Fracture in tension-compression-asymmetry solids via phase field modeling. (English) Zbl 1442.74025
Summary: Many nonlinear elastic materials exhibit an asymmetric response when loaded in tension versus compression. In this paper, we study the fracture of such nonlinearly elastic materials with tension-compression asymmetry by virtue of phase field modeling. An additive decomposition of strain energy is utilized and extended to account for the modulus difference between tension and compression. This strain energy decomposition is demonstrated for both a neo-Hookean model and an Odgen model. The decomposition is proven to be consistent with the basic requirements of thermodynamics and is important for fracture modeling under the stress states with both compression and tension. The implementation of the phase field model with both decomposed energy of neo-Hookean and Odgen model is given. The implemented model is capable of capturing the tension-compression asymmetry of nonlinear elastic solids. It also can model crack initiation and propagation efficiently especially when the material undergoes both tension and compression, demonstrated through several typical specimens with pre-set crack. The stress fields around the crack tip break the classical law of singularity for the elastic solids with tension-compression symmetry \((\sigma \propto r^{- 0 . 5})\), which is greatly influenced by the ratio of tensile modulus to compressive modulus. The hardening of the material can delay the fracture with more diffusive fracture process zone, demonstrated by the modified Odgen model. The proposed approach shows a great potential to predict the damage behavior of the materials with tension-compression asymmetry at finite strain.
Keywords:
nonlinear elastic materials; fracture; phase field; tension-compression asymmetry; soft tissueReferences:
| [1] | Miller, K.; Chinzei, K., Constitutive modelling of brain tissue: experiment and theory, J. Biomech., 30, 11-12, 1115-1121 (1997) |
| [2] | Thibault, K. L.; Margulies, S. S., Age-dependent material properties of the porcine cerebrum: effect on pediatric inertial head injury criteria, J. Biomech., 31, 12, 1119-1126 (1998) |
| [3] | Prange, M. T.; Margulies, S. S., Regional, directional, and age-dependent properties of the brain undergoing large deformation, J. Biomech. Eng., 124, 2, 244-252 (2002) |
| [4] | Miller, K.; Chinzei, K., Mechanical properties of brain tissue in tension, J. Biomech., 35, 4, 483-490 (2002) |
| [5] | Velardi, F.; Fraternali, F.; Angelillo, M., Anisotropic constitutive equations and experimental tensile behavior of brain tissue, Biomech. Model. Mechanobiol., 5, 1, 53-61 (2006) |
| [6] | Rashid, B.; Destrade, M.; Gilchrist, M. D., Mechanical characterization of brain tissue in compression at dynamic strain rates, J. Mech. Behav. Biomed. Mater., 10, 1, 23-38 (2012) |
| [11] | Alexander, M. P., Mild traumatic brain injury: pathophysiology, natural history, and clinical management, Neurology, 45, 7, 1253-1260 (1995) |
| [13] | Faul, M.; Xu, L.; Wald, M.; Coronado, V.; Dellinger, A., Traumatic brain injury in the united states: national estimates of prevalence and incidence, Injury Prevention, 16, Suppl 1, A268 (2010) |
| [14] | Young, L.; Rule, G. T.; Bocchieri, R. T.; Walilko, V., When physics meets biology: Low and high-velocity penetration, blunt impact, and blast injuries to the brain, Front. Neurol., 6, 89 (2015) |
| [15] | Hemphill, M.; Dauth, S.; Chung Jong, Y.; Dabiri, B., Traumatic brain injury and the neuronal microenvironment: A potential role for neuropathological mechanotransduction, Neuron, 85, 6, 1177-1192 (2015) |
| [16] | Eyal, B. K.; Scimone, M. T.; Estrada, J. B.; Christian, F., Strain and rate-dependent neuronal injury in a 3D in vitro compression model of traumatic brain injury, Sci. Rep., 6, 30550 (2016) |
| [17] | Ambartsumyan, S., The axisymmetric problem of circular cylindrical shell made of materials with different stiffness in tension and compression, Izv. Akad. Nauk SSSR Mekh., 4, 4, 77-85 (1965) |
| [18] | Ambartsumyan, S.; Khachatryan, A., The basic equations of the theory of elasticity for materials with different tensile and compressive stiffness, Mech. Solids, 1, 2, 29-34 (1966) |
| [19] | Ambartsumyan, S., Elasticity Theory of Different Moduli (1986), China Railway Publishing House: China Railway Publishing House Beijing |
| [20] | Du, Z.; Guo, X., Variational principles and the related bounding theorems for bi-modulus materials, J. Mech. Phys. Solids, 73, 183-211 (2014) · Zbl 1349.74293 |
| [21] | Du, Z.; Zhang, Y.; Zhang, W.; Guo, X., A new computational framework for materials with different mechanical responses in tension and compression and its applications, Int. J. Solids Struct., 100, 54-73 (2016) |
| [22] | Francfort, G. A.; Marigo, J. J., Revisiting brittle fracture as an energy minimization problem, J. Mech. Phys. Solids, 46, 8, 1319-1342 (1998) · Zbl 0966.74060 |
| [23] | Miehe, C.; Hofacker, M.; Welschinger, F., A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits, Comput. Methods Appl. Mech. Engrg., 199, 45, 2765-2778 (2010) · Zbl 1231.74022 |
| [24] | Miehe, C.; Welschinger, F.; Hofacker, M., Thermodynamically consistent phase-field models of fracture: variational principles and multi-field FE implementations, Internat. J. Numer. Methods Engrg., 83, 10, 1273-1311 (2010) · Zbl 1202.74014 |
| [25] | Amor, H.; Marigo, J. J.; Maurini, C., Regularized formulation of the variational brittle fracture with unilateral contact: Numerical experiments, J. Mech. Phys. Solids, 57, 8, 1209-1229 (2009) · Zbl 1426.74257 |
| [26] | Larsen, C. J.; Ortner, C., Existence of solutions to a regularized model of dynamic fracture, Math. Models Methods Appl. Sci., 20, 07, 1021-1048 (2010) · Zbl 1425.74418 |
| [27] | Larsen, C. J., Models for Dynamic Fracture Based on Griffith’s Criterion, 131-140 (2010), Springer Netherlands |
| [28] | Bourdin, B.; Larsen, C. J.; Richardson, C. L., A time-discrete model for dynamic fracture based on crack regularization, Int. J. Fract., 168, 168, 133-143 (2010) · Zbl 1283.74055 |
| [29] | Borden, M. J.; Verhoosel, C. V.; Scott, M. A.; Hughes, T. J.R.; Landis, C. M., A phase-field description of dynamic brittle fracture, Comput. Methods Appl. Mech. Engrg., 217, 1, 77-95 (2012) · Zbl 1253.74089 |
| [30] | Hofacker, M.; Miehe, C., A phase field model of dynamic fracture: Robust field updates for the analysis of complex crack patterns, Internat. J. Numer. Methods Engrg., 93, 3, 276-301 (2013) · Zbl 1352.74022 |
| [31] | Tang, S.; Zhang, G.; Guo, T. F.; Guo, X.; Liu, W. K., Phase field modeling of fracture in nonlinearly elastic solids via energy decomposition, Comput. Methods Appl. Mech. Engrg., 347, 477-494 (2019) · Zbl 1440.74038 |
| [32] | Miehe, C.; GöKtepe, S.; Lulei, F., A micro-macro approach to rubber-like materials-part i: the non-affine micro-sphere model of rubber elasticity, J. Mech. Phys. Solids, 52, 11, 2617-2660 (2004) · Zbl 1091.74008 |
| [33] | Tang, S.; Greene, S.; Liu, W. K., Two-scale mechanism-based theory of nonlinear viscoelasticity, J. Mech. Phys. Solids, 60, 2, 199-226 (2012) · Zbl 1244.74098 |
| [34] | Li, Y.; Tang, S.; Kröger, M.; Liu, W. K., Molecular simulation guided constitutive modeling on finite strain viscoelasticity of elastomers, J. Mech. Phys. Solids, 88, 204-226 (2016) |
| [35] | Rosati, L.; Valoroso, N., A return map algorithm for general isotropic elasto/viscoplastic materials in principal space, Internat. J. Numer. Methods Engrg., 60, 2, 461-498 (2004) · Zbl 1060.74520 |
| [36] | Tang, S.; Yang, Y.; Peng, X. H.; Liu, W. K.; Huang, X. X.; Elkhodary, K., A semi-numerical algorithm for instability of compressible multilayered structures, Comput. Mech., 56, 1, 63-75 (2015) · Zbl 1329.74289 |
| [37] | Miehe, C.; Schänzel, L. M.; Ulmer, H., Phase field modeling of fracture in multi-physics problems. part i. balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids, Comput. Methods Appl. Mech. Engrg., 294, 1, 449-485 (2015) · Zbl 1423.74838 |
| [38] | Miehe, C.; Schänzel, L. M., Phase field modeling of fracture in rubbery polymers. part i: Finite elasticity coupled with brittle failure, J. Mech. Phys. Solids, 65, 5, 93-113 (2014) · Zbl 1323.74012 |
| [39] | Hocine, N. A.; Abdelaziz, M. N.; Imad, A., Fracture problems of rubbers: J-integral estimation based upon \(\eta\) factors and an investigation on the strain energy density distribution as a local criterion, Int. J. Fract., 117, 1, 1-23 (2002) |
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