private static Tensor2D ComputeStrainTensor(Matrix2by2 displacementGradient)
{
double exx = displacementGradient[0, 0];
double eyy = displacementGradient[1, 1];
double exy = 0.5 * (displacementGradient[0, 1] + displacementGradient[1, 0]);
return(new Tensor2D(exx, eyy, exy));
}
private void ComputeInteractionIntegrals(XContinuumElement2D element, Vector standardNodalDisplacements,
Vector enrichedNodalDisplacements, double[] nodalWeights, TipCoordinateSystem tipSystem,
out double integralMode1, out double integralMode2)
{
integralMode1 = 0.0;
integralMode2 = 0.0;
foreach (GaussPoint naturalGP in element.JintegralStrategy.GenerateIntegrationPoints(element))
{
// Nomenclature: global = global cartesian system, natural = element natural system,
// local = tip local cartesian system
EvalInterpolation2D evaluatedInterpolation =
element.Interpolation.EvaluateAllAt(element.Nodes, naturalGP);
CartesianPoint globalGP = evaluatedInterpolation.TransformPointNaturalToGlobalCartesian();
Matrix constitutive =
element.Material.CalculateConstitutiveMatrixAt(naturalGP, evaluatedInterpolation);
// State 1
Matrix2by2 globalDisplacementGradState1 = element.CalculateDisplacementFieldGradient(
naturalGP, evaluatedInterpolation, standardNodalDisplacements, enrichedNodalDisplacements);
Tensor2D globalStressState1 = element.CalculateStressTensor(globalDisplacementGradState1, constitutive);
Matrix2by2 localDisplacementGradState1 = tipSystem.
TransformVectorFieldDerivativesGlobalCartesianToLocalCartesian(globalDisplacementGradState1);
Tensor2D localStressTensorState1 = tipSystem.
TransformTensorGlobalCartesianToLocalCartesian(globalStressState1);
// Weight Function
// TODO: There should be a method InterpolateScalarGradient(double[] nodalValues) in EvaluatedInterpolation
// TODO: Rewrite this as a shapeGradients (matrix) * nodalWeights (vector) operation.
var globalWeightGradient = Vector2.CreateZero();
for (int nodeIdx = 0; nodeIdx < element.Nodes.Count; ++nodeIdx)
{
globalWeightGradient.AxpyIntoThis(
evaluatedInterpolation.ShapeGradientsCartesian.GetRow(nodeIdx), // Previously: GetGlobalCartesianDerivativesOf(element.Nodes[nodeIdx])
nodalWeights[nodeIdx]);
}
Vector2 localWeightGradient = tipSystem.
TransformScalarFieldDerivativesGlobalCartesianToLocalCartesian(globalWeightGradient);
// State 2
// TODO: XContinuumElement shouldn't have to pass tipCoordinate system to auxiliaryStates.
// It would be better to have CrackTip handle this and the coordinate transformations. That would also
// obey LoD, but a lot of wrapper methods would be required.
AuxiliaryStatesTensors auxiliary = auxiliaryStatesStrategy.ComputeTensorsAt(globalGP, tipSystem);
// Interaction integrals
double integrandMode1 = ComputeJIntegrand(localWeightGradient, localDisplacementGradState1,
localStressTensorState1, auxiliary.DisplacementGradientMode1,
auxiliary.StrainTensorMode1, auxiliary.StressTensorMode1);
double integrandMode2 = ComputeJIntegrand(localWeightGradient, localDisplacementGradState1,
localStressTensorState1, auxiliary.DisplacementGradientMode2,
auxiliary.StrainTensorMode2, auxiliary.StressTensorMode2);
integralMode1 += integrandMode1 * evaluatedInterpolation.Jacobian.DirectDeterminant * naturalGP.Weight;
integralMode2 += integrandMode2 * evaluatedInterpolation.Jacobian.DirectDeterminant * naturalGP.Weight;
}
}
public AuxiliaryStatesTensors(Matrix2by2 displacementGradientMode1, Matrix2by2 displacementGradientMode2,
Tensor2D strainTensorMode1, Tensor2D strainTensorMode2,
Tensor2D stressTensorMode1, Tensor2D stressTensorMode2)
{
this.DisplacementGradientMode1 = displacementGradientMode1;
this.DisplacementGradientMode2 = displacementGradientMode2;
this.StrainTensorMode1 = strainTensorMode1;
this.StrainTensorMode2 = strainTensorMode2;
this.StressTensorMode1 = stressTensorMode1;
this.StressTensorMode2 = stressTensorMode2;
}
public Tensor2D EvaluateAt(XContinuumElement2D element, NaturalPoint point,
Vector standardDisplacements, Vector enrichedDisplacements)
{
EvalInterpolation2D evaluatedInterpolation = element.Interpolation.EvaluateAllAt(element.Nodes, point);
Matrix2by2 displacementGradient = element.CalculateDisplacementFieldGradient(
point, evaluatedInterpolation, standardDisplacements, enrichedDisplacements);
Matrix constitutive =
element.Material.CalculateConstitutiveMatrixAt(point, evaluatedInterpolation);
return(element.CalculateStressTensor(displacementGradient, constitutive));
}
public Tensor2D EvaluateAt(XContinuumElement2D element, NaturalPoint point,
Vector standardDisplacements, Vector enrichedDisplacements)
{
EvalInterpolation2D evaluatedInterpolation = element.Interpolation.EvaluateAllAt(element.Nodes, point);
Matrix2by2 displacementGradient = element.CalculateDisplacementFieldGradient(
point, evaluatedInterpolation, standardDisplacements, enrichedDisplacements);
double strainXX = displacementGradient[0, 0];
double strainYY = displacementGradient[1, 1];
double strainXY = 0.5 * (displacementGradient[0, 1] + displacementGradient[1, 0]);
return(new Tensor2D(strainXX, strainYY, strainXY));
}
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;
}
private static double ComputeJIntegrand(Vector2 weightGrad, Matrix2by2 displGrad1, Tensor2D stress1,
Matrix2by2 displGrad2, Tensor2D strain2, Tensor2D stress2)
{
// Unrolled to greatly reduce mistakes. Alternatively Einstein notation products could be implementated
// in Tensor2D (like the tensor-tensor multiplication is), but still some parts would have to be unrolled.
// Perhaps vector (and scalar) gradients should also be accessed by component and derivative variable.
double strainEnergy = stress1.MultiplyColon(strain2);
double parenthesis0 = stress1.XX * displGrad2[0, 0] + stress1.XY * displGrad2[1, 0]
+ stress2.XX * displGrad1[0, 0] + stress2.XY * displGrad1[1, 0] - strainEnergy;
double parenthesis1 = stress1.XY * displGrad2[0, 0] + stress1.YY * displGrad2[1, 0]
+ stress2.XY * displGrad1[0, 0] + stress2.YY * displGrad1[1, 0];
return(parenthesis0 * weightGrad[0] + parenthesis1 * weightGrad[1]);
}
private (Tensor2D strain, Tensor2D stress) ComputeStrainStress(XContinuumElement2D element, NaturalPoint gaussPoint,
EvalInterpolation2D evaluatedInterpolation, Vector standardNodalDisplacements,
Vector enrichedNodalDisplacements)
{
Matrix constitutive =
element.Material.CalculateConstitutiveMatrixAt(gaussPoint, evaluatedInterpolation);
Matrix2by2 displacementGradient = element.CalculateDisplacementFieldGradient(gaussPoint, evaluatedInterpolation,
standardNodalDisplacements, enrichedNodalDisplacements);
double strainXX = displacementGradient[0, 0];
double strainYY = displacementGradient[1, 1];
double strainXYtimes2 = displacementGradient[0, 1] + displacementGradient[1, 0];
double stressXX = constitutive[0, 0] * strainXX + constitutive[0, 1] * strainYY;
double stressYY = constitutive[1, 0] * strainXX + constitutive[1, 1] * strainYY;
double stressXY = constitutive[2, 2] * strainXYtimes2;
return(new Tensor2D(strainXX, strainYY, 0.5 * strainXYtimes2), new Tensor2D(stressXX, stressYY, stressXY));
}
// Computes stresses directly at the nodes. The other approach is to compute them at Gauss points and then extrapolate
private IReadOnlyDictionary <XNode, Tensor2D> ComputeNodalStressesOfElement(XContinuumElement2D element,
Vector freeDisplacements, Vector constrainedDisplacements)
{
Vector standardDisplacements = dofOrderer.ExtractDisplacementVectorOfElementFromGlobal(element,
freeDisplacements, constrainedDisplacements);
Vector enrichedDisplacements =
dofOrderer.ExtractEnrichedDisplacementsOfElementFromGlobal(element, freeDisplacements);
IReadOnlyList <NaturalPoint> naturalNodes = element.Interpolation.NodalNaturalCoordinates;
var nodalStresses = new Dictionary <XNode, Tensor2D>();
for (int i = 0; i < element.Nodes.Count; ++i)
{
EvalInterpolation2D evaluatedInterpolation =
element.Interpolation.EvaluateAllAt(element.Nodes, naturalNodes[i]);
Matrix2by2 displacementGradient = element.CalculateDisplacementFieldGradient(
naturalNodes[i], evaluatedInterpolation, standardDisplacements, enrichedDisplacements);
Matrix constitutive =
element.Material.CalculateConstitutiveMatrixAt(naturalNodes[i], evaluatedInterpolation);
nodalStresses[element.Nodes[i]] = element.CalculateStressTensor(displacementGradient, constitutive);
}
return(nodalStresses);
}
/// <summary>
/// Attention: The input vector field is differentiated w.r.t. the polar cartesian system coordinates.
/// The output vector field is differentiated w.r.t. the local cartesian system coordinates. However the
/// representations of both vector fields (aka the coordinates of the vectors) are in the local cartesian
/// coordinate system.
/// </summary>
/// <param name="gradient">A 2x2 matrix, for which: Row i is the gradient of the ith component of the vector
/// field, thus: gradient = [Fr,r Fr,theta; Ftheta,r Ftheta,theta],
/// where Fi,j is the derivative of component i w.r.t. coordinate j</param>
/// <returns></returns>
public Matrix2by2 TransformVectorFieldDerivativesLocalPolarToLocalCartesian(Matrix2by2 gradient)
{
return(gradient * inverseJacobianPolarToLocal);
}
/// <summary>
///
/// </summary>
/// <param name="gradient">A 2x2 matrix, for which: Row i is the gradient of the ith component of the vector
/// field, thus: gradient = [Fx,x Fx,y; Fy,x Fy,y],
/// where Fi,j is the derivative of component i w.r.t. coordinate j</param>
/// <returns></returns>
public Matrix2by2 TransformVectorFieldDerivativesGlobalCartesianToLocalCartesian(Matrix2by2 gradient)
{
return(RotationMatrixGlobalToLocal * (gradient * TransposeRotationMatrixGlobalToLocal));
}
