Numerical solution for plasticity models using consistency bisection and a transformed-space closest-point return: a nongradient solution method

Abstract

A new approach is presented for computing the return in numerical solutions for computational plasticity models that ensures convergence through bisection of the plastic consistency parameter, while using a transformed-space closest-point return based on a geometric search that eliminates the need to compute gradients of the yield function or a consistent tangent operator. Numerical solution of the governing equations for computational plasticity is highly-nontrivial for complex constitutive laws. In particular for geomaterials, a predictive model may account for nonlinear elasticity, shear strength that depends nonlinearly on pressure and Lode angle, and nonlinear evolution models for internal variables such as porosity or pore pressure. Traditional gradient-based integration methods may perform poorly when the hardening laws are highly nonlinear or when the yield function has an ill-defined or cumbersome gradient because of high curvature, vertices, or complicated functional form. The application of this new approach to geomaterial modeling is described, along with verification benchmarks.

Appendices

Appendix 1: Proof of the transformed-space closest-point return

From Simo and Hughes [30], “For an associative flow rule, $\varvec{\sigma }^{\text {n+1}}$ is the closest point projection onto the yield surface of the trial elastic stress $\varvec{\sigma }^{\text {trial}}$ in the inner product induced by the compliance tensor ${\mathbb {C}}^{-1}$”, where the energy norm is defined

$$\begin{aligned} \Vert \sigma \Vert _{{\mathbb {C}}^{-1}}:=\sqrt{\sigma :{\mathbb {C}}^{-1}:\sigma }. \end{aligned}$$

(33)

This is equated to an $L_2$ norm of some transformed tensor $\sigma ^\star ={\mathbb {C}}^{-1/2}:\sigma$ so that

$$\begin{aligned} \Vert \sigma \Vert _{{\mathbb {C}}^{-1}}^2=\sigma :{\mathbb {C}}^{-1}:\sigma =\sigma ^\star :\sigma ^\star \end{aligned}$$

(34)

Writing ${\mathbb {C}}^{-1/2}$ in terms of its spectral decomposition, the transformed tensor is defined in terms of eigenvalues $\lambda _k$ and eigenprojectors ${\mathbb {P}}_k$

$$\begin{aligned} \sigma ^\star = \sum \limits _{k=1}^{m}\frac{1}{\sqrt{\lambda _k}}{\mathbb {P}}_k:\sigma , \end{aligned}$$

(35)

where m is the number of distinct unique eigenprojectors.

Under certain restrictions [14], an isotropic material has an isotropic elastic tangent stiffness, implying that it can be additively decomposed

$$\begin{aligned} {\mathbb {C}}^{-1} = \frac{1}{3K}{\mathbb {P}}^\text {iso} + \frac{1}{2G}{\mathbb {P}}^\text {symdev} \end{aligned}$$

(36)

so that

$$\begin{aligned} {\mathbb {C}}^{-1/2} = \frac{1}{\sqrt{3K}}{\mathbb {P}}^\text {iso} + \frac{1}{\sqrt{2G}}{\mathbb {P}}^\text {symdev} \end{aligned}$$

(37)

where ${\mathbb {P}}^\text {iso}$ and ${\mathbb {P}}^\text {symdev}$ are the isotropic and symmetric-deviatoric fourth-order eigenprojectors. Specifically,

$$\begin{aligned}&{\mathbb {P}}_{ijkl}^\text {iso} = \frac{1}{3}(\delta _{ij}\delta _{kl}) \end{aligned}$$

(38a)

$$\begin{aligned}&{\mathbb {P}}_{ijkl}^\text {symdev} = \frac{1}{2}(\delta _{ik} \delta _{jl}+\delta _{il}\delta _{jk})-\frac{1}{3}\delta _{ij}\delta _{kl} \end{aligned}$$

(38b)

The stress tensor may always be decomposed as

$$\begin{aligned} \sigma = r \mathbf {E}_r + z \mathbf {E}_z, \end{aligned}$$

(39)

where ${\mathbf {E}}_z = \mathbf {I}/\sqrt{3}$ and $\mathbf {E}_r = \sigma ^\text {dev}/r$. Therefore, the transformed tensor is

$$\begin{aligned} \sigma ^\star&= {\mathbb {C}}^{-1/2}:\sigma \nonumber \\&= \left( \frac{1}{\sqrt{3K}}{\mathbb {P}}^\text {iso} + \frac{1}{\sqrt{2G}}{\mathbb {P}}^\text {symdev}\right) :(r \mathbf {E}_r + z \mathbf {E}_z) \end{aligned}$$

(40)

Using the identities ${\mathbb {P}}^\text {iso}:\mathbf {E}_r = \mathbf {0},\,{\mathbb {P}}^\text {iso}:\mathbf {E}_z = \mathbf {E}_z,\,{\mathbb {P}}^\text {symdev}:\mathbf {E}_z = \mathbf {0}$, and ${\mathbb {P}}^\text {symdev}:\mathbf {E}_r = \mathbf {E}_r$, the transformed stress can be written

$$\begin{aligned} \sigma ^\star&= \frac{z}{\sqrt{3K}}{\mathbb {P}}^\text {iso}:\mathbf {E}_z + \frac{r}{\sqrt{2G}}{\mathbb {P}}^\text {symdev}:\mathbf {E}_r \nonumber \\&= \frac{z}{\sqrt{3K}}\mathbf {E}_z + \frac{r}{\sqrt{2G}}\mathbf {E}_r. \end{aligned}$$

(41)

Equating this to $\sigma ^\star = z^\star \mathbf {E}_z + r^\star \mathbf {E}_r$, the scaling of the axis is then $r^\star = \frac{1}{\sqrt{2G}}r$ and $z^\star = \frac{1}{\sqrt{3K}}z$.

To obtain the same change in angles, it is sufficient to transform a single axis to produce the same stretch ratio, giving

$$\begin{aligned} r'=\sqrt{\frac{3K}{2G}}r, \text { and } z'=z \end{aligned}$$

(42)

or

$$\begin{aligned} r'=r, \text { and } z'=\sqrt{\frac{2G}{3K}}z \end{aligned}$$

(43)

Throughout this manuscript, the former relationship Eq. (42) has been applied to scale the r invariant for a 2-D solution, or to scale the stress deviator for the 3-D and full-stress solutions, though it may be more efficient to use Eq. (43), as it would result in fewer multiplication operators to scale the isotropic stress rather than the stress deviator.

Appendix 2: Formulation of the simplified yield surface

This section gives the formulation for the simplified constitutive model described in Sect. 2.2.2 and Fig. 5. With this construction, stress space is divided into three regions (I) where the closest point is the vertex, (II) where the closest point is on the linear Drucker–Prager surface, (III) where the closest point is on the elliptical cap.

The ellipse is centered at $(z^a,0)$, and passes though $(z^x,0)$ with principal axis aligned with the $r'$ and $z'$ axis, aspect ratio $\mathcal {R} =\frac{a}{b}$, and tangent to the linear Drucker–Prager surface at the point $\left\{ z^{\kappa },r^{\kappa }\right\}$. The values of the vertex position $\left( z^v\right)$ and cap position $\left( z^x\right)$ are known, as is the slope of the linear surface $(\beta )$ and the aspect ratio of the ellipse, $\mathcal {R}$.

The axis of the ellipse $(a \text { and } b)$, the coordinates of the branch point $\left( z^{\kappa },r^{\kappa }\right)$, and definitions for the three return regions, can be expressed in terms of the known values.

The coordinates of the branch point:

$$\begin{aligned} z^{\kappa }= & {} \frac{\beta z^v}{\sqrt{\beta ^2+\mathcal {R} ^2}}-\frac{\beta z^x}{\sqrt{\beta ^2+\mathcal {R} ^2}}+z^x \end{aligned}$$

(44a)

$$\begin{aligned} r^{\kappa }= & {} \frac{\beta \left( \sqrt{\beta ^2+\mathcal {R} ^2}-\beta \right) \left( z^v-z^x\right) }{\sqrt{\beta ^2+\mathcal {R} ^2}} \end{aligned}$$

(44b)

The z-coordinate of the apex:

$$\begin{aligned} z^a=\frac{\beta \left( \sqrt{\beta ^2+\mathcal {R} ^2}-\beta \right) \left( z^v-z^x\right) +\mathcal {R} ^2 z^x}{{\mathcal {R}}^2} \end{aligned}$$

(45)

The principal axis of the ellipse:

$$\begin{aligned} a= & {} \frac{\beta \left( \sqrt{\beta ^2+\mathcal {R} ^2}-\beta \right) \left( z^v-z^x\right) }{\mathcal {R} ^2} \end{aligned}$$

(46a)

$$\begin{aligned} b= & {} \frac{\beta \left( \sqrt{\beta ^2+\mathcal {R} ^2}-\beta \right) \left( z^v-z^x\right) }{\mathcal {R}} \end{aligned}$$

(46b)

A piecewise function $g\left( r',z',z^X,z^{v},\beta '\right)$ is defined at every point to be the distance in transformed and shifted stress space to the closest point on the yield surface.

In region I this is simply the distance to the vertex:

$$\begin{aligned} g^I=\sqrt{\left( z-z^v\right) ^2+r^2} \end{aligned}$$

(47)

In region II this is the distance from a point to a line.

$$\begin{aligned} g^{\text {II}}=\frac{r+\beta \left( z-z^v\right) }{\sqrt{1+\beta ^2}} \end{aligned}$$

(48)

In region III this is the distance from a point to an ellipse. This can be solved for analytically, but doing so requires evaluating the solution for a quartic polynomial, which is subject to significant numerical error. The error can be imaginary, which is problematic for implementation in a numerical simulation. Instead a simple numerical algorithm computes the closest point on the elliptical curve. For simplicity we present the algorithm in the coordinate system of the ellipse and consider a return from a point $(x_0,y_0)$ in the first quadrant.

The point on the ellipse is represented parametrically in terms of the angle $\theta$ as $x=a\cos (\theta )$ and $y=b\sin (\theta )$, and we seek to find the value $\theta$ to minimize the length $l(\theta )=\sqrt{\left( x_0-x(\theta )\right) {}^2+\left( y_0-y(\theta )\right) {}^2}$. In the first quadrant, the solution will lie in the range $0\le \theta \le \pi /2$, and can be efficiently computed with a Newton’s method.

Figure 5 shows the ideal flow function contours for the simplified yield surface, which allow a steepest descent to the closest point on the yield surface. Since the value of the flow function is the distance to the closest point, the updated stress for a nonhardening return can be evaluated exactly in a single step as

$$\begin{aligned} \varvec{\sigma }^{\text {n+1}} =\varvec{\sigma }^{\text {trial}}-\nabla g\left( \varvec{\sigma }^{\text {trial}} \right) g\left( \varvec{\sigma }^{\text {trial}} \right) . \end{aligned}$$

(49)

Appendix 3: Pseudocode for the computing the updated stress and plastic strain for a generalized 2-D nonhardening return

Inputs

• $\varvec{\sigma }^{\text {trial}}$, Trial stress tensor

• $\varvec{\sigma }^\text {n}$, Stress tensor at start of substep

• $\delta \varvec{\varepsilon }$, Increment in total strain (tensor)

• $X$, Hydrostatic compressive strength

• $\zeta$, Trace of isotropic backstress

• $K$, Elastic tangent bulk modulus

• $G$, Elastic tangent shear modulus

Outputs

• $\varvec{\sigma }^{\text {n+1}}$, Stress tensor from nonhardening return

• $\delta \varvec{\varepsilon }^\text {p,n+1}$, Increment in plastic strain (tensor) for the nonhardening return

Pseudocode

• 1. Compute invariants, for the trial stress $\left( I_1^\text {trial},J_2^\text {trial},\mathbf {S}^\text {trial} \right)$, as well as for an interior point, which we will take as $\left\{ I_1^0=\zeta +\frac{1}{2} \left( X + I_1^\mathbf{peak } \right) ,J_2^0=0\right\}$

• 2. Transform the trial and interior points as follows, where $\beta$ defines the degree of non-associativity.

  • $z^{\text {trial}\star }=\frac{I_1^\text {trial}}{\sqrt{3}}$

  • $r^{\text {trial},\star } = \sqrt{2J_2^\text {trial}}\beta \sqrt{\frac{3K}{2G}}$

  • $z^{0 \star } = \frac{I_1^0}{\sqrt{3}}$

  • $r^{0,\star } = \sqrt{2J_2^0}\beta \sqrt{\frac{3K}{2G}}$

• 3. Perform bisection in transformed space between $(z^{0\star },r^{0\star })$ and $(z^{\text {trial}\star },r^{\text {trial}\star })$ to find the new point on the yield surface $(z^{\text {new}\star },r^{\text {new}\star })$

• 4. Perform a rotation of $\left\{ z^{\text {new}\star },r^{\text {new}\star }\right\}$ about $\left\{ z^{\text {Trial}\star },r^{\text {Trial}\star }\right\}$ until a new internal point is found, set this as $\left\{ z^{0 \star },r^{0 \star }\right\}$

  • (a) $n=0$

  • (b) $n=n+1$

  • (c) $\theta =\frac{\pi }{2}(-1)^{n+1} \left( \frac{1}{2}\right) ^{\text {Floor}\left( \frac{n+1}{2}\right) }$

  • (d) $\left( \begin{array}{c} z^{\text {test}\star } \\ z^{\text {test}\star } \\ \end{array} \right) \!=\![Q]\cdot \left[ \left( \begin{array}{c} z^{\text {new}\star } \\ r^{\text {new}\star } \\ \end{array} \right) \!-\!\left( \begin{array}{c} z^{\text {trial}\star } \\ r^{\text {trial}\star } \\ \end{array} \right) \right] +\left( \begin{array}{c} z^{\text {trial}\star } \\ r^{\text {trial}\star } \\ \end{array} \right)$, where $[Q]=\left( \begin{array}{cc} \cos \theta &{} \quad -\sin \theta \\ \sin \theta &{} \quad \cos \theta \\ \end{array} \right)$

    • IF the transformed yield function is plastic when evaluated at $(z^{\text {test}\star },r^{\text {test}\star })$, GOTO[4b]

    • ELSE, Set $\left\{ z^{0 \star },r^{0 \star }\right\} =\left\{ z^{\text {new}\star },r^{\text {new}\star }\right\}$, GOTO[5]

• 5. Test for convergence:

  • IF $\theta \le \text {TOL},\,\hbox {Solution is converged}$, GOTO[6]

  • ELSE $\theta >\text {TOL},\,\hbox {Not converged}$, GOTO[3]

• 6. Solution converged, compute untransformed updated stress:

  • $I_1^{\text {new}}=\sqrt{3} z^{\text {new}\star }$

  • $\sqrt{J_2^{\text {new}}}=\frac{(2 G) r^{\text {new}\star }}{\sqrt{2} \beta (3 K)}$

  • $\varvec{\sigma }^{\text {n+1}} =\frac{1}{3}I_1^\text {trial} \mathbf {I} + \frac{\sqrt{J_2^{\text {new}}}}{\sqrt{J_2^{\text {trial}}}}\mathbf {S}^\text {trial}$

• 7. Compute increment in plastic strain for return: $\delta \varvec{\varepsilon }^p =\delta \varvec{\varepsilon }- \left( \mathbb {C} ^{-1}\right) :\left( \varvec{\sigma }^{\text {n+1}}-\varvec{\sigma }^\text {n} \right)$