Pith. sign in

REVIEW 3 major objections 5 minor 63 references

A hybrid s-version isogeometric strategy for dynamic crack propagation in 2D and 3D problems

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Replacing only the global mesh with B-splines makes global–local coupling integrands continuous, so standard Gauss quadrature suffices and cuts dynamic crack-analysis integration points by roughly 81% in 2D and 95.6% in 3D benchmarks.

desk verdict A solid, honest incremental advance: B-spline global basis removes the s-method's coupling-integration bottleneck, with strong benchmark verification; the main gap is that quadrature robustness is shown empirically, not theoretically. read the letter →

arxiv 2608.05589 v1 pith:SFLM57DS submitted 2026-08-06 cs.CE physics.comp-ph

Tianyu He , Kosei Kurosaki , Naoki Morita , Naoto Mitsume , Kazuki Shibanuma This is my paper · ORCID
classification cs.CEphysics.comp-ph MSC 74S0565N3074R10
keywords hybrids-versionisogeometricanalysisdynamiccrackpropagationglobal-localcouplingB-splinebasisfunctionsLagrangelocalmeshstressintensityfactorGaussquadrature
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes a hybrid s-version isogeometric analysis (hS-IGA) for dynamic crack propagation: B-spline basis functions in the coarse global mesh, Lagrange elements in the local crack mesh. The B-spline global basis removes the derivative discontinuities that, in the conventional Lagrange-based s-method, force recursive subdivision of integration domains. As a result, the coupling terms can be evaluated with standard 3×3 or 3×3×3 Gauss quadrature, preserving the global–local modelling advantages while cutting integration points by approximately 81% in the two-dimensional dynamic benchmark and 95.6% in the three-dimensional dynamic benchmark. The paper verifies this on stationary and propagating straight cracks in 2D and circular cracks in 3D, showing accurate dynamic stress intensity factors and local stresses, comparable degrees-of-freedom efficiency to the s-method, and reduced sensitivity near the local-domain boundary.

What carries the argument

The load-bearing object is the hybrid basis assignment: quadratic B-spline basis functions in tensor-product form in the global IGA mesh, and Lagrange basis functions in the local mesh. The B-splines' higher inter-element continuity makes the first derivatives of the global basis, and hence the coupling integrands in the stiffness and mass coupling matrices $\mathbf{K}_{GL}$, $\mathbf{K}_{LG}$, $\mathbf{M}_{GL}$, $\mathbf{M}_{LG}$, continuous inside local elements. This makes fixed $3\times 3$ or $3\times 3\times 3$ Gauss quadrature accurate without recursive subdivision, while the Lagrange local mesh keeps the crack as a $C^0$-but-not-$C^1$ line and permits nodal force release, moving local mesh updates, domain-integral $J$ evaluation, and nodal-force-based local stress post-processing.

What would settle it

Compute the coupling stiffness and mass entries for a local element whose support intersects a global knot span and compare the fixed 3×3 or 3×3×3 Gauss result against a highly subdivided high-order reference quadrature; if the relative error exceeds the level needed to keep the dynamic stress intensity factor and local stress within the reported few percent for some valid overlap configuration, the efficiency claim would not transfer outside the tested benchmarks.

Watch

Extended reading notes

Core claim

The central discovery is that the global approximation's inter-element continuity, not the local mesh, is the cause of the s-method's coupling-integration bottleneck. Replacing the global Lagrange basis with quadratic B-splines, which are $C^1$ continuous across knot spans, makes the coupling integrands continuous even when a local element spans several global elements. The local mesh remains Lagrange-based, which naturally represents the $C^0$-but-not-$C^1$ crack-tip displacement field and supports the nodal-force-release technique, local mesh update, and fracture post-processing. With this split, standard Gauss quadrature suffices without recursive subdivision; benchmark comparisons against Broberg's and Sneddon's solutions show accuracy comparable to conventional FEM and the s-method while integration points drop by roughly 81% in 2D dynamic propagation and 95.6% in 3D dynamic propagation.

Load-bearing premise

The assumption that fixed 3×3 or 3×3×3 Gauss quadrature over local elements that span several global IGA elements integrates the coupling integrands accurately enough; the paper demonstrates this on the chosen benchmarks but provides no quadrature-error bound, and the recommended mesh parameters are derived from the same benchmark sweeps.

Editorial extensions

If this is right

  • If the central claim is correct, dynamic crack propagation analyses can retain the s-method's degrees-of-freedom advantage while eliminating the coupling-integration bottleneck, with roughly 81% fewer integration points in the 2D dynamic benchmark and 95.6% fewer in the 3D dynamic benchmark.
  • The near-crack fracture quantities, dynamic stress intensity factor and local stress, match analytical or reference solutions within a few percent in both 2D and 3D, including a wide crack-velocity range from 200 to 1500 m/s.
  • The accuracy of local stress becomes governed mainly by the local element size $h_L$ rather than the global element size $h_G$, removing the mesh-interface sensitivity near $\Gamma_{GL}$ that appears in the conventional s-method.
  • The parameter conditions identified from the benchmark sweeps, such as $r_{GL} \geq 4$, $a_L = 2.5 h_G$, $l_L = 1.2 h_G$ (or $\sqrt{2} h_G$ in 3D), and $H_L = 1.8 h_G$, provide a starting recipe for configuring future hS-IGA crack analyses.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the continuity argument is general, the same hybrid basis assignment could reduce integration cost in any superposition or overlay method with non-matching meshes, not only fracture, by using spline bases on the coarser background mesh.
  • The empirical parameter sweeps suggest a quadrature-accuracy threshold; a systematic error estimate for Gauss quadrature of products of B-splines and Lagrange functions would show whether the savings transfer to curved crack fronts, elastoplasticity, or adaptive local meshes.
  • The paper restricts verification to linear-elastic planar cracks; the extension to 3D elastoplastic fracture that the conclusion mentions would test whether the continuity benefit survives material nonlinearity and more complex crack-front geometry.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a hybrid s-version isogeometric strategy (hS-IGA) for dynamic crack propagation analyses, in which the global mesh is discretized with quadratic B-spline basis functions while the local crack-domain mesh retains standard Lagrange basis functions. The motivation is that the global B-spline basis removes the derivative discontinuities that, in the conventional Lagrange-based s-method, make the global-local coupling integrands discontinuous inside local elements and force recursive subdivision for numerical integration. The formulation is presented in Sections 2 and 3, and verification is carried out on two-dimensional stationary and dynamic straight-crack benchmarks and three-dimensional stationary and dynamically propagating circular-crack benchmarks. The results are compared with the standard FEM, the conventional s-method, and available analytical or high-fidelity FEA reference solutions. The headline quantitative claims are that the proposed method reduces the number of integration points required for coupling integration by approximately 81% in the two-dimensional dynamic benchmark and 95.6% in the three-dimensional dynamic benchmark relative to the conventional s-method, while preserving comparable accuracy in the dynamic stress intensity factor and local stress.

Significance. The proposed method is potentially valuable: it combines the modelling advantages of the s-method (independent global and local meshes, localized refinement, natural crack representation by Lagrange elements) with the smoother global approximation offered by IGA. If the central claim holds, the method removes a serious practical bottleneck of the conventional s-method without requiring a full IGA local discretization, which would be attractive for fracture-oriented global-local simulations. The verification is extensive and, on the reported benchmarks, convincing: convergence studies are shown for displacement error, the fine-mesh FEA reference histories are checked against Broberg's analytical DSIF and local-stress solutions with maximum errors below 0.9% and 2.6%, and comparisons against both the conventional s-method and standard FEM are systematic. The paper also makes several honest limitations explicit, including restriction to planar linear-elastic crack problems.

major comments (3)
  1. [§3.2.2, Eqs. (46)-(47)] The continuity argument for the coupling integrands establishes only C0 continuity, not the polynomial smoothness that fixed Gauss quadrature would need for exactness. For quadratic B-splines with single-multiplicity interior knots, the first derivatives entering B_G are continuous but piecewise linear, so a stiffness coupling integrand containing B_G^T D B_L has a kink (a discontinuity in its derivative) when a local element crosses a global knot. Fixed 3x3 or 3x3x3 Gauss-Legendre quadrature over that local element has no a priori error estimate for such an integrand, and the error depends on where the kink falls relative to the Gauss points. The verification in Sections 4 and 5 is empirical and is performed on the same benchmark families used to select the recommended mesh conditions. I recommend adding a quadrature-convergence study for representative worst-case configurations (e.g., a local element whose global-knot kink is near a Gauss point, and a local element spanning a global knot on the crack path) or providing an explicit quadrature error estimate; without this, the strong claim that recursive subdivision can be omitted entirely is not fully supported for problems outside the calibrated benchmark set.
  2. [§4.2.2 and §5.2.2] The recommended parameter conditions (r_GL=4, a_L=2.5h_G, l_L=1.2h_G in 2D and sqrt(2)h_G in 3D, H_L=1.8h_G) are obtained from systematic sweeps on the same benchmarks that are later used to demonstrate accuracy. Consequently, the quoted accuracy ranges (within 1.8% for DSIF and 4.0% for local stress in 2D; within 1.3% and 2.5% in 3D) are partly a measure of how well the method fits the calibration benchmarks rather than of predictive performance on new problems. I recommend adding at least one out-of-sample configuration (a different domain geometry, crack-front curvature, or loading history) or a sensitivity analysis around the recommended parameters, so that the claimed generality of the method is not conflated with benchmark-specific tuning.
  3. [§4.2.2, §5.2.2, Figs. 33 and 49] Computational efficiency is assessed exclusively by counting integration points, not by wall-clock time. The hS-IGA method replaces recursive subdivision with a per-point global parametric mapping (Newton-Raphson iteration) and a global-element identification step; these operations also have a cost. A reduction in integration-point count is therefore not automatically equivalent to a proportional reduction in assembly time. The wording in the abstract and conclusions ('computationally efficient', 'computational cost') should be qualified either by adding measured assembly or total solve times or by explicitly stating that integration-point count is used as a proxy for assembly cost.
minor comments (5)
  1. [§4.1.2, Fig. 18] The L-shaped convergence curves under fixed h_L are described but not explained; a sentence interpreting why the global mesh begins to limit the attainable accuracy would help the reader.
  2. [Eq. (54)] The relative L2 error norm is written as a ratio of integral expressions; please check that the notation matches the intended definition of a norm ratio and clarify the denominator in the text.
  3. [Abstract and §4.2.2, §5.2.2] The integration-point reductions of 81% and 95.6% are quoted without stating the reference model (the conventional s-method with the same global-to-local element-size ratio). This context is given later in the text and should be stated in the abstract as well.
  4. [General] No data or code availability statement is included; providing the meshes, parameter sets, and reference FEA histories would materially improve reproducibility of the verification results.
  5. [Figs. 16, 26-29, 43-46] Several convergence and parameter-sweep figures use similar marker styles and are difficult to read in grayscale; increasing marker size and using distinct line styles would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the hS-IGA central claim is structural and validated against independent analytical and reference solutions.

full rationale

The derivation chain is not circular. The core claim is that replacing the Lagrange global basis with quadratic B-splines removes discontinuities in the coupling integrands (Eqs. (46)-(47), (49)-(50)) because B-spline first derivatives are C0 across knot spans with multiplicity 1 (Section 3.2.1). This is a structural property of B-splines, not an input equivalent to the claimed result. The accuracy of the resulting 3x3/3x3x3 Gauss quadrature is assessed empirically against Broberg's analytical solution, Sneddon's exact solution, and FEA references validated against Broberg; these are independent of the method's fitted values. The mesh-condition parameters (r_GL=4, a_L=2.5h_G, l_L=1.2h_G or sqrt(2)h_G, H_L=1.8h_G) are tuned on the same benchmarks, which is a mild in-sample calibration concern rather than a circular reduction: the evaluated DSIF and local stress are not defined in terms of these parameters, and the qualitative efficiency improvement (no recursive subdivision) follows from the integrand-continuity argument rather than from the tuning. The paper does not import any load-bearing uniqueness theorem from self-citations; self-citations [12,13,25,26] supply background, prior reference-generation procedures, and conventional s-method settings, none of which define the central claim. The absence of a quadrature error estimate for worst-case knot-overlap configurations is a correctness and robustness limitation, not circularity.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard B-spline continuity theory plus domain assumptions inherited from fracture mechanics and prior s-method implementations. The main user-chosen parameters are mesh sizes and local-domain dimensions selected by benchmark sweeps; no new physical entities are introduced.

free parameters (9)
  • J-integral radial selection radius R_J = 1.5 h_L
    Chosen based on preliminary investigations [13] to localize the domain integral; a user-supplied discretization parameter.
  • J-integral crack-front window width W_J = h_L(z') (3D)
    Chosen based on preliminary investigations [13]; determines the DSIF evaluation domain width in the crack-front direction.
  • Global-to-local element size ratio r_GL = >= 4
    Minimum ratio found by parameter sweeps in Sections 4.2.2 and 5.2.2 to avoid instabilities in DSIF and local stress.
  • Local crack length a_L = 2.5 h_G
    Parameter sweeps in Sections 4.2.2 and 5.2.2 show accuracy is governed by a_L/h_G.
  • Local ligament length l_L = 1.2 h_G (2D), sqrt(2) h_G (3D)
    Parameter sweeps; the 3D value is tied to the diagonal length of a global element in the x'-z' plane.
  • Local domain height H_L = 1.8 h_G
    Parameter sweeps in Sections 4.2.2 and 5.2.2 identify this as the condition for accurate evaluation of DSIF and local stress.
  • Rayleigh damping coefficient beta_R = 2.57 h_L sqrt(rho/E)
    Taken from prior work [38] to suppress numerical oscillations in nodal force release; affects the dynamic solution.
  • HHT time integration parameter alpha_HHT = -0.02
    Set based on preliminary evaluation; introduces small numerical dissipation.
  • Local element size h_L = 0.05 mm (2D), 0.04 mm (3D)
    Chosen so local stress is evaluated at x'=0.2 mm, the characteristic distance from steel fracture studies.
assumptions (6)
  • domain assumption The exact Mode I asymptotic displacement field (Eq. 52) and Sneddon's solution (Eqs. 57-59) can be imposed as Dirichlet boundary conditions on a finite target domain to represent an infinite cracked body.
    Used in the stationary benchmarks; the approximation error is assumed small but is not bounded in the paper.
  • domain assumption Broberg's analytical solution (Appendix B) is the correct reference for the dynamic straight-crack and circular-crack benchmarks.
    Used to validate the FEA reference histories and the hS-IGA results.
  • standard math A B-spline of degree p with single interior knots is C^{p-1} continuous, so quadratic B-splines are C1 across global element boundaries.
    This is the basis for the claim that coupling integrands are continuous; standard Cox-de Boor theory.
  • domain assumption The nodal force release technique with linear force decay (Eq. 7) and mesh update reproduces the physical crack propagation process with prescribed velocity V = h_L / Delta t.
    Inherited from prior s-method work [12,13,38]; not re-verified in this paper.
  • domain assumption Rayleigh damping with alpha_R=0 and beta_R from Eq. (A24) stabilizes the solution without materially changing DSIF and local stress.
    Coefficients are taken from previous work [38]; no sensitivity study is included here.
  • domain assumption The local stress at x'=0.2 mm ahead of the crack tip is a meaningful fracture quantity and can be compared against Broberg or reference FEA values.
    Motivated by brittle crack arrest studies [10,27,50,51].

how reviews work

0 comments
Cite this review

Pith. "Pith review of A hybrid s-version isogeometric strategy for dynamic crack propagation in 2D and 3D problems." pith.science (2026). https://pith.science/paper/SFLM57DS

@misc{pith2026260805589,
  author       = {Pith},
  title        = {Pith review of: A hybrid s-version isogeometric strategy for dynamic crack propagation in 2D and 3D problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SFLM57DS}},
  note         = {Machine review of arXiv:2608.05589}
}
read the original abstract

A hybrid s-version of isogeometric analysis (hS-IGA) strategy is proposed for accurate and efficient evaluation of near-crack fracture quantities in dynamic crack propagation analysis. The strategy retains the global-local superposition framework of the conventional s-method, while introducing B-spline basis functions only into the global discretisation and preserving a Lagrange-based local mesh in the crack domain. This hybrid formulation is motivated by the continuity-related bottleneck in the global-local coupling integration of the conventional Lagrange-based s-method, and by the need to retain a Lagrange-based local mesh for crack representation and post-processing of the dynamic stress intensity factor (DSIF) and local stress. The resulting formulation removes discontinuities in the coupling integrands caused by the global approximation and enables accurate coupling integration by standard Gauss quadrature without recursive subdivision. The proposed strategy is verified using two-dimensional stationary and dynamic straight-crack problems against the standard finite element method and the conventional s-method, and is further assessed using three-dimensional stationary and dynamically propagating circular-crack problems against the conventional s-method. Results show that the proposed hS-IGA strategy accurately evaluates the DSIF and local stress while retaining the global-local modelling advantages of the s-method. It also substantially reduces the number of integration points required for coupling integration, by approximately 81% in the two-dimensional dynamic benchmark and 95.6% in the three-dimensional dynamic benchmark relative to the conventional s-method. These results demonstrate that the proposed hS-IGA framework provides an accurate and efficient global-local strategy for dynamic crack propagation analyses requiring reliable evaluation of near-crack fracture quantities.

Figures

Figures reproduced from arXiv: 2608.05589 by the authors.

Figure 1
Figure 1. Global and local meshes for a crack propagation analysis based on the s-method. 2.2. Crack representation This section introduces the conventional s-method framework for modelling dynamic crack propagation. The present study considers two-dimensional straight-crack problems and three-dimensional circular-crack problems subjected to remote tensile loading under Mode I conditions. These configurations represent fundam… view at source ↗
Figure 2
Figure 2. Crack representation in the conventional s-method framework: (a) two-dimensional mesh configuration and node sets 𝒮ୈ and 𝒮୒; (b1) three-dimensional mesh configuration; (b2) node sets 𝒮ୈ and 𝒮୒ on the symmetry plane; and (c) deformation of the global and local meshes. Dynamic crack propagation is modelled using the nodal force release technique [12,13,38,39] applied to the local mesh [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 3
Figure 3. Nodal force release technique applied to the local mesh: (a) crack-front node sets in a three-dimensional local mesh at the beginning and end of a propagation step; and (b) nodal force release process within one step, illustrated in a two-dimensional section. Following the previous studies [12,13,38], a linear nodal force release assumption is adopted within each step. Specifically, the nodal force at the nodes corr… view at source ↗
Figures from the paper (46 more)
Figure 4
Figure 4. Figure 4: Local mesh update during dynamic crack propagation: (a) two-dimensional straight-crack problem; and (b) three-dimensional circular-crack problem. 2.3. Evaluation of stress intensity factor and local stress This section presents the methodology for evaluating the stress…
Figure 5
Figure 5. Figure 5: Integration domain for J-integral-based evaluation of the stress intensity factor. 2.3.2. Evaluation of local stress The local stress is another fracture quantity considered in the present study and is associated with the local fracture stress criterion for brittle cra…
Figure 6
Figure 6. Figure 6: illustrates representative linear and quadratic Lagrange basis functions assembled over adjacent elements, together with their first derivatives. As shown in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Schematic illustration of a discontinuous coupling integrand within a local element in the conventional Lagrange-based s-method. To address this issue, previous implementations typically employ recursive subdivision [13,19], as schematically shown in [PITH_FULL_IMAGE:…
Figure 8
Figure 8. Figure 8: Schematic illustration of recursive subdivision for coupling integration in the conventional s-method [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
Figure 9
Figure 9. Figure 9: (c). These considerations suggest that, for the purpose of representing a dynamically propagating crack tip in the local mesh, a standard Lagrange discretisation is more appropriate than a B-spline-based IGA discretisation. Higher continuity, making crack-tip represent…
Figure 10
Figure 10. Figure 10: illustrates the quadratic (𝑝ൌ 2) B-spline basis functions over multiple elements together with their first derivatives. As shown in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Schematic illustration of continuous coupling integration in the proposed hS-IGA strategy. By contrast, for the basis functions adopted in the local mesh, the continuity-related issue discussed in Section 3.1(i) does not arise in the same manner. Moreover, as discusse…
Figure 12
Figure 12. Figure 12: Schematic illustration of the mapping from the physical coordinates of a local integration point to the parametric coordinates of the corresponding global IGA element. 3.3. Crack representation The crack representation procedure introduced in Section 2.2 is retained i…
Figure 13
Figure 13. Figure 13: Schematic illustration of crack representation in the proposed hS-IGA strategy using a representative two-dimensional setting. 4. Verification and discussion: two-dimensional crack problems This section systematically verifies the proposed hS-IGA strategy through two-…
Figure 14
Figure 14. Figure 14: Two-dimensional stationary crack benchmark problem: (a) problem schematic; (b) target domain for the numerical analysis. 4.1.2. Verification of the two-dimensional stationary crack problem The global and local meshes used in the verification, together with the corresp…
Figure 15
Figure 15. Figure 15: Global and local mesh configurations and associated boundary conditions used in the verification analyses for the two-dimensional stationary crack problem. Convergence studies of the relative 𝐿ଶ error norm, 𝑒௅మ, defined in Eq. (54), were conducted for the global￾to-lo…
Figure 16
Figure 16. Figure 16: Convergence results of the 𝐿ଶ error norm under fixed 𝑟ୋ୐: (a) comparison with the standard FEM; and comparison with the conventional Lagrange-based s-method for (b1) 𝑟ୋ୐ ൌ 4, and (b2) 𝑟ୋ୐ ൌ 8. To clarify the principal computational advantage of the proposed strategy, …
Figure 17
Figure 17. Figure 17: Relationship between the relative 𝐿ଶ error norm and the number of integration points for the hS-IGA and the conventional Lagrange-based s-method under: (a) 𝑟ୋ୐ ൌ 4, (b) 𝑟ୋ୐ ൌ 8. To investigate the respective influences of ℎୋ and ℎ୐ for the proposed hS-IGA strategy, 𝑒௅…
Figure 18
Figure 18. Figure 18: When ℎୋ is fixed, the convergence curves are downward-convex in the low-DOF range, and their slopes gradually stabilise as the DOF increases. In contrast, under fixed-ℎ୐settings, the convergence curves exhibit an L￾shaped trend. These results indicate that the local-m…
Figure 19
Figure 19. Figure 19: Evaluation of the dimensionless tensile stress field and its difference from the exact solution for four mesh conditions: (a) ℎୋ ൌ 2 11 ⁄ , ℎ୐ ൌ 1 40 ⁄ , (b) ℎୋ ൌ 2 11 ⁄ , ℎ୐ ൌ 1 80 ⁄ , (c) ℎୋ ൌ 2 41 ⁄ , ℎ୐ ൌ 1 40 ⁄ , (d) ℎୋ ൌ 2 41 ⁄ , ℎ୐ ൌ 1 80 ⁄ . Based on the metho…
Figure 20
Figure 20. Figure 20: Normalised local stress ahead of the crack tip under two crack-tip mesh resolutions: (a) 1/40, (b) 1/80. To further examine the applicability of the proposed hS-IGA strategy to local stress evaluation under different mesh settings, analyses are conducted under three c…
Figure 21
Figure 21. Figure 21: Normalised local stress in front of the crack tip: (a) fixed 𝑟ୋ୐ ൌ 4, (b) fixed ℎୋ ൌ 2 21 ⁄ , (c) fixed ℎ୐ ൌ 1 80 ⁄ . 0.8 0.9 1.0 1.1 1.2 0.0 0.1 0.2 0.3 0.4 0.5 Distance from the crack tip 𝒙𝐜 [-] Norm. local stress 𝝈𝒚𝒚 𝝈𝐞𝐱ሺ𝐬ሻ 𝒚𝒚 ⁄ [-] Local domain boundary 𝛤ୋ୐ 0.8 0.…
Figure 22
Figure 22. Figure 22: Two-dimensional dynamic crack benchmark problem: (a) problem schematic; (b) target domain for the numerical analysis. As in the stationary benchmark in Section 4.1, the ideal dynamic crack propagation in an infinite plate is reproduced by prescribing Dirichlet boundar…
Figure 23
Figure 23. Figure 23: Schematic of the finite element model for generating the reference histories of the displacement field. The accuracy of the FEA model used to generate the boundary displacement histories is assessed by comparing the dynamic stress intensity factor, 𝐾୍ ୊୉ሺୢሻ , and the …
Figure 24
Figure 24. Figure 24: Comparison of the normalised DSIF and local stress obtained by the FEA with Broberg’s analytical solutions for the two-dimensional dynamic crack problem. 4.2.2. Verification of the two-dimensional dynamic crack problem The global and local meshes, together with the as…
Figure 25
Figure 25. Figure 25: Global and local mesh configurations and associated boundary conditions used in the verification analyses for the two-dimensional dynamic crack problem. With the analysis framework defined above, the next step is to identify the local domain and mesh conditions that e…
Figure 26
Figure 26. Figure 26: Influence of the global-to-local element-size ratio 𝑟ୋ୐ (𝑎ൌ 10 mm, 𝑎୐ ൌ 2.5ℎୋ, 𝑙୐ ൌ 1.2ℎୋ, 𝐻୐ ൌ 1.8ℎୋ): (a) Histories of dynamic stress intensity factor; (b) local stress distributions in crack propagation direction; (c) histories of local stress at 𝑥′ ൌ 0.2 mm (ൌ 4ℎ୐…
Figure 27
Figure 27. Figure 27: Influence of the crack length in the local domain 𝑎୐ (𝑎ൌ 10 mm, 𝑟ୋ୐ ൌ 8, 𝑙୐ ൌ 1.2ℎୋ, 𝐻୐ ൌ 1.8ℎୋ): (a) Histories of dynamic stress intensity factor; (b) local stress distributions in crack propagation direction; (c) histories of local stress at 𝑥′ ൌ 0.2 mm (ൌ 4ℎ୐). 0.8…
Figure 28
Figure 28. Figure 28: Influence of the ligament length in the local domain 𝑙୐ (𝑎ൌ 10 mm, 𝑟ୋ୐ ൌ 8, 𝑎୐ ൌ 2.5ℎୋ, 𝐻୐ ൌ 1.8ℎୋ): (a) Histories of dynamic stress intensity factor; (b) local stress distributions in crack propagation direction; (c) histories of local stress at 𝑥′ ൌ 0.2 mm (ൌ 4ℎ୐). …
Figure 29
Figure 29. Figure 29: Influence of the height of the local domain 𝐻୐ (𝑎ൌ 10 mm, 𝑟ୋ୐ ൌ 8, 𝑎୐ ൌ 2.5ℎୋ, 𝑙୐ ൌ 1.2ℎୋ): (a) Histories of dynamic stress intensity factor; (b) local stress distributions in crack propagation direction; (c) histories of local stress at 𝑥′ ൌ 0.2 mm (ൌ 4ℎ୐). 0.8 0.9 1…
Figure 30
Figure 30. Figure 30: Verification for two-dimensional dynamic crack problem over a wide range of crack velocities: (a) Normalised dynamic stress intensity factor; (b) normalised local stress at 𝑥′ ൌ 0.2 mm (ൌ 4ℎ୐). To further assess the effectiveness of the proposed strategy, crack propag…
Figure 31
Figure 31. Figure 31: Standard FEM model for comparison in the two-dimensional dynamic crack problem: (a) Mesh and boundary conditions; (b) close-up of A. 0.0 0.5 1.0 1.5 2.0 0 500 1,000 1,500 Norm. DSIF 𝑲ሺ𝐝ሻ 𝐈 𝑲𝐁ሺ𝐝ሻ 𝐈 ൗ [-] Crack velocity [m/s] 0.0 0.5 1.0 1.5 2.0 0 500 1,000 1,500 Norm. …
Figure 32
Figure 32. Figure 32: Accuracy comparison between the hS-IGA, the conventional s-method and the standard FEM for two￾dimensional dynamic crack propagation analysis: (a) normalised dynamic stress intensity factor; (b) normalised local stress at 𝑥′ ൌ 0.2 mm (ൌ 4ℎ୐). To demonstrate the comput…
Figure 33
Figure 33. Figure 33: Computational cost comparison between the hS-IGA, the conventional s-method and the standard FEM for two-dimensional dynamic crack propagation analysis: (a) degrees of freedom; and (b) number of integration points required for the analyses. 5. Verification and discuss…
Figure 34
Figure 34. Figure 34: Three-dimensional stationary crack benchmark problem. The exact solution of the tensile stress distribution ahead of the crack front, 𝜎௬௬ ୣ୶ሺୱሻ , which is used to verify the local stress 𝜎௬௬, was also provided by Sneddon [58] and is expressed as 𝜎௬௬ ୣ୶ሺୱሻ ห ௬ୀ଴ ൌ𝜎ஶ ൬ …
Figure 35
Figure 35. Figure 35: Global and local meshes and their boundary conditions in the three-dimensional stationary crack problem: (a) 𝑦𝑧 view; (b) 𝑥𝑧 view. Convergence studies of the relative 𝐿ଶ error norm, 𝑒௅మ, defined in Eq. (54), were conducted for the global￾to-local element-size ratios 𝑟…
Figure 36
Figure 36. Figure 36: Convergence results of the 𝐿ଶ error norm under fixed 𝑟ୋ୐: (a) proposed hS-IGA strategy for different 𝑟ୋ୐; and comparison with the conventional Lagrange-based s-method for (b1) 𝑟ୋ୐ ൌ 4, and (b2) 𝑟ୋ୐ ൌ 8. To clarify the principal computational advantage of the proposed …
Figure 37
Figure 37. Figure 37: Relationship between the relative 𝐿ଶ error norm and the number of integration points for the hS-IGA and the conventional Lagrange-based s-method under: (a) 𝑟ୋ୐ ൌ 4, (b) 𝑟ୋ୐ ൌ 8. 0.0001 0.001 0.01 0.1 1 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 Degrees of freedom Relativ…
Figure 38
Figure 38. Figure 38: Representative deformed configurations, magnified views around the crack front, and dimensionless 𝜎௬௬ fields for the three-dimensional stationary circular-crack problem (scale factor ൌ 10; 𝜎௬௬ fields are displayed only in the local meshes): (a) ℎୋ ൌ 2/25, ℎ୐ ൌ 1/48, D…
Figure 39
Figure 39. Figure 39: (b) presents the average value of 𝜎௬௬/𝜎௬௬ ୣ୶ሺୱሻ along the 𝑥ᇱ -direction. The error bars indicate the range from the minimum to maximum values along the 𝑧ᇱ -direction. Both methods accurately evaluate the local stress, and errors of less than 2.0% are obtained when the…
Figure 40
Figure 40. Figure 40: Three-dimensional dynamic crack benchmark problem. As in the two-dimensional dynamic problem discussed in Section 4.2.1, the exact solutions of the displacement and stress fields for the dynamically propagating circular-crack problem have not yet been clarified. There…
Figure 41
Figure 41. Figure 41: Global and local meshes and their boundary conditions in the three-dimensional dynamic crack propagation problem: (a) 𝑦𝑧 view; (b) 𝑥𝑧 view [PITH_FULL_IMAGE:figures/full_fig_p045_41.png]
Figure 42
Figure 42. Figure 42: Representative deformed configurations, magnified crack-front views, and 𝜎௬௬ fields for the three￾dimensional dynamic crack propagation analysis at 𝑉ൌ 500 m/s (𝑟ୋ୐ ൌ 8, 𝑎୐ ൌ 2.5ℎୋ, 𝑙୐ ൌ √2ℎୋ, and 𝐻୐ ൌ 1.8ℎୋ; scale factor ൌ 400; 𝜎௬௬ fields are displayed only in the loc…
Figure 43
Figure 43. Figure 43: Influence of the global-to-local element-size ratio 𝑟ୋ୐ (𝑎ൌ 4 mm, 𝑎୐ ൌ 2.5ℎୋ, 𝑙୐ ൌ √2ℎୋ, 𝐻୐ ൌ 1.8ℎୋ): (a) Dynamic stress intensity factor distributions along the crack front; (b) local stress distributions in crack propagation direction; (c) local stress distributions…
Figure 44
Figure 44. Figure 44: Influence of the crack length in the local domain 𝑎୐ (𝑎ൌ 4 mm, 𝑟ୋ୐ ൌ 8, 𝑙୐ ൌ √2ℎୋ, 𝐻୐ ൌ 1.8ℎୋ): (a) Dynamic stress intensity factor distributions along the crack front; (b) local stress distributions in crack propagation direction; (c) local stress distributions at 𝑥′…
Figure 45
Figure 45. Figure 45: Influence of the ligament length in the local domain 𝑙୐ (𝑎ൌ 4 mm, 𝑟ୋ୐ ൌ 8, 𝑎୐ ൌ 2.5ℎୋ, 𝐻୐ ൌ 1.8ℎୋ): (a) Dynamic stress intensity factor distributions along the crack front; (b) local stress distributions in crack propagation direction; (c) local stress distributions a…
Figure 46
Figure 46. Figure 46: Influence of the height of the local domain 𝐻୐ (𝑎ൌ 4 mm, 𝑟ୋ୐ ൌ 8, 𝑎୐ ൌ 2.5ℎୋ, 𝑙୐ ൌ √2ℎୋ): (a) Dynamic stress intensity factor distributions along the crack front; (b) local stress distributions in crack propagation direction; (c) local stress distributions at 𝑥′ ൌ 0.2…
Figure 47
Figure 47. Figure 47: Verification for three-dimensional dynamic crack problem over a wide range of crack velocities: (a) Normalised dynamic stress intensity factor; (b) normalised local stress at 𝑥′ ൌ 0.2 mm (ൌ 5ℎ୐). To further assess the effectiveness of the proposed strategy in the thre…
Figure 48
Figure 48. Figure 48: Accuracy comparison between the hS-IGA and the conventional s-method for three-dimensional dynamic crack propagation analysis at representative propagation stages: (a) normalised dynamic stress intensity factor; (b) normalised local stress at 𝑥′ ൌ 0.2 mm (ൌ 5ℎ୐). This…
Figure 49
Figure 49. Figure 49: Computational efficiency comparison for the three-dimensional dynamic crack problem: cumulative number of integration points required by the proposed hS-IGA strategy and the conventional s-method at representative propagation stages. 0.0 0.5 1.0 1.5 2.0 0 20 40 60 80 …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

63 extracted references · 51 canonical work pages

  1. [1]

    S. Aoki, K. Kishimoto, M. Sakata, Finite element computation of dynamic stress intensity factor for a rapidly propagating crack using Ĵ-integral, Comput. Mech. 2 (1987) 54–6 2. https://doi.org/10.1007/BF00282044

    work page doi:10.1007/bf00282044 1987

  2. [2]

    Kishimoto, S

    K. Kishimoto, S. Aoki, M. Sakata, Dynamic stress intensity factors using and finite element method, Eng. Fract. Mech. 13 (1980) 387–394. https://d oi.org/10.1016/0013-7944(80)90067-3

    work page doi:10.1016/0013-7944(80)90067-3 1980

  3. [3]

    Nishioka, S.N

    T. Nishioka, S.N. Atluri, On the computation of mixed-mode for a dynamically propagating crack, using path- independent integrals, Eng. Fract. Mech. 20 (1984) 193–208. https://doi.org/10.1016/0013-7944(84)90128- 0

    work page doi:10.1016/0013-7944(84)90128- 1984

  4. [4]

    C . J . J i h , C . T . S u n , E v a l u a t i o n o f a f i n i t e e l e m e n t b a s e d c r a c k - c l o s u r e m e t h o d f o r c a l c u l a t i n g s t a t i c a n d dynamic strain energy release rates, Eng. Fract. Mech. 37 (1990) 313–322. https://doi.org/10.1016/0013- 7944(90)90043-G

    doi:10.1016/0013- 1990

  5. [5]

    Prabel, S

    B. Prabel, S. Marie, A. Combescure, Using the X-FEM method to model the dynamic propagation and arrest of cleavage cracks in ferritic steel, Eng. Fract. Mech. 75 (2008) 2984–3009. https://doi.org/10.1016/j.engfracmech.2008.01.008

    work page doi:10.1016/j.engfracmech.2008.01.008 2008

  6. [6]

    Z. Wang, L. Ma, H. Yu, L. Wu, Dynamic stress inte nsity factors for homogeneous and non-homogeneous materials using the interaction integral method, Eng. Fract. Mech. 128 (2014) 8–21. https://doi.org/10.1016/j.engfracmech.2014.06.002

    work page doi:10.1016/j.engfracmech.2014.06.002 2014

  7. [7]

    H. Deng, B. Yan, T. Okabe, A new path-independent interaction integral for dynamic stress intensity factors of cracked structures, Int. J. Solids Struct. 243 (2022) 111559. https://doi.org/10.1016/j.ijsolstr.2022.111559

    arXiv 2022

  8. [8]

    Imachi, S

    M. Imachi, S. Tanaka, M. Ozdemir, T.Q. Bui, S. Oterkus, E. Oterkus, Dynamic crack arrest analysis by ordinary state-based peridynamics, Int. J. Fract. 221 (2020) 155–169. https://doi.org/ 10.1007/s10704-019-00416-3

    work page doi:10.1007/s10704-019-00416-3 2020

Show all 63 references
  1. [9]

    Panchadhara, P.A

    R. Panchadhara, P.A. Gordon, Application of peridynamic stress intensity factors to dynamic fracture initiation and propagation, Int. J. Fract. 201 (2016) 81–96. https://doi.org/10 .1007/s10704-016-0124-8

    2016

  2. [10]

    Yanagimoto, K

    F. Yanagimoto, K. Shibanuma, K. Suzuki, T. Matsum oto, S. Aihara, Local stress in the vicinity of the propagating cleavage crack tip in ferritic steel, Mater. Des. 144 (2018) 361–373. https://doi.org/10.1016/j.matdes.2018.02.037

    2018 doi

  3. [11]

    S.M.A. Khan, N. Merah, M.J. Adinoyi, 3D effects on crack front core regions, stress intensity factors and crack initiation angles, Int. J. Solids Struct. 50 (2013) 1449–1459. https://doi.org/10.1 016/j.ijsolstr.2013.01.019

    2013

  4. [12]

    Kishi, Y

    K. Kishi, Y. Takeoka, T. Fukui, T. Matsumoto, K. Suzuki, K. Shibanuma, Dynamic crack propagation analysis based on the s-version of the finite element method, Comput. Methods Appl. Mech. Eng. 366 (2020) 113091. https://doi.org/10.1016/j.cma.2020.113091

    2020

  5. [13]

    Shibanuma, K

    K. Shibanuma, K. Kishi, T. He, N. Morita, N. Mitsume, T. Fukui, S-version finite element strategy for accurately evaluating local stress in the vicinity of dy namically propagating crack front in 3D solid, Comput. Methods Appl. Mech. Eng. 399 (2022) 115374. https://doi.org/10....

    2022

  6. [14]

    Yanagimoto, T

    F. Yanagimoto, T. He, K. Shibanuma, The state-of-art of studies on brittle crack a rrest in steel, Eng. Fract. Mech. (2025) 111132. https://doi.org/10.1016/j.engfracmech.2025.111132

    2025

  7. [15]

    Gupta, J.P

    P. Gupta, J.P. Pereira, D.-J. Kim, C.A. Duarte, T. Ea son, Analysis of three-dimensional fracture mechanics problems: A non-intrusive approach using a generalized finite element method, Eng. Fract. Mech. 90 (2012) 41–64. https://doi.org/10.1016/ j.engfracmech.2012.04.014

    2012

  8. [16]

    Guidault, O

    P.-A. Guidault, O. Allix, L. Champaney, C. Cornuault, A multiscale extended finite element method for crack propagation, Comput. Methods Appl. Mech. Eng. 197 (2008) 381–399. https://doi.org/10.1016/j.cma.2007.07.023

    2008 doi

  9. [17]

    T. He, K. Kishi, N. Morita, N. Mitsume, F. Yanagimoto, Y.J. Kim, K. Shibanuma, Strategy for simulating high- 59 speed crack propagation in 3D-plate structures based on S-version FEM, Int. J. Mech. Sci. 274 (2024). https://doi.org/10.1016/j.ijmecsci.2024.109261

    2024

  10. [18]

    T. He, N. Morita, N. Mitsume, F. Yanagimoto, K. Suzuki, Y.-J. Kim, K. Shibanuma, Application phase analysis for predicting high-speed crack propagation/arrest in 3D thick plates using S-version FEM, Eng. Fract. Mech. 322 (2025) 111173. https://doi.org/10.1016/j.engfracmech.2025.111173

    2025

  11. [19]

    Okada, C.T

    H. Okada, C.T. Liu, T. Ninomiya, Y. Fukui, N. Kumaza wa, Analysis of particulate composite materials using an element overlay technique, CMES - Computer Mode ling in Engineering and Sciences 6 (2004) 333–347. https://doi.org/10.3970/cmes.2004.006.333

    2004 doi

  12. [20]

    Hughes, J.A

    T.J.R. Hughes, J.A. Cottrell, Y. Bazilevs, Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement, Comput. Methods A ppl. Mech. Eng. 194 (2005) 4135–4195. https://doi.org/10.1016/j.cma.2004.10.008

    2005 doi

  13. [21]

    Hughes, A

    T.J.R. Hughes, A. Reali, G. Sangalli, Efficient quad rature for NURBS-based isogeometric analysis, Comput. Methods Appl. Mech. Eng. 199 (2010) 301–31 3. https://doi.org/10.1016/j.cma.2008.12.004

    2010 doi

  14. [22]

    Schillinger, L

    D. Schillinger, L. Dedè, M.A. Scott, J.A. Evans, M.J. Borden, E. Rank, T.J.R. Hughes, An isogeometric design- through-analysis methodology based on adaptive hier archical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces, Comput. Meth ods Appl. Mech. En...

    2012 doi

  15. [23]

    Nguyen, C

    V.P. Nguyen, C. Anitescu, S.P.A. Bordas, T. Rabczuk, Isogeometric analysis: An overview and computer implementation aspects, Math. Comput. Simul. 117 (2015) 89–116. https://doi.org/10.1016/j.matcom.2015.05.008

    2015 doi

  16. [24]

    Cottrell, T.J.R

    J.A. Cottrell, T.J.R. Hughes, A. Reali, Studies of refinement and continuity in isogeometric structural analysis, Comput. Methods Appl. Mech. Eng. 196 (2007) 4160–4183. https://doi.org/10.1016/j.cma.2007.04.007

    2007 doi

  17. [25]

    Magome, N

    N. Magome, N. Morita, S. Kaneko, N. Mitsume, Higher-continuity s-version of finite element method with B- spline functions, J. Comput. Phys. 497 (2024) 112593. https://doi.org/10.1016/j.jcp.2023.112593

    2024

  18. [26]

    Tsuchiyama, Y

    Y. Tsuchiyama, Y. Sunaoka, H. Okada, Y. Otoguro, A me thod of overlaying models of isogeometric analysis (IGA) for modeling localized features of structure and its accuracy, Mechanical Engineering Journal 11 (2024) 24–00173. https://doi.org/10.1299/mej.24-00173

    2024 doi

  19. [27]

    Shibanuma, F

    K. Shibanuma, F. Yanagimoto, T. Namegawa, K. Suzuki, S. Aihara, Brittle crack propagation/arrest behavior in steel plate - Part II: Experime nts and model validation, Eng. Fr act. Mech. 162 (2016) 341–360. https://doi.org/10.1016/j.engfracmech.2016.02.053

    2016 doi

  20. [28]

    Fish, The s-version of the finite element method, Comput

    J. Fish, The s-version of the finite element method, Comput. Struct. 43 (1992) 539–547. https://doi.org/10.1016/0045-7949(92)90287-A

    1992 doi

  21. [29]

    Y. Yusa, H. Okada, Y. Yumoto, Three-Dimensional Elastic Analysis of a Structure with Holes Using Accelerated Coupling-Matrix-Free Iterative s-Version FE M , I n t . J . C o m p u t . M e t h o d s 1 5 ( 2 0 1 8 ) 1 8 5 0 0 3 6 . https://doi.org/10.1142/S0219876218500366

    doi

  22. [30]

    T. He, N. Mitsume, F. Yasui, N. Morita, T. Fukui, K. Shibanuma, Strategy for accurately and efficiently modelling an internal traction-fr ee boundary based on the s-version finite element method: Problem clarification and solutions verification, Comput . Methods Appl. Mech. En...

    2023

  23. [31]

    O k a d a , S

    H . O k a d a , S . E n d o h , M . K i k u c h i , A p p l i c a t i o n o f s - V e r s i o n F i n i t e E l e m e n t M e t h o d t o T w o - D i m e n s i o n a l Fracture Mechanics Problems, Jour nal of Solid Mechanics and Materials Engineering 1 (2007) 699–710. https://...

    2007 doi

  24. [32]

    Y. Yusa, J. Okamoto, D. Toyama, H. Okada, Analysis of a many-hole problem using coupling-matrix-free iterative s-version FEM with multiple local meshes, Mechanical Engineering Journal 5 (2018) 18-00264-18– 60 00264. https://doi.org/10.1299/mej.18-00264

    2018 doi

  25. [33]

    Yumoto, Y

    Y. Yumoto, Y. Yusa, H. Okada, Element subdivision te chnique for coupling-matrix-free iterative s-version FEM and investigation of sufficient element subdivision, Mechanical Engineering Journal 3 (2016) 16-00361- 16–00361. https://doi.org/ 10.1299/mej.16-00361

    2016 doi

  26. [34]

    Yumoto, Y

    Y. Yumoto, Y. Yusa, H. Okada, An s-version finite el ement method without generation of coupling stiffness matrix by using iterative technique, Mechanical Engineering Journal 3 (2016) 16-00001-16–00001. https://doi.org/10.1299/mej.16-00001

    2016 doi

  27. [35]

    S. Tu, N. Morita, T. Fukui, K. Shibanuma, The s-vers ion finite element method for non-linear material problems, Appl. Math. Model. 126 (2024) 287–30 9. https://doi.org/10.1016/j.apm.2023.10.040

    2024 doi

  28. [36]

    Okada, S

    H. Okada, S. Endoh, M. Kikuchi, On fracture analysis using an element overlay technique, Eng. Fract. Mech. 72 (2005) 773–789. https://doi.org/ 10.1016/j.engfracmech.2004.05.003

    2005 doi

  29. [37]

    T. Ooya, S. Tanaka, H. Okada, On the Linear Dependen cies of Interpolation Functions in s-Version Finite Element Method, J. Comput. Sci. Technol. 3 (2009) 124–135. https://doi .org/10.1299/jcst.3.124

    2009 doi

  30. [38]

    Yanagimoto, K

    F. Yanagimoto, K. Shibanuma, Y. Nishioka, Y. Shirai, K. Suzuki, T. Matsumoto, Local stress evaluation of rapid crack propagation in finite element analyses , Int. J. Solids Struct. 144–145 (2018) 66–77. https://doi.org/10.1016/j.ijsolstr.2018.04.014

    2018 doi

  31. [39]

    Kuna, Finite Elements in Fracture Mechanics Theory-Numerics-Applications, Springer, 2010

    M. Kuna, Finite Elements in Fracture Mechanics Theory-Numerics-Applications, Springer, 2010

    2010

  32. [40]

    Kawabata, Y

    T. Kawabata, Y. Nishizono, S. Aihara, Brittle crack pr opagation behavior in a memb er subjected to bending load, Theoretical and Applied Fractu re Mechanics 92 (2017) 266–275. https://doi.org/10.1016/j.tafmec.2017.09.005

    2017 doi

  33. [41]

    Kawabata, T

    T. Kawabata, T. Inoue, T. Tagawa, T. Fukui, Y. Takashima, K. Shibanuma, S. Aihara, Historical review of research on brittle crack propagatio n arresting technology for large weld ed steel structures developed in Japan with the application of Kca paramete rs, Marine Structures ...

    2020

  34. [42]

    T a k a h a s h i , K

    K . T a k a h a s h i , K . A r a k a w a , D e p e n d e n c e o f c r a c k a c c e l e r a t i o n o n t h e d y n a m i c s t r e s s - i n t e n s i t y f a c t o r i n polymers, Exp. Mech. 27 (1987) 195–19 9. https://doi.org/10.1007/BF02319474

    1987 doi

  35. [43]

    Arakawa, T

    K. Arakawa, T. Mada, K. Takahashi, Correlations amon g dynamic stress intensity factor, crack velocity and acceleration in brittle fracture, Int. J. Fract. 105 (2000) 311–320. https://doi.org/10.1023/A:1007654002732

    2000 doi

  36. [44]

    Kalthoff JF, Beinert J, Winkler S, Measurements of dynamic stress intensity factors for fast running and arresting cracks in double-cantilever-beam specimens, ASTM STP627 (1977) 161–176

    1977

  37. [45]

    N., A method for determining dynamic stress intensity factors from COD measurement at the notch mouth in dynamic tear testing, Eng

    Nishioka T., Atluri S. N., A method for determining dynamic stress intensity factors from COD measurement at the notch mouth in dynamic tear testing, Eng. Fract. Mech. 16 (1982) 333–339

    1982

  38. [46]

    Arakawa, D

    K. Arakawa, D. Nagoh, K. Takahashi, Crack velocity and acceleration effects on dynamic stress intensity factor in polymers, Int. J. Fract. 83 (1997) 305–313. https://do i.org/10.1023/A:1007387417517

    1997 doi

  39. [47]

    Arakawa, T

    K. Arakawa, T. Mada, K. Takahashi, On the correlations among dynamic stress intensity factor, crack velocity and acceleration in fast fracture, Nihon Kikai Gakkai Ronbunshu, A Hen/Transactions of the Japan Society of Mechanical Engineers, Part A 66 (2000) 883–887. https://doi....

    2000 doi

  40. [48]

    Bertram Broberg, Cracks and Fracture, Elsevi er, 1999

    K. Bertram Broberg, Cracks and Fracture, Elsevi er, 1999. https://doi.org/10.1016/B978-0-12-134130- 5.X5000-4

    1999 doi

  41. [49]

    Anderson, Fracture mechanics, Fourth Edi, CRC Press, 2017

    T.L. Anderson, Fracture mechanics, Fourth Edi, CRC Press, 2017

    2017

  42. [50]

    Shibanuma, F

    K. Shibanuma, F. Yanagimoto, T. Namegawa, K. Suzuki, S. Aihara, Brittle crack propagation/arrest behavior in steel plate - Part I: Model formulation, Eng. Fract. Mech. 162 (2016) 324–340. https://doi.org/10.1016/j.engfracmech.2016.02.054. 61

    2016 doi

  43. [51]

    Shibanuma, F

    K. Shibanuma, F. Yanagimoto, K. Suzuki, S. Aihara, Brittle crack propagation/arrest behavior in steel plate – Part III: Discussions on arrest design, Eng. Fract. Mech. 190 (2018) 104–119. https://doi.org/10.1016/j.engfracmech.2017.12.004

    2018 doi

  44. [52]

    Yanagimoto, T

    F. Yanagimoto, T. Hemmi, Y. Suzuki, Y. Takashima, T. Kawabata, K. Shibanuma, Contribution of grain size to resistance against cleavage crack propagation in ferritic steel, Acta Mater. 177 (2019) 96–106. https://doi.org/10.1016/j.actamat.2019.06.038

    2019 doi

  45. [53]

    Shibanuma, Y

    K. Shibanuma, Y. Suzuki, K. Kiriyama, K. Suzuki, H. Shirahata, A model of cleavage crack propagation in a BCC polycrystalline solid based on the extended finite element method, Acta Mater. 176 (2019) 232–241. https://doi.org/10.1016/j.actamat.2019.07.013

    2019 doi

  46. [54]

    Okada, H

    H. Okada, H. Kawai, K. Araki, A virtual crack closure-integral method (VCCM) to compute the energy release rates and stress intensity factors based on quadratic tetrahedral finite elements, Eng. Fract. Mech. 75 (2008) 4466–4485. https://doi.org/10.10 16/j.engfracmech.2008.04.014

    2008

  47. [55]

    Bouchbinder, J

    E. Bouchbinder, J. Fineberg, M. Marder, Dynamics of Simple Cracks, Annu. Rev. Condens. Matter Phys. 1 (2010) 371–395. https://doi.org/10.114 6/annurev-conmatphys-070909-104019

    2010

  48. [56]

    Freund, Dynamic Fracture Mechanic s, Cambridge University Press, 1990

    L.B. Freund, Dynamic Fracture Mechanic s, Cambridge University Press, 1990. https://doi.org/10.1017/CBO9780511546761

    1990 doi

  49. [57]

    M.-C. Hsu, C. Wang, F. Xu, A.J. Herrema, A. Krishnam urthy, Direct immersogeome tric fluid flow analysis using B-rep CAD models, Comput. Aided Geom. Des. 43 (2016) 143–158. https://doi.org/10.1016/j.cagd.2016.02.007

    2016 doi

  50. [58]

    Sneddon, The distribution of stress in the neighbourhood of a crack in an elastic solid, Proc

    I.N. Sneddon, The distribution of stress in the neighbourhood of a crack in an elastic solid, Proc. R. Soc. Lond. A Math. Phys. Sci. 187 (1946) 229–260. https://doi.org/10.1098/rspa.1946.0077

    1946

  51. [59]

    Shibanuma, T

    K. Shibanuma, T. Utsunomiya, Reformulation of XF EM based on PUFEM for solving problem caused by blending elements, Finite Elements in Analysis and Design 45 (2009). https://doi.org/10.1016/j.finel.2009.06.007

    2009 doi

  52. [60]

    Shibanuma, T

    K. Shibanuma, T. Utsunomiya, S. Aihara, An explicit application of partition of unity approach to XFEM approximation for precise reproduction of a priori knowledge of solution, Int. J. Numer. Methods Eng. 97 (2014) 551–581. https://doi.org/10.1002/nme.4593

    2014 doi

  53. [61]

    Tada, P.C

    H. Tada, P.C. Paris, G.R. Irwin, The Stress Analysis of Cracks Handbook, Third Edition, ASME Press, 2000. https://doi.org/10.1115/1.801535

    2000 doi

  54. [62]

    Fardis, Seismic design, assessment and retrofitting of concrete buildings, Springer, 2009

    M.N. Fardis, Seismic design, assessment and retrofitting of concrete buildings, Springer, 2009

    2009

  55. [63]

    Hilber, T.J.R

    H.M. Hilber, T.J.R. Hughes, R.L. Taylor, Improved nume rical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. Dyn. 5 (1977) 283–292. https://doi.org/10.1002/eqe.4290050306

    1977 doi

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.