The cohesive zone model(CZM)has been used widely and successfully in fracture propagation,but some basic problems are still to be solved.In this paper,artificial compliance and discontinuous force in CZM are investiga...The cohesive zone model(CZM)has been used widely and successfully in fracture propagation,but some basic problems are still to be solved.In this paper,artificial compliance and discontinuous force in CZM are investigated.First,theories about the cohesive element(local coordinate system,stiffness matrix,and internal nodal force)are presented.The local coordinate system is defined to obtain local separation;the stiffness matrix for an eight-node cohesive element is derived from the calculation of strain energy;internal nodal force between the cohesive element and bulk element is obtained from the principle of virtual work.Second,the reason for artificial compliance is explained by the effective stiffnesses of zero-thickness and finite-thickness cohesive elements.Based on the effective stiffness,artificial compliance can be completely removed by adjusting the stiffness of the finite-thickness cohesive element.This conclusion is verified from 1D and 3D simulations.Third,three damage evolution methods(monotonically increasing effective separation,damage factor,and both effective separation and damage factor)are analyzed.Under constant unloading and reloading conditions,the monotonically increasing damage factor method without discontinuous force and healing effect is a better choice than the other two methods.The proposed improvements are coded in LS-DYNA user-defined material,and a drop weight tear test verifies the improvements.展开更多
A constitutive model to describe the behavior of rubber from low to high strain rates is presented.For loading,the primary hyperelastic behavior is characterized by the six parameter Ogden’s strain-energy potential o...A constitutive model to describe the behavior of rubber from low to high strain rates is presented.For loading,the primary hyperelastic behavior is characterized by the six parameter Ogden’s strain-energy potential of the third order.The rate-dependence is captured by the nonlinear second order BKZ model using another five parameters,having two relaxation times.For unloading,a single parameter model has been presented to define Hysteresis or continuous damage,while Ogden’s two term model has been used to capture Mullin’s effect or discontinuous damage.Lastly,the Feng-Hallquist failure surface dictates the ultimate failure for element deletion.The proposed model can accurately predict the response of rubber using a limited set of experimental data.The model has been validated here for the case of rubber but can be extended to a wide range of polymers.展开更多
The classical phase field model has wide applications for brittle materials,but nonlinearity and inelasticity are found in its stress-strain curve.The degradation function in the classical phase field model makes it a...The classical phase field model has wide applications for brittle materials,but nonlinearity and inelasticity are found in its stress-strain curve.The degradation function in the classical phase field model makes it a linear formulation of phase field and computationally attractive,but stiffness reduction happens even at low strain.In this paper,generalized polynomial degradation functions are investigated to solve this problem.The first derivative of degradation function at zero phase is added as an extra constraint,which renders higher-order polynomial degradation function and nonlinear formulation of phase field.Compared with other degradation functions(like algebraic fraction function,exponential function,and trigonometric function),this polynomial degradation function enables phase in[0,1](should still avoid the first derivative of degradation function at zero phase to be 0),so there is noconvergence problem.The good and meaningful finding is that,under the same fracture strength,the proposed phase field model has a larger length scale,which means larger element size and better computational efficiency.This proposed phase field model is implemented in LS-DYNA user-defined element and user-defined material and solved by the Newton-Raphson method.A tensile test shows that the first derivative of degradation function at zero phase does impact stress-strain curve.Mode I,mode II,and mixed-mode examples show the feasibility of the proposed phase field model in simulating brittle fracture.展开更多
The phase field method for fracture integrates the Griffith theory and damage mechanics approach to predict crack initiation and propagation within one framework.It replaced the discrete representation of crack by dif...The phase field method for fracture integrates the Griffith theory and damage mechanics approach to predict crack initiation and propagation within one framework.It replaced the discrete representation of crack by diffusive damage and solved it based on a minimization of the global energy storage functional.As a result,no crack tracking topology is needed,and complex crack shapes can be captures without user intervention.However,it is also reported to have an inconsistency between the predicted fracture toughness and the material strength.Recently,a novel energetic degradation function was proposed in literature to handle this issue.This research does some further modifications to the global energy storage functional so that Newton's method can be directly used to solve the energy minimization.With the new energy form,direct implementation of the length-scale independent phase field method into finite element packages like LS-DYNA becomes possible.This paper presents the framework and details of implementing the length-scale independent phase field method into LS-DYNA through a user-defined element and material subroutine.Several numerical examples are presented to compare with the experiment crack shape.Most importantly,this paper is one of the first ones to quantitatively predict accurate force response compared to experiments.These examples verify the accuracy of the new energy form and implementation.展开更多
文摘The cohesive zone model(CZM)has been used widely and successfully in fracture propagation,but some basic problems are still to be solved.In this paper,artificial compliance and discontinuous force in CZM are investigated.First,theories about the cohesive element(local coordinate system,stiffness matrix,and internal nodal force)are presented.The local coordinate system is defined to obtain local separation;the stiffness matrix for an eight-node cohesive element is derived from the calculation of strain energy;internal nodal force between the cohesive element and bulk element is obtained from the principle of virtual work.Second,the reason for artificial compliance is explained by the effective stiffnesses of zero-thickness and finite-thickness cohesive elements.Based on the effective stiffness,artificial compliance can be completely removed by adjusting the stiffness of the finite-thickness cohesive element.This conclusion is verified from 1D and 3D simulations.Third,three damage evolution methods(monotonically increasing effective separation,damage factor,and both effective separation and damage factor)are analyzed.Under constant unloading and reloading conditions,the monotonically increasing damage factor method without discontinuous force and healing effect is a better choice than the other two methods.The proposed improvements are coded in LS-DYNA user-defined material,and a drop weight tear test verifies the improvements.
文摘A constitutive model to describe the behavior of rubber from low to high strain rates is presented.For loading,the primary hyperelastic behavior is characterized by the six parameter Ogden’s strain-energy potential of the third order.The rate-dependence is captured by the nonlinear second order BKZ model using another five parameters,having two relaxation times.For unloading,a single parameter model has been presented to define Hysteresis or continuous damage,while Ogden’s two term model has been used to capture Mullin’s effect or discontinuous damage.Lastly,the Feng-Hallquist failure surface dictates the ultimate failure for element deletion.The proposed model can accurately predict the response of rubber using a limited set of experimental data.The model has been validated here for the case of rubber but can be extended to a wide range of polymers.
文摘The classical phase field model has wide applications for brittle materials,but nonlinearity and inelasticity are found in its stress-strain curve.The degradation function in the classical phase field model makes it a linear formulation of phase field and computationally attractive,but stiffness reduction happens even at low strain.In this paper,generalized polynomial degradation functions are investigated to solve this problem.The first derivative of degradation function at zero phase is added as an extra constraint,which renders higher-order polynomial degradation function and nonlinear formulation of phase field.Compared with other degradation functions(like algebraic fraction function,exponential function,and trigonometric function),this polynomial degradation function enables phase in[0,1](should still avoid the first derivative of degradation function at zero phase to be 0),so there is noconvergence problem.The good and meaningful finding is that,under the same fracture strength,the proposed phase field model has a larger length scale,which means larger element size and better computational efficiency.This proposed phase field model is implemented in LS-DYNA user-defined element and user-defined material and solved by the Newton-Raphson method.A tensile test shows that the first derivative of degradation function at zero phase does impact stress-strain curve.Mode I,mode II,and mixed-mode examples show the feasibility of the proposed phase field model in simulating brittle fracture.
文摘The phase field method for fracture integrates the Griffith theory and damage mechanics approach to predict crack initiation and propagation within one framework.It replaced the discrete representation of crack by diffusive damage and solved it based on a minimization of the global energy storage functional.As a result,no crack tracking topology is needed,and complex crack shapes can be captures without user intervention.However,it is also reported to have an inconsistency between the predicted fracture toughness and the material strength.Recently,a novel energetic degradation function was proposed in literature to handle this issue.This research does some further modifications to the global energy storage functional so that Newton's method can be directly used to solve the energy minimization.With the new energy form,direct implementation of the length-scale independent phase field method into finite element packages like LS-DYNA becomes possible.This paper presents the framework and details of implementing the length-scale independent phase field method into LS-DYNA through a user-defined element and material subroutine.Several numerical examples are presented to compare with the experiment crack shape.Most importantly,this paper is one of the first ones to quantitatively predict accurate force response compared to experiments.These examples verify the accuracy of the new energy form and implementation.
