A pseudo-elastic material damage model, part II: Application to the scaling of specimen size, material behavior and loading rate

A failure criterion is presented which relates the strain energy density of the material to both yielding and fracture. Cumulative material damage throughout a structural component may be monitored and the relative influence of yielding and stable crack growth assessed. The criterion is demonstrated, using finite element analysis, for center cracked panel specimens differing by material toughness values. From crack growth increment predictions using the uniaxial stress-strain behavior of the material, the criterion predicts the critical value of the strain energy density factor S_c governing crack instability.     

Effects of fillers on fracture performance of thermoplastics: Strain energy density criterion

This paper presents the experimental results on the fracture performance of filled thermoplastics. The emphasis is put on verification of the validity of different fracture criteria. The effects of two- and three-dimensional fillers on the fracture toughness of a representative thermoplastic, polypropylene, are analyzed. It has been found that classical fracture mechanics do not properly describe the fracture behavior of these composites. The strain energy density theory provides a more appropriate criterion for predicting fracture. On the macroscopic scale, the addition of fillers leads to a reduction in the critical strain energy density of thermoplastics. However, on the microscopic level fillers enhance a more wide spread crack-growth and failure by fracture becomes more stable. The material is therefore less prone to shatter in service. This effect of fillers is interpreted in terms of damage development, induced by the debonding at the matrix/fillers interface. A better interfacial adhesion reduces the microscopic damage and the critical increment of crack growth prior to instability. The results explain the negative effect of coupling agent on the impact resistance observed in practice.     

Fracture of piezoelectric materials: energy density criterion

In this paper, the concept of energy density factor S for piezoelectric materials is presented. In addition to the mechanical energy the electrical energy is included as well. The direction of crack initiation is assumed to occur when S_min reaches a critical value S_cr that can be used as an intrinsic materials parameter and is independent of the crack geometry and loading. The result agrees with empirical evidence qualitatively and explains rationally the effect of applied electric field on fracture strength: positive electric fields decrease the apparent fracture toughness of piezoelectric materials while negative electric fields increase it.     

Crack initiation behavior of functionally graded piezoelectric material: prediction by the strain energy density criterion

Transient response of a functionally graded piezoelectric medium is considered for a through crack under the mixed-mode in-plane mechanical and electric load. Integral transforms and dislocation density functions are employed to reduce the problem to singular integral equations. The energy density factor criterion is applied to obtain the maximum of the minimum energy density factor. This determines the direction of crack initiation. Numerical results display the effects of material constants, loading combination parameter, mechanical loading angle and material gradient parameter on the possible fracture behavior.     

A two-scale damage model with material length

The Note presents the formulation of a class of two-scale damage models involving a micro-structural length. A homogenization method based on asymptotic developments is employed to deduce the macroscopic damage equations. The damage model completely results from energy-based micro-crack propagation laws, without supplementary phenomenological assumptions.We show that the resulting two-scale model has the property of capturing micro-structural lengths. When damage evolves, the micro-structural length is given by the ratio of the surface density of energy dissipated during the micro-crack growth and the macroscopic damage energy release rate per unit volume of the material.The use of fracture criteria based on resistance curves or power laws for sub-critical growth of micro-cracks leads to quasi-brittle and, respectively, time-dependent damage models. To cite this article: C. Dascalu, C. R. Mecanique 337 (2009).     

Thermal-stress induced phenomena in two-component material： part I

The paper deals with analytical fracture mechanics to consider elastic thermal stresses acting in an isotropic multi-particle-matrix system. The multi-particle-matrix system consists of periodically distributed spherical particles in an infinite matrix. The thermal stresses originate during a cooling process as a consequence of the difference αm - αp in thermal expansion coefficients between the matrix and the particle, αm and αp, respectively. The multi-particle-matrix system thus represents a model system applicable to a real two-component material of a precipitation-matrix type. The infinite matrix is imaginarily divided into identical cubic cells. Each of the cubic cells with the dimension d contains a central spherical particle with the radius R, where d thus corresponds to inter-particle distance. The parameters R, d along with the particle volume fraction v = v（R, d） as a function of R, d represent microstructural characteristics of a twocomponent material. The thermal stresses are investigated within the cubic cell, and accordingly are functions of the microstructural characteristics. The analytical fracture mechanics includes an analytical analysis of the crack initiation and consequently the crack propagation both considered for the spherical particle （q = p） and the cell matrix （q = m）. The analytical analysis is based on the determination of the curve integral Wcq of the thermal-stress induced elastic energy density Wq. The crack initiation is represented by the determination of the critical particle radius Rqc = Rqc（V）. Formulae for Rqc are valid for any two-component mate- rial of a precipitate-matrix type. The crack propagation for R 〉 Rqc is represented by the determination of the function fq describing a shape of the crack in a plane perpendicular     

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.     

Damage in a viscoplastic material part I: Cavity growth

The exact velocity, stress and strain rate fields around a spheroidal cavity in an infinite linear viscoplastic compressible matrix are derived analytically by the ‘three function approach’. The perturbation of the velocity field due to the cavity is the superposition of three independent modes, inducing homothetic growth, pure distortion and both volume and shape changes, respectively. This solution is then used to investigate the velocity field around a spheroidal cavity in a nonlinear viscous compressible material by means of a variational principle. The behaviour of such damaged linear and nonlinear materials will be discussed in a forthcoming companion paper.The importance of the reference strain, while studying void growth in a compressible material, is emphasized. If the axial strain is chosen as a reference, void growth is found to be enhanced at low triaxiality ratios, but lowered at high triaxiality ratios in a compressible matrix relative to an incompressible one. Finally, the transition from a power law to a linear law with intercept, at increasing strain rates, is shown to reduce damage growth rate.     

Crack energy density and energy release rate for piezoelectric material

In this work, the concept of crack energy density (CED) was extended so as to be applied to piezoelectric material and its fundamental matters and properties were studied, and taking the knowledge about it into consideration, energy release rate for the material was newly derived. The definitions of CED, its mechanical and electrical contribution are given first and their path independent expressions are derived through the electromechanical energy conservation law. Subsequently, the loading path dependence of mechanical and electrical CEDs is discussed in detail. Some supplementary quantities related to CED and energy release rate are also defined and their path independent expressions are given. Energy release rate is derived through two opposite limit procedures, and the relations between energy release rate and other parameters are elicited through the discussions based on the fundamental properties of energy release rate.     

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.     

Mode-I crack projection procedure using the strain energy density criterion

A practice used in linear elastic fracture mechanics is the projection of a crack onto a plane normal to the principal tensile stress axes for computing the stress intensity factor K_I. The minimum strain-energy criterion is applied for different crack configurations with the introduction of a safety factor S_i which is the ratio of the strain energy density factor of the projected crack and that of the original crack. Numerous crack configurations are investigated to illustrate the degree of conservativeness of the crack projection procedure.     

Direction of crack growth initiation in roller contact: strain energy density criterion

Small defects or cracks near the surface of roller contact could spread and lead to failure at large. Their growth behavior depends on the rolling load, size and orientation of the initial defects, and material property in addition to friction at the contacting surfaces. Stress intensity factors K₁and K₂ are obtained for three different crack types near the surface between the roller and contacting solid. Various possible directions of crack growth initiation are obtained as the different roller loads are moved relative to the crack. The results are indicative of railway failure observed in service and are helpful to future studies on subcritical and/or critical crack growth.     

A constitutive-microdamage model to simulate hypervelocity projectile-target impact,material damage and fracture

A set of constitutive-microdamage equations are presented that can model shock compression and the microdamage and fracture that can evolve following hypervelocity impact. The equations are appropriate for polycrystalline metals. For impact at a projectile velocity of 6.0 km/s, numerical simulations are preformed that describe the impact of spherical soda-lime glass projectiles with aluminum 1100 rectangular target plates. Three ratios of the projectile diameter to the target thickness are chosen for the simulations, providing a wide range of damage features. The simulated impact damage is compared with experimental damage of corresponding test specimens, illustrating the capability of the model.     

Strain energy density bounds for linear anisotropic elastic materials

Upper and lower bounds are presented for the magnitude of the strain energy density in linear anisotropic elastic materials. One set of bounds is given in terms of the magnitude of the stress field, another in terms of the magnitude of the strain field. Explicit algebraic formulas are given for the bounds in the case of cubic, transversely isotropic, hexagonal and tetragonal symmetry. In the case of orthotropic symmetry the explicit bounds depend upon the solution of a cubic equation, and in the case of the monoclinic and triclinic symmetries, on the solution of sixth order equations.     

Reliability assessment of a bi-material notch: Strain energy density factor approach

Joints of different materials have many applications in structural engineering and microelectronics. In the present contribution the joint is modelled as a bi-material notch. The singular stress field near the notch tip is investigated. Depending on the notch geometry and materials, the stress field can have one or two singularities. It is shown that to study the problem of a crack onset at the notch, both terms have to be taken into account. Criteria for the direction and for crack nucleation are formulated. The approach utilizes the knowledge of the strain energy density factor distribution in a bi-material notch vicinity.     

The strain energy density failure criterion applied to notched elastic solids

Application of the strain energy density failure criterion is made to plane notch problems, where the crack now becomes a special case of a more generalized approach to failure. The specific case considered is that of the plane elliptical cavity under remote tension and compression. Both failure loads and fracture trajectories are discussed. It is shown that an additional characteristic dimension provides satisfactory agreement of the theory with available data. Finally, known characteristics of fracture trajectories from a notch tip are shown to be predicted for unstable fracture conditions.     

A failure criterion based on material instability

Scaling laws for adiabatic shear bands are used to parameterize a model that is suitable for introducing shear damage within engineering calculations. One-dimensional solutions to the governing equations for a single shear band provide laws that connect the driving deformation, the imperfections, and the physical characteristics of the material to the process of stress collapse [International Journal of Plasticity 8 (1992) 583, Mechanics of Materials 17 (1994) 215]. The current model uses homogeneous material response and the scaling laws to anticipate the correct timing beyond the maximum stress at which stress collapse should occur. The model is implemented into a finite element code for wave propagation and used in the analysis of boundary value problems that are dominated by shear failure. Finally, implications of the model for simulations of material failure are discussed.     

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.     

Characterization of strain-induced damage in composites based on the dissipated energy density Part III. General material constitutive relation

An approach to characterizing failure behavior and degree of load induced internal damage in composite materials and structures is formulated in Part I of this work. It is based on a systematic experimental procedure to observe the response of composite materials subjected to multiaxial load environment. The energy dissipated by internal failure mechanisms is employed as a measure of internal damage and is characterized by an energy dissipation function, which is identified by means of a deconvolution procedure using data provided by NRL's automated in-plane loader testing machine.Part II of this work will display the dissipated energy density distributions in composite specimens that are used for the in-plane loader machine and naval structures, while Part III presents a general theory that includes the derivation for the constitutive behavior of the damaged composites.