A smeared crack modeling framework accommodating multi-directional fracture at finite strains

Abstract

A generic smeared crack modeling framework predicated on the deformation gradient decomposition (DGD) approach is proposed for use in dynamic fracture problems at finite strains, accommodating failure along multiple mutually orthogonal fracture planes embedded within an independently defined bulk material model. Within this constitutive framework, the traction equilibrium conditions imposed at each failure surface are used to determine the associated crack displacements stored as internal state variables. In general, the enforcement of interfacial equilibrium entails the implicit solution of a non-linear system of equations within the constitutive update procedure. However, if inertial effects arising due to the relative motion of the fractured material are incorporated within the model, the traction equilibrium conditions are shown to give rise to corresponding dynamic equations of motion governing the time-evolution of the crack opening displacements. For dynamic problems, an explicit time-integration procedure is devised to efficiently update the material state, subject to a set of internal frictionless contact constraints to prevent material inversion. The efficacy of the proposed modeling framework is investigated through several benchmark dynamic fracture problems run within the explicit finite element code DYNA3D.

A simplified evaluation of integral products of Heaviside functions

Recall From Eq. (26) the expression for the total internal kinetic energy of the cracked continuum idealization, restated below:

$$\begin{aligned} T^d = \frac{\rho }{8} \sum _{a,b=1}^{N_{\text {cracks}}} (\dot{\tilde{{\mathbf {u}}}}_a \cdot \dot{\tilde{{\mathbf {u}}}}_b) \frac{1}{|\varOmega |} \int _{\varOmega } {\mathscr {H}} ({\mathbf {N}}_a) {\mathscr {H}} ({\mathbf {N}}_b) \, dV. \end{aligned}$$

(70)

The primary difficulty in evaluating the above concerns the computation of the integrals $\int _{\varOmega } {\mathscr {H}} ({\mathbf {N}}_a) {\mathscr {H}} ({\mathbf {N}}_b) \, dV$ containing products of Heaviside functions. Herein, the aforementioned integral products are evaluated approximately by assuming an idealized geometric distribution of material within the RVE domain $\varOmega$.

In particular, consider the case where $\varOmega$ is approximated as an ellipsoidal domain obtained through an affine mapping ${\mathbf {J}}$ from a reference sphere with unit volume. Let the volume of the ellipsoidal domain correspond to $| \varOmega | = \det ({\mathbf {J}})$. The directed area $d {\mathbf {a}}_c = {\mathbf {n}}_c \, da$ of any medial crack plane passing through the reference sphere with circular area $da = \root 3 \of {\frac{9 \pi }{16}}$ and unit normal ${\mathbf {n}}_c$ is transformed via Nanson’s relation into the directed area $d {\mathbf {A}}_c = {\mathbf {N}}_c \, dA_c$ of a corresponding medial crack plane passing though the ellipsoidal domain with area $dA_c = | \varGamma _c |$ and normal ${\mathbf {N}}_c$, i.e.

$$\begin{aligned} d {\mathbf {A}}_c = \text {cof} ({\mathbf {J}}) \cdot d {\mathbf {a}}_c . \end{aligned}$$

(71)

Referring to Fig. 13, the volume of an ellipsoidal wedge (denoted $|W_{12}|$) formed by the intersection of $\varOmega$ with two half spaces ${\mathbf {X}} \cdot {\mathbf {N}}_1 \ge 0$ and ${\mathbf {X}} \cdot {\mathbf {N}}_2 \ge 0$ may be computed as the transformed volume of the corresponding spherical wedge defined in the reference sphere, such that

$$\begin{aligned} |W_{12}| = \frac{\pi - \cos ^{-1} ({\mathbf {n}}_1 \cdot {\mathbf {n}}_2)}{2 \pi } \, |\varOmega |, \end{aligned}$$

(72)

A reasonable approximation to the above over the complete range of values for the inner product ${\mathbf {n}}_1 \cdot {\mathbf {n}}_2 \in \left[ -1,+1\right]$ is given by

$$\begin{aligned} |W_{12}| \approx \frac{1 + {\mathbf {n}}_1 \cdot {\mathbf {n}}_2}{4} \, |\varOmega |. \end{aligned}$$

(73)

Fig. 13

Two-dimensional depiction of the ellipsoidal idealization for $\varOmega$ with principal semi-diameters $r_i$; the ellipsoidal wedge $W_{12}$ is defined by the two unit normals ${\mathbf {N}}_1$ and ${\mathbf {N}}_2$, such that a corresponding spherical wedge may be defined between ${\mathbf {n}}_1$ and ${\mathbf {n}}_2$ in the reference sphere

Given the chosen idealization for the RVE domain, the Heaviside integral products may be directly evaluated via

$$\begin{aligned} \int _{\varOmega } {\mathscr {H}} ({\mathbf {N}}_a) {\mathscr {H}} ({\mathbf {N}}_b) \, dV = 4 |W_{ab}| - |\varOmega |. \end{aligned}$$

(74)

Exploiting the approximation given by equation (73), the above reduces to a bilinear product expressed in terms of the metric tensor ${\mathbf {G}}$ defined in Sect. 2.1:

$$\begin{aligned} \int _{\varOmega } {\mathscr {H}} ({\mathbf {N}}_a) {\mathscr {H}} ({\mathbf {N}}_b) \, dV \approx \frac{16}{9} \frac{{\mathbf {N}}_a}{\ell _a} \cdot {\mathbf {G}} \cdot \frac{{\mathbf {N}}_b}{\ell _b} \, |\varOmega |. \end{aligned}$$

(75)