Numerical simulation of dynamic fracture using finite elements with embedded discontinuities

Abstract

This paper presents the extension of some finite elements with embedded strong discontinuities to the fully transient range with the focus on dynamic fracture. Cracks and shear bands are modeled in this setting as discontinuities of the displacement field, the so-called strong discontinuities, propagating through the continuum. These discontinuities are embedded into the finite elements through the proper enhancement of the discrete strain field of the element. General elements, like displacement or assumed strain based elements, can be considered in this framework, capturing sharply the kinematics of the discontinuity for all these cases. The local character of the enhancement (local in the sense of defined at the element level, independently for each element) allows the static condensation of the different local parameters considered in the definition of the displacement jumps. All these features lead to an efficient formulation for the modeling of fracture in solids, very easily incorporated in an existing general finite element code due to its modularity. We investigate in this paper the use of this finite element formulation for the special challenges that the dynamic range leads to. Specifically, we consider the modeling of failure mode transitions in ductile materials and crack branching in brittle solids. To illustrate the performance of the proposed formulation, we present a series of numerical simulations of these cases with detailed comparisons with experimental and other numerical results reported in the literature. We conclude that these finite element methods handle well these dynamic problems, still maintaining the aforementioned features of computational efficiency and modularity.

Elastic damage to crack transition in a coupled non-local implicit discontinuous Galerkin/extrinsic cohesive law framework

“…In this section, the problem of calculating the TSL of a cohesive law from a damage model is solved by particularizing and resolving the equations derived in Section 2.2.3 to the case of an exponential non-local elastic damage model (14) developed in Section 2.1.2.…”

“…As an illustration example we consider the short glass-fiber-reinforced polypropylene material whose properties are given in the benchmark study in [43]. The considered material has a Young modulus E of 3.2 GPa, and follows a power damage evolution law (14), with κ i = 0.011, κ C = 0.5, α = 5.0, and with β = 0.75. When damage is high, the characteristic length of this material is √ 2 mm according to the reference [43].…”

“…In the weak form (65-66), the stress tensor σ is obtained from the elastic damage model reported in Section 2.1.2, with the damage evolution (14). The traction on cracked surfaces follows the governing equation (50) constructed in Section 2.3 from the strength and energy defined in Section 2.2.4.…”

“…On the other hand, when dealing with discontinuous approaches as the CZM, the FE discretization requires to be modified to account for a description of the crack. The three most popular methods to represent the crack are (i) the extended finite element method (xFEM) [12,13], (ii) the embedded localization method (EFEM) [14], and (iii) the use of interface elements. For the two first approaches, the crack can be represented in an arbitrary existing FE mesh through global or local enrichment and can also integrate a CZM [15].…”

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

“…The local strain field ε ℓ = G (c) ξ is approximated by defining a 'compatibility operator' G (c) acting on local element parameters ξ which define the geometric separation within the finite element B h e as shown in more detail in [3]. The locally introduced internal degrees of freedom ξ are statically condensed from the global set of equations thereby retaining the original degrees of freedom.Problems of fracture involving stationary and growing discontinuities can be very efficiently modeled with the above approach as shown in [3][4][5][6][7]. However for the phenomenon of crack branching where multiple cracks develop out of a single propagating crack the application of this approach is limited.…”