Fig. 1 A continuum body in the reference configuration where a fracture is captured in a ( a ) discrete and ( b ) diffuse representation More about this image found in A continuum body in the reference configuration where a fracture is capture...

Fig. 2 A schematic for the domain J-integral in a sharp crack setting in the reference configuration More about this image found in A schematic for the domain J-integral in a sharp crack setting in the refer...

Fig. 3 A rectangular domain with an edge-crack in plane strain conditions subject to ( a ) uniform loading and ( b ) triangular-type loading More about this image found in A rectangular domain with an edge-crack in plane strain conditions subject ...

Fig. 4 The result of the phase-field simulation for the edge-crack sample under uniform loading in the current (top rows) and reference (bottom rows) configurations for viscosity parameter with a value of ( a ) 0, ( b ) 2.5, and ( c ) 5. Each row shows the first four time-s... More about this image found in The result of the phase-field simulation for the edge-crack sample under un...

Fig. 5 Load–displacement curves for four simulations with different artificial viscosity parameters (range from 0 to 5) for the rectangular domain with an edge-crack under uniform loading. The edge-crack in the initial state is represented diffusely. More about this image found in Load–displacement curves for four simulations with different artificial vis...

Fig. 6 The result of the phase-field simulation for the edge-crack sample under triangular-type loading with no artificial viscosity at steps 1, 30, 50, and 70 from left to right. Note that at the initial state, the crack is represented diffusely. More about this image found in The result of the phase-field simulation for the edge-crack sample under tr...

Fig. 7 For an edge-crack sample with triangular-type loading, domain J-integral evaluations with respect to measured crack length for different values of imposed critical energy release rate ( G c range from 0.25 to 1). The dashed lines show the imposed G c values. More about this image found in For an edge-crack sample with triangular-type loading, domain J-integral ev...

Fig. 8 Plots of the q functions utilized in the domain J-integral for four different cases with crack length, circle center location, and circle radii as (1) a ~ = 0.2 , O = 0.5 , r 1 = 0.4 , r 2 = 0.47 ; (2) a ~ = 0.35 , O = 0.35 , ... More about this image found in Plots of the q functions utilized in the domain J-integral for four d...

Fig. 9 Evaluation of the domain J-integral during crack propagation for an edge-crack sample under triangular-type loading with G c = 0.5 corresponding to the cases shown in Fig. 8 More about this image found in Evaluation of the domain J-integral during crack propagation for an edge-cr...

Fig. 10 The effect of artificial viscosity parameter on ( a ) force–displacement curve and ( b ) energy release rate as obtained from the domain J-integral. The sample is an edge-crack sample with a diffuse representation of the crack at the initial state under triangular-type load... More about this image found in The effect of artificial viscosity parameter on ( a ) force–displac...

Fig. 11 Crack propagation for an edge-crack sample under triangular-type loading for four consecutive time-steps after the onset of crack propagation, depicted for ( a ) discrete and ( b ) diffuse cracks. The artificial viscosity parameter is set at η = 0.5 . More about this image found in Crack propagation for an edge-crack sample under triangular-type loading fo...

Fig. 12 A comparison of the energy release rate during crack propagation for an edge-crack sample under triangular-type loading, where the crack is initially represented in a discrete or diffuse fashion. The artificial viscosity parameter is set at η = 0.5 . More about this image found in A comparison of the energy release rate during crack propagation for an edg...

Fig. 13 Damage function utilized in the GED framework plotted using the values c = 0.99 and γ = 20 More about this image found in Damage function utilized in the GED framework plotted using the values c...

Fig. 14 The result of GED simulations for an edge-crack sample with an initially diffuse crack under triangular-type loading and with no artificial viscosity. Snapshots correspond to steps 1, 33, 37, and 75 from left to right. More about this image found in The result of GED simulations for an edge-crack sample with an initially di...

Fig. 15 The effect of artificial viscosity on the crack propagation in GED modeling for four consecutive time-steps after the onset of crack propagation for an edge-crack sample under triangular-type loading. Note that at the initial state, the crack is represented diffusely. More about this image found in The effect of artificial viscosity on the crack propagation in GED modeling...

Fig. 16 The effect of artificial viscosity parameter on the energy release rate for an edge-crack sample under triangular-type loading utilizing the GED framework More about this image found in The effect of artificial viscosity parameter on the energy release rate for...