Application of the strain energy density criterion to ductile fracture

The strain energy density criterion due to Sih is used to predict fracture loads of two thin plates subjected to large elastic-plastic deformation. The prediction is achieved with a finite element analysis which is based on Hill's variational principle for incremental deformations capable of solving gross yielding problems involving arbitrary amounts of deformation. The computed results are in excellent agreement with those obtained in Sih's earlier analysis and with an experimental observation.     

On a 2D hydro-mechanical lattice approach for modelling hydraulic fracture

A 2D lattice approach to describe hydraulic fracturing is presented. The interaction of fluid pressure and mechanical response is described by Biot's theory. The lattice model is applied to the analysis of a thick-walled cylinder, for which an analytical solution for the elastic response is derived. The numerical results obtained with the lattice model agree well with the analytical solution. Furthermore, the coupled lattice approach is applied to the fracture analysis of the thick-walled cylinder. It is shown that the proposed lattice approach provides results that are independent of the mesh size. Moreover, a strong geometrical size effect on nominal strength is observed which lies between analytically derived lower and upper bounds. This size effect decreases with increasing Biot's coefficient.     

A phase-field modeling approach of hydraulic fracture in saturated porous media

Continuum porous media theories, extended by a diffusive phase-field modeling (PFM) approach, introduce a convenient and efficient tool to the simulation of hydraulic fracture in fluid-saturated heterogeneous materials. In this, hydraulic- or tension-induced fracture occurs in the solid phase. This leads to permanent local changes in the permeability, the volume fractions of the constituents as well as the interstitial-fluid flow. In this work, the mechanical behaviors of the multi-field, multi-phase problem of saturated porous media, such as the pore-fluid flow and the solid-skeleton deformation, are described using the macroscopic Theory of Porous Media (TPM). To account for crack nucleation and propagation in the sense of brittle fracture, the energy-minimization-based PFM procedure is applied, which approximates the sharp edges of the crack by a diffusive transition zone using an auxiliary phase-field variable. Furthermore, the PFM can be implemented in usual continuum finite element packages, allowing for a robust solution of initial-boundary-value problems (IBVP). For the purpose of validation and comparison, simulations of a two-dimensional IBVP of hydraulic fracture are introduced at the end of this research paper.     

A variational approach to the fracture of brittle thin films subject to out-of-plane loading

We address the problem of fracture in homogenous linear elastic thin films using a variational model. We restrict our attention to quasi-static problems assuming that kinetic effects are minimal. We focus on out-of-plane displacement of the film and investigate the effect of bending on fracture. Our analysis is based on a two-dimensional model where the thickness of the film does not need to be resolved. We derive this model through a formal asymptotic analysis. We present numerical simulations in a highly idealized setting for the purpose of verification, as well as more realistic micro-indentation experiments.     

The strain energy density ratio criterion for predicting cracking direction in composite materials

The strain energy density ratio criterion for predicting cracking direction incomposite materials is proposed.The Tsai-Hill criterion and Norris criterion ofcomposite materials are extended to predict the cracking direction in composites.Thethree criteria are used to analyse the crack propagation problem of the unidirectionalfibre composite sheet with various fibre directions.The predicted results are comparedwith those of the existing normal stress ratio criterion and strain energy densitycriterion.     

Simulation of proppant transport at intersection of hydraulic fracture and natural fracture of wellbores using CFD-DEM

Proppants transport is an advanced technique to improve the hydraulic fracture phenomenon, in order to promote the versatility of gas/oil reservoirs. A numerical simulation of proppants transport at both hydraulic fracture (HF) and natural fracture (NF) intersection is performed to provide a better understanding of key factors which cause, or contribute to proppants transport in HF–NF intersection. Computational fluid dynamics (CFD) in association with discrete element method (DEM) is used to model the complex interactions between proppant particles, host fluid medium and fractured walls. The effect of non-spherical geometry of particles is considered in this model, using the multi-sphere method. All interaction forces between fluid flow and particles are considered in the computational model. Moreover, the interactions of particle–particle and particle–wall are taken into account via Hertz–Mindlin model. The results of the CFD-DEM simulations are compared to the experimental data. It is found that the CFD-DEM simulation is capable of predicting proppant transport and deposition quality at intersections which are in agreement with experimental data. The results indicate that the HF–NF intersection type, fluid velocity and NF aperture affect the quality of blockage occurrence, presenting a new index, called the blockage coefficient which indicates the severity of the blockage.     

Use of the energy criterion of fracture to determine the shape of a slightly curved crack

An asymptotic formula is obtained for the total-energy increment during quasistatic growth of a semiin finite crack in an anisotropic elastic plane under complex loading. It is assumed that the shear loads are much larger than the tearing loads. The shape of the slightly curved crack was determined using the Griffith criterion in two versions: global and local. It is shown, in particular, that the first version leads to an improbable result. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 119–130, September–October, 2006.     

A review of the volume-based strain energy density approach applied to V-notches and welded structures

A large bulk of experimental data from static tests of sharp and blunt V-notches and from fatigue tests of welded joints are presented in an unified way by using the mean value of the Strain Energy Density (SED) over a given finite-size volume surrounding the highly stressed regions. When the notch is blunt, the control area assumes a crescent shape and R₀ is its width as measured along the notch bisector line. In plane problems, when cracks or pointed V-notches are considered, the volume becomes a circle or a circular sector, respectively. The radius R₀ depends on material fracture toughness, ultimate tensile strength and Poisson’s ratio in the case of static loads; it depends on the fatigue strength Δσ_A of the butt ground welded joints and the Notch Stress Intensity Factor (NSIF) range ΔK₁ in the case of welded joints under high cycle fatigue loading (with Δσ_A and ΔK₁ valid for 5 × 10⁶ cycles).Dealing with welded joints characterised by a plate thickness greater than 6 mm, the final synthesis based on SED summarises nine hundred data taken from the literature while a new synthesis from spot-welded joints under tension and shear loading, characterised by a limited thickness of the main plate, is presented here for the first time (more than two hundred data).Dealing with static tests, about one thousand experimental data as taken from the recent literature are involved in the synthesis. The strong variability of the non-dimensional radius R/R₀, ranging from about zero to about 1000, makes the check of the approach based on the mean value of the SED severe.     

A natural neighbour method based on Fraeijs de Veubeke variational principle for materially non-linear problems

Abstract The natural neighbour method can be considered as one of many variants of the meshless methods. In the present paper, a new approach based on the Fraeijs de Veubeke （FdV） functional, which is initially developed for linear elasticity, is extended to the case of geometrically linear but materially non-linear solids. The new approach provides an original treatment to two classical problems： the numerical evaluation of the integrals over the domain A and the enforcement of boundary conditions of the type ui = hi on Su. In the absence of body forces （Fi = 0）, it will be shown that the calculation of integrals of the type fA .dA can be avoided and that boundary conditions of the type ui = hi on Su can be imposed in the average sense in general and exactly if hi is linear between two contour nodes, which is obviously the case for tTi = O.     

Dual-parameter elastic-plastic fracture criterion of metals and alloys in the plane strain case

In recent years, some investigators discussed the applicability of the HRR theory for engineering materials based on the results of numerical analyses and experimental studies. In the present paper, the finite element method is employed to analyze the crack tip fields of the engineering elastic-plastic material with a variety of geometry configurations of cracked specimens from elastic state to intensely general yielded state in the plane strain case. The results indicate that the HRR theory loses its validity of application for engineering elastic-plastic materials in the plane strain case. The reasons for this are analyzed. A dual-parameter fracture criterion is suggested for this case.     

Fracture behavior of a nickel-base single crystal superalloy as predicted by the strain energy density criterion

Results from compact tension specimens and scanning electronic microscopy are used to study the fracture characteristics of nickel-base single crystal superalloy DD3. Finite element analyses are also made to better understand the outcome. The experiments showed that the initiation and growth of cracks were dependent on the crystallographic orientation of specimens, and the fracture surfaces coincided with the slip planes. Three-dimensional stresses near the crack tip are obtained. The strain energy density criterion is applied to predict the fracture of the superalloy. Agreement with experiments is good.     

Magnetoelastic formulation of soft ferromagnetic elastic problems with collinear cracks: energy density fracture criterion

Magnetoelasticity is applied to solve collinear crack problems for soft ferromagnetic materials in two dimension. Complex functions are used for reducing the problem to the solution of a system of singular integral equations. The energy density factors are derived for determining how an off-axis magnetic field without mechanical load would influence the direction of crack initiation. The critical conditions are also determined for the case when both magnetic and mechanical load are present.     

Gas-condensate flow in the vicinity of a hydraulic fracture

The problem of gas-condensate flow in the vicinity of a production well with a hydraulic fracture is considered. In the matrix, the flow is assumed to be three-dimensional, and at the fracture, it is assumed to be two-dimensional. It is shown that, for steady-state flow, the problem is split into a physicochemical problem (of phase transitions) and a filtration problem (of determining the pressure field). Numerical solutions are constructed for a rectangular fracture with finite and infinite conductivities. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 128–136, May–June, 2008.     

The brittle fracture criterion based on the maximum tensile stress on the surface of blunt crack tip

The brittle fracture criterion is developed for a blunted crack. The curvature radius of the blunt crack tip is suggested as a characteristic length for brittle materials, and then the fracture toughness of the brittle materials can be determined from the cohesion strength and the characteristic length of the materials.     

Prediction of mixed-mode fracture based on an elasto–plastic energy balance criterion

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. This fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, however, the G _max criterion has been modified in order to accommodate the consideration of plastic strain energy. This modified criterion is extended to study the fatigue crack growth characteristics of mixed-mode cracks. To predict crack propagation due to fatigue loads, a new elasto–plastic energy model is presented. This new model includes the effects of material properties such as strain hardening exponent n, yield strength σ _y , and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

Three-dimensional ductile fracture analysis with a hybrid multiresolution approach and microtomography

A modified-JIC test on CT (compact tension) specimens of an alloy (Ti-Modified 4330 steel) was carried out. The microstructure (primary and secondary inclusions) in the fracture process zone and fracture surface are reconstructed with a microtomography technique. The zig-zag fracture profile resulting from nucleation of microvoid sheets at the secondary population of inclusions is observed. Embedding the experimentally reconstructed microstructure into the fracture process zone, the ductile fracture process occurring at different length scales within the microstructure is modeled by a hybrid multiresolution approach. In combination with the large scale simulation, detailed studies and statistical analysis show that shearing of microvoids (the secondary population of voids) determines the mixed mode zig-zag fracture profile. The deformation in the macro and micro zones along with the interaction between them affects the fracture process. The observed zig-zag fracture profile in the experiment is also reasonably captured. Simulations can provide a more detailed understanding of the mechanics of the fracture process than experiments which is beneficial in microstructure design to improve performance of alloys.     

Damage growth by debonding in a single fibre metal matrix composite: Elastoplasticity and strain energy density criterion

A displacement-based finite element-based numerical approach has been employed to study the damage growth in a unidirectional SiC/Al composite containing a pre-existing crack along the fibre/matrix interface. The composite is modeled as a two-material cylinder subjected to uniform displacement. A detailed analysis is made for the stress field in the vicinity of the debond crack tip. This approach incorporates an elastic-plastic analysis combined with a strain energy density criterion to predict debonded crack growth direction, extended stable growth and final termination. The influence of contact taking place between the debonded surfaces is also considered. It is shown that such surface contact leads to reduced stress and strain fields around the crack tip, while the extent of reduction is increased with debonding length. By combining the reduced stress field with the strain energy density criterion, a limiting value for the debonding extension can be calculated for the critical applied displacement that led to fibre fracture.     

A simple variational approach to a converse of the Lagrange-Dirichlet theorem

Communicated by K. Kirchgässner     

An indentation technique for estimating the energy density as fracture toughness with Berkovich indenter for ductile bulk materials

A technique is proposed to estimate the energy density as fracture toughness for ductile bulk materials with an indentation system equipped with a Berkovich indenter based on the theory of plastic deformation energy transforming into the indentation energy of fracture. With progressive increase of penetration loads, the material damage is exhibited on the effective elastic modulus. A quadratic polynomial relationship between the plastic penetration depth and penetration load, and an approximate linear relationship between logarithmic plastic penetration depth and logarithmic effective elastic modulus are exhibited by indentation investigation with Berkovich indenter. The parameter of damage variable is proposed to determine the critical effective elastic modulus at the fracture point. And the strain energy density factor is calculated according to the equations of penetration load, plastic penetration depth and effective elastic modulus. The fracture toughness of aluminum alloy and stainless steel are evaluated by both indentation tests and K_IC fracture toughness tests. The predicted S_cr values of indentation tests are in good agreement with experimental results of CT tests.