Quasi-static Crack Growth Research Articles

A hybrid experimental/numerical technique to extract cohesive fracture properties for mode-I fracture of quasi-brittle materials

We propose a hybrid technique to extract cohesive fracture properties of a quasi-brittle (not exhibiting bulk plasticity) material using an inverse numerical analysis and experimentation based on the optical technique of digital image correlation (DIC). Two options for the inverse analysis were used—a shape optimization approach, and a parameter optimization for a potential-based cohesive constitutive model, the so-called PPR (Park-Paulino-Roesler) model. The unconstrained, derivative free Nelder-Mead algorithm was used for optimization in the inverse analysis. The two proposed schemes were verified for realistic cases of varying initial guesses, and different synthetic and noisy displacement field data. As proof of concept, both schemes were applied to a Polymethyl-methacrylate (PMMA) quasi-static crack growth experiment where the near tip displacement field was obtained experimentally by DIC and was used as input to the optimization schemes. The technique was successful in predicting the applied load-displacement response of a four point bend edge cracked fracture specimen.

A quasi‐static crack growth simulation based on the singular ES‐FEM

AbstractIn the edge‐based smoothed finite element method (ES‐FEM), one needs only the assumed displacement values (not the derivatives) on the boundary of the edge‐based smoothing domains to compute the stiffness matrix of the system. Adopting this important feature, a five‐node crack‐tip element is employed in this paper to produce a proper stress singularity near the crack tip based on a basic mesh of linear triangular elements that can be generated automatically for problems with complicated geometries. The singular ES‐FEM is then formulated and used to simulate the crack propagation in various settings, using a largely coarse mesh with a few layers of fine mesh near the crack tip. The results demonstrate that the singular ES‐FEM is much more accurate than X‐FEM and the existing FEM. Moreover, the excellent agreement between numerical results and the reference observations shows that the singular ES‐FEM offers an efficient and high‐quality solution for crack propagation problems. Copyright © 2011 John Wiley & Sons, Ltd.

Fractal modeling of the J–R curve and the influence of the rugged crack growth on the stable elastic–plastic fracture mechanics

In this paper, fractal geometry is used to modify the Griffith–Irwin–Orowan classical energy balance. Crack fractal geometry is introduced in the elastic–plastic fracture mechanics by means of the Eshelby–Rice J-integral and the influence of the ruggedness of the crack surface on the quasistatic crack growth is evaluated. It is shown that the rising of the J–R curve correlates to the topological ruggedness dimension of the crack surface. Results from fracture experiments are shown to be very well fitted with the proposed model, which is shown to be a unifying approach for fractal models currently used in fracture mechanics.

A two-scale approach for propagating cracks in a fluid-saturated porous material

An extension to a finite strain framework of a two-scale numerical model for propagating crack in porous material is proposed to model the fracture in intervertebral discs. In the model, a crack is described as a propagating cohesive zone by exploiting the partition-of-unity property of finite element shape functions. At the micro-scale, the flow in the cohesive crack is modelled as viscous fluid using Stokes' equations which are averaged over the cross section of the cavity. At the macro-scale, identities are derived to couple the local momentum and the mass balance to the governing equations for a saturated porous material. The resulting discrete equations are nonlinear due to the cohesive constitutive equations and the geometrically nonlinear kinematic relations. A Newton-Raphson iterative procedure is used to consistently linearise the derived system while a Crank-Nicholson scheme takes care of the time integration of the system. The derived model is used to analyse a quasi-static crack growth in confined compression under tensile loading.

Quasistatic crack growth in finite elasticity with Lipschitz data

We extend the recent existence result of Dal Maso and Lazzaroni (Ann Inst H Poincaré Anal Non Linéaire 27:257–290, 2010) for quasistatic evolutions of cracks in finite elasticity, allowing for boundary conditions and external forces with discontinuous first derivatives.

Effect of switching stresses on domain evolution during quasi-static crack growth in a ferroelastic single crystal

The present paper demonstrates the effect of switching stresses on domain evolution and fracture toughening during quasi-static crack growth in elastically isotropic ferroelastic single crystals with transversally isotropic ferroelastic strains. With a simple switching algorithm and crack propagation procedure, domain evolution is simulated in an exemplary material with semi-infinite crack under mode I loading, starting from a mono-domain configuration. Domain reorientation is found to be strongly affected by switching stresses, which therefore have to be considered in the context of domain evolution modelling and fracture toughening. Before the onset of crack growth a needle-like domain is formed at the tip of the stationary crack, but this does not effect the crack tip stress intensity factor. Elongation of this domain during the onset of crack growth causes a large increase of the fracture toughness. Domain separation in a later stage results in toughness reduction. The subsequent domain evolution indicates a periodic formation of needle-like domains as observed in soft ferroelastic materials.

Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [G. Dal Maso, C. Zanini: Quasi-static crack growth for a cohesive zone model with prescribed crack path. Proc. Roy. Soc. Edinburgh Sect. A, 137A (2007), 253–279.] is recovered. In this case, the convergence of the discrete time approximations is improved.

Quasistatic crack growth in finite elasticity with non-interpenetration

We present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking into account the non-interpenetration condition.

The asymptotic field of mode I quasi-static crack growth on the interface between a rigid and a pressure-sensitive material

A mechanical model of the quasi-static interface of a mode I crack between a rigid and a pressure-sensitive viscoelastic material was established to investigate the mechanical characteristic of ship-building engineering bi-materials. In the stable growth stage, stress and strain have the same singularity, ie (σ, e) ∞ r−1/(n−1). The variable-separable asymptotic solutions of stress and strain at the crack tip were obtained by adopting Airy’s stress function and the numerical results of stress and strain in the crack-tip field were obtained by the shooting method. The results showed that the near-tip fields are mainly governed by the power-hardening exponent n and the Poisson ratio ν of the pressure-sensitive material. The fracture criterion of mode I quasi-static crack growth in pressure-sensitive materials, according to the asymptotic analyses of the crack-tip field, can be viewed from the perspective of strain.

SGBEM modeling of fatigue crack growth in particulate composites

The symmetric-Galerkin boundary element method (SGBEM) has previously been employed to model 2-D crack growth in particulate composites under quasi-static loading conditions. In this paper, an initial attempt is made in extending the simulation technique to analyze the interaction between a growing crack and clusters of perfectly bonded particles in a brittle matrix under cyclic loading conditions. To this end, linear elastic fracture mechanics and no hysteresis are assumed. Of particular interest is the role clusters of inclusions play on the fatigue life of particulate composites. The simulations employ a fatigue crack growth prediction tool based upon the SGBEM for multiregions, a modified quarter-point crack-tip element, the displacement correlation technique for evaluating stress intensity factors, a Paris law for fatigue crack growth rates, and the maximum principal stress criterion for crack-growth direction. The numerical results suggest that this fatigue crack growth prediction tool is as robust as the quasi-static crack growth prediction tool previously developed. The simulations also show a complex interplay between a propagating crack and an inclusion cluster of different densities when it comes to predicting the fatigue life of particulate composites with various volume fractions.

Frequency Domain Structural Synthesis Applied to Quasi-Static Crack Growth Modeling

Quasi-static crack growth in a composite beam was modeled using the structural synthesis technique along with a finite element model. The considered crack was an interface crack in the shear mode (i.e. mode II), which occurs frequently in the scarf joint of composite structures. The analysis model was a composite beam with an edge crack at the midplane of the beam subjected to a three-point bending load. In the finite element model, beam finite elements with translational degrees of freedom only were used to model the crack conveniently. Then, frequency domain structural synthesis (substructure coupling) was applied to reduce the computational time associated with a repeated finite element calculation with crack growth. The quasi-static interface crack growth in a composite beam was predicted using the developed computational technique, and its result was compared to experimental data. The computational and experimental results agree well. In addition, the substructure-based synthesis technique showed the significantly improved computational efficiency when compared to the conventional full analysis.

Frequency Domain Structural Synthesis Applied to Quasi-Static Crack Growth Modeling

Quasi-static crack growth in a composite beam was modeled using the structural synthesis technique along with a finite element model. The considered crack was an interface crack in the shear mode (i.e. mode II), which occurs frequently in the scarf joint of composite structures. The analysis model was a composite beam with an edge crack at the midplane of the beam subjected to a three-point bending load. In the finite element model, beam finite elements with translational degrees of freedom only were used to model the crack conveniently. Then, frequency domain structural synthesis (substructure coupling) was applied to reduce the computational time associated with a repeated finite element calculation with crack growth. The quasi-static interface crack growth in a composite beam was predicted using the developed computational technique, and its result was compared to experimental data. The computational and experimental results agree well. In addition, the substructure-based synthesis technique showed the significantly improved computational efficiency when compared to the conventional full analysis.

Ductile–brittle transitions in the fracture of plastically-deforming, adhesively-bonded structures. Part I: Experimental studies

Rate effects for adhesively-bonded joints in steel sheets failing by mode-I fracture and plastic deformation were examined. Three types of test geometries were used to provide a range of crack velocities between 0.1 and 5000 mm/s: a DCB geometry under displacement control, a wedge geometry under displacement control, and a wedge geometry loaded under impact conditions. Two fracture modes were observed: quasi-static crack growth and dynamic crack growth. The quasi-static crack growth was associated with a toughened mode of failure; the dynamic crack growth was associated with a more brittle mode of failure. The experiments indicated that the fracture parameters for the quasi-static crack growth were rate independent, and that quasi-static crack growth could occur even at the highest crack velocities. Effects of rate appeared to be limited to the ease with which a transition to dynamic fracture could be triggered. This transition appeared to be stochastic in nature, it did not appear to be associated with the attainment of any critical value for crack velocity or loading rate. While the mode-I quasi-static fracture behavior appeared to be rate independent, an increase in the tendency for dynamic fracture to be triggered as the crack velocity increased did have the effect of decreasing the average energy dissipated during fracture at higher loading rates.

Cracks in Complex Bodies: Covariance of Tip Balances

In complex bodies, actions due to substructural changes alter (in some cases drastically) the force driving the tip of macroscopic cracks in quasi-static and dynamic growth, and must be represented directly. Here it is proven that tip balances of standard and substructural interactions are covariant. In fact, the former balance follows from the Lagrangian density’s requirement of invariance with respect to the action of the group of diffeomorphisms of the ambient space to itself, the latter balance accrues from an analogous invariance with respect to the action of a Lie group over the manifold of substructural shapes. The evolution equation of the crack tip can be obtained by exploiting invariance with respect to relabeling the material elements in the reference place. The analysis is developed by first focusing on general complex bodies that admit metastable states with substructural dissipation of viscous-like type inside each material element. Then we account for gradient dissipative effects that induce nonconservative stresses; the covariance of tip balances in simple bodies follows as a corollary. When body actions and boundary data of Dirichlet type are absent, the standard variational description of quasi-static crack growth is simply extended to the case of complex materials.

A plasticity framework for (linear elastic) fracture mechanics

An incremental description of (linear elastic) fracture mechanics is presented which shows a perfect analogy with the plasticity theory. After comparing (in terms of angle of propagation, of safe-equilibrium domains, and of energy dissipation) several criteria for crack propagation, the formulation of a generic criterion stemming from the associated plasticity theory is presented and its feature discussed. The analogy between plasticity and quasi-static crack growth leads also to a new algorithm for crack propagation for an arbitrary number of cracks in multi-connected materials, which is driven by the increment of external actions. Stability of crack path under mode I loading is finally analyzed.

A higher order scheme for two-dimensional quasi-static crack growth simulations

An efficient scheme for the simulation of quasi-static crack growth in two-dimensional linearly elastic isotropic specimens is presented. The crack growth is simulated in a stepwise manner where an extension to the already existing crack is added in each step. In a local coordinate system each such extension is represented as a polynomial of some, user specified, degree, n. The coefficients of the polynomial describing an extension are found by requiring that the mode II stress intensity factor is equal to zero at certain points of the extension. If a crack grows from a pre-existing crack so that a kink develops, the leading term describing the crack shape close to the kink will, in a local coordinate system, be proportional to x 3/2. We therefore allow the crack extensions to contain such a term in addition to the monomial terms. The discontinuity in the crack growth direction at a kink, the kink angle, is determined by requiring that the mode II stress intensity factor should be equal to zero for an infinitesimal extension of the existing crack. To implement the scheme, accurate values of the stress intensity factors and T-stress are needed in each step of the simulation. These fracture parameters are computed using a previously developed integral equation of the second kind.

A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.

Subcritical stable growth of a penny-shaped crack in an aging viscoelastic body with cylindrical anisotropy

The paper studies the subcritical stable growth of a penny-shaped mode I crack in an aging linear viscoelastic body with cylindrical anisotropy under tensile loads applied at infinity. The equations of quasistatic crack growth are derived using the Volterra principle for a modified Dugdale model and the critical crack opening displacement as a failure criterion. The kernel of the operator that describes the viscoelastic crack opening is found using operator-valued continued fractions. As an example, the subcritical stable growth of a crack in a specific material is studied numerically

Quasi-static crack growth for a cohesive zone model with prescribed crack path

In this paper we study the quasi-static crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes.

Stress Field Evaluation for a Propagating Crack in a Thin Glass Plate

An instantaneous phase-stepping technique using a CCD camera equipped with a pixelated micro-retarder array is applied to the study of quasi-static crack growth in a quenched thin glass plate. The distributions of the principal direction as well as the principal stress diference around a growing crack tip are obtained by instantaneous phase-stepping photoelasticity. Then, not only the mixed-mode stress intensity factors but also the T-stress are evaluated from the distribution of the principal stress difference, and they are validated using the reconstructed phase maps. By applying the proposed method to the sucsessive images, the time-variations of the fracture mechanics parameters are evaluated quantitatively. Results show that the proposed instantaneous phase-stepping technique is effective for the study of the crack growth behavior in a thin glass plate