = [6,6,6,6], & !< number of active twin systems
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potential = [6,6,6,6] !< all the potential twin systems
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real, dimension(3) :: &
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direction, normal
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real, dimension(3,24) :: normal_vector, direction_vector
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real, dimension(3,3,24) :: SchmidMatrix, corresponcenceMatrix
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real, dimension(24) :: characteristicShear
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real :: cOverA = 1.6235
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real :: pi = 3.14159274
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real, dimension(8,24) :: &
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system = reshape(real([&
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! <-10.1>{10.2} systems, shear = (3-(c/a)^2)/(sqrt(3) c/a)
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! tension in Co, Mg, Zr, Ti, and Be; compression in Cd and Zn
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-1, 0, 1, 1, 1, 0, -1, 2, & !
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0, -1, 1, 1, 0, 1, -1, 2, &
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1, -1, 0, 1, -1, 1, 0, 2, &
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1, 0, -1, 1, -1, 0, 1, 2, &
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0, 1, -1, 1, 0, -1, 1, 2, &
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-1, 1, 0, 1, 1, -1, 0, 2, &
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! <11.6>{-1-1.1} systems, shear = 1/(c/a)
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! tension in Co, Re, and Zr
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-1, -1, 2, 6, 1, 1, -2, 1, &
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1, -2, 1, 6, -1, 2, -1, 1, &
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2, -1, -1, 6, -2, 1, 1, 1, &
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1, 1, -2, 6, -1, -1, 2, 1, &
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-1, 2, -1, 6, 1, -2, 1, 1, &
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-2, 1, 1, 6, 2, -1, -1, 1, &
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! <10.-2>{10.1} systems, shear = (4(c/a)^2-9)/(4 sqrt(3) c/a)
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! compression in Mg
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1, 0, -1, -2, 1, 0, -1, 1, &
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0, 1, -1, -2, 0, 1, -1, 1, &
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-1, 1, 0, -2, -1, 1, 0, 1, &
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-1, 0, 1, -2, -1, 0, 1, 1, &
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0, -1, 1, -2, 0, -1, 1, 1, &
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1, -1, 0, -2, 1, -1, 0, 1, &
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! <11.-3>{11.2} systems, shear = 2((c/a)^2-2)/(3 c/a)
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! compression in Ti and Zr
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1, 1, -2, -3, 1, 1, -2, 2, &
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-1, 2, -1, -3, -1, 2, -1, 2, &
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-2, 1, 1, -3, -2, 1, 1, 2, &
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-1, -1, 2, -3, -1, -1, 2, 2, &
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1, -2, 1, -3, 1, -2, 1, 2, &
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2, -1, -1, -3, 2, -1, -1, 2 &
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]),shape(system))
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real, dimension(3,3), parameter :: &
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I3 = real(reshape([&
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1, 0, 0, &
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0, 1, 0, &
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0, 0, 1 &
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],shape(I3))) !< 3x3 Identity
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integer :: &
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a, & !< index of active system
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p, & !< index in potential system matrix
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f, & !< index of my family
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s, & !< index of my system in current family
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f1, s1, e1, i, j, k !< indices for similar loops
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!-----------------------------------------------------------
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!> Normal vector to twin plane and direction vector
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!> of the twin
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!-----------------------------------------------------------
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a = 0
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do f = 1, size(active,1) !< Active Twin Modes
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do s = 1, active(f) !< Active twin systems
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a = a + 1
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p = sum(potential(1:f-1))+s
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! direction [uvtw]->[3u/2 (u+2v)*sqrt(3)/2 w*(p/a)])
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direction = [ system(1,p)*1.5, &
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(system(1,p)+2.0*system(2,p))*sqrt(0.75), &
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system(4,p)*cOverA ]
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! plane (hkil)->(h (h+2k)/sqrt(3) l/(p/a))
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normal = [ system(5,p), &
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(system(5,p)+2.0*system(6,p))/sqrt(3.0), &
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system(8,p)/cOverA ]
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normal_vector(1:3,a) = normal /norm2(normal)
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direction_vector(1:3,a) = direction / norm2(direction)
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end do
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end do
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!-----------------------------------------------------------
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!> Magnitude of Characteristic shear for twinning modes
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!-----------------------------------------------------------
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do f1 = 1,size(active,1) !< Active twin modes
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s1 = sum(active(:f1-1)) + 1
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e1 = sum(active(:f1))
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select case(f1)
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case (1)
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characteristicShear(s1:e1) = (3.0-cOverA**2)/sqrt(3.0)/cOverA
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case (2)
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characteristicShear(s1:e1) = 1.0/cOverA
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case (3)
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characteristicShear(s1:e1) = (4.0*cOverA**2-9.0)/sqrt(48.0)/cOverA
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case (4)
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characteristicShear(s1:e1) = 2.0*(cOverA**2-2.0)/3.0/cOverA
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end select
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enddo
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!> Write results for characteristic shear
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write(6,*)'characteristic shear, for [1, 0, -1, 1],(-1, 0, 1, 2)'
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write(6,*)characteristicShear(4)
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write(6,*)'characteristic shear, for [-1, -1, 2, 6],(1, 1, -2, 1)'
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write(6,*)characteristicShear(7)
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write(6,*)'characteristic shear, for [1, 0, -1, -2],(1, 0, -1, 1)'
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write(6,*)characteristicShear(13)
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write(6,*)'characteristic shear, for [1, 1, -2, -3],(1, 1, -2, 2)'
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write(6,*)characteristicShear(19)
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!---------------------------------------------------------------
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!> SchmidMatrix = Outer product of direction and normal vectors.
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!---------------------------------------------------------------
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do i = 1, sum(active)
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forall(j=1:3, k=1:3) &
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SchmidMatrix(j,k,i) = direction_vector(j,i) * &
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normal_vector(k,i)
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enddo
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!--------------------------------------------------------------
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!> Correspondence Matrix = Reorientation * Shear
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!--------------------------------------------------------------
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do i = 1, sum(active)
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corresponcenceMatrix(1:3,1:3,i) = matmul(math_axisAngleToR &
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(normal_vector(1:3,i),pi),&
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I3+characteristicShear(i) &
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*SchmidMatrix(1:3,1:3,i))
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enddo
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!> Write results for Correspondence Matrix
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write(6,*)'correspondence matrix for [1, 0, -1, 1],(-1, 0, 1, 2)'
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write(6,*)corresponcenceMatrix(1:3,1:3,4)
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write(6,*)'oxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxox'
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write(6,*)'correspondence matrix for [-1, -1, 2, 6],(1, 1, -2, 1)'
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write(6,*)corresponcenceMatrix(1:3,1:3,7)
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write(6,*)'oxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxox'
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write(6,*)'correspondence matrix for [1, 0, -1, -2],(1, 0, -1, 1)'
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write(6,*)corresponcenceMatrix(1:3,1:3,13)
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write(6,*)'oxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxoxox'
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write(6,*)'correspondence matrix for [1, 1, -2, -3],(1, 1, -2, 2)'
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write(6,*)corresponcenceMatrix(1:3,1:3,19)
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end program corresponcence_matrix
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\end{minted}
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