Global cracking elements: a novel tool for Galerkin-based approaches simulating quasi-brittle fracture

• Fast strain softening in quasi-brittle materials produces a very small localized failure zone.Source: §1 · p. 2
• Existing approaches use continuum, discrete, or particle/meshless frameworks alongside fracture, damage, or equivalent theories.The cited examples include peridynamics and lattice models.Source: §1 · p. 2
• The paper evaluates numerical methods through questions about implementation ease, reliability, robustness, and efficiency.The criteria are posed for quasi-brittle fracture simulation and answered yes or no.Source: §1 · p. 2

• GCEM's discretization dependency is described as negligible relative to the smeared-crack approach, and examples use irregular meshes of different element sizes.Source: §5 · pp. 2–4
• A relatively coarse mesh can be used with GCEM, unlike the phase-field method.Source: §5 · p. 3
• GCEM is a Strong-Discontinuity-embedded Approach that does not need remeshing or precise crack-tip stress descriptions.The cited comparison includes phase-field, XFEM, and NMM methods.Source: §5 · p. 3
• Disconnected element-wise cracking segments represent crack paths without a crack-tracking strategy; local orientation determination provides self-propagating cracks.Source: §5 · p. 3
• The GCEM stiffness matrix is symmetric and sparse, supporting the paper's characterization of its numerical efficiency as reasonably high.Source: §5 · p. 3
• A standard Q9 element becomes a global cracking element by using its displacement degrees of freedom to represent local crack openings.Source: §5 · p. 3
• GCEM is reported as more numerically stable and efficient than CEM, requiring fewer iteration steps while retaining self-propagating cracking elements.Source: §5 · p. 3
• The numerical examples include an L-shaped panel test and disk tests with initial slots.Source: §5 · p. 3

2.1. Traction-separation law

• A mixed-mode traction-separation law relates normal and parallel crack-opening tractions; it sets Gf,0 = 0.001 Gf and ζ0 = 2 Gf,0/ft.The maximum opening ζmx is updated at the end of each loading step when it exceeds ζ0.Source: §2.1 · p. 4

2.2. Kinematics

• The displacement field combines a regular part with a Heaviside jump across the discontinuity, while the differentiable function ϕ(x) defines the crack transition and lc = V/A sets the characteristic length.Here A is the area of an equivalent crack surface passing through the element center; lc depends on element shape and crack orientation, not crack position.Source: §2.2 · p. 6
• After a crack appears, the stress/strain state at the center Gauss point represents the whole element, using equivalence between discrete and embedded-model forces.The equivalence relations are stated to hold throughout the cracking element, not only at its center.Source: §2.2 · pp. 6–7
• Because ϕ(x) need not be known, unlike in XFEM and some other SDA formulations, the approach is easier to implement and requires fewer assumptions.Source: §2.2 · p. 6

2.3. Global Cracking Element

• The formulation replaces a nonsymmetric matrix with many zero entries, which can destabilize iteration, by a symmetric matrix; the final term is retained to avoid stress locking.Source: §2.3 · pp. 9–10
• The GCEM Newton–Raphson iteration needs fewer steps than the CEM iteration because its updates are formulated at the element level.The text says this efficiency difference will be verified in the numerical investigation.Source: §2.3 · p. 10
• The cracking-element formulation is recast in matrix form as a pseudo-Q9, replacing the center node’s displacement degrees of freedom with element-level crack-opening degrees of freedom.The authors also note that the formulation can be extended to enriched crack openings, introducing extra crack-opening degrees of freedom only when needed.Source: §2.3 · p. 10

2.4. Initiation and propagation of cracks

• The crack search checks the propagation region first, expands it when new cracking elements appear, and checks the new root-search region when propagation stalls, capturing initiation and propagation.Uncracked elements are divided between the two regions according to whether they share a boundary with a cracking element.Source: §2.4 · pp. 11–12
• The procedure is simpler than the one used in the CEM framework.Source: §2.4 · p. 12
• The domain is first discretized with Q9 elements, which are converted to pseudo-Q9 cracking elements when a crack appears.Source: §2.4 · p. 12
• The Q9 center-node degrees of freedom are borrowed to indicate element-wise crack openings.Source: §2.4 · p. 12
• The borrowed degrees of freedom can be returned only under a strict condition: eq = 0 and the cracking element is not adjacent to another cracking element.Source: §2.4 · p. 12
• No degrees of freedom were returned in the numerical examples, so no Q9-to-cracking-element transformation was reversed.Source: §2.4 · p. 12
• Figure 4 presents the calculation procedure within one Newton–Raphson iteration step.Source: Figure 4 · p. 13

3. NUMERICAL EXAMPLES

• The numerical examples use plane-stress assumptions, with irregular meshes of different element sizes.Single-slot disk tests use both coarse and fine meshes, while double-slot tests use fine meshes only.Source: §3 · p. 13, §3.2 · p. 15

3.1. L-shaped panel

• GCEM requires fewer Newton–Raphson iterations than CEM in the L-shaped panel test.The force-displacement curves also show good agreement with XFEM and CEM results.Source: §3.1 · p. 14, Figure 8 · p. 15
• L-shaped panel results for mixed-mode crack openings compare well across the different meshes considered.The plotted force-displacement curves compare GCEM with experimental results [35], XFEM, and CEM [1].Source: §3.1 · p. 14, Figure 7 · p. 14

3.2. Brazilian disk tests with single or double slots

• For single-slot disk tests, mesh dependence is very small and results with coarser meshes are good.Force-displacement curves show similar peak loads and load histories across meshes; GCEM also captures initiation, propagation, and minor cracks at α = 60°.Source: §3.2 · pp. 15–16, Figure 13 · p. 17

• Global Cracking Elements are formally similar to Q9 elements, which makes them easy to implement in the FEM.Source: §4 · p. 21
• The GCEM needs no nodal or element-wise enrichment or crack-tracking strategy, and reliable results can be obtained with relatively coarse meshes and negligible mesh dependency.Source: §4 · p. 21
• The GCEM captures crack initiation and propagation and produces reliable mixed-mode crack openings; numerical tests indicate reliability, efficiency, and robustness.Source: §4 · p. 21
• Presently, the GCEM is limited to quadrilateral elements with nonlinear displacement interpolation, which are more complex than linear-interpolation elements.Source: §4 · p. 21
• The method avoids precise descriptions of crack tips, while still capturing multiple crack growth, a difficult task even for methods that track actual tips.Source: §4 · p. 21
• The GCEM captures multiple crack growth, although its element-wise crack openings do not permit direct tracking of actual crack tips.Source: §4 · p. 21