Rank Two Scalar Tools
Description
This is a set of functions to compute scalar quantities such as invariants and components in specified directions from rank-2 tensors such as stress or strain. These functions are not directly invoked in the input file, but are called by several other classes such as: RankTwoCylindricalComponent, RankTwoDirectionalComponent, RankTwoCartesianComponent, RankTwoInvariant, RankTwoScalarAux, and RankTwoAux.
The scalar quantities that can be computed include:
Axial Stress
AxialStress calculates the scalar value of a Rank-2 tensor, , in the direction of the axis specified by the user. The user should give the starting point, , and the end point, which define the axis.
(1) where is the normalized direction vector for the axis defined by the points and .
Direction
Direction calculates the scalar value of a Rank-2 tensor, , in the direction selected by the user as shown by Eq. (2): (2) where is the direction vector specified in the input file.
Effective Strain Increment
Effective plastic strain or effective creep strain, which are computed as integrals over the history of the inelastic strain as can be computed with the help of the effectiveStrain method. The integration of the effective increment is performed in Rank Two Invariant, yielding the effective strain.
Hoop Stress in Cylindrical System
HoopStress calculates the value of a Rank -2 tensor along the hoop direction of a cylinder, shown in Eq. (3). The cylinder is defined with a normal vector from the current position to the cylinder surface and a user specified axis of rotation.The user defines this rotation axis with a starting point, , and the end point, .
(3) where is the hoop direction normal, defined as (4) where is the current sampling position point, and is the direction normal to the plane defined by the cylinder axis of rotation vector and the direction normal to the axis of rotation at the current position .
Hoop Stress in Spherical System
HoopStress calculates the value of a Rank -2 tensor along the tangential direction of a sphere, shown in Eq. (5). The spherical system is defined by the center point . The radial direction at current point is calculated as . The tangential plane at the Point is given as . Any vector that passes through on this plane is tangential to the spherical surface. To find a point on the tangential plane, we can freely set the values of two coordinates and the solve for last one using the equation of the plane. For example, we set and where is the norm of the radial direction vector. Then the is calculated as . The tangential vector is defined as .
(5)
Hydrostatic Stress
Hydrostatic calculates the hydrostatic scalar of a Rank-2 tensor, , as shown in Eq. (6).
(6)
Invariant Values
First Invariant
FirstInvariant calculates the first invariant of the specified Rank-2 tensor, , according to Eq. (7) from Malvern (1969). (7)
Second Invariant
SecondInvariant finds the second invariant of the Rank-2 tensor, , as shown in Eq. (8). This method is defined in Hjelmstad (2007). (8)
Third Invariant
ThirdInvariant computes the value of the Rank-2 tensor, , third invariant as given in Eq. (9) from Malvern (1969).
(9) where is the Rank-3 permutation tensor.
L2 Norm
L2Norm calculates the L2 normal of a Rank-2 tensor, , as shown in Eq. (10).
(10)
Maximum Shear Stress
MaxShear calculates the maximum shear stress for a Rank-2 tensor, as shown in Eq. (11). (11)
Principal Values
Maximum Principal Quantity
MaxPrincipal calculates the largest principal value for a symmetric tensor, using the calcEigenValues method from the Rank Two Tensor utility class.
Middle Principal Quantity
MidPrincipal finds the second largest principal value for a symmetric tensor, using the calcEigenValues method from the Rank Two Tensor utility class.
Minimum Principal Quantity
MinPrincipal computes the smallest principal value for a symmetric tensor, using the calcEigenValues method from the Rank Two Tensor utility class.
Radial Stress in Cylindrical System
RadialStress calculates the scalar component for a Rank-2 tensor, , in the direction of the normal vector from the user-defined axis of rotation, as shown in Eq. (12). (12) where is the direction normal to the plane defined by the cylinder axis of rotation vector and the direction normal to the axis of rotation at the current position .
Radial Stress in Spherical System
RadialStress calculates the scalar component for a Rank-2 tensor, , in the direction of the normal vector from the user-defined center point, as shown in Eq. (13). (13) where is the direction defined by center point and current position $P^c.
Stress Intensity
StressIntensity calculates the stress intensity for a Rank-2 tensor, as shown in Eq. (14). (14)
This quantity is useful in evaluating whether a Tresca failure criteria has been met and is two times the MaxShear quantity.
Triaxiality Stress
TriaxialityStress finds the ratio of the hydrostatic measure, , to the von Mises measure, , as shown in Eq. (15). As the name suggests, this scalar measure is most often used for stress tensors. (15) where is the deviatoric tensor of the Rank-2 tensor .
##Volumetric Strain
VolumetricStrain computes the volumetric strain, defined as (16) where is the change in volume and is the original volume.
This calculation assumes that the strains supplied as input() are logarithmic strains, which are by definition is the current length and is the original length of a line segment in a given direction.The ratio of the volume change of a strained cube to the original volume is thus : (17) This is the value computed as the volumetric strain.
This calculation assumes that the supplied Rank-2 tensor is a logarithmic strain, which is the strain quantity computed for finite strain calculations. The small-strain equivalent of this calculation would be (19) which assumes that engineering strains are supplied and ignores higher-order terms. There is currently no option to compute this small-strain form of the volumetric strain because at small strains, the differences between the finite strain form used and the small strain approximation is small.
Von Mises Stress
VonMisesStress calculates the vonMises measure for a Rank-2 tensor, as shown in Eq. (18). This quantity is usually applied to the stress tensor. (18) where is the deviatoric tensor of the Rank-2 tensor .