private void ComputeDisplacementDerivatives(TipJacobians polarJacobians, CommonValues val,
out Matrix2by2 displacementGradientMode1, out Matrix2by2 displacementGradientMode2)
{
// Temporary values and derivatives of the differentiated quantities. See documentation for their derivation.
double E = material.HomogeneousYoungModulus;
double v = material.HomogeneousPoissonRatio;
double vEq = material.HomogeneousEquivalentPoissonRatio;
double k = (3.0 - vEq) / (1.0 + vEq);
double a = (1.0 + v) / (E * Math.Sqrt(2.0 * Math.PI));
double b = val.sqrtR;
double b_r = 0.5 / val.sqrtR;
// Mode 1
{
// Temporary values that differ between the 2 modes
double c1 = val.cosThetaOver2 * (k - val.cosTheta);
double c2 = val.sinThetaOver2 * (k - val.cosTheta);
double c1_theta = -0.5 * c2 + val.cosThetaOver2 * val.sinTheta;
double c2_theta = 0.5 * c1 + val.sinThetaOver2 * val.sinTheta;
// The vector field derivatives w.r.t. to the local polar coordinates.
// The vector components refer to the local cartesian system though.
var polarGradient = Matrix2by2.CreateFromArray(
new double[, ] {
{ a *b_r *c1, a *b *c1_theta }, { a *b_r *c2, a *b *c2_theta }
});
// The vector field derivatives w.r.t. to the local cartesian coordinates.
displacementGradientMode1 =
polarJacobians.TransformVectorFieldDerivativesLocalPolarToLocalCartesian(polarGradient);
}
// Mode 2
{
double paren1 = 2.0 + k + val.cosTheta;
double paren2 = 2.0 - k - val.cosTheta;
double c1 = val.sinThetaOver2 * paren1;
double c2 = val.cosThetaOver2 * paren2;
double c1_theta = 0.5 * val.cosThetaOver2 * paren1 - val.sinThetaOver2 * val.sinTheta;
double c2_theta = -0.5 * val.sinThetaOver2 * paren2 + val.cosThetaOver2 * val.sinTheta;
// The vector field derivatives w.r.t. to the local polar coordinates.
// The vector components refer to the local cartesian system though.
var polarGradient = Matrix2by2.CreateFromArray(
new double[, ] {
{ a *b_r *c1, a *b *c1_theta }, { a *b_r *c2, a *b *c2_theta }
});
// The vector field derivatives w.r.t. to the local cartesian coordinates.
displacementGradientMode2 =
polarJacobians.TransformVectorFieldDerivativesLocalPolarToLocalCartesian(polarGradient);
}
}
/// <summary>
///
/// </summary>
/// <param name="tipCoordinates">Coordinates of the crack tip in the global cartesian system.</param>
/// <param name="tipRotationAngle">Counter-clockwise angle from the O-x axis of the global cartesian system to
/// the T-x1 axis of the local corrdinate system of the tip (T being the tip point)</param>
public TipCoordinateSystem(CartesianPoint tipCoordinates, double tipRotationAngle)
{
this.RotationAngle = tipRotationAngle;
double cosa = Math.Cos(tipRotationAngle);
double sina = Math.Sin(tipRotationAngle);
RotationMatrixGlobalToLocal = Matrix2by2.CreateFromArray(new double[, ] {
{ cosa, sina }, { -sina, cosa }
});
TransposeRotationMatrixGlobalToLocal = RotationMatrixGlobalToLocal.Transpose();
localCoordinatesOfGlobalOrigin = -1 * (RotationMatrixGlobalToLocal * Vector2.Create(tipCoordinates.X, tipCoordinates.Y));
DeterminantOfJacobianGlobalToLocalCartesian = 1.0; // det = (cosa)^2 +(sina)^2 = 1
}
public TipJacobians(TipCoordinateSystem tipSystem, PolarPoint2D polarCoordinates)
{
this.tipSystem = tipSystem;
double r = polarCoordinates.R;
double cosTheta = Math.Cos(polarCoordinates.Theta);
double sinTheta = Math.Sin(polarCoordinates.Theta);
inverseJacobianPolarToLocal = Matrix2by2.CreateFromArray(new double[, ]
{
{ cosTheta, sinTheta }, { -sinTheta / r, cosTheta / r }
});
inverseJacobianPolarToGlobal = inverseJacobianPolarToLocal * tipSystem.RotationMatrixGlobalToLocal;
}
