Damage Zone Model Research Articles

Softening of concrete in compression — Localization and size effects

This paper presents a constitutive macromechanical model for concrete in compression. The model, named the Compressive Damage Zone (CDZ) model, is based on the hypothesis of compressive failure within a zone of limited length. In this damage zone, the failure mode is a combination of distributed axíal splitting and localized deformation. Within the damage zone the post-peak behaviour is described by means of two curves, one related to the distributed splitting cracking and one to the localized deformation. The compressive strain caused by the axial splitting cracks is assumed to be proportional to the tensile fracture energy G F. It is shown through compressive tests of specimens of different length and width that the proposed failure hypothesis is realistic.

Evaluation of elastic-plastic fracture in aluminum panels

An experimental and numerical study of failure in 6061-T6 aluminum panels was completed. The experimental effort included generation of G R curves and collection of data from static fracture tests on thin aluminum panels with cracks and notches. The numerical effort centered on predicting the crack opening displacement as a function of applied load, predicting applied load at failure, and predicting the nature of failure: either yield or fracture. A finite element model, referred to in this paper as the damage zone model (DZM), incorporates a cohesive stress zone adjacent to a crack or notch. The predicted failure loads using the DZM were within 9.0% of experimental values for yield failure. Predicted failure loads were within 11.0% of experimental failure loads when stable crack growth preceded unstable fracture. In all cases, the DZM predicted the nature of the final failure.

Ductile crack analysis by a cohesive damage zone model

A cohesive damage zone model of the Dugdale-Barenblatt type is presented to explore the effects of microscopic damage on a macroscopic crack in ductile materials. A semi-infinite macrocrack in a power-law plastic material under antiplane shear (mode III) is considered. The microscopic damage is assumed to occur in the form of dispersed microcracks which are localized into a narrow strip directly ahead of a macrocrack tip. By using the analytical result of the J-integral for a periodic array of collincar cracks in a power-law solid and by applying a homogenization technique, the effective mechanical property of the damage zone is obtained. Then, the effect of damage on the fracture initiation toughness is analyzed. Under the small-scale damage condition and by the use of an approximate and constant average shear stress within the damage zone, the size of the damage zone at the initiation of macrocrack growth is estimated. For small-scale damage and small-scale yielding, a nonlinear first order differential equation for the crack resistance curve is derived, where the hypothesis of constant crack tip opening angle (CTOA) during the stable crack growth is adopted. Numerical results are presented and discussed to reveal the effects of microscopic damage on the resistance and tearing modulus of a macrocrack.

Mixed-Mode Fracture in Plastically Deforming Materials

A finite element method was developed for evaluating an angled crack in a thin plate. In the spirit of the Dugdale and Barenbaltt damage zone models, the method superposes cohesive stresses in a damage zone onto a linear elastic fracture analysis. The method uses incremental changes in the boundary conditions along the damage zone to model loss in stiffness due to yielding and stable crack growth. A crack tip opening displacement criterion is used for predicting stable crack growth and unstable crack growth. The resulting nonlinear analysis is used to predict the applied loads at failure. Numerical results compared favorably with experimental values.

Strength of carbon/epoxy laminates with countersunk hole

The aim of this study is to predict the static strength of carbon/epoxy laminates with countersunk hole. Also, three-point bend (TPB) specimens with the same lay-up were analysed. For this purpose, the notched strength of the laminates was analysed by a damage zone model (DZM), where damage around the notch is represented by an ‘equivalent crack’ with cohesive forces acting between the crack surfaces. The DZM requiring only basic properties of the laminate such as unnotched tensile strength, δ 0 , fracture energy, G c ∗ , and stiffnesses of the laminate. However, the complex geometry around the countersunk hole implies that both δ 0 and G c ∗ will vary in this area, and in order to avoid this problem an approximate geometry of the countersunk hole is used in the DZM-calculations. With this approximation, good agreement between experimental and calculated strength was observed for the laminates with countersunk hole. This was also the case for the TPB specimens.

Notched strength of long- and short-fibre reinforced polyamide

A test programme has been conducted to evaluate the tensile strength of injection moulded plates with machined and moulded-in notches. The long-fibre compound, Verton, is consistently less notch sensitive than the short-fibre compound, Maranyl. Moulded-in holes appear to have several advantages over machined holes: they yield up to 25% higher strength and less sensitivity to plate thickness and injection speed. Five existing laminate fracture models are applied to the machine-notched plates; the average stress criterion and the damage zone model give the most accurate predictions.

Effects of filler particles on abrasive wear of elastomer-based composites

The effect of filler particulates on the abrasive wear of rubbers was analyzed in a model composite of silicone rubbers filled with glass spheres or glass rods. Depending on the interactions between filler particles and hard abrasive asperities, remarkably different wear features were observed for different sizes of the same filler. For large fillers, where the load-supporting action is provided by the fillers, the wear of the individual filler particle is important during the abrasion of the filled elastomer; as the filler fraction increases, the abrasion is dominated by the wear of the filler particles. When the fillers are comparable in size with the asperity spacing, a mechanism of filler “pull-out” is observed which triggers an accelerated wear rate. For small fillers, where no load-supporting action is offered by the particles, the damage zone model can be used to describe the abrasion. On increasing the aspect ratio of the particles, from spheres to short fibers, the critical filler fraction decreases significantly. The wear resistance of filled rubbers can be improved dramatically by increasing the interface adhesion between the glass fillers and the rubber matrix.

In-plane shear failure analysis of notched composites

An in-plane shear failure analysis capability has been included in the Damage Zone Model (DZM), assuming a linearly softening shear failure behavior. Since in-plane shear tests do not result in a constant shear stress distribution throughout the net section of a specimen, a new approach was developed to determine the DZM parameters. This approach involves testing specimens with two different notch geometries. DZM failure analyses, assuming a net section shear failure, were performed on double V-notched T300/914C graphite/epoxy [0 α/90 β/±45 γ] laminates. Small and large coupon specimens, with sharp and constant-radius notch roots, were tested with an Iosipescu and a rigid double-rail shear testing fixture, respectively. The chosen notch geometries did not result in explicit DZM shear properties. Only approximate values are established for each laminate. Furthermore, X-ray radiographs revealed that failure did not always occur across the specimen net section. Therefore, the determined DZM shear strength and fracture energy properties represent only minimum values. Nevertheless, strength predictions based on the DZM properties averaged from all test geometries were within ±15% of the experimental results.

Failure analysis of a shear loaded graphite/epoxy beam containing an irregular cutout

Strength analysis with the Damage Zone Model (DZM) and the Point Stress Criterion (PSC) on a T300/914C graphite/epoxy beam under pure shear loading are compared to experimental results. The beam represents a thick walled wing spar section. The spar web consists of a [±45/(±45/90 2) 2/±45 2/0 2] s laminate (longitudinal) lay-up and contains an off-center semisymmetric cutout. The tested beams experienced both tensile and some compressive failure extending from each end of the cutout at a +45- and −45-degree angle, respectively. No explicit critical failure mode could be determined, although tensile damage was more extensive. Based on the assumption that tensile failure at a +45-degree angle was the dominant failure mode, strength predictions underestimated the shear strength of the beam. DZM by −5.9% and PSC by −18.4%. In contrast to PSC, DZM is a progressive failure model which, as for PSC, requires only one finite element computation. This is enabled through the simplified manner in which DZM accounts for the formation and growth of damage. The DZM failure analysis predicts an almost simultaneous tensile failure progressing from opposite sides of the cutout.

Notch sensitivity of thermoset and thermoplastic laminates loaded in tension

Inplane tensile fracture of unnotched and notched thermoset graphite-epoxy and thermoplastic graphite-PEEK composite laminates is examined. Both fibre-dominated quasi-isotropic and matrix dominated ±45 angle-ply layups were investigated. Classical lamination theory predictions of elastic and strength properties of unnotched specimens are compared with experiments. Several notched geometries, i.e. centre-notched, double-edge notched and open-hole specimens subjected to tensile loading to fracture were examined. The notched strength of the quasi-isotropic laminates was analysed by a damage zone model, where damage around the notch is represented by an “equivalent crack” with cohesive force acting between the crack surfaces. Good agreement between experimental and calculated strength was observed for the graphite-epoxy laminates which failed in a collinear manner. For the graphite-PEEK laminates discrepancies between predicted and experimental strength are related to observed deviations from collinear crack growth. The angle-ply graphite-PEEK laminates showed larger notch sensitivity than the corresponding graphite-epoxy, probably due to less degree of stress relieving damage formation around the notch.

PVC sandwich core materials: Mode I fracture toughness

The fracture of a cracked PVC sandwich core material has been investigated. Mode I fracture toughness is evaluated using linear elastic fracture mechanics (LEFM), nonlinear fracture mechanics and a damage zone model. For the crack sizes and structural dimensions relevant to sandwich constructions, LEFM is not applicable according to the ASTM standard requirements for metals. However, the results show that the deviation between nonlinear analyses and LEFM is relatively small. Therefore, K Ic and G Ic can still be used for engineering purposes. It is also found that the fracture toughness depends on the average cell size.

On the sensitivity of the damage zone model for linearly softening materials

The fracture model for softening materials called the Damage Zone Model (DZM), is found to be basically insensitive to small errors in the constituents. However, the constituent G ∗ c (the apparent fracture energy) is deduced from an experimentally determined notched strength by using DZM. This may cause large errors in G ∗ c if the notched strength is measured from a specimen with a ratio of crack length to material ductility below a critical value. A test procedure to ensure limited errors in G ∗ c is proposed based on numerical results.

Notch sensitivity of linearly softening materials exhibiting tensile fracture

Notch sensitivity diagrams are numerically derived for tensile specimens of linearly softening materials, exhibiting tensile fracture. The notch types are center crack, single and double edge notch, and center hole. Computations are based on Finite Element modeling of the specimen geometry and the Damage Zone Model (DZM) for the linear softening fracture process and static strength prediction. The diagrams are presented in a non-dimensional format, representing failures ranging from linear elastic fracture to net section failure at the ultimate strength level. Furthermore, the diagrams reveal the effect of the notch and specimen size, and the stiffness, fracture energy and strength of the material.

Wedge notch sensitivity of linearly softening materials

The strength of double edged wedge notched tensile specimens of linear elastic linearly softening materials is examined numerically using the Damage Zone Model (DMZ). Failure assessment diagrams are given revealing the dependence of the notched strength on the material strength and ductility, notch size, specimen width and the wedge angle.

Fracture analysis of center cracked infinite plates of linearly softening materials

A crack in an infinite plate is described as a mixed boundary value problem based on the Damage Zone Model. The boundary value problem is treated by means of Fourier transformation. The resulting integral equations are solved numerically to give stresses and displacements. It is shown that the maximum applied stress before failure can be found by an eigenvalue calculation. The dependence of the maximum applied stress on the crack length and the material parameters is given.

Tensile Fracture of Laminates with Cracks

In this paper a method, called the Damage Zone Model (DZM), is used for predicting strength of composites with through-the-thickness cracks. The DZM is based on the two fundamental parameters unnotched tensile strength (σ0) and apparent fracture energy (G*c). The damage zone, developed at a notch in the composite, is modelled as a crack with cohesive forces acting on the crack surfaces. Redistribution of stresses and change in stiffness is accounted for in the model. For comparison, strengths are also calculated by semi-empirical methods such as the inherent flaw and the point stress criteria. Experimental results for three point bend (TPB), single edge notch (SEN) and compact tension (CT) quasi-isotropic carbon/epoxy specimens are presented. Some results for specimens made from randomly oriented short glass fiber/polyester specimens are also discussed. The damage zone model is shown to accurately predict fracture load, load- deformation behaviour and damage zone sizes in these types of laminates.

Sensitivity analysis of the damage zone model

In this paper a sensitivity study of the Damage Zone Model (DZM) is presented. In the DZM the complex failure state developed in the damage zone at a notch in a composite laminate is modelled as a crack with cohesive forces acting on the crack surfaces. The behaviour of the cohesive forces is assumed to follow an a priori-determined stress-crack opening curve, which is based on the two fundamental parameters: unnotched tensile strength ( σ 0) and apparent fracture energy ( G ∗ c ). The sensitivity in predicted notched strength of specimens with a circular hole, a rectangular hole and cracks is investigated for variations in σ 0 and G ∗ c . Furthermore, an assessment of required finiteelement meshes to obtain convergence is made.

Stacking sequence effects on fracture of notched carbon fibre/epoxy composites

The purpose of this study was to investigate the ability of the so-called damage zone model (DZM) to predict the influence of stacking sequence on the strength of notched carbon fibre/epoxy composites. The DZM is in essence based on the unnotched tensile strength, σ 0 , and the apparent fracture energy, G c ∗ , and the damage zone is modelled as a crack with cohesive forces acting on the crack surfaces. The DZM predicts fracture loads for three-point bend (TPB) specimens and specimens with circular holes quite accurately. As an attempt to explain the difference in strengths, the damage zone extension in the TPB specimens with different stacking sequence was examined.