Abstract
The study of the mechanical action between a punch and a cracked substrate has some theoretical guidance for the material protection. So the coupling problem of a cracked semi-infinite harmonic substrate under the action of a rigid flat punch is studied. The mixed boundary value problem is transformed into the Riemann-Hilbert boundary value problem by applying the complex-variable method, and then converted into singular integral equation for a numerical solution. The stress intensity factors at the contact ends and crack tips and the Piola stresses of whole harmonic material can be expressed as complex functions. The results indicate that the stressed state of harmonic solid near the crack tip and contact ends have similar features as those in linear elastic solids. The crack causes an obvious impact on the stress distributions near the contact region. The study provides theoretical guidance for analyzing the damaged problems of some soft materials under small deformation.
Similar content being viewed by others
Data availability
There is no data to be shared.
References
Akinola A (2003) On complex variable method in elasticity. J Appl Math Comput 12(1–2):183–198
Broitman E (2017) Indentation hardness measurements at macro-, micro-, and nanoscale: a critical overview. Tribol Lett 65(23):1–18
Chen Z, Dong QB, Bai XY, Zhou K (2023) A dislocation-based model for shear cracks in arbitrary orientations under contact loading. Eng Fract Mech 289:1–21
Dag S, Apatay T, Guler MA, Gulgrc M (2012) A surface crack in a graded coating subjected to sliding frictional contact. Eng Fract Mech 80:72–91
Dong QB, Chen Z, Bai XY, Wei J, Zhou K (2022) A model for fretting contact of layered materials with interfacial cracks. Theoret Appl Fract Mech 122:1–14
Hasebe N (2022) Analytical solution of an orthotropic elasticity for a frictional rigid flat punch on a half plane with an oblique edge crack using a mapping function. Mech Res Commun 125(103955):1–18
Heiduschke K (2023) Nonlinear logarithmic hyperelasticity with isotropy in the initial state. J Nonlinear Sci 33(47):1–34
Hossain M, Steinmann P (2013) More hyperelastic model for rubber-like materials: consistent tangent operators and comparative study. J Mech Behav Mater 22(1–2):27–50
Kim HK (2015) Analysis of edge crack behaviour influenced by non-symmetrical contact tractions and bulk tension. Fatigue Fract Eng Mater Struct 38(5):551–562
Korobeynikov SN, Larichkin AY, Rotanova TA (2022) Hyperelasticity models extending Hooke’s law from small to moderate strains and experimental verification of their scope of application. Int J Solids Struct 252:1–17
Kumar P, Mahanty M, Singh AK, Chattopadhyay (2021) Analytical study on stress intensity factor due to the propagation of Griffith crack in a crystalline monoclinic layer subjected to punch pressure. Fatigue Fract Eng Mater Struct 44:475–478
Kurashige M (1971) Two-dimensional crack problem for initially Neo-Hookean solid. Z Angew Math Mech 51:145–147
Lv CF, Li M, Xiao JL, Jung IH, Wu J, Huang YG (2013) Mechanics of tunable hemispherical electronic eye camera systems that combine rigid device elements with soft elastomers. J Appl Mech 80:1–7
Lv SH, Meng LC, Qiu J, Qi LQ, Shi Y, Gao CF (2022) Square indentation on a soft elastomer layer with finite thickness. Acta Mech 233:2161–2172
Ma HL, Zhou YT, Li X, Ding SH (2023a) Analysis of a cracked neo-Hookean substrate with initial stress under a rigid punch. Eng Fract Mech 292:1–19
Ma YY, Zhou YT, Yang J, Zhao XF, Ding SH (2023b) Interface crack behaviors disturbed by Love waves in a 1D hexagonal quasicrystal coating-substrate structure. Z Angew Math Mech 74(61):1–11
Mohamed BR, Sami EB, Malek C (2013) An embedded crack in a functionally graded orthotronic coating bonded to a homogeneous substrate under a frictional Hertzian contact. Int J Solids Struct 50:3898–3910
Omar A, Abdeljalil T, Bouazza B, Hamid Z, Michel PF (2018) Method of fundamental solutions and high order algorithm to solve nonlinear elastic problems. Eng Anal Bound Elem 89:25–35
Pei PY, Shi Y, Yang G, Gao CF (2018) Fracture analyses of soft materials with hard inclusion. J Appl Mech 85:1–9
Ru CQ (1997) Finite strain singular field near the tip of a crack terminating at a material interface. Math Mech Solids 2:49–73
Ru CQ (2002) On complex-variable formulation for finite plane elastostatics of harmonic materials. Acta Mech 156:219–234
Rubio-Gonzalez C (2001) Elastodynamic analysis of the finite punch and finite crack problems in orthotropic materials. Int J Fract 112:355–378
Shimizu K, Nagai T, Shintake J (2021) Dielectric elastomer fiber actuators with aqueous electrode. Polymers 13:1–13
Signh AK, Signh AK (2022) Dynamic stress concentration of a smooth moving punch influenced by a shear wave in an initially stressed dry sandy layer. Acta Mech 233:1757–1768
Sun Y, Zhou WX, Xin SJ, Yang FQ (2021) Capillary-induced deformation of an initially stressed neo-Hookean solid: a sessile liquid droplet. Mech Res Commun 113(103688):1–6
Varley E, Comberbatch E (1980) Finite deformation of elastic materials surrounding cylindrical holes. J Elast 10(4):341–405
Wang GF, Schiavone P, Ru CQ (2005a) Harmonic shapes in finite elasticity under nonuniform loading. J Appl Mech 72:691–694
Wang GF, Schiavone P, Ru CQ (2005b) Surface instability of a Semi-infinite harmonic solid under van der Waals attraction. Acta Mech 180:1–10
Wang GF, Wang TJ, Schiavone P (2007) The contact problem in a compressible hyperelastic material. J Appl Mech 74:829–831
Xing C, Yu TT, Sun YL, Wang YX (2023) An adaptive phase-field model with variable-node elements for fracture of hyperelastic materials at large deformations. Eng Fract Mech 281:1–19
Zhang CX, Zhang BW, Zhou YT, Ding SH (2023) Continuous contact problem of interaction between two arbitrarily positioned flat stamps on the thermoelectric material. Acta Mech 234(10):4719–4732
Zheng Y, Hu YH, Cai SQ (2019) Contact mechanics of a gel under constrained swelling. J Mech Phys Solids 124:427–445
Acknowledgements
This work was supported by The National Natural Science Foundation of China (12272195, 12262033, 12272269, 12062021, and 12062022), Ningxia Hui Autonomous Region Science and Technology Innovation Leading Talent Training Project (2020GKLRLX01), and the Natural Science Foundation of Ningxia (2023AAC02003, 2022AAC03012).
Author information
Authors and Affiliations
Contributions
Ma finishes the major derivation, draw the pictures and write the paper; Zhou, Wang, Li and Ding provide the guidances of the theorical methods; Zhou, Li and Ding provide the financial supports.
Corresponding authors
Ethics declarations
Competing interest
The authors affirm that they have no known competing financial interests or personal relation that could have appeared to impact the work reported in this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Ma, H., Zhou, Y., Wang, X. et al. Analysis of a cracked harmonic substrate under a rigid punch. Int J Fract (2024). https://doi.org/10.1007/s10704-024-00782-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10704-024-00782-7