A cohesive zone model for fatigue crack growth allowing for crack retardation

A damage-based cohesive model is developed for simulating crack growth due to fatigue loading. The cohesive model follows a linear damage-dependent traction–separation relation coupled with a damage evolution equation. The rate of damage evolution is characterized by three material parameters corresponding to common features of fatigue behavior captured by the model, namely, damage accumulation, crack retardation and stress threshold. Good agreement is obtained between finite element solutions using the model and fatigue test results for an aluminum alloy under different load ratios and for the overload effect on ductile 316 L steel.     

A rate-dependent damage model for brittle materials based on the dominant crack

A rate-dependent, continuum damage model is developed for brittle materials under dynamic loading. This model improves on the approach (ISOSCM) of [Addessio, F.L., Johnson, J.N., 1990. A constitutive model for the dynamic response of brittle materials. Journal of Applied Physics 67, 3275–3286] in several respects. (1) A new damage surface is found by applying the generalized Griffith instability criterion to the dominant crack (having the most unstable orientation), rather than by averaging the instability condition over all crack orientations as done previously. The new surface removes a discontinuity in the damage surface in ISOSCM when the pressure changes sign. (2) The strain due to crack opening is more consistent with crack mechanics, with only the tensile principal stresses contributing to the crack opening strain. This is achieved by incorporating a projection operator in the equation for the crack opening strain. One consequence of incorporating the projection operator is a prediction of shear dilatancy, which is not accounted for in ISOSCM. (3) The evolution of damage, which is based on the energy-release rate for the dominant crack, has a physical basis, whereas in the previous approach the damage growth rate was assumed to be an exponential function of the distance from the stress state to the damage surface without specific physical justification.An implicit algorithm has been developed so that a larger time step can be used than with the explicit algorithm used in ISOSCM. The numerical results of a silicon carbide (SiC) ceramic under several loading paths (hydrostatic tension/compression, uniaxial strain, uniaxial stress, and shear) and strain rates are presented to illustrate the main features of the model.     

A penny-shaped cohesive crack model for material damage

A novel micromechanics based damage model is proposed to address failure mechanism of defected solids with randomly distributed penny-shaped cohesive micro-cracks (Barenblatt–Dugdale type). Energy release contribution to the material damage process is estimated in a representative volume element (RVE) under macro hydrostatic stress state. Macro-constitutive relations of RVE are derived via self-consistent homogenization scheme, and they are characterized by effective nonlinear elastic properties and a class of pressure sensitive plasticity which depends on crack opening volume fraction and Poisson’s ratio. Several distinguished features of the present model are compared with Gurson model and Gurson–Tvergaard–Needleman (GTN) model, showing that the proposed model can better capture material degradation and catastrophic failure due to cohesive micro-crack growth and coalescence.     

High strain-rate crack growth in rate-dependent plastic solids

At high crack velocities in metallic materials nearly all plastic strain accumulates at very high strain-rates, typically in the range 10³ s^?1 to 10⁵ s^?1. At these rates, dislocation motion is limited by dynamic lattice effects and the plastic strain-rate increases approximately linearly with stress. The problem for a crack growing at high velocity is posed for steady-state, small scale yielding in elastic/rate-dependent plastic solids. A general expression is derived for the near-tip stress intensity factor in terms of the remote intensity factor, or equivalently for the near-tip energy release-rate in terms of the overall release-rate. An approximate calculation of the plastic strain-rates provides this relation in analytical form. Imposition of the condition that the near-tip energy release-rate be maintained at a critical value provides a propagation equation for the growing crack. A single, nondimensional combination of material constants emerges as the controlling parameter. Implications for dynamic crack propagation are discussed.     

Subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone

A recent experimental study has demonstrated the attainability of intersonic shear crack growth along weak planes in otherwise homogeneous, isotropic, linear elastic solids subjected to remote loading conditions (Rosakis et al., Science 284 (5418) (1999) 1337). The relevant experimental observations are summarized briefly here and the conditions governing the attainment of intersonic crack speeds are examined. Motivated by experimental observations, subsonic and intersonic mode II crack propagation with a rate-dependent cohesive zone is subsequently analyzed. A cohesive law is assumed, wherein the cohesive shear traction is either a constant or varies linearly with the local sliding rate. Complete decohesion is assumed to occur when the crack tip sliding displacement reaches a material-specific critical value. Closed form expressions are obtained for the near-tip fields. With a cohesive zone of finite size, it is found that the dynamic energy release rate is finite through out the intersonic regime. Crack tip stability issues are addressed and favorable speed regimes are identified. The influence of shear strength of the crack plane and of a rate parameter on crack propagation behavior is also investigated. The isochromatic fringe patterns predicted by the analytical solution are compared with the experimental observations of Rosakis et al. (1999) and comments are made on the validity of the proposed model.     

A refined cohesive zone model that accounts for inertia of cohesive zone of a moving crack

Existing cohesive zone models assume that actual fracture zone of non-zero mass can be modeled by a line segment (cohesive zone) with no mass and inertia. In the present work, a simplified mass-spring model is presented to study inertia effect of cohesive zone on a mode-I steady-state moving crack. It is showed that fracture energy predicted by the present model increases dramatically when a finite limiting crack speed is approached. Reasonable agreement with known experiments indicates that the present model has the potential to catch the inertia effect of cohesive zone which has been ignored in existing cohesive zone models and better simulate dynamic fracture at high crack speed.     

Analysis of crack growth and crack-tip plasticity in ductile materials using cohesive zone models

Crack initiation and crack growth resistance in elastic plastic materials, dominated by crack-tip plasticity are analyzed with the crack modeled as a cohesive zone. Two different types (exponential and bilinear) of cohesive zone models (CZMs) have been used to represent the mechanical behavior of the cohesive zones. In this work, it is suggested that different forms of CZMs (e.g., exponential, bilinear) are the manifestations of different micromechanisms-based inelastic processes that participate in dissipating energy during the fracture process and each form is specific to each material system. It is postulated that the total energy release rate comprises the plastic dissipation rate in the bounding material and the separation energy rate within the fracture process zone, the latter is determined by CZMs. The total energy release rate then becomes a function of the material properties (e.g., yield strength, strain hardening exponent) and cohesive properties of the fracture process zone (e.g., cohesive strength and cohesive energy), and the form of cohesive zone model (CZM) that determines the rate of energy dissipation in the forward and wake regions of the crack. The effects of material parameters, cohesive zone parameters as well as the form/shape of CZMs in predicting the crack growth resistance and the size of plastic zone (SPZ) surrounding the crack tip are systematically examined. It is found that in addition to the cohesive strength and cohesive energy, the form (shape) of the traction–separation law of CZM plays a very critical role in determining the crack growth resistance (R-curve) of a given material. It is further observed that the shape of the CZM corresponds to inelastic processes active in the forward and wake regions of the crack, and has a profound influence on the R-curve and SPZ.     

Thermodynamic-based cohesive zone healing model for self-healing materials

This work presents a thermodynamic-based cohesive zone framework to model healing in materials that tend to self-heal. The nominal, healing and effective configurations of continuum damage-healing mechanics are extended to represent cohesive zone configurations. To incorporate healing in a cohesive zone model, the principle of virtual power is used to derive the local static/dynamic macroforce balance and the boundary traction as well as the damage and healing microforce balances. A thermodynamic framework for constitutive modeling of damage and healing mechanisms of cracks is used to derive the evolution equations for the damage and healing internal state variables. The effects of temperature, resting time, crack closure, history of healing and damage, and level of damage on the healing behavior of the cohesive zone are incorporated. The proposed model promises solid basis for understanding the self-healing phenomena in self-healing materials.     

A damage model for crack initiation and growth

A damage accumulation model is presented for the study of the problem of crack initiation and stable growth in an elastic-plastic material. A centre-cracked specimen subjected to a uniform stress perpendicular to the crack plane is considered. A coupled stress and failure analysis is performed by using a finite element computer program based on J₂-plasticity theory in conjunction with the strain energy density theory. After initial yielding, each material element follows a different equivalent uniaxial stress-strain behavior depending on the amount of energy dissipation by permanent deformation. A host of uniaxial stress-strain curves constituting parts of the same stress-strain curve were assigned to material elements for each increment of loading. The path-dependent nature of the onset of crack initiation and growth was revealed. The proposed model predicts faster crack growth rates than those obtained on the basis of a single uniaxial stress-strain curve and is closer to experimental observation.     

Inverse parameter identification of cohesive zone model for simulating mixed-mode crack propagation

Inverse analysis is widely applied to the identification of material properties or model parameters. In order to improve the computational efficiency of the inverse method based on the genetic algorithm, an interpolation scheme upon the response surface constructed by the finite element simulation has been adopted in this paper. Meanwhile, a gradual homogenization treatment scheme has also been presented to improve the convergence of the inverse method based on the Kalman filter algorithm. Both methods are proven effective in dealing with the single-objective inverse problem. However, literature studies show that the adoption of multiple types of experimental information is useful to improve the accuracy of inverse analysis. In this case, it turns into a multiple-objective inverse problem. Our practice proved that the above-mentioned two methods might not yield a proper result if the sensitivity issue of different types of information is not considered. Therefore, another multi-objective inverse method, in combination of the above two optimization algorithms and a weight-estimating scheme that can consider such sensitivity, has been further presented. Finally, by using a mixed-mode crack propagation simulation and two types of experimental information (loading-displacement response curve and crack path profile), the parameters of the cohesive zone model were inversely identified and its simulation results are in good agreement with the experiment.     

Thermal softening induced plastic instability in rate-dependent materials

A linear perturbation analysis is performed for a class of rate-dependent materials, such as the Johnson-Cook model, in which the rate contribution to the stress can be separated from that of the plastic strain and temperature and in which the temperature rises adiabatically. The analysis is facilitated by perturbing both the rate of momentum equation and the momentum equation. An identical material stability/instability criterion is deduced from the characteristic spectral equations for one-dimensional deformation, one-dimensional shearing, and general three-dimensional field equations, and thus shows that the instability derived here is a material constitutive instability.The criteria indicate that the materials become unstable once the thermal softening overcomes the strain hardening, regardless of the strain rate. The strain rate enters the criteria through its effects on the accumulated temperature and the current stress. Based on the criterion, the three-dimensional instability surface is established in the space of plastic strain, plastic strain rate, and temperature. Instability surface is shown as a material property and independent of deformation histories or modes. Both necking and shear banding are simulated to validate the excellent predictive capability of the criterion.     

Modelling of crack growth by fracture damage zone

The strain energy density (SED) criterion is applied for analyzing the full range of mixed mode fracture from tensile to shear loading. A fracture damage zone (FDZ) local to the crack tip is defined and discussed in connection with the influence of crack geometry, loading and local material property. The size of FDZ tends to change continuously from statically to cyclically applied load conditions. It can be estimated from the uniaxial mechanical properties of the material. Both experimental and analytical results are examined for subcritical crack growth under static loading that depends on the type steel structures the fracture behavior of which could be represented by a single curve for the given specimen geometry.     

Steady growth of a crack with a rate and temperature sensitive cohesive zone

The steady-state dynamic propagation of a crack in a heat conducting elastic body is numerically simulated. Specifically, a mode III semi-infinite crack with a nonlinear temperature dependent cohesive zone is assumed to be moving in an unbounded homogeneous linear thermoelastic continuum. The numerical results are obtained via a semi-analytical technique based on complex variables and integral transforms. The relation between the thermo-mechanical properties of the failure zone and the resulting crack growth regime are investigated. The results show that temperature dependent solutions are substantially different from purely mechanical ones in that their existence and stability strongly depends on the cohesive zone thermal properties.     

Pseudo plane stress mode II crack growth in plastic materials

The near tip field of mode II crack that grows in thin bodies with power hardening or perfectly plastic behavior is analyzed. It is shown that for power hardening behavior, the pseudo plane stress field possesses the logarithm singularity, i.e. σ (ln r)²/(n−1), (ln r)^{2n/(n − 1)}, where r is the distance from the crack tip, n the hardening exponent is σⁿ. When n → ∞ the solution reduced to that for the perfectly plastic case.     

A thermomechanical cohesive zone model for bridged delamination cracks

The coupled thermomechanical numerical analysis of composite laminates with bridged delamination cracks loaded by a temperature gradient is described. The numerical approach presented is based on the framework of a cohesive zone model. A traction-separation law is presented which accounts for breakdown of the micromechanisms responsible for load transfer across bridged delamination cracks. The load transfer behavior is coupled to heat conduction across the bridged delamination crack. The coupled crack-bridging model is implemented into a finite element framework as a thermomechanical cohesive zone model (CZM). The fundamental response of the thermomechanical CZM is described. Subsequently, bridged delamination cracks of fixed lengths are studied. Values of the crack tip energy release rate and of the crack heat flux are computed to characterize the loading of the structure. Specimen geometries are considered that lead to crack opening through bending deformation and buckling delamination. The influence of critical mechanical and thermal parameters of the bridging zone on the thermomechanical delamination behavior is discussed. Bridging fibers not only contribute to crack conductance, but by keeping the crack opening small they allow heat flux across the delamination crack to be sustained longer, and thereby contribute to reduced levels of thermal stresses. The micro-mechanism based cohesive zone model allows the assessment of the effectiveness of the individual mechanisms contributing to the thermomechanical crack bridging embedded into the structural analysis.     

Asymptotic fields for dynamic crack growth in non-associative pressure sensitive materials

Asymptotic near-tip fields are analyzed for a plane strain Mode I crack propagating dynamically in non-associative elastic–plastic solids of the Drucker–Prager type with an isotropic linear strain hardening response. Eigen solutions are obtained over a range of material parameters and crack speeds, based on the assumption that asymptotic solutions are variable-separable and fully continuous. A limiting speed, beyond which a tendency to slope discontinuity in angular distributions of stresses and velocities is detected, is found to deviate from the associative models. At low strain-hardening rates, the onset of the plastic potential corner zone ahead of the crack-tip imposes another limit to the crack speed. Correspondingly, those limits imply the limits to the degree of non-associativity at a given crack speed. In addition, a tendency to slope discontinuity in the angular radial stress distribution sets another limit on the non-associativity at vanishing hardening rates.     

An improved atomistic simulation based mixed-mode cohesive zone law considering non-planar crack growth

A novel and improved atomistic simulation based cohesive zone law characterizing interfacial debonding is developed which explicitly accounts for the non-planarity of the crack propagation. Group of atoms in the simulation constituting cohesive zones which are used to obtain local stress and crack opening displacement data are determined dynamically during the non-planar crack growth as they cannot be determined apriori. The methodology is used to study the debonding of Σ5 (2 1 0)/[0 0 1] symmetric tilt grain boundary interface in a Cu bicrystal under several mixed mode loading conditions. Simulations show that such bicrystalline specimen exhibits three types of energy dissipative mechanisms – shear coupled GB migration (SCM) away from the crack-tips, change in spacial orientation of GB structural units rendering highly disordered grain boundary near the crack tips and brittle intergranular fracture. Which combination of these three deformation mechanism will be active influencing the degree of non-planarity of the crack propagation at various stages of loading depends on the loading mode-mixity. As the ratio of shear component of the loading parallel to the GB plane and normal to the tilt axis with respect to the normal loading increases (thereby increasing the mode-mixity), overall strain-to-failure also increases and SCM tends to become the dominant deformation mechanism. Through this framework, analytical functional forms and parameters describing cohesive laws for both normal and shear traction as a function of the mode-mixity of the loading and crack opening displacement are predicted.     

复合材料桥纤维拔出的动态裂纹模型

在无限大正交各向异性体弹性平面上对复合材料桥纤维平行自由表面的内部中央裂纹提出了桥纤维拔出的动态裂纹模型。通过复变函数将其转化为Reimann-Hilbert混合边界值问题。求得了裂纹在坐标原点受载荷Px/t、Px2/t作用的解析解。利用这一解析解可通过迭加原理求得任意复杂问题的解。     

On plastic anisotropy of constitutive model for rate-dependent single crystal

An algorithm for single crystals was developed and implemented to simulate plastic anisotropy using a rate-dependent slip model. The proposed procedure was a slightly modified form of single crystal constitutive model of Sarma and Zacharia. Modified Euler method, together with Newton-Raphson method was used to integrate this equation which was stable and efficient. The model together with the developed algorithm was used to study three problems. First, plastic anisotropy was examined by simulating the crystal deformation in tension and plane strain compression, respectively. Secondly, the orientation effect of some material parameters in the model and applied strain rate on plastic anisotropy for single crystal also is investigated. Thirdly, the influence of loading direction on the active slip system was discussed.