Research On The Fracture Of Nonlinear Elastic Solids Based On Phase Field Method Via Energy Decomposition

The prediction of the failure mechanism of nonlinear elastic materials is of great significance for engineering applications.The prediction of crack propagation in rubber products such as tires,seals,conveyor belts,and building foundation insulation is an important issue.In addition,from the microelectronics of household appliances to aerospace components,the failure prediction of ploymers has also received much attention.Phase-field models for fracture problems have attracted considerable attention in recent years,which are capable of tracking the discontinuities numerically,and also produce complex crack patterns in many applications.A phase-field model for a general nonlinearly elastic material is proposed using a novel additive decomposition of strain energy.This decomposition has two parts:one is principal stretch related and the other solely composed of volumetric deformation,which accounts for different behaviors of fracture in tension and compression.We construct the Lagrangian by integrating the split energies and the separation energy from phase-field approximation for discrete cracks.A coupled system of equations is also derived that governs the deformation of the body and the evolution of phase field.The capability and performance of the proposed model are demonstrated in several representative examples.Our results show that the predicted fracture surfaces are in good agreement with experimental observations.Compared with the previous models in which the energy is simply split into the isochoric and volumetric parts,the present model is numerically more robust and effective in simulating sharp cracks.Secondly,study on the fracture of tension-compression asymmetry nonlinear elastic solids by phase field model.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.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.and this strain energy decomposition is important for fracture modeling under the stress states with both compression and tension.The stress fields around the crack tip break the classical law of singularity for the elastic solids with tension-compression symmetry(σ∝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.Finally,a phase field model with energy decomposition previously developed is extended for hydrogel.To be consistent with the phase field model for rubber,we first reformulate the free energy of hydrogels with initial swelling that takes into account the initial volume fraction of water.The free energy（for intact hydrogels）is then modified to be suited for phase field modeling.The model parameters are calibrated by experimental tests performed under uniform compression without the phase field damage.A typical water-containing soft solid,hydrogel,is fabricated.Uniaxial tension,compression and three-point bending experiments on hydrogel blocks are carried out under plane strain conditions.These experiments were simulated by two different phase filed models.Predications of the two model are qualitatively and quantitatively compared with our experimental results.The simulation results show that the phase field model can capture the experimentally observed sequence of deformation and fracture at finite strains.It is also found that the energy decomposition is a key for the robust modeling of experimental fracture involving the compression.The role of the water played in the toughness of the gel is also revealed.Compared with the specimen dominated by the tensile loading,the specimens in our experiments（compression and three-point bending）experience a large portion of compression.For the phase field model in which the energy is simply split into the isochoric and volumetric parts,the damage（in terms of the phase-field variable p）is not very localized.It should be noted that the energy decomposition of this model just distinguish bulk compressions.Another phase field model considers fibre compressions（possible distinct compressions among the three principal stretches）.Therefore,the fracture damage zone is more localized and the method has more robust numerical stability.