Non-differentiable energy minimization for cohesive fracture

Abstract

An energy minimization approach to initially rigid cohesive fracture is proposed, whose key feature is a term for the energy stored in the interfaces that is nondifferentiable at the origin. A consequence of this formulation is that there is no need to define an activation criterion as a separate entity from the traction-displacement relationship itself. Instead, activation happens automatically when the load reaches a critical level because the minimizer of the potential no longer occurs at the 0-displacement level. Thus, the activation computation necessary in previous initially rigid formulations is now replaced by the computation of a minimizer of a nondifferentiable objective function. This immediately makes the method more amenable to implicit time stepping, since the activation criterion no longer interacts with the nonlinear solver for the next time step. A novel extension of the functional to the dynamic case is presented. The optimization problem is solved by a continuation (homotopy) method used in conjunction with an augmented Lagrangian and a trust region minimization algorithm to find the minimal energy configuration. Because the approach eliminates the need for an activation criterion, the algorithm sidesteps the complexities of time-discontinuity and traction-locking previously observed in relation to initially rigid models.

Appendix

Here, the equations for the smoothed versions of the potential functions $\phi$ and $\theta$ used in the continuation steps are given. A description of $\phi$ is as follows. Function $\phi '$ is specified by imposing the following requirements on the knots and coefficients of $\phi '$:

$$\begin{aligned}&\phi '''_{(-\infty ,\kappa _1)} = 0, \\&\kappa _1 = 0, \\&\phi _1'(\kappa _1) = 0, \\&\phi _1''(\kappa _1) = 0, \\&\kappa _2 = \hat{{\delta }}_c, \\&\phi '(\kappa _2) = \hat{\sigma }_c, \\&\phi '''_{(\kappa _2,\kappa _3)} = -\hat{\sigma }_c/(\gamma \hat{{\delta }}_c^2), \\&\phi ''(\kappa _3) = -\hat{\sigma }_c/\hat{{\delta }}_{u}, \\&\phi '''_{(\kappa _3,\kappa _4)} = 0, \\&\phi '''_{(\kappa _4,\kappa _5)} = \hat{\sigma }_c/(\gamma \hat{{\delta }}_c^2), \\&\phi '(\kappa _5) = 0, \\&\phi ''(\kappa _5) = 0, \\&\phi '''_{(\kappa _5,\infty )} = 0. \end{aligned}$$

In these formulas $\hat{{\delta }}_c$ is another parameter defined to be $\gamma \hat{{\delta }}_{u}$. It is the abscissa at which $\phi '$ attains the value $\hat{\sigma }_c$ as seen from the fifth and sixth equations. It can be demonstrated via explicit but slightly tedious algebra that these constraints (plus the hypothesis that $\phi '$ is $C^1$ at all knots) uniquely specify all knots and coefficients of $\phi '$.

Let us further explain these constraints as follows. First, since $\phi '$ is piecewise quadratic, $\phi '''$ is piecewise constant and discontinuous. The notation $\phi '''_{(\cdot ,\cdot )}$ used in the above formulas refers to the constant value of $\phi '''$ in the interval $(\cdot ,\cdot )$. The first six equations completely specify $\phi '$ on the interval $[\kappa _1,\kappa _2]$ (i.e., on $[0,\hat{{\delta }}_c]$) as a quadratic function with constant and linear coefficients equal to 0 and quadratic coefficient equal to $\sigma _c/\hat{{\delta }}_c^2$. The second derivative of $\phi '$ tends to $\infty$ for small $\gamma$ in the three intervals $[\kappa _1,\kappa _2]$, $[\kappa _2,\kappa _3]$, and $[\kappa _4,\kappa _5]$ indicating sharp bends in the function plot. Over the interval $[\kappa _2,\kappa _3]$, $\phi '''$ is zero and $\phi '$ is decreasing linearly with slope $-\sigma _c/\hat{{\delta }}_{u}$.

Thus, in the final continuation step when $\gamma$ is very small and $\hat{{\delta }}_{u}={{\delta }}_{u}$, the function $\phi '$ is very close to a piecewise linear discontinuous function in which there is a jump from 0 to $\sigma _c$ at ${{\delta }}=0$, then a linearly descending branch with slope $-\sigma _c/{{\delta }}_\mathrm{u}$ over the interval $(0,{{\delta }}_\mathrm{u})$ decreasing to 0, and finally the constant zero function over the interval $({{\delta }}_{u},\infty )$.

For a positive value of $\gamma$, it is specified that $\phi '(0)=0$. This means that the function $\phi ({{\delta }})$ is a differentiable function of ${\varvec{\updelta }}$, where, as explained earlier, ${\varvec{\updelta }}=({{\delta }}_n,{{\delta }}_2,{{\delta }}_3)$. The reason is that although the square-root appearing in (8) yields a nondifferentiable function at the origin (similar to the fact that the function $\mathbf{x}\mapsto \Vert \mathbf{x}\Vert$ is nondifferentiable at $\mathbf{x}=\mathbf{0}$), upon composing this function with $\phi$ such that $\phi '(0)=\phi ''(0)=0$, one obtains a $C^2$ function due to the chain rule. On the other hand, for $\gamma =0$, $\phi '(0)>0$, so the potential is nondifferentiable.

The function $\theta '$ is also $C^1$ piecewise quadratic. Its second derivative (i.e., the third derivative of $\theta$) is the following piecewise constant function:

$$\begin{aligned}\theta '''({{\delta }})= \left\{ \begin{array}{ll} 0 &{} \text{ for } {{\delta }}< \zeta _1, \\ 1/\bar{\gamma }&{} \text{ for } {{\delta }}\in (\zeta _1,\zeta _2), \\ 0 &{} \text{ for } {{\delta }}\in (\zeta _2,\zeta _3), \\ -1/\bar{\gamma }&{} \text{ for } {{\delta }}\in (\zeta _3,\zeta _4), \\ 0 &{} \text{ for } {{\delta }}> \zeta _4 \end{array} \right. \end{aligned}$$

The additional equations that uniquely determine $\theta '$ are as follows:

$$\begin{aligned}&\zeta _4 = 0, \\&\theta '(0) = 0, \\&\theta ''(0) = 0, \\&\theta '(\zeta _3) =\mu /{{\delta }}_b, \\&\zeta _3-\zeta _2 = {{\delta }}_b, \\&\theta '(\zeta _1)= 0. \end{aligned}$$

It can be checked that as $\bar{\gamma }$ and ${{\delta }}_b$ tend to 0, the function $\theta '$ tends to the discontinuous function that is the constant $-\mu$ for negative values of ${{\delta }}$ and 0 for positive values. In particular, all four knots $\zeta _1,\ldots ,\zeta _4$ tend to 0.