Expression For Energy Release Rate Research Articles

Mode I fracture toughness determination and environmental durability evaluation of adhesive bonds by wedge test

This paper presents a study of a method for determining the fracture toughness of adhesive bonds under Mode I loading using the wedge test. The slope of adhesively bonded specimen arms at loading points was used to provide an effective crack length for the wedge test specimen. A simple expression was derived based on beam theory for the Mode I energy release rate. Finite element analyses were conducted on wedge specimens with a range of material and geometrical properties. Good agreement was found between the energy release rates from the beam theory-based expression and the virtual crack closure technique (VCCT). A comparison of various energy release rate expressions showed that the present expression was more accurate than the other expressions available in the literature. The results of this study suggest that the environmental durability of adhesive bonds with composite adherends as well as metallic adherends can be assessed in terms of their fracture toughness using the present method. Experiments under hot and humid conditions were performed on wedge specimens with adherends made of carbon fiber reinforced polymer (CFRP) composite laminate and titanium alloy, and the fracture toughness results were obtained.

A micromechanics‐based approach to damage propagation criterion in viscoelastic fractured materials regarded as homogenized media

AbstractThis paper aims to formulate a damage propagation criterion in microfractured viscoelastic materials, relying upon a micromechanics reasoning together with thermodynamics concepts. The fracture density is regarded as damage parameter at macroscopic scale. The equivalent behavior of the heterogeneous material (solid matrix + fractures) is first formulated within the framework of viscoelastic homogenization theory. In this context, relevant relationships relating local fields to macroscopic fields are derived, thus allowing a clear micromechanical interpretation of quantities involved in the upscaling process, such as the residual or viscous strains. Based on thermodynamic concepts, the energy dissipation and the free energy of the homogenized viscoelastic material are deduced at the macroscopic scale. The formulation of an energetic‐based criterion for damage propagation in viscoelastic fractured materials is then achieved by viewing the macroscopic energy release rate as the thermodynamic force responsible for propagation. Due to the delayed deformation component, the formulation is time‐dependent. Since it is formulated directly at the homogenized material level, the main advantage of the approach developed in this work is the rigorous determination of the energy release rate expression, without neglecting any residual term. In the last part of the paper, several numerical applications are performed to illustrate the main features of the modeling and to provide comparison with available simplified formulations. Finally, the proposed damage propagation criterion is applied to give qualitative insights on fracturing process of sedimentary layered rocks at geological times scale viewed as a long‐term mechanical damage problem, emphasizing the viscosity effects in preventing fracture propagation.

Peeling behavior of a film on inner surface of a tube

Adhesion behavior of elastic films is a common phenomenon in many biological processes and industrial applications. Current film peeling theories and adhesion property testing methods are mostly limited to flat peeling cases. In this paper, peeling behaviors of elastic films on tubular surfaces are studied through theoretical analysis and finite element simulation. Based on the theory of non-linear elasticity, mechanical behaviors of cylindrical films under finite deformation are investigated and the total strain energy is obtained through dimensional analysis. Furthermore, an analytical expression for energy release rate of an axisymmetric crack is derived. By balancing with the density of energy, the length of crack varies linearly with the axial stretch. Finite element method is adopted to simulate the peeling process of a film and the results agree well with theoretical predictions. Our peeling method, which decouples the influence of surface frictions that commonly cannot be ignored in conventional one degree of freedom peeling setups, represents a new, useful, and easy-to-implement strategy to study elastic film peeling behaviors. Beyond this, the model and the results are also useful for measuring adhesion properties of soft films.

A generalized model for dynamic mixed‐mode fracture via state‐based peridynamics

AbstractA generalized model is developed to investigate dynamic crack propagation in isotropic solids under mixed‐mode I/II conditions using state‐based peridynamics. The critical stretch and the critical strain energy release rate (ERR) are related within the state‐based peridynamic framework to construct a computational model capable of capturing fracture energy of the kinked cracks. A novel formulation is presented to predict crack growth trajectory and pattern by combining the traditional expression of ERR and the peridynamic states of the crack opening and sliding displacements. The proposed model is used to predict dynamic fracture behavior in polymethyl methacrylate (PMMA) and soda‐lime glass using various test specimens, including cracked semi‐circular bending (SCB), cracked rectangular plate, and single edge‐notched tensile (SENT) specimens, and under different dynamic loading conditions. The developed model is examined against the numerical and experimental data available in the literature, and a very good agreement is observed.

Fracture mechanics solutions and operative formulae for isotropic bi-material layers with large elastic mismatch

Following previous works in the literature on the fracture of layered composites, semi-analytical expressions for energy release rate and mode-mixity phase angle in isotropic bi-material layers subjected to arbitrary end loadings are extended to materials characterized by a large elastic mismatch described by Dundurs’ parameters |α| ≤ 0.99 and |β| ≤ 0.4. The solutions depend on elementary bending, axial and shear loads acting at the crack tip cross section, elastic constants and relative thickness of the layers, and dimensionless coefficients describing the fracture parameters of single elementary loads. The coefficients associated to the shear forces are presented in the literature only for combinations with β = 0; here they are determined numerically for different values β ≠ 0 to describe materials used in current applications. Simplified fitting formulae having general validity, as polynomials, which highlight the dependence on Dundurs’ parameters and are practical to support laboratory tests, are derived for layers with mid-thickness delaminations and thin layers on semi-infinite substrates. Applications to some mixed mode fracture specimens highlight the effects of β ≠ 0 on the fracture parameters and the limitations of estimates based on the commonly adopted assumption β = 0. As expected, the results for mid-thickness delaminations show that such effects are stronger on mode-mixity phase angle than on energy release rate.

Symmetric arrangements of cracks with perturbed symmetry: Extremal properties of perturbed configurations

Perturbations of symmetric crack configurations produce mixed-mode conditions at crack tips, in cases of pure mode I (or II) conditions in unperturbed configurations. This may lead to the increase of stress intensity factors and energy release rates, so that their maximums are reached in somewhat perturbed configurations (even if the perturbations increase spacing between crack tips) thus making them energetically preferable. Тhe effect of perturbations on the direction of crack kinking is also examined, in particular, whether the kinking drives the crack system farther away from the original configuration or back towards it. These considerations may be related to the experimentally observed tendency of cracks to “avoid each other” Melin (1983). They also shed some light on the crack coalescence problems. Closed-form approximate expression for the maximal energy release rate at the crack kink tip is presented. The results of Panasiuk et al (1976), Cotterell and Rice (1980), and Amestoy and Leblond (1992) are also discussed.

Free vibration of the double tapered cracked beam

This study presents the free vibration analysis of a double tapered beam having linearly varying both thickness and width, by using finite element and component mode synthesis methods. To determine the natural frequency and mode shape of the double tapered cracked beam, the stiffness and mass matrices of the beam have been obtained. The crack in the beam is modeled as a massless spring, and the beam is divided into two subcomponents from the crack section. The stiffness of spring has been derived from the linear elastic fracture mechanics theory as the inverse of the compliance matrix calculated using stress intensity factors and strain energy release rate expressions. It has been observed that natural frequencies and mode shapes vary depending on the location of the crack, the depth of the crack and the aspect ratios of the beam. The results of the present study and those in the literature are compared and a great deal of consistency has been found.

Strength and toughness of adhesion of soft materials measured in lap shear

Adhesion of soft materials is commonly characterized by strength and toughness. Their applicability, however, is often questionable even for widely used test methods. Here we study lap shear using a combination of experiment, theory, and computation. Lap shear has long been used to measure adhesion strength of soft materials, but our experiments show that the measured adhesion strength is constant only when the sample is thin and short enough. When the sample thickness exceeds the fractoadhesive length, the corner of the sample concentrates stress, and the adhesion strength decreases as the sample thickness increases. When the sample length exceeds the shear lag length, the backing layer deforms appreciably, much of the sample does not carry shear stress, and the adhesion strength decreases as the sample length increases. Furthermore, we derive an analytical expression for energy release rate for a crack long compared to the sample thickness. For samples thicker than the fractoadhesive length, we show that the adhesion toughness is constant, independent of the sample length. These findings will guide the development of the methods to characterize strength and toughness of adhesion.

An end notched flexure specimen model for mode-II fracture of high-temperature superconductor tapes with thermal effect

End notched flexure (ENF) experiment using three-point bending beam specimen is a very convenient test configuration to evaluate the mode-II fracture toughness of composites. High-temperature superconductor (HTS) tapes are multi-layered composite that have been widely used in magnets and cables. HTS tapes are manufactured at room temperature ([Formula: see text][Formula: see text]300 K) but are used in the liquid-nitrogen temperature (77 K) environment. The large temperature change leads to large shear stress which may cause delamination of HTS tapes. In this paper, a closed-form expression of energy release rate (and therefore the stress intensity factor) for mode-II fracture of HTS tapes with thermal effect is developed based on the end notched flexure specimen. Results show that HTS tapes in working condition are prone to delaminating. Increasing the thickness of Hastelloy layer can reduce the energy release rate of mode-II cracking thus increase the interface strength. In the design of HTS tapes, the silver and copper layers are used to enhance the stretch of the HTS tapes. While silver and copper layers will increase the energy release rate for mode-II crack. This fact indicates that thicker silver and copper layers are not always good from the fracture mechanics point of view. The results of thermomechanical model and fracture criterion model of this research are useful for determining mode-II fracture toughness of superconducting composites for high-temperature applications.

A Theoretical Closed Form Expression for Threshold Energy Release Rate of Environment-assisted Cracking of SiO<sub>2</sub> Taken into Account the Effects of pH

A Theoretical Closed Form Expression for Threshold Energy Release Rate of Environment-assisted Cracking of SiO<sub>2</sub> Taken into Account the Effects of pH

Post-buckling induced delamination propagation of composite laminates with bi-nonlinear properties and anti-penetrating interaction effects

This paper presents a unique theoretical analysis for the multiple failures of in-plane loaded composite laminates containing a through-width delamination. Under the action of the in-plane load, the delaminated laminates undergo first post buckling failure, then delamination propagation. The anti-penetrating contact effect occurred at the interfaces of the delamination is identified by analyzing the local transverse deformation and is incorporated into the multiple failure stages. Based on von Karman nonlinear geometric relationships, the post-buckling equilibrium path of the delaminated laminates characterized by the in-plane load vs deflection amplitude curves is solved. Expression of energy release rate is derived from variational principles as the summation of the products of shear forces and deflection slopes at the delamination fronts. With the available solutions from post-buckling analysis, threshold to initiate the delamination to propagate is computed according to Griffith fracture criterion and is calculated for different delamination conditions by developing Matlab program. It is found that the delaminated laminate undergoes a consistently macroscopic deflection with microscopic local buckles on the interfaces of the delamination. For all cases, Mode I delamination propagation can never occur due to the contact behavior. Instead, Mode II delamination propagation prevails regardless of the delamination size and position. The analysis presented in this paper accounts for nonlinear geometric deformation, nonlinear constitutive behavior and anti-penetrating interaction effects and identifies multiscale deformation modes involved in different failure mechanisms, thereby demonstrates the latest research idea for the current frontier problems.

Complete analytical solutions for double cantilever beam specimens with bi-linear quasi-brittle and brittle interfaces

In this work we develop a complete analytical solution for a double cantilever beam (DCB) where the arms are modelled as Timoshenko beams, and a bi-linear cohesive-zone model (CZM) is embedded at the interface. The solution is given for two types of DCB; one with prescribed rotations (with steady-state crack propagation) and one with prescribed displacement (where the crack propagation is not steady state). Because the CZM is bi-linear, the analytical solutions are given separately in three phases, namely (i) linear-elastic behaviour before crack propagation, (ii) damage growth before crack propagation and (iii) crack propagation. These solutions are then used to derive the solutions for the case when the interface is linear-elastic with brittle failure (i.e. no damage growth before crack propagation) and the case with infinitely stiff interface with brittle failure (corresponding to linear-elastic fracture mechanics (LEFM) solutions). If the DCB arms are shear-deformable, our solution correctly captures the fact that they will rotate at the crack tip and in front of it even if the interface is infinitely stiff. Expressions defining the distribution of contact tractions at the interface, as well as shear forces, bending moments and cross-sectional rotations of the arms, at and in front of the crack tip, are derived for a linear-elastic interface with brittle failure and in the LEFM limit. For a DCB with prescribed displacement in the LEFM limit we also derive a closed-form expression for the critical energy release rate, G_c. This formula, compared to the so-called ‘standard beam theory’ formula based on the assumptions that the DCB arms are clamped at the crack tip (and also used in standards for determining fracture toughness in mode-I delamination), has an additional term which takes into account the rotation at the crack tip. Additionally, we provide all the mentioned analytical solutions for the case when the shear stiffness of the arms is infinitely high, which corresponds to Euler–Bernoulli beam theory. In the numerical examples we compare results for Euler–Beronulli and Timoshenko beam theory and analyse the influence of the CZM parameters.

The Double Drum Peel (DDP) test: A new concept to evaluate the delamination fracture toughness of cylindrical laminates

Standard delamination tests of monolithic composites prescribe configurations where the crack is a symmetry plane for both overall geometry and stacking sequence, ensuring a controlled mode ratio. These normalized configurations do not enable testing of curved specimens, like those manufactured by filament winding. Here, we propose a new concept for the delamination testing of cylindrical laminates, the Double Drum Peel, related to the peel tests used for adhesives or thin-films debonding. A global energy analysis, including all sources of energy release and dissipation, provides the expression of the critical strain energy release rate. As in the classical peel test, the energy dissipated by mechanisms other than delamination should be accounted for to determine intrinsic interface properties. The local mode mixity is evaluated based on analytical results on the classical peel test. Tests using carbon-peek rings manufactured by laser assisted tape placement are presented to illustrate the potential of the DDP.

Dynamic interfacial fracture of a thin-layered structure

To calculate the dynamic energy release rate of a crack is important for understanding a structure’s fracture behavior under transient or varying loads, such as impact and cyclic loads, when the inertial effect can be significant. In this work, a method is proposed to derive an analytic expression for the dynamic energy release rate of a stationary crack under general applied displacement. An asymmetric double cantilever beam with one very thin layer is considered as a special case, with vibration superimposed onto a constant displacement rate applied to the free end. The resulting expression for dynamic energy release rate is verified using the finite-element method (FEM) in conjunction with the virtual crack closure technique. The mode-mixity of the dynamic energy release rate is also calculated. The predicted total dynamic energy release rate and its components, GI and GII, are both in close agreement with results from FEM simulations.

Theoretical formulation of double scalar damage variables

The predictive utility of a damage model depends heavily on its particular choice of a damage variable, which serves as a macroscopic approximation in describing the underlying micromechanical processes of microdefects. In the case of spatially perfectly randomly distributed microcracks or microvoids in all directions, isotropic damage model is an appropriate choice, and scalar damage variables were widely used for isotropic or one-dimensional phenomenological damage models. The simplicity of a scalar damage representation is indeed very attractive. However, a scalar damage model is of somewhat limited use in practice. In order to entirely characterize the isotropic damage behaviors of damaged materials in multidimensional space, a system theory of isotropic double scalar damage variables, including the expressions of specific damage energy release rate, the coupled constitutive equations corresponding to damage, the conditions of admissibility for two scalar damage effective tensors within the framework of the thermodynamics of irreversible processes, was provided and analyzed in this study. Compared with the former studies, the theoretical formulations of double scalar damage variables in this study are given in the form of matrix, which has many features such as simpleness, directness, convenience and programmable characteristics. It is worth mentioning that the above-mentioned theoretical formulations are only logically reasonable. Owing to the limitations of time, conditions, funds, etc. they should be subject to multifaceted experiments before their innovative significance can be fully verified. The current level of research can be regarded as an exploratory attempt in this field.

Energy release rate of beam-type fracture specimens with hygrothermal influence

Three interface joint models for calculating energy release rate are extended to include hygrothermal effect with consideration of uniform or nonuniform changes of temperature or moisture through layer thickness in a cracked bilayer beam system using both J-integral and correction methods. The method of J-integral leads to a general expression of total energy release rate in terms of the crack-tip moments and forces, while the correction method gives explicit formulas of total energy release rate with inclusion of a separate energy release rate term subjected to only mechanical loads. Several bilayer beam fracture specimens subjected to transverse load or equivalent bending moment and thermal load are considered to validate accuracy of two methods in connection with three interface joint models, and the results show that the flexible joint model produces the best predictions for all the specimens. A parameter study is conducted to evaluate influences of thickness ratio, modulus ratio of upper and lower layers, change of temperature, and temperature gradient on the energy release rate. All the factors show a pronounced influence on the energy release rate, and a special attention is paid to evaluate effect of temperature change and temperature gradient on the energy release rate of bilayer beams. The moisture change and its gradient have the same effect and formulations on the energy release rate of bilayer beams as those of temperature, and the moisture portion of derivations and its effect are thus not given in detail.

Calculation of total energy release rate in post-local buckling delamination of composite laminates

In the present article, the variational energy principle is used to derive the expression for energy release rate in buckled composite laminate containing through-the-width delamination, subjected to in-plane strains. Boundary conditions are clamped at both edges. Buckling and post-buckling solutions are obtained and expressions for critical buckling load and post-buckling deflection have been developed. A through-the-width delamination model has been considered and formulations are based on higher order shear deformation theory. The effects of considering the higher order shear deformation theory on equivalent bending rigidity, buckling load, and energy release rate have been investigated. Finally, the results of current study have been compared with the results of finite element method analysis by Abaqus/CAE and those available in the literature.

How does surface tension affect energy release rate of cracks loaded in Mode I?

The role of surface stress on fracture of elastic solids has been studied by two group of researchers with different conclusions. One group concludes that surface stress has no effect on energy release rate except that the singularity of the crack tip field is stronger than the usual inverse square root singularity dictated by linear elastic fracture mechanics. The other group concludes that the singularity of the stress field reduces to a logarithmic singularity, thus implying that the local energy release rate is zero. In this letter we resolve this paradox by examining the solution of a special case where surface stress is isotropic and independent of surface strain. We show that surface tension resists crack growth by lowering the applied energy release rate while retaining the inversely square root singularity of the elastic crack tip field. We demonstrate this idea by solving a perturbation problem where the capillary length is much smaller than the crack length. A closed form expression for the local energy release rate is obtained in the limit of small surface tension.

Effect of initial T-stress on stress intensity factor for a crack in a thin pre-stressed layer

Fracture of a pre-stressed thin elastic layer is studied and modelled as bending of double clamped beams with axial force. Using the beam theory, the strain energy in bending state of the beam with an axial force is derived. Explicit expressions for energy release rate and stress intensity factor (SIF) for a crack with initial transverse stress (T-stress) are obtained. Results show that tensile (compressive) initial T-stress decreases (increases) the SIF and impedes (promotes) crack growth. The influence of compressive initial stress is greater than that of tensile initial stress in the same magnitude.

Finite element method for the vibration of cracked beams with varying cross section

AbstractVibration analysis of cracked beams having linearly varying cross‐sections both in thickness and width was investigated. A computer program using the finite element method has been written to find the dynamical characteristics (natural frequencies and mode shapes) of the cracked beam. The cracked section in the beam has been modeled by a massless spring whose flexibility depens on the local flexibility induced by the crack. The stiffness of spring has been derived from the linear elastic fracture mechanics theory as the inverse of the compliance matrix calculated using stress intensity factors and strain energy release rate expression. Some examples have been given to explain the proposed method and investigate the effects of the depth and location of cracks on the natural frequencies and mode shapes. The results of current study and those in the literature are compared and good agreements have been found. Consequently it is showed that proposed method is reliable and simple. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)