Steady-state Energy Release Rate Research Articles

Analytical solution on interfacial reliability of 3-D through-silicon-via (TSV) containing dielectric liner

Interfacial reliability is a challenging issue in through-silicon-via (TSV) technique. To accurately investigate the interfacial reliability of TSV, this paper developed an analytical solution approach, in which the effects of the liner are considered. The validity of the analytical solution is executed by comparison with finite element simulation results. Results show that two approaches have good agreement, with a deviation within 10%, illustrating the validity of the analytical solution developed in this study. Then, using the developed analytical solution, the effects of via diameter, the liner thickness, and the liner materials of TSV on interfacial reliability are investigated with the steady-state energy release rate (ERR). Analytical results show that the steady-state ERR is not only determined by the coefficient of thermal expansion (CTE) mismatch between adjacent materials, but also affected by the products (E×CTE2) of Young’s modulus (E) and CTE2 of the liner. Liner materials with lower E×CTE2 values will lead to lower steady-state ERR. Additionally, the combined effects of copper via diameter and liner thickness on ERR declare that the ERR highly depends on copper via diameter.

Critical cooling rates to avoid transient-driven cracking in thermal barrier coating (TBC) systems

This paper presents critical cooling rates to avoid cracking in thermal barrier coatings (TBCs) driven by thermal transients. A complete thermomechanical model is presented for multilayers; it determines the history of temperature, deformation and stress distributions in the layers, as well as the steady-state energy release rate (ERR) for delamination for all possible crack locations. The model is used to analyze bilayers over a broad range of properties and cooling rates; critical cooling rates are identified that distinguish scenarios in which the transient delamination driving force is higher than that associated with the fully cooled state. Implications for coating the durability are discussed.

Controlling the Chemical Composition of the Interface Thermally Grown Oxides to Prolong the Lifespan of Thermal Barrier Coatings Based on Interfering Aluminum Diffusion Mechanism

This study places great emphasis on the relationship between the TGO chemical composition and the TGO growth rate and the mechanical property of thermal barrier coatings upon isothermal oxidation. The results indicate that the control of the θ-Al2O3α-Al2O3 phase transformation can have significant effects on the subsequent isothermal oxidation during the earlier stage of isothermal oxidation. The formation of (Al, Cr)2O3 oxide sub-layer originated from the textured and larger grain of α-Al2O3 can influence the potential TBCs lifetime. Additionally, the potential longer oxidation lifetime of TBCs that subjected to the appropriate vacuum heat pre-treatment can be attributed to the larger absorptive energy of the TGO area prior to its fully microcrack imperfection propagation. Most importantly, TBCs delamination was determined by the steady-state energy release rate in ceramic topcoat and TGO, the TGO and the interface, as well as the formation and development of the microstructure and composition of imperfections in the vicinity of the interface.

A Finite-Deformation Mechanics Theory for Kinetically Controlled Transfer Printing

The widely used steady-state energy release rate G = F/w is extended to account for the elastic energy of deformed compliant stamps, e.g., low-modulus poly(dimethyl siloxane) (PDMS). An analytical expression for the energy release rate is obtained to quantify interfacial adhesion strength in tape peeling tests, and to analyze the dynamics of kinetically controlled transfer printing. The critical delamination velocity to separate retrieval and printing is related to the critical energy release rate and the tensile stiffness of the stamp. Experimental results validate the analytical expression established by the mechanics model.

Effect of elastic constants on crack channeling in a thin film bonded to an orthotropic substrate

A channeling crack confined in an orthotropic film bonded to an orthotropic substrate under a steady-state condition is investigated. The problem is formulated based on a modified Stroh formalism and an orthotropy rescaling technique, in order to determine the necessary material parameters required to describe the steady-state energy release rate. A closed form of the energy release rate is obtained with the exception of the normalized energy release rate for the transformed bimaterial structure that consists of the orthotropic film and isotropic substrate. The normalized energy release rates for the transformed problem are shown to depend on only four material parameters and are numerically calculated using finite element analyses. The periodic channels in the film layer of the bimaterial structure are also considered. The steady-state energy release rates for the periodic channeling cracks are obtained as a function of the ratio of the film thickness to the crack spacing for various combinations of the material parameters.

Stochastic modeling of delamination growth in unidirectional composite DCB specimens using cohesive zone models

Delamination between plies is the most common failure mode in composite laminates that can cause fiber breakage and reduction in life of the composite. Currently Mode-I interlaminar fracture toughness or critical energy release rate of a composite is measured using double cantilever beam (DCB) with unidirectional composites (ASTM D5528). Unlike metals, the energy plot of a DCB specimen shows increase in energy release rate with increase in crack length. Also during testing the steady state energy release rate from each of the samples of the same batch shows a lot of variation due to fiber cross-over bridging that occurs only in unidirectional composites. In this study, a probabilistic Cohesive Zone Model (CZM) is developed to capture steady state energy release rate variations of 51mm crack size DCB specimens based on unidirectional composite (IM7/977-3) test data. Then using the probabilistic CZM, the energy release rate variations of 76.2mm crack size DCB model are predicted and compared to the test results. The predictions showed good agreement with the experiments suggesting a probabilistic CZM is capable of simulating the delamination process in unidirectional composites.

The effect of elevated temperature exposure on the fracture toughness of solid wood and structural wood composites

Fracture toughness of wood and wood composites has traditionally been characterized by a stress intensity factor, an initiation strain energy release rate (G init) or a total energy to fracture (G f). These parameters provide incomplete fracture characterization for these materials because the toughness changes as the crack propagates. Thus, for materials such as wood, oriented strand board (OSB), plywood and laminated veneer lumber (LVL), it is essential to characterize the fracture properties during crack propagation by measuring a full crack resistant or R curve. This study used energy methods during crack propagation to measure full R curves and then compared the fracture properties of wood and various wood-based composites such as, OSB, LVL and plywood. The effect of exposure to elevated temperature on fracture properties of these materials was also studied. The steady-state energy release rate (G SS) of wood was lower than that of wood composites such as LVL, plywood and OSB. The resin in wood composites provides them with a higher fracture toughness compared to solid lumber. Depending upon the internal structure of the material, the mode of failure also varied. With exposure to elevated temperatures, G SS for all materials decreased while the failure mode remained the same. The scatter associated with conventional bond strength tests, such as internal bond and bond classification tests, renders any statistical comparison using those tests difficult. In contrast, fracture tests with R curve analysis may provide an improved tool for characterization of bond quality in wood composites.

Delamination resistance of thermal barrier coatings containing embedded ductile layers

Micromechanical models are developed to explore the effect of embedded metal layers upon thermal cycling delamination failure of thermal barrier coatings (TBCs) driven by thickening of a thermally grown oxide (TGO). The effects of reductions in the steady-state (i.e. maximum) energy release rate (ERR) controlling debonding from large interface flaws and decreases in the thickening kinetics of TGO are investigated. The models are used to quantify the dependence of the ERR and delamination lifetime upon the geometry and constitutive properties of metal/TBC/TGO multilayers. Combinations of multilayer properties are identified which maximize the increase in delamination lifetime. It is found that even in the absence of TGO growth rate effects, the delamination lifetime of TBC systems with weak TGO/bond coat interfaces can be more than doubled by replacing 10–20% of the ceramic TBC layer with a metal whose ambient temperature yield stress is in the ∼100–200MPa range.

Prediction of environmental degradation of closed adhesive joints using data from open-faced specimens

A framework was established to predict the fracture toughness of degraded closed DCB (CDCB) joints of a toughened adhesive–aluminum system using fracture data obtained from accelerated degradation tests on open-faced joints. The exposure index ( EI), the time integral of water concentration, was calculated at all points in the closed joints using the water diffusion properties of the adhesive. The fracture toughness of the closed joints was then predicted from these calculated EIs by making reference to previously reported fracture toughness data from open-faced DCB (ODCB) specimens degraded to various EI levels. To verify the predictions, fracture experiments and analyses were carried out for closed DCB joints degraded at 60 °C–95% relative humidity (RH) and 60 °C–82% RH conditions. The failure mode of both closed and open DCBs was cohesive in the adhesive layer. Good agreement was observed between the predicted steady-state critical strain energy release rate ( G cs ) values and the experimentally measured G cs values for the degraded closed DCB joints. The results showed that the accelerated open-faced methodology, which significantly reduces the exposure time to reach a given level of degradation, can be used to predict the durability of degraded closed joints used in service conditions. It was also shown that at a given temperature, the knowledge of the degradation behavior at one RH level could be extended to other levels of RH with an acceptable accuracy using the fact that fracture degradation at a given temperature is a unique function of EI, independent of the RH exposure history that gives rise to EI. The results are applicable to other laminated systems where degradation of the bonding layer is a failure mode of concern.

Impact of Near-Surface Thermal Stresses on Interfacial Reliability of Through-Silicon Vias for 3-D Interconnects

Continual scaling of on-chip wiring structures has brought significant challenges for materials and processes beyond the 32-nm technology node in microelectronics. Recently, 3-D integration with through-silicon vias (TSVs) has emerged as an effective solution to meet the future interconnect requirement. Among others, thermomechanical reliability is a key concern for the development of TSV structures used in die stacking as 3-D interconnects. This paper examines the effects of thermally induced stresses on the interfacial reliability of TSV structures. First, 3-D distribution of the thermal stress near the TSV and the wafer surface is analyzed. Using a linear superposition method, a semianalytic solution is developed for a simplified structure consisting of a single TSV embedded in a silicon (Si) wafer. The solution is verified for relatively thick wafers by comparing to numerical results from finite element analysis (FEA). The stress analysis suggests interfacial delamination as a potential failure mechanism for the TSV structure. An analytical solution is then obtained for the steady-state energy release rate as the upper bound for the interfacial fracture driving force, while the effect of crack length is evaluated numerically by FEA. With these results, the effects of the TSV dimensions (e.g., via diameter and wafer thickness) on the interfacial reliability are elucidated. Furthermore, the effects of via material properties and dielectric buffer layers are discussed.

Effect of geometry on the fracture behavior of lead-free solder joints

Copper bars were soldered along their length with a thin layer of lead-free Sn3.0Ag05.Cu alloy under standard surface mount processing conditions to prepare double cantilever beam (DCB) specimens. The geometry of the DCBs was varied by changing the thickness of the solder layer and the copper bars. These specimens were then fractured under mode-I and two mixed-mode loading conditions. The initiation strain energy release rate, G ci , increased with the relative fraction of mode-II, but was unaffected by the changes in either the substrate stiffness or the solder layer thickness. However, the steady-state strain energy release rate, realized after several millimeters of crack growth, was found to increase with the solder layer thickness at the various mode ratios. The crack path was found to be influenced by mode ratio of loading and followed a path that maximizes the von Mises strain rather than maximum principal stress. Finally, some preliminary results indicated that the loading rate significantly affects the G ci .

Hygrothermal degradation of two rubber-toughened epoxy adhesives: Application of open-faced fracture tests

The absorption/desorption properties of two commercial, toughened epoxy adhesive systems were evaluated gravimetrically, and by X-ray photoelectron spectroscopy (XPS) and dynamic mechanical thermal analysis (DMTA). Fracture tests on degraded open-faced DCB specimens showed that these two adhesive systems have very different degradation behaviors. The steady-state critical strain energy release rate, G cs , of an adhesive system 1 decreased rapidly with an exposure time in various hot-wet environments, reaching a relatively low value that was stable for over one year, while that of adhesive system 2 remained unchanged for more than one and a half years. A degradation mechanism which accounts for the different characteristics of the two adhesive systems was proposed. A model of fracture toughness degradation, analogous to Fick’s law, was then used to characterize the fracture toughness loss in an adhesive system 1, and the effects of temperature, RH and water concentration were evaluated. The results illustrate the wide variation in water absorption behaviors that can exist among toughened epoxy adhesives, and show how these differences relate to the degradation of fracture strength. The data were also used to assess the applicability of an exposure index (EI), defined as the integral of relative humidity over time, as a means of characterizing an aging history. The fracture strength degradation was measured after aging to achieve a range of EI values, and it was found that the strength loss was independent of the time–humidity path for sufficiently large EI.

Effect of adhesive thickness on fatigue and fracture of toughened epoxy joints – Part II: Analysis and finite element modeling

The effect of bondline thickness on the fatigue and fracture of aluminum adhesive joints bonded using a rubber-toughened epoxy adhesive was studied using finite element analysis. The fatigue data of Part I examined the dependence of the fatigue threshold and cyclic crack growth rate on the adhesive thickness under both mode-I and mixed-mode loading. The fracture data of Part I illustrated the relation between the adhesive thickness and the quasi-static crack initiation and steady-state critical strain energy release rates. These experimental trends are explained in terms of the effects of the adhesive thickness and the applied strain energy release rate on the stress distribution in the bondline, the stress triaxiality at the crack tip, and the plastic zone size in the adhesive layer.

Effect of adhesive thickness on fatigue and fracture of toughened epoxy joints – Part I: Experiments

The effect of bondline thickness, from 130 μm to 790 μm, on the fatigue and quasi-static fracture behavior of aluminum joints bonded using a toughened epoxy adhesive was studied experimentally under mode-I (DCB) and mixed-mode (ADCB) loading. Under mode-I loading, it was found that the fatigue threshold energy release rate, G th , decreased for very thin bondlines, while under mixed-mode loading, the G th changed very little with bondline thickness. In both cases, the effect of bondline thickness was more pronounced at higher crack growth rates. For quasi-static fracture, the effect of adhesive thickness on the energy release rate for the onset of fracture from the fatigue threshold, G c 0 , was similar to that found for the fatigue threshold; however, the steady-state energy release rate, G c s , increased linearly with increasing bondline thickness.

Fracture R-curve of a toughened epoxy adhesive as a function of irreversible degradation

Open-faced double cantilever beam (DCB) specimens of a toughened epoxy-aluminum adhesive system were degraded over a relatively wide range of temperature, relative humidity (RH) and exposure time, dried and tested to characterize the irreversible evolution of the mixed-mode fracture resistance curves ( R-curves). The water diffusion properties of the bulk adhesive were modeled using an earlier sequential dual Fickian (SDF) model for the same adhesive in order to predict the adhesive water content. Three temporal stages of degradation possessing different R-curve and fracture surface characteristics were observed. In general, the steady-state critical strain energy release rate ( G cs ), the rate of toughening ( dG cr / da) and the length of the rising part of the R-curve decreased with increasing exposure temperature, RH and water concentration, while the initiation G c ( G ci ) remained unchanged. It is hypothesized that crack initiation is governed by the properties of the epoxy matrix and that the toughening action of rubber particles does not become appreciable until after a certain amount of crack extension (more than about 50 μm in the present case). The irreversible degradation of fracture toughness was found to be insensitive to the phase angle, which simplifies the construction of the fracture toughness envelope for a given level of degradation. These effects were incorporated into a new R-curve degradation model which has an application in the R-curve prediction for closed joints having nonuniform degradation.

Experimental assessment of bonded FRP-to-steel interfaces

The use of bonded fibre-reinforced polymer (FRP) materials for the repair of civil infrastructure is becoming well established. The majority of applications are ‘bond critical'; that is, the dominate limit state is affected by the FRP debonding from the substrate. Considerable research has been conducted on the behaviour of FRP bonded to a concrete substrate; however, there is a growing interest in using FRP bonded to a steel substrate. In the former case, bond behaviour is dominated by cohesive failure in the concrete substrate; in the latter, bond is dominated by adhesive behaviour. As a result of not needing to consider substrate failure in the case of FRP-to-steel applications, the FRP may be utilised much more efficiently, provided bond can be maintained. This paper briefly reviews factors affecting bond of FRP-to-steel and presents a relatively simple laboratory test method for assessing bond performance. The method is well suited to comparing performance of different applications, surface preparations, environmental exposures and so on, but also yields quantitative data in the form of the steady-state energy release rate useful for modelling and designing FRP-to-steel bonded interfaces.

Fracture load predictions and measurements for highly toughened epoxy adhesive joints

A method to predict the ultimate strength of adhesive joints has been evaluated for the quasi-static loading of a variety of cracked-lap shear (CLS) and single-lap shear (SLS) joints bonded with a high-strength, toughened epoxy adhesive. The adhesive strength was experimentally characterized in terms of the steady-state critical strain energy release rate, G c s , as a function of the loading phase angle, using double cantilever beam (DCB) joints. Comparing the calculated energy release rate using the J-integral with the G c s at the corresponding phase angle, the ultimate failure load in the fracture joint was predicted and compared with experimental results. When the toughening of the adhesive during subcritical crack growth (i.e. its R-curve behavior) was considered in the analysis, good agreement between the predicted and experimental failure loads was achieved, both for joints made with aluminum or steel adherends. The initial condition at the end of joint overlap (fillet or precrack) did not affect the ultimate joint strength because of the significant amount of subcritical crack growth.

Analyses of steady-state interface fracture of elastic multilayered beams under four-point bending

The closed-form solution for the steady-state interface energy release rate of elastic multilayered beams subjected to four-point bending is presented. The beam can have an arbitrary number of layers and fracture can occur at any interface. Specific results are calculated for alternate layered systems to elucidate the essential trends of the dependences of the normalized steady-state interface energy release rate on the thickness ratio and the modulus ratio for each interface.

Evaluation of toughness by finite fracture mechanics from crack onset strain of brittle coatings on polymers

Crack onset strain measurements of a confined layer in tension provide the means for layer toughness estimation. The procedure can be simplified if steady-state conditions prevail starting from the commencement of crack propagation, an assumption frequently employed in energy release rate evaluation. It is demonstrated, by numerical analysis of experimental data, that an estimate of the defect size in the film is needed in order to reliably evaluate its fracture toughness from the crack onset strain. Only if microcracks of sufficient size are present in the brittle layer, the steady-state energy release rate at the crack onset strain can be identified with layer toughness. Otherwise, the toughness estimate obtained by such a procedure is likely to be non-conservative.

Mechanics of tunnelling cracks in trilayer elastic materials in tension

In this work, the crack driving force for a tunnelling crack in a thin brittle layer confined by dissimilar thick, and more compliant, elastic layers is considered at tensile loading. The steady-state energy release rate is evaluated using distributed dislocation technique and series representation of the complex potentials for an isotropic trimaterial. Evolution of the energy release rate with the crack length is studied by means of FEM. The 3D FEM simulations for tunnel cracks suggest that the ERR can represented by a universal relation (mastercurve) in suitably normalised co-ordinates. An analytical approximation of the ERR mastercurve is obtained as a function of crack length, cracking layer thickness, and a non-dimensional steady-state ERR.