Three-dimensional crack propagation with distance-based discontinuous kernels in meshfree methods

Abstract

Distance fields are scalar functions defining the minimum distance of a given point in the space from the boundary of an object. Crack surfaces are geometric entities whose shapes can be arbitrary, often described with ruled surfaces or polygonal subdivisions. The derivatives of distance functions for such surfaces are discontinuous across the surface, and continuous all around the edges. These properties of the distance function were employed to build intrinsic enrichments of the underlying mesh-free discretisation for efficient simulation of three-dimensional crack propagation, removing the limitations of existing criteria (reviewed in this paper). Examples show that the proposed approach is able to introduce quite convoluted crack paths. The incremental nature of the developed approach does not require re-computation of the enrichment for the entire crack surface as advancing crack front extends incrementally as a set of connected surface facets. The concept is based on purely geometric representation of discontinuities thus addressing only the kinematic aspects of the problem, such to allow for any constitutive and cohesive interface models to be used.

Explicit expression for \(\min \) and \(\max \) functions

It is useful to define explicit expressions for the $min$ and $max$ functions, where the $min$ ($max$) function is the minimum (maximum) between two functions $f:\mathbb{R }^k\rightarrow \mathbb{R }$ and $g:\mathbb{R }^k\rightarrow \mathbb{R }$ with $k=1,2,3$

$$\begin{aligned} m\left( f(\mathbf{x}),g(\mathbf{x})\right)&= \min \left( f(\mathbf{x}),g(\mathbf{x})\right) = \left\{ \begin{array}{ll} f(\mathbf{x}) &{} \mathrm{if} \; f(\mathbf{x})-g(\mathbf{x})\le 0 \\ g(\mathbf{x}) &{} \mathrm{if} \; f(\mathbf{x})-g(\mathbf{x}) > 0 \end{array}\right. \nonumber \\ \end{aligned}$$

(51)

$$\begin{aligned} M\left( f(\mathbf{x}),g(\mathbf{x})\right)&= \max \left( f(\mathbf{x}),g(\mathbf{x})\right) = \left\{ \begin{array}{ll} g(\mathbf{x}) &{} \mathrm{if} \; f(\mathbf{x})-g(\mathbf{x})\le 0 \\ f(\mathbf{x}) &{} \mathrm{if} \; f(\mathbf{x})-g(\mathbf{x}) > 0 \end{array}\right. \nonumber \\ \end{aligned}$$

(52)

Equations (51) and (52) can be rewritten using the Heaviside function $\mathcal H (\mathbf{x})$

$$\begin{aligned} m(f,g) = g \, \mathcal H (f-g) + f \, \mathcal H (g-f) \end{aligned}$$

(53)

$$\begin{aligned} M(f,g) = f \, \mathcal H (f-g) + g \, \mathcal H (g-f) \end{aligned}$$

(54)

with the obvious commutative property

$$\begin{aligned} m(f,g) = m(g,f) \quad M(f,g) = M(g,f) \end{aligned}$$

(55)

Particular cases of Eqs. (53) and (54) are the positive and negative part of a function (Figs. 30, 31)

$$\begin{aligned} f_+ = M(f,0) \end{aligned}$$

(56)

$$\begin{aligned} f_- = m(f,0) \end{aligned}$$

(57)

and

$$\begin{aligned} f(\mathbf{x}) = f_+(\mathbf{x}) + f_-(\mathbf{x}) = M(f,0) + m(f,0) \end{aligned}$$

(58)

The functions in Eqs. (53) and (54) can be generalized to

$$\begin{aligned} m(a,b,f,g) = b \, \mathcal H (f-g) + a \, \mathcal H (g-f) \end{aligned}$$

(59)

$$\begin{aligned} M(a,b,f,g) = a \, \mathcal H (f-g) + b \, \mathcal H (g-f) \end{aligned}$$

(60)

where m assigns function a whenever $f\ge g$ and b otherwise; M instead assigns function b whenever $f \ge g$ and a otherwise. The following property holds true:

$$\begin{aligned} \min (f,g) = m(f,g,f,g)\quad \max (f,g) = M(f,g,f,g)\nonumber \\ \end{aligned}$$

(61)

Equation (61) allows the calculation of the derivatives of the $\min$ and $\max$ functions:

$$\begin{aligned} \dfrac{\partial \min (f,g)}{\partial x} = m\left( \dfrac{\partial f}{\partial x},\dfrac{\partial g}{\partial x},f,g\right) \end{aligned}$$

(62)

$$\begin{aligned} \dfrac{\partial \max (f,g)}{\partial x} = M\left( \dfrac{\partial f}{\partial x},\dfrac{\partial g}{\partial x},f,g\right) \end{aligned}$$

(63)

Fig. 30

$\max$ and $\min$ functions with $f(x)=\cos (x)$ (dash-dotted thin line) and $g(x) = \sin (x)$ (dashed thick line))

Fig. 31

Derivatives of $\min$ and $\max$ functions with $f(x)=\cos (x)$ and $g(x) = \sin (x)$; continuous line $\min$ and $\max$ functions, dashed line derivatives