Mixed-mode Bending Research Articles

In-situ characterization of interface delamination by a new miniature mixed mode bending setup

A new miniature mixed mode bending (MMMB) setup for in-situ characterization of interface delamination in miniature multi-layer structures was designed and realized. This setup consists of a novel test configuration to accomplish the full range of mode mixities and was specially designed with sufficiently small dimensions to fit in the chamber of a scanning electron microscope (SEM) or under an optical microscope for detailed real-time fracture analysis during delamination. Special care was taken to minimize the effects of friction, the influence of gravity, and non-linearities due to the geometry of the setup. The performance of the setup was assessed using specially-designed test samples supported by finite element analyses. Delamination experiments conducted on homogeneous bilayer samples in mode I and mixed mode loading were visualized with a scanning electron microscope and showed the formation of small micro cracks ahead of the crack tip followed by crack bridging and a full crack, thereby demonstrating the advantages of in-situ testing to reveal the microscopic delamination mechanism.

Fracture simulation of CFRP laminates in mixed mode bending

This paper analyses the progressive mixed mode delamination failure in unidirectional and multidirectional composite laminates using fracture experiments, finite element (FE) simulations and an analytical solution. The numerical model of the laminate is described as an assembly of damageable layers and bilinear interface elements subjected to mixed mode bending. The analytical approach is used to estimate the total mixed mode and decomposed fracture energies for laminates with different stacking sequences, which is also validated through experiments. It is concluded that the interlaminar fracture toughness of multidirectional laminates is considerably higher than that of the unidirectional ones. The effect of initial interfacial stiffness and element size is studied and it is also shown that their value must not exceed a definite limit for the numerical simulations to converge. The model can also be further extended to simulate the mixed mode fracture in hybrid fiber metal laminates.

A phantom node formulation with mixed mode cohesive law for splitting in laminates

A phantom node method with mixed mode cohesive law is proposed for the simulation of splitting in laminates. With this method, a discontinuity in the displacement field can be modeled at arbitrary locations. The micromechanical phenomenon that splitting cracks grow parallel to the fiber, is incorporated on the mesolevel, i.e., in the homogenized ply, by setting the direction of the crack propagation equal to the fiber direction. A new mixed mode cohesive law is introduced for increased robustness of the incremental-iterative solution procedure. The model is validated with mixed mode bending tests, and its utility is illustrated with examples for a single ply and for a laminate.

熱残留応力を考慮した層間き裂進展に関するエネルギー解放率：Ｃｒａｃｋ Ｔｉｐ Ｅｌｅｍｅｎｔ法による定式化

Crack tip element (CTE) method is applied to the formulation of energy release rate associated with interfacial crack growth of laminates with arbitrary layups and residual thermal stresses using the Timochenko beam model. Special attention is paid to the energy release rates of mixed mode bending (MMB) test, and mode-I and mode-II energy release rates are formulated including residual thermal stresses. The results derived are verified by the comparison to finite element analysis, and the effect of residual thermal stresses on the mode mixity of the MMB test is discussed.

On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization

The mixed mode bending specimen originally developed for mixed mode delamination fracture characterization of unidirectional composites has been extended to the study of debond propagation in foam cored sandwich specimens. The compliance and strain energy release rate expressions for the mixed mode bending sandwich specimen are derived based on a superposition analysis of solutions for the double cantilever beam and cracked sandwich beam specimens by applying a proper kinematic relationship for the specimen deformation combined with the loading provided by the test rig. This analysis provides also expressions for the global mode mixities. An extensive parametric analysis to improve the understanding of the influence of loading conditions, specimen geometry and mechanical properties of the face and core materials has been performed using the derived expressions and finite element analysis. The mixed mode bending compliance and energy release rate predictions were in good agreement with finite element results. Furthermore, the numerical crack surface displacement extrapolation method implemented in finite element analysis was applied to determine the local mode mixity at the tip of the debond.

Analysis of mixed-mode interlaminar fracture and damage behavior of GFRP woven laminates at cryogenic temperatures

The objective of this work is to investigate the interlaminar fracture and damage behavior of glass fiber reinforced polymer (GFRP) woven laminates loaded in a mixed-mode bending (MMB) apparatus at cryogenic temperatures. The finite element analysis (FEA) is used to determine the mixed-mode interlaminar fracture toughness of MMB specimen at room temperature (RT), liquid nitrogen temperature (77 K) and liquid helium temperature (4 K). A FEA coupled with damage is also employed to study the damage distributions within the MMB specimen and to examine the effect of damage on the mixed-mode energy release rate. The technique presented can be efficiently used for characterization of mixed-mode interlaminar fracture and damage behavior of woven laminate specimens at cryogenic temperatures.

A unified potential-based cohesive model of mixed-mode fracture

A generalized potential-based constitutive model for mixed-mode cohesive fracture is presented in conjunction with physical parameters such as fracture energy, cohesive strength and shape of cohesive interactions. It characterizes different fracture energies in each fracture mode, and can be applied to various material failure behavior (e.g. quasi-brittle). The unified potential leads to both intrinsic (with initial slope indicators to control elastic behavior) and extrinsic cohesive zone models. Path dependence of work-of-separation is investigated with respect to proportional and non-proportional paths—this investigation demonstrates consistency of the cohesive constitutive model. The potential-based model is verified by simulating a mixed-mode bending test. The actual potential is named PPR (Park–Paulino–Roesler), after the first initials of the authors’ last names.

Mixed-mode interlaminar fracture and damage characterization in woven fabric-reinforced glass/epoxy composite laminates at cryogenic temperatures using the finite element and improved test methods

The present research examines experimentally and analytically the mixed-mode interlaminar fracture and damage behavior of glass fiber reinforced polymer (GFRP) woven laminates at cryogenic temperatures. The mixed-mode bending (MMB) tests were performed with the improved test apparatus, at room temperature, liquid nitrogen temperature (77 K) and liquid helium temperature (4 K). The energy release rates at the onset of delamination crack propagation were evaluated for the woven GFRP specimens using both beam theory and finite element analysis. The fracture surfaces were also examined to verify the fracture mechanisms. In addition, the initiation and growth of damage in the specimens were predicted by a damage analysis, and the damage effect on the mixed-mode interlaminar fracture properties at cryogenic temperatures was explored.

Effective Simulation of Delamination in Aeronautical Structures Using Shells and Cohesive Elements

A cohesive element for shell analysis is presented. The element can be used to simulate the initiation and growth of delaminations between stacked, non-coincident layers of shell elements. The procedure to construct the element accounts for the thickness offset by applying the kinematic relations of shell deformation to transform the stiffness and internal force of a zero-thickness cohesive element such that interfacial continuity between the layers is enforced. The procedure is demonstrated by simulating the response and failure of the Mixed Mode Bending test and a skin-stiffener debond specimen. In addition, it is shown that stacks of shell elements can be used to create effective models to predict the inplane and delamination failure modes of thick components. The results indicate that simple shell models can retain many of the necessary predictive attributes of much more complex 3D models while providing the computational efficiency that is necessary for design.

Mixed mode I + II interlaminar fracture of carbon/epoxy laminates

This paper reports a study of the mixed mode I + II fracture of carbon/epoxy unidirectional and multidirectional specimens with symmetric, quasi-symmetric and markedly asymmetric legs. A beam model was developed for predicting the compliance and total strain energy release rate, G. Its predictions agreed well with three-dimensional finite element models and with experimental data of double cantilever beam, end-notched flexure and mixed-mode bending tests. The mode mix was determined from a virtual crack closure technique limit value analysis, in order to overcome ambiguity for delaminations between differently oriented plies. Initiation G c values versus mode II ratio G II/ G plots had practically identical shapes, showing a linear G c increase with G II/ G ⩾ 25%. Multidirectional specimens with quasi-symmetric and markedly asymmetric legs had similar G c values.

Energy release rate for interlaminar cracks in graded laminates

In this study, the energy release rate for interlaminar cracks in a laminate beam, graded along the depth, has been derived using beam theory. The correction for accounting crack-tip flexibility, used for conventional composites, is incorporated into the derived expressions and the variation of the energy release rate as a function of crack location and elastic gradient is investigated for the load configurations of double cantilever beam (DCB), end notched flexure (ENF) and mixed mode bending (MMB). The results of the study indicate that the energy release rate calculated using the derived expressions is in very good agreement with that obtained through finite element analysis (FEA) whereas the mode partitioning calculated from the beam theory analysis was not in agreement with that obtained from FEA. As the position of the crack varies along the depth of the beam, there is a considerable change in the energy release rate and mode partitioning. Further, it is shown that by proper load configuration, pure mode (mode-I or mode-II) conditions can be achieved for crack located at any depth along the beam. The effect of the gradation type and its strength on the energy release rate and mode partitioning is also investigated. It was observed that the energy release rate is strongly sensitive to both the strength and type of gradation, whereas, the mode partitioning is influenced more by the gradation strength than by the gradation type.

Mechanical Reliability of a Segmented Silicon Layer on Ultra-Thin Polyimide Substrates

Substrate transfer technology for SOI and non-SOI single-crystalline silicon wafers was demonstrated allowing for high-performance low-power RF applications. 3D deformable electronics could be realized by vertically thinning and laterally partitioning of the silicon substrate on sub-millimeter scale. By varying the partition dimensions and the geometry of connecting bridges, the level of acceptable deformations can be controlled. The targeted applications of this technology are wireless ID tags and sensor networks. The mechanical properties such as crack, interfacial delamination are critical to reach the flexible substrate. In this contribution, results of our work on mechanical reliability issues of poly- and single crystalline silicon on ultra-thin polyimide substrates are presented. To improve reliability, square and hexagonal segmentation with different size is applied to the silicon layer before it is transferred onto an ultra-thin polyimide substrate using wafer-to-wafer substrate transfer technique based on a temporary glass carrier. Generation of cracks within the silicon and dielectric layers is then studied under controlled bending and tensile loads. The formation of cracks is studied experimentally using specially for this purpose designed bending and tensile tools. Ultra-thin interfacial delamination are also focused in this work by experimental and FE simulation method. A new test setup is designed for mixed mode bending testing which has capacity to observation specimen and recording the crack length and crack opening by microscope. The critical energy release depended on mixed angle can be reach by combination experimental data with FEM simulation.

Correction method for evaluation of interfacial fracture toughness of DCB, ENF and MMB specimens with residual thermal stresses

The energy release rates associated with crack growth in multilayered double cantilever beam (DCB), three-point end notched flexure (ENF), and mixed-mode bending (MMB) specimens, including the effects of residual thermal stresses, are derived as special cases of Nairn’s formulation [Nairn JA. On the calculation of energy release rates for cracked laminates with residual stresses. Int J Fract 2006;139:267–93]. The difference between the true toughness and apparent toughness attributable to the consequence of ignoring residual stresses can be calculated for a given specimen geometry and mechanical properties based on classical lamination theory. The true fracture toughness is obtainable using the apparent fracture toughness and the present correction method. Numerical examples of DCB, ENF, and MMB tests are presented, and the calculated difference in the energy release rates based on the present correction method is compared to those obtained using FEM to verify the correction method.

Characterization of semiconductor interfaces using a modified mixed mode bending apparatus

This research deals with the experimental assessment of the strength of bi-material interfaces as a function of mode mixity, focusing on two dimensional problems. A modified mixed mode bending apparatus is designed and tested, which can be used to measure small forces involved in the delamination of semiconductor packaging materials. Using this setup, it is possible to measure interface strength over nearly the full range of mode mixities using a single specimen design. A finite element model is used to determine interface strength and mode mixity. As an example, the combined numerical-experimental procedure is applied to the interface between copper lead frame (LF) and epoxy molding compound (MCE). A remarkable result is that a double cantilever beam (DCB) test of this interface does not yield the lowest possible interface strength, meaning that it can not be used as a worst case test.

A mixed-mode damage model for delamination growth applied to a new woven composite

In this paper, delamination initiation and propagation of reinforced fibrous composite laminate, in single and mixed mode, are numerically predicted by taking into account the existence of imperfect interfaces between the plies. The bonding conditions between layers are characterised by jumps in displacements which are proportional to the traction stresses. In order to describe the interface damage an approach based on the indirect use of fracture mechanics considering a softening stress-relative displacements law is presented. The accuracy of the predictions is evaluated in single-mode delamination tests, in the mixed-mode bending test, and in a structural configuration consisting of the debonding of a new woven laminated composite reinforced by particles of date cores for orthopedic use. The interface is regarded as being a whole of several interfacial bonds. Each bond is supposed to be made up of three stiffnesses acting in the three directions.

Improved analysis of unidirectional composite delamination specimens

Generally, the delamination testing of composite specimens involves the analytical determination of the compliance and the energy release rate for the actual configuration. In this work an improved analysis for unidirectional composite delamination specimens is presented, which is based on the results of the last 30 years. The main goal of the present model is the improvement of the global mode decomposition method including four mechanical deformations apart from simple beam theory: Winkler–Pasternak foundation, transverse shear, Saint–Venant effect and crack tip shear deformation, respectively. These effects are investigated separately using a general loading scheme and at the final stage their results are superimposed. The derived model is applied to the mixed-mode I/II single-cantilever beam, the single-leg bending and the universal mixed-mode bending specimens. Beside the present solution three other ones are adopted to compare the results for the compliance and energy release rate of the mentioned specimens. Although the present model shows similar results to the most accurate finite element model, it is simpler and helps us to understand better the mechanisms of the delamination process.

Interlaminar fracture of multidirectional glass/epoxy laminates under mixed-mode I + II loading

The mixed-mode I + II interlaminar fracture of multidirectional glass/epoxy laminates is investigated. Mixed-mode bending (MMB) tests were performed on specimens with delaminations in 0/θ-type interfaces, with θ varying from 0 to 90°. Preliminary three-dimensional finite-element analyses validated the beam theory model (BTM) used for analysing experimental data. The compliances measured are in a good agreement with BTM predictions. The total critical energy release rate Gc varies linearly with the mode II ratio GII/G, although some discrepancies are observed in the high-mode II results for the 0/45 and 0/90 specimens.

Characterization of the interlaminar fracture toughness of a laminated carbon/epoxy composite

Delamination between layers is an important problem in applications of fiber reinforced composite laminates. Tests were carried out to determine the interlaminar fracture toughness of AS4/3501-6 (carbon/epoxy) composite laminates using mixed-mode bending tests. Analysis of the test specimens in terms of mode I and mode II energy release rates showed good agreement between methods based on beam equations, compliance measurements, and detailed finite element analyses. The results showed that the critical mode I energy release rate for delamination decreased monotonically with increasing mode II loading. This is in contrast to some results in the literature. Various analytic representations of the mode interaction from the literature were compared, and shown to fit the data with reasonable accuracy.

Analysis of the mixed-mode end load split delamination test

Composite delaminations are commonly characterized using the double cantilever beam test for mode I, the end-notched flexure test or the end load split test for mode II and the mixed-mode bending test for mixed-mode. For all these tests, the mode mix remains constant and does not vary with the crack length. However, in the mixed-mode end load split test (MMELS), the delamination propagates under a varying mode mix that depends on the crack extension, which is a more realistic scenario. The MMELS test has been previously analysed by different researchers but the resulting expressions are not equivalent. A more accurate alternative analysis of the test, based on the finite element method and the virtual crack closure technique, is used in the present work for comparison. The results are compared to the predictions of approaches presented in the literature and significant findings are found for materials characterization using the MMELS test.

A correction to the analytical solution of the mixed-mode bending (MMB) problem

This paper addresses the analytical solution of the mixed-mode bending (MMB) problem. The first published solutions used a load separation in pure mode I and mode II and were applied for a crack length less than the beam half-span, a⩽L. In later publications, the same mode separation was used in deriving the analytical solution for crack lengths bigger than the beam half-span, a>L. In this paper it is shown that this mode separation is not valid when a>L and in some cases may lead to very erroneous results. The correct mode separation and the corresponding analytical solutions, when a>L, are presented. Results, of force vs. displacement and force vs. crack length graphs, obtained using the existing formulation and the corrected formulation are compared. A finite element solution, which does not use mode separation, is also presented.