Micromechanical consideration of tensile crack behavior based on virtual internal bond in contrast to cohesive stress

A modified version of the virtual internal bond model (VIB) is presented. This involves the introduction of a R-bond restricting the relative rotation freedom of pairwise mass particle. Such a modification allows the VIB model to consider arbitrary values of the Poisson ratio. A linear elastic cohesive law considering both the R-bond and L-bond are assumed. The constitutive relationship is derived using the Cauchy–Born rules. The derived constitutive associates the bond stiffness with the Young’s modulus and Poisson ratio of materials. This gives the bond stiffness in terms of the Young’s modulus and Poisson ratio of materials.The modified VIB model is then used to analyze the tensile crack behavior. In contrast to the cohesive stress method, the deformation-governed concept will be used. The local materials failure is assumed to coincide with the reduction of the bond density due to the local deformation rather than by the local cohesive stress. A phenomenological relationship between the bond density and the deformation is established. The criterion which is applied to determined crack initiation and propagation is built into the constitutive model. As an example, the method is used to study the crack initiation and propagation behavior under tensile loading.     

Finite deformation continuum model for single-walled carbon nanotubes

A continuum-based model for computing strain energies and estimating Young’s modulus of single-walled carbon nanotubes (SWCNTs) is developed by using an energy equivalence-based multi-scale approach. A SWCNT is viewed as a continuum hollow cylinder formed by rolling up a flat graphite sheet that is treated as an isotropic continuum plate. Kinematic analysis is performed on the continuum level, with the Hencky (true) strain and the Cauchy (true) stress being employed to account for finite deformations. Based on the equivalence of the strain energy and the molecular potential energy, a formula for calculating Young’s modulus of SWCNTs is derived. This formula, containing both the molecular and continuum scale parameters, directly links macroscopic responses of nanotubes to their molecular structures. Sample numerical results show that the predictions by the new model compare favorably with those by several existing continuum and molecular dynamics models.     

Delamination effects on cracked steel members reinforced by prestressed composite patch

Prestressed composite patch bonded on cracked steel section is a promising technique to reinforce cracked details or to prevent fatigue cracking on steel structural elements. It introduces compressive stresses that produce crack closure effect. Moreover, it modifies the crack geometry by bridging the crack lips and reduces the stress range at crack tip. Fatigue tests were performed on notched steel plate reinforced by CFRP strips as a step toward the validation of crack patching for fatigue life extension of riveted steel bridges. A debond crack in the adhesive–plate interface was observed by optical technique. Debond crack total strain energy release rate is computed by the modified virtual crack closure technique. A parametric analysis is performed in order to investigate the influence of some design parameters such as the composite patch Young’s modulus, the adhesive thickness and the pretension level on the adhesive–plate interface debond.     

Effect of microstructure on fracture of brittle materials: Unified approach

A theoretical approach to the fracture of brittle solids based on crack opening displacement and energy rate criterion is presented. The approach allows for the prediction of elastic (Young’s modulus) and fracture (fracture strength and thermal shock) response of a brittle material containing spherical pores and polycrystalline solids containing anisotropic residual stresses.     

On the determination of the Young’s modulus of thin films using indentation tests

The main difficulty with the characterization of thin coatings using depth-sensing indentation tests is related to the determination of the contributions of the substrate and the film to the measured properties. In this study, three-dimensional numerical simulations of the Vickers hardness test are used in order to examine the influence of the elastic and plastic properties of the substrate and the film on the composite’s Young’s modulus results. The hardness of the film is equal to or higher than the substrate hardness. A study of the stress distributions and the indentation geometry of composites, film/substrate, was performed, taking into account the relative mechanical properties of the film and substrate. In addition, stress evolution during indentation was studied, in order to quantify the critical indentation depth under which the substrate is not elastically deformed. The accurate evaluation of the Young’s modulus of the films using weight functions is also examined: some of these have previously been proposed and one was introduced for this study. Two different fitting procedures were used to compare the results obtained from eight fictive film/substrate combinations using six weight functions. The first procedure, commonly used, considers the substrate’s modulus as a known parameter in the fitting process. In the second, the film and the substrate’s modulus are considered as unknown variables that are calculated simultaneously during the fitting process. The validity of the conclusions obtained using the fictive materials was checked by applying the weight functions to four real composites.     

Three-dimensional modeling of the mechanical property of linearly elastic open cell foams

Three-dimensional Voronoi models are developed to investigate the mechanical behavior of linearly elastic open cell foams. Dependence of the Young’s modulus, Poisson’s ratio and bulk modulus of the foams on the relative density is evaluated through finite element analysis. Obtained results show that in the low density regime the Young’s modulus and bulk modulus of random Voronoi foams can be well represented by those of Kelvin foams, and are sensitive to the geometric imperfections inherent in the microstructure of foams. In contrast, the compressive plateau stress of the foams is less sensitive to the imperfections. Failure surface of the foams subject to multi-axial compression is determined and is found to comply with the maximum compressive principal stress criterion, consistent with available experimental observations on polymer foams. Numerical results also show that elastic buckling of cell edges at microscopic level is the dominant mechanism responsible for the compressive failure of elastic open cell foams.     

Identification of Dynamic (Young’s and Shear) Moduli of a Structural Adhesive Using Modal Based Direct Model Updating Method

In this paper, mechanical characteristics (Young’s modulus and shear modulus) of an adhesive are identified using modal based direct model updating method and experimental modal data. The results show that both Young’s and shear moduli of adhesive are frequency dependent. Also, it is demonstrated that the thickness and length of the adhesive-line have influence on these properties. All experiments and subsequent identifications are conducted both in bending and shear modes, and it has been shown that the shear modulus of adhesive is more sensitive to length and thickness variations. The repeatability and consistency of method is proved through repeating the process several times and with different adherends.     

Mechanical degradation of microelectronics solder joints under current stressing

Understanding the mechanical degradation of microelectronic solder joints under high electric current stressing is an important step to develop a damage mechanics model in order to predict the reliability of a solder joint under such loading. In this paper, the experiment results for flip chip solder joints under high current stressing are reported. Nano-indentation tests suggest that mechanical property, e.g. Young’s modulus, degrades in the localized area where void nucleates during current stressing. The experiments also show that thermomigration due to the thermal gradient within solder joint caused by joule heating is significant during current stressing. A three-dimensional coupled thermal electrical finite element analysis shows the existence of a significant thermal gradient in solder joint during current stressing.     

Evolution of properties in hydration of cements—A numerical study

The present work develops a numerical method for analysis of the microstructure and property evolution in the hydration of the cement. A time-dependent micro-mechanical model is established to investigate the microstructure development and the effective property evolution of the cement paste, while the input parameters of the model are based on experimental data. It is assumed that the cement paste composite consists of the anhydrous cement particles, cement gel and pores. The cement particles have a periodically spatial array and are wrapped by the cement gel. The Young’s modulus and Poisson’s ratio of the cement paste are calculated by direct average method and two-scale expansion method. The comparisons between the numerical results and experimental data show that this model can simulate the evolution of the microstructure and properties during the hydration of the cements quite satisfactorily.     

Strain energy release rate for interfacial cracks in hybrid beams

The finite element modeling and fracture mechanics concept were used to study the interfacial fracture of a FRP-concrete hybrid structure. The strain energy release rate of the interfacial crack was calculated by the virtual crack extension method. It is shown that the crack growth has three phases, namely, cracking initiation, stable crack growth and unstable crack propagation. The effects of geometric and physical parameters of the hybrid beam on the energy release rate were considered. These parameters include Young’s moduli of the FRP, the concrete and the adhesive, thickness of the FRP plate and adhesive, and the distance of FRP plate end from the beam end. The numerical results show that the energy release rate of the interfacial crack is influenced considerably by these parameters. The present investigation can contribute to the mechanism understanding and engineering design of the hybrid structures.     

Adhesive fracture of a lap shear joint

In many respects, adhesive and cohesive fractures are similar. It has been demonstrated in both cases that a Griffith-type energy balance can often be used to predict failure, e.g., crack growth. The only essential difference involves the interpretation of the energy required to create new (adhesive or cohesive) surface area. In the cohesive case, the specific fracture energy γ _c is that required to create a new surface in the same material, while in the adhesive case, the specific fracture energy γ _a is the energy per unit area required to separate different materials. The mechanical analysis, including a stress analysis to determine the strain energy and energy balance in principle remains unchanged. Generally speaking adhesive-bonded joints involve sharp corners or other “singularities” between adjacent materials which act as stress concentrators, particularly if a crack or other sharp imperfection is present or arises at such a location. The Griffith energy approach circumvents the problem of just how large this mathematically infinite stress must be to initiate failure. Recently, this method had been successfully applied to a number of different adhesive geometries; this paper discusses the case of a single-lap shear joint. This geometry is important because the lap-joint test is a common method for comparing adhesive strengths; in addition, the configuration itself is often used in engineering practice. Adhesive fracture is, therefore, compared on the basis of both energy and maximum stress criteria. Experimental data suggest the former to yield more accurate predictions.     

Effect of CNT length and CNT-matrix interphase in carbon nanotube (CNT) reinforced composites

The load transfer mechanisms and effective moduli of single-walled nanotube (SWNT) reinforced composites are studied using a continuum model. A “critical” fiber length is defined for full load transfer by numerically evaluating the strain-energy-changes for different fiber lengths. The effective longitudinal Young’s modulus and bulk modulus of the composite are derived. The effect of the interphase is also discussed. The results indicate the fiber length is critical both to the load transfer efficiency and effective moduli of the composite. The SWNT-matrix interphase plays an important role in load transfer efficiency but affects the effective moduli only slightly.     

An analytically-numerical approach for the analysis of an interface crack with a contact zone in a piezoelectric bimaterial compound

An interface crack with an artificial contact zone at the right-hand side crack tip between two dissimilar finite-sized piezoelectric materials is considered under remote mixed-mode loading. To find the singular electromechanical field at the crack tip, an asymptotic solution is derived in connection with the conventional finite element method. For mechanical loads, the stress intensity factors at the singular points are obtained. As a particular case of this solution, the contact zone model (in Comninou’s sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are derived. The dependencies of the contact zone lengths on external load coefficients are illustrated in graphical form. For a particular case of a short crack with respect to the dimensions of the bimaterial compound, the numerical results are compared to the exact analytical solutions, obtained for a piezoelectric bimaterial plane with an interface crack.     

Very Large Poisson’s Ratio with a Bounded Transverse Strain in Anisotropic Elastic Materials

In a recent paper by Ting and Chen [18] it was shown by examples that Poisson’s ratio can have no bounds for all anisotropic elastic materials. With the exception of cubic materials, the examples presented involve a very large transverse strain. We show here that a very large Poisson’s ratio with a bounded transverse strain exists for all anisotropic elastic materials. The large Poisson’s ratio with a bounded transverse strain occurs when the axial strain is in the direction very near or at the direction along which Young’s modulus is very large. In fact the transverse strain has to be very small for the material to be stable. If the non-dimensionalized Young’s modulus is of the order δ⁻¹, where δ is very small, the axial strain, the transverse strain and Poisson’s ratio are of the order δ, δ^1/2 and δ^−1/2, respectively. Mathematics Subject Classifications (2000) 74B05, 74E10.T.C.T. Ting: Professor Emeritus of University of Illinois at Chicago and Consulting Professor of Stanford University.     

An extension of the J-integral to piezoelectric materials

This paper provides an analysis of the crack propagation criterion for a thin piezoelectric plate with a symmetry of order six. On the basis of Gol’denveizer’s asymptotic integration method or Destuynder’s unidirectional zoom technique, we obtain an extension of the purely mechanical J-integral to piezoelectric materials, with a dependence of the gradient of energy of the plate only on zeroth order terms of asymptotic expansions.     

Initiation and stable growth of penny-shaped crack in aging viscoelastic transversally isotropic material

Considered is the long-term cracking of an aging transversally isotropic material containing a Mode I penny-shaped crack under remotely applied tensile stress. The aging material properties are described by the Boltzmann–Volterra’s linear theory for integral operators with non-difference kernels. It applied to wood, concrete, some polymers and rocks. Only the symmetric case is considered where the crack lies in the plane of isotropy. The modified Leonov–Panasyuk–Dugdale’s crack model is used with a constant process zone assuming that the critical opening displacement is the fracture criterion. Volterra’s principle is applied to derive the equations of subcritical crack growth. Numerical calculations are made for subcritical crack growth for the specific example of transversally isotropic material simulating the behavior of reinforced concrete.     

Crack-tip stress fields in functionally graded materials with linearly varying properties

Crack-tip stress fields for a stationary crack along or inclined to the direction of property gradation in functionally graded materials (FGMs) are obtained through an asymptotic analysis coupled with Westergaard’s stress function approach. The elastic modulus of the FGM is assumed to vary linearly along the gradation direction. The first six terms for a crack along the direction of property variation and first four terms for a crack inclined to the direction of property variation in the expansion of the stress field are derived to explicitly bring out the influence of nonhomogeneity on the structure of the stress field. Using these stress fields, contours of constant maximum shear stress and constant out of plane displacement are generated and the effect of inclination of property gradation direction on these contours is discussed. The strain energy density criterion is applied to obtain critical conditions for crack initiation and the effect of property gradation is discussed. It is shown that the materials with varying properties can offer more resistance to crack propagation and will suppress crack growth in some situations.     

Stress analysis of a functional graded material plate with a circular hole

This paper is to study the two-dimensional stress distribution of a functional graded material plate (FGMP) with a circular hole under arbitrary constant loads. With using the method of piece-wise homogeneous layers, the stress distribution of the functional graded material plate having radial arbitrary elastic properties is derived based on the theory of the complex variable functions. As examples, numerical results are presented for the FGMPs having given radial Young’s modulus or Poisson’s ratio. It is shown that the stress is greatly reduced as the radial Young’s modulus increased, but it is less influenced by the variation of the Poisson’s ratio. Moreover, it is also found that the stress level varies when the radial Young’s modulus increased in different ways. Thus, it can be concluded that the stress around the circular hole in the FGMP can be effectively reduced by choosing the proper change ways of the radial elastic properties.     

The anti-plane solution for the edge cracks originating from an arbitrary hole in a piezoelectric material

In this paper, a general and simple way was found to solve the problem of an arbitrary hole with edge cracks in transversely isotropic piezoelectric materials based on the complex variable method and the method of numerical conformal mapping. Firstly, the approximate mapping function which maps the outside of the arbitrary hole and the cracks into the outside of a circular hole is derived after a series of conformal mapping process. Secondly, based on the assumption that the surface of the cracks and hole is electrically impermeable and traction-free, the approximate expressions for the complex potential, fields intensity factors and energy release rates are presented, respectively. Thirdly, under the in-plane electric loading together with the out-plane mechanical loading, the influences of the hole size, crack length and mechanical/electric loading on the fields intensity factors and energy release rates are analyzed. Finally, some particular holes with edge cracks are studied in numerical analysis. The result shows that, the mechanical loading always promotes crack growth, while the electric loading may retard crack growth.     

A brittle material with tunable elasticity: Crêpe paper

We study the mechanical response, and tearing features of crêpe paper, a two-dimensional, very anisotropic material, with one direction much less stiff than the other one. Depending on how the soft direction has been pre-stretched or not, the apparent Young modulus of the material can be varied over a broad range, while its fracture energy remains unaltered. The classical tearing concertina problem shows that a macroscopic measurement (the shape of the teared region) provides a direct access to the fracture properties of the material (effective Young's modulus, and fracture energy). The overall discussion is conducted in the frame of Griffith's theory of fracture.