The Influence of Crack-Face Normal and Shear Stress Loading on Hydraulic Fracture-Tip Singular Plastic Fields

Abstract

We investigated the singular plastic fields at the crack tip of a fracture that is loaded with normal and shear loads due to a viscous flow in a hydraulic fracturing. The level of the expected shear load in comparison with the normal load is examined. The lubrication flow and plastic deformation were decoupled assuming that the relation between applied shear load and normal load follows a linear friction-type relation. This assumption allows to investigate extreme bounds of the solution. Both the applied normal and shear loads are assumed to exhibit singular behavior near the tip which is consistent at the fracture surfaces with the plastic singular stress fields that are investigated. The fractured material is assumed to obey a non-associative Drucker–Prager solid with power law hardening response. The singular values and the corresponding fields were determined over a range of material parameters. For both von Mises material and associative Drucker–Prager material, we found that the level of singularity is given by 1/n where n is the power coefficient of the hardening relation. This level of singularity is stronger than the HRR value, 1/(n + 1), which has been determined for traction free crack surfaces. We found that the shear loading does not influence the level of singularity but it changes the shape of the developed plastic zones with the emergence of a boundary layer near the fracture surface. Deviation from material associativity produces consistent small increases in the level of singularity. The near-tip stress, strain and displacement profiles are illustrated for a few representative cases.

Appendix I: Numerical Solution

The system of governing Eqs. (33) and (40), along with boundary conditions (41), is solved numerically by a discretization procedure using the Galerkin finite element method (Strang and Fix 1973). The first step of this scheme is to discretize the space domain replacing the original set of PDE with a set of ODE. Considering the nonlinearity of equation system, the solution is obtained with Newton’s method. The system can be put in a compact form:

$$ R\left( {\tilde{\sigma }_{rr} ,\tilde{\sigma }_{\theta \theta } ,\tilde{\sigma }_{r\theta } ,\tilde{u},\tilde{v}} \right) = 0 $$

(45)

where R is the vector of residuals. Newton’s method is an iterative method where, given an initial approximation y(0) to the solution y, the (m + 1) th approximation is obtained from the m th by solving

$$ J(y(m))[y_{(m + 1)} - y_{(m)} ] = - R\left( {y_{(m)} } \right) $$

(46)

where J is the Jacobian matrix which contains the partial derivatives of the residuals R with respect to the unknowns, i.e., $(J_{ij} = {{\partial R_{i} } \mathord{\left/ {\vphantom {{\partial R_{i} } {\partial y_{j} }}} \right. \kern-0pt} {\partial y_{j} }})$. The derivatives are evaluated at y = y (m). The integer m is the iteration counter.

The unknown variables are expressed in terms of their nodal values as

$$ \tilde{\sigma }_{rr} = \sum\limits_{j = 1}^{N} {\tilde{\sigma }_{rr}^{j} } \varPhi^{j} \quad \tilde{\sigma }_{\theta \theta } = \sum\limits_{j = 1}^{N} {\tilde{\sigma }_{\theta \theta }^{j} } \varPhi^{j} \quad \tilde{\sigma }_{r\theta } = \sum\limits_{j = 1}^{N} {\tilde{\sigma }_{r\theta }^{j} } \varPhi^{j} \quad \tilde{u} = \sum\limits_{j = 1}^{N} {\tilde{u}^{j} } \varPhi^{j} \quad \tilde{v} = \sum\limits_{j = 1}^{N} {\tilde{v}^{j} } \varPhi^{j} $$

(47)

where Φ^j are the approximation functions (basis functions) over the element. For these approximations the Lagrange quadratic interpolation functions were used.

The Galerkin-weighted residuals of the system equations are formed by, respectively, multiplying them by each basis function Φ^ι in turn and then integrating over the entire space domain [0,π]. At each step of Newton’s iteration Eq. (46) can be written in an expanded form

$$ \left| {\begin{array}{*{20}c} {\partial R_{i}^{1} /\partial \tilde{\sigma }_{rr} } & {\partial R_{i}^{1} /\partial \tilde{\sigma }_{\theta \theta } } & {\partial R_{i}^{1} /\partial \tilde{\sigma }_{r\theta } } & {\partial R_{i}^{1} /\partial \tilde{u}} & {\partial R_{i}^{1} /\partial \tilde{v}} \\ {\partial R_{i}^{2} /\partial \tilde{\sigma }_{rr} } & {\partial R_{i}^{2} /\partial \tilde{\sigma }_{\theta \theta } } & {\partial R_{i}^{2} /\partial \tilde{\sigma }_{r\theta } } & {\partial R_{i}^{2} /\partial \tilde{u}} & {\partial R_{i}^{2} /\partial \tilde{v}} \\ {\partial R_{i}^{3} /\partial \tilde{\sigma }_{rr} } & {\partial R_{i}^{3} /\partial \tilde{\sigma }_{\theta \theta } } & {\partial R_{i}^{3} /\partial \tilde{\sigma }_{r\theta } } & {\partial R_{i}^{3} /\partial \tilde{u}} & {\partial R_{i}^{3} /\partial \tilde{v}} \\ {\partial R_{i}^{4} /\partial \tilde{\sigma }_{rr} } & {\partial R_{i}^{4} /\partial \tilde{\sigma }_{\theta \theta } } & {\partial R_{i}^{4} /\partial \tilde{\sigma }_{r\theta } } & {\partial R_{i}^{4} /\partial \tilde{u}} & {\partial R_{i}^{4} /\partial \tilde{v}} \\ {\partial R_{i}^{5} /\partial \tilde{\sigma }_{rr} } & {\partial R_{i}^{5} /\partial \tilde{\sigma }_{\theta \theta } } & {\partial R_{i}^{5} /\partial \tilde{\sigma }_{r\theta } } & {\partial R_{i}^{5} /\partial \tilde{u}} & {\partial R_{i}^{5} /\partial \tilde{v}} \\ \end{array} } \right|^{m} \left| {\begin{array}{*{20}c} {\delta \tilde{\sigma }_{rr} } \\ {\delta \tilde{\sigma }_{rr} } \\ {\delta \tilde{\sigma }_{rr} } \\ {\delta \tilde{u}} \\ {\delta \tilde{v}} \\ \end{array} } \right|^{(m + 1)} = - \,\left| {\begin{array}{*{20}c} {R_{i}^{1} } \\ {R_{i}^{2} } \\ {R_{i}^{3} } \\ {R_{i}^{4} } \\ {R_{i}^{5} } \\ \end{array} } \right|^{m} $$

(48)

The derivation of Galerkin-weighted residuals R_i and their partial derivatives $(J_{ij} = {{\partial R_{i} } \mathord{\left/ {\vphantom {{\partial R_{i} } {\partial y_{j} }}} \right. \kern-0pt} {\partial y_{j} }})$ which form the elements of the Jacobian matrix is given below.

The Galerkin-weighted residuals and the corresponding (i-th row, j-th column) element of Jacobian of Eq. (33a) are given by

$$ R_{i}^{1} = \int\limits_{{0}}^{{\pi}} {\left[ {(s + 1)\tilde{\sigma }_{rr} + \tilde{\sigma }^{\prime}_{r\theta } - \tilde{\sigma }_{\theta \theta } } \right]} \varPhi^{i} {\text{d}}\theta = 0 $$

(49)

$$ \frac{{\partial R_{i}^{1} }}{{\partial \tilde{\sigma }_{rr} }} = \int\limits_{0}^{\pi } {(s + 1)} \varPhi^{i} \varPhi^{j} {\text{d}}\theta \quad \frac{{\partial R_{i}^{1} }}{{\partial \tilde{\sigma }_{\theta \theta } }} = - \int\limits_{0}^{\pi } {\varPhi^{i} \varPhi^{j} } {\text{d}}\theta \quad \frac{{\partial R_{i}^{1} }}{{\partial \tilde{\sigma }_{r\theta } }} = \int\limits_{0}^{\pi } {\varPhi^{i} \frac{{\partial \varPhi^{j} }}{\partial \theta }} {\text{d}}\theta \quad \frac{{\partial R_{i}^{1} }}{{\partial \tilde{u}}} = 0\quad \frac{{\partial R_{i}^{1} }}{{\partial \tilde{v}}} = 0 $$

The Galerkin-weighted residual and the corresponding (i-th row, j-th column) element of Jacobian of Eq. (33b) are given by

$$ R_{i}^{2} = \int\limits_{0}^{\pi } {\left[ {(s + 2)\tilde{\sigma }_{r\theta } + \tilde{\sigma }^{\prime}_{\theta \theta } } \right]\varPhi^{i} {\text{d}}\theta } = 0 $$

(50)

$$ \frac{{\partial R_{i}^{2} }}{{\partial \tilde{\sigma }_{rr} }} = 0\quad \frac{{\partial R_{i}^{1} }}{{\partial \tilde{\sigma }_{\theta \theta } }} = \int\limits_{0}^{\pi } {\varPhi^{i} \frac{{\partial \varPhi^{j} }}{\partial \theta }} {\text{d}}\theta \quad \frac{{\partial R_{i}^{2} }}{{\partial \tilde{\sigma }_{r\theta } }} = \int\limits_{0}^{\pi } {(s + 2)} \varPhi^{i} \varPhi^{j} {\text{d}}\theta \quad \frac{{\partial R_{i}^{2} }}{{\partial \tilde{u}}} = 0\quad \frac{{\partial R_{i}^{2} }}{{\partial \tilde{v}}} = 0 $$

The Galerkin-weighted residual and the corresponding (i-th row, j-th column) element of Jacobian of Eq. (40c) are given by

$$ R_{i}^{3} = \int\limits_{0}^{\pi } {\left\{ {\tilde{u^{\prime}} + ns\tilde{v} - \frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\left[ {\frac{{3\tilde{\sigma }_{r\theta } }}{{\tilde{q}}}} \right]} \right\}} \varPhi^{i} {\text{d}}\theta $$

(51)

$$ \frac{{\partial R_{i}^{3} }}{{\partial \tilde{\sigma }_{rr} }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{rr} }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{rr} }}}}{{\tilde{g}^{2} }}\left[ {\frac{{3\tilde{\sigma }_{r\theta } }}{{\tilde{q}}}} \right] + \frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{3\tilde{\sigma }_{r\theta } \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{rr} }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{3} }}{{\partial \tilde{\sigma }_{\theta \theta } }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{\theta \theta } }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{\theta \theta } }}}}{{\tilde{g}^{2} }}\left[ {\frac{{3\tilde{\sigma }_{r\theta } }}{{\tilde{q}}}} \right] + \frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{3\tilde{\sigma }_{r\theta } \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{\theta \theta } }}}}{{\tilde{q}^{2} }}} \right\}\varPhi^{j} } \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{3} }}{{\partial \tilde{\sigma }_{r\theta } }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{r\theta } }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{r\theta } }}}}{{\tilde{g}^{2} }}\left[ {\frac{{3\tilde{\sigma }_{r\theta } }}{{\tilde{q}}}} \right] - \frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{3\tilde{q} - 3\tilde{\sigma }_{r\theta } \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{r\theta } }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

where

$$ \begin{aligned} \frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{rr} }} & = \left( {\frac{3}{{1 - \eta^{2} /9}}} \right)^{1/2} \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{{\left[ {\left( {\frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{2}} \right)^{2} + \sigma_{r\theta }^{2} } \right]^{1/2} }} \\ \frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{\theta \theta } }} & = - \left( {\frac{3}{{1 - \eta^{2} /9}}} \right)^{1/2} \frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{{\left[ {\left( {\frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{2}} \right)^{2} + \sigma_{r\theta }^{2} } \right]^{1/2} }} \\ \frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{r\theta } }} & = \left( {\frac{3}{{1 - \eta^{2} /9}}} \right)^{1/2} \frac{{\sigma_{r\theta } }}{{\left[ {\left( {\frac{{\sigma_{rr} - \sigma_{\theta \theta } }}{2}} \right)^{2} + \sigma_{r\theta }^{2} } \right]^{1/2} }} \\ \end{aligned} $$

$$ \begin{aligned} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{rr} }} & = \left( {1 - \frac{\mu \eta }{9}} \right)\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{rr} }} + \frac{\mu }{2} \\ \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{\theta \theta } }} & = \left( {1 - \frac{\mu \eta }{9}} \right)\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{\theta \theta } }} + \frac{\mu }{2} \\ \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{r\theta } }} & = \left( {1 - \frac{\mu \eta }{9}} \right)\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{r\theta } }} \\ \end{aligned} $$

and

$$ \begin{aligned} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{rr} }} & = \left( {1 - \frac{{\eta^{2} }}{9}} \right)\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{rr} }} + \frac{\eta }{2} \\ \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{\theta \theta } }} & = \left( {1 - \frac{{\eta^{2} }}{9}} \right)\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{\theta \theta } }} + \frac{\eta }{2} \\ \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{r\theta } }} & = \left( {1 - \frac{{\eta^{2} }}{9}} \right)\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{r\theta } }} \\ \end{aligned} $$

$$ \frac{{\partial R_{i}^{3} }}{{\partial \tilde{u}}} = \int\limits_{0}^{\pi } {\frac{1}{2}} \frac{{\partial \varPhi^{j} }}{\partial \theta }\varPhi^{i} {\text{d}}\theta \quad \frac{{\partial R_{i}^{3} }}{{\partial \tilde{v}}} = \int\limits_{0}^{\pi } {\frac{1}{2}} ns\varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

The Galerkin-weighted residual and the corresponding (i-th row, j-th column) element of Jacobian of Eq. (40a) are given by

$$ R_{i}^{4} = \int\limits_{0}^{\pi } {\left\{ {(ns + 1)\tilde{u} - \frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\left[ {\frac{\eta }{2} + \frac{3}{{2\tilde{q}}}\frac{{\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } }}{2}} \right]} \right\}} \varPhi^{i} {\text{d}}\theta $$

(52)

$$ \frac{{\partial R_{i}^{4} }}{{\partial \tilde{\sigma }_{rr} }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{rr} }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{rr} }}}}{{\tilde{g}^{2} }}\left[ {\frac{3}{{2\tilde{q}}}\frac{{\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } }}{2} + \frac{\eta }{2}} \right] - \frac{3}{4}\frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{\tilde{q} - (\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } )\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{rr} }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{4} }}{{\partial \tilde{\sigma }_{\theta \theta } }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{\theta \theta } }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{\theta \theta } }}}}{{\tilde{g}^{2} }}\left[ {\frac{3}{{2\tilde{q}}}\frac{{\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } }}{2} + \frac{\eta }{2}} \right] + \frac{3}{4}\frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{\tilde{q} + (\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } )\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{\theta \theta } }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{4} }}{{\partial \tilde{\sigma }_{r\theta } }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{r\theta } }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{r\theta } }}}}{{\tilde{g}^{2} }}\left[ {\frac{3}{{2\tilde{q}}}\frac{{\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } }}{2} + \frac{\eta }{2}} \right] + \frac{3}{4}\frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{(\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } )\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{r\theta } }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{4} }}{{\partial \tilde{u}}} = \int\limits_{0}^{\pi } {(ns + 1)\varPhi^{j} } \varPhi^{i} {\text{d}}\theta \quad \frac{{\partial R_{i}^{4} }}{{\partial \tilde{v}}} = 0 $$

The Galerkin-weighted residual and the corresponding (i-th row, j-th column) element of Jacobian of Eq. (40b) are given by

$$ R_{i}^{5} = \int\limits_{0}^{\pi } {\left\{ {\tilde{u} + \tilde{v}^{\prime} - \frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\left[ {\frac{\eta }{2} + \frac{3}{{2\tilde{q}}}\frac{{\tilde{\sigma }_{\theta \theta } - \tilde{\sigma }_{rr} }}{2}} \right]} \right\}} \varPhi^{i} {\text{d}}\theta $$

(53)

$$ \frac{{\partial R_{i}^{5} }}{{\partial \tilde{\sigma }_{rr} }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{rr} }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{rr} }}}}{{\tilde{g}^{2} }}\left[ {\frac{3}{4}\frac{{(\tilde{\sigma }_{\theta \theta } - \tilde{\sigma }_{rr} )}}{{\tilde{q}}} + \frac{\eta }{2}} \right] + \frac{3}{4}\frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{\tilde{q} - (\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } )\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{rr} }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{5} }}{{\partial \tilde{\sigma }_{\theta \theta } }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{\theta \theta } }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{\theta \theta } }}}}{{\tilde{g}^{2} }}\left[ {\frac{3}{4}\frac{{(\tilde{\sigma }_{\theta \theta } - \tilde{\sigma }_{rr} )}}{{\tilde{q}}} + \frac{\eta }{2}} \right] - \frac{3}{4}\frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{\tilde{q} + (\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } )\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{\theta \theta } }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{5} }}{{\partial \tilde{\sigma }_{r\theta } }} = \int\limits_{0}^{\pi } {\left\{ { - \frac{{(n + 1)\tilde{\sigma }_{\text{e}}^{n} \frac{{\partial \tilde{\sigma }_{\text{e}} }}{{\partial \tilde{\sigma }_{r\theta } }}\tilde{g} - \tilde{\sigma }_{\text{e}}^{n + 1} \frac{{\partial \tilde{g}}}{{\partial \tilde{\sigma }_{r\theta } }}}}{{\tilde{g}^{2} }}\left[ {\frac{3}{4}\frac{{(\tilde{\sigma }_{\theta \theta } - \tilde{\sigma }_{rr} )}}{{\tilde{q}}} + \frac{\eta }{2}} \right] - \frac{3}{4}\frac{{\tilde{\sigma }_{\text{e}}^{n + 1} }}{{\tilde{g}}}\frac{{(\tilde{\sigma }_{rr} - \tilde{\sigma }_{\theta \theta } )\frac{{\partial \tilde{q}}}{{\partial \tilde{\sigma }_{r\theta } }}}}{{\tilde{q}^{2} }}} \right\}} \varPhi^{j} \varPhi^{i} {\text{d}}\theta $$

$$ \frac{{\partial R_{i}^{5} }}{{\partial \tilde{u}}} = \int\limits_{0}^{\pi } {\varPhi^{j} } \varPhi^{i} {\text{d}}\theta \quad \frac{{\partial R_{i}^{5} }}{{\partial \tilde{v}}} = \int\limits_{0}^{\pi } {\frac{{\partial \varPhi^{j} }}{\partial \theta }} \varPhi^{i} {\text{d}}\theta $$

The set of linear Eqs. (48) can be solved repeatedly until the iteration converges to the solution, $y = \tilde{\sigma }_{rr} ,\tilde{\sigma }_{\theta \theta } ,\tilde{\sigma }_{r\theta } ,\tilde{u},\tilde{v}$ of equation set (45). Convergence is achieved when the Euclidean norm of the solution update, δy, approximately vanishes. For the unknown singularity s is we use the HRR value as an initial guess to the problem. Convergence is achieved and for values of s that deviate from the correct value of the singularity, but the eigenvectors exhibit oscillatory behavior from node to node. This observation was used to search for the correct singularity by moving s to the direction that minimizes the oscillation. The results and efficiency of the algorithm were tested in earlier studies by Papanastasiou and Durban (2001) for the case of von Mises and associative Drucker–Prager materials where the singularity is a priori known to be the HRR singularity.