Traction-separation Relation Research Articles

Analytical solution for double cantilever beam based on cohesive zone model considering shear deformation

The analytical method can not only solve the deflection of double cantilever beam (DCB) specimens based on cohesive zone model (CZM), but also inversely solve the cohesive constitutive parameters through using only the experimental load displacement data. Where, the governing equation of the Euler-Bernoulli beam based on elastic foundation model (EB-EF) was usually used. Because the governing equation of the Timoshenko beam based on elastic foundation model (EB-EF) is difficult to be applied directly to CZM with the nonlinear constitutive relations. To tackle this problem, a new governing equation of the Timoshenko Beam based on Cohesive Zone Model (TB-CZM) is derived in this work and it is suitable for any traction-separation relations. The validity of the derived analytical solutions is verified by numerical CZM results. The effect of shear deformation is studied within the range of material properties and geometric dimensions in the literature. Furthermore, the cohesive parameters are obtained by inversely solving the derived governing equation. Valid cohesive parameters can be obtained by using the proposed method when the input parameters exist some degrees of uncertainty.

Mode I cohesive law of birch wood-biobased adhesive systems

ABSTRACT The development and assessment of bio-based wood adhesives face challenges due to the limitations of conventional test procedures, particularly in predicting adhesive performance in real-world applications. This study investigated the mode I cohesive law of two birch wood-biobased adhesive systems, comparing them with conventional fossil-fuel-based systems. The experimental approach, developed during the ‘90s at Lund University, involves direct measurement of the traction-separation relation and allows evaluation of post-peak behaviour and fracture mechanical property characterisation. SEM analysis confirmed fracture development within the bond line for all tests. The birch–fish-adhesive bonds demonstrated peak stress of approx. 4 MPa, with serrated fracture surfaces and favourable fracture properties (specific fracture energy approx. 370 Nm/m2, brittleness approx. 40 GPa/m). Birch wood bonded with lignin-based adhesive showed lower, yet reasonable, levels of peak stress (approx. 3 MPa) but less favourable fracture properties (specific fracture energy approx. 110 Nm/m2, brittleness approx. 90 GPa/m), suggesting room for further optimisation.

Numerical and experimental crack-tip cohesive zone laws with physics-informed neural networks

The cohesive zone law represents the constitutive traction versus separation response along the crack-tip process zone of a material, which bridges the microscopic fracture process to the macroscopic failure behavior. Elucidating the exact functional form of the cohesive zone law is a challenging inverse problem since it can only be inferred indirectly from the far-field in experiments. Here, we construct the full functional form of the cohesive traction and separation relationship along the fracture process zone from far-field stresses and displacements using a physics-informed neural network (PINN), which is constrained to satisfy the Maxwell-Betti's reciprocal theorem with a reciprocity gap to account for the plastically deforming background material. Our numerical studies simulating crack growth under small-scale yielding, mode I loading, show that the PINN is robust in inversely extracting the cohesive traction and separation distributions across a wide range of simulated cohesive zone shapes, even for those with sharp transitions in the traction-separation relationships. Using the far-field elastic strain and residual elastic strain measurements associated with a fatigue crack for a ZK60 magnesium alloy specimen from synchrotron X-ray diffraction experiments, we reconstruct the cohesive traction-separation relationship and observe distinct regimes corresponding to transitions in the micromechanical damage mechanisms.

Investigation of a secondary bonded pultrusion composite laminate containing a ply-drop, Part 2: A novel two-dimensional hybrid fracture discrete element for producing debond failure

This is the second of two papers in which a novel numerical method to predict the debond failure of a secondary bonded pultrusion laminate is presented. In Part 1 of this study, experimental work was described which is used here in the development, calibration and validation of the numerical model. The structure investigated in this work may be represented by a bonded composite laminate with external ply-drops (PDs).In this part of the study, a fully coupled mixed mode cohesive zone model is developed which relies upon a traction-separation relation and the virtual crack closure technique to obtain the mode mixity. These are used to develop a hybrid fracture discrete element (HFDE). Fracture toughness tests, as well as tests on PD specimens are used to calibrate the model. Based on these tests, run-arrest fracture criteria are defined. The model is validated by comparing test results for two other PD specimen types to those obtained with the HFDE and finite element analyses.

Temperature-dependent R-curve and traction-separation relation in mode-I fracture of GFRP laminates

The R-curve and fiber bridging phenomenon in mode-I fracture of glass-fiber reinforced laminates at different temperatures are investigated in this study, aiming to reveal their changes with temperature. The mode-I fracture experiments are carried out by adopting double cantilever beam (DCB) configuration at −55 ℃, 23 ℃ and 80 ℃. Fiber bridging is observed during the tests. The R-curve and bridging traction are quantitatively analyzed, from which the relationship between the R-curve and fiber bridging phenomenon, and temperature is obtained. It is found that fiber bridging effect is enhanced with the increase of temperature. The bridging traction of specimens tested at 80 ℃ is significantly higher than that at −55 ℃ and 23 ℃. An R-curve model considering both temperature and fiber bridging effects is proposed. In addition, bilinear and tri-linear traction-separation relations (TSLs) are utilized to establish a numerical model for the simulation of delamination growth behavior with the consideration of the temperature-dependent effect on the mechanical properties of composite materials. When using the bilinear TSL, the fiber bridging is considered by integrating the resulted R-curve into finite element model via a user-defined USDFLD subroutine. Effects of initial interface stiffness, interface strength and viscosity coefficient on simulated results are numerically investigated. Finally, applicability of the established numerical models is illustrated by comparisons between the simulations and the test results.

Bending of fusion-bonded thermoplastic single lap joints

The assembly of thermoplastic composites is one of the most important issues for their reliable use in the aerospace and automotive industries. Assembly through co-curing is a process that enables simultaneous bonding and curing of adherends and is a preferable option for these industries. This paper presents a study of fusion-bonded, co-cured single lap joints with thermoplastic adherends that were subjected to four-point bending. Processing of the joint created a 45° notch at one edge of the overlap. The effect of this notch on the strength of the joint was assessed in tension and compression by flipping the joint. Compared to samples without such notches, experiments indicated that the tensile stresses in the vicinity of the notch decreased the maximum load and applied displacements at failure initiation by about 30 % and 33 %, respectively. In order to better understand this behavior, the joints were also analyzed using finite element simulations. The initial failure was predicted using the Maximum Stress Theory. Both experimental and numerical results were consistent up to the initial failure. The analysis did not account for the subsequently observed interlaminar delamination in the adherends. This type of failure would most likely be approached using measured interlaminar traction-separation relations in a cohesive zone modelling approach.

Reconstruction and prediction of Mode-I cohesive law using artificial neural network

Cohesive zone model (CZM) is a widely used tool for simulating both static and fatigue damage propagation for bonded area in composite materials. Under static loading, the damage propagation behavior is governed by a cohesive law (CL), the shape of which is an important property especially when non-linear damage mechanisms such as fiber bridging is involved. In this work, a CZM driven by artificially neural network (ANN) for simulating mode-I damage propagation is proposed, where the traction-separation relationship is calculated by a multi-layer perceptron (MLP). More importantly, a reconstruction method is proposed to extract the CL through the training of the neural network using a simple load-displacement (P–U) relation of a double cantilever beam (DCB) as the only inputs, without the need to measure the crack opening displacement (COD). The proposed method is first validated using finite element generated virtual experimental data with various CL shapes before being applied to experimental data, where good correlations are obtained, proving the effectiveness of the proposed method.

Mineral asperities reinforce nacre through interlocking and friction-like sliding

While the surface asperities of mineral platelets are widely believed to play important roles in stiffening, strengthening, and toughening nacre, their effects have not been thoroughly investigated. Here, a computationally efficient bar-spring model is adopted to simulate, as platelets with multiple interfacial asperities slide over each other, the tensile force versus elongation behaviors as well as the effective mechanical properties such as modulus, strength, and work-to-fracture in nacre or nacre-like composites. The model employs an effective cohesive law derived from a micromechanical model based on the kinematic and deformation analysis of a single pair of contacting asperities to characterize the traction-separation relationship during the asperity inter-climbing. Strikingly, we find that the mineral asperities and resulting interfacial roughness can elevate the composites’ strength and toughness by up to 2–3 orders of magnitude through a combination of mechanical interlocking and multimodal friction-like mechanisms. Of particular interest is that the asperity-induced strengthening and toughening mechanisms are insensitive to the asperity shapes such as ellipse, hyperbolic cosine, cosine, and parabola. These findings may provide useful guidelines for developing advanced engineering composites with nacre-inspired interface designs.

Fracture behavior of composite piezoelectric intelligent structures formed by pasting MFC based on serial multi-scale method

The composite piezoelectric intelligent structures formed by combining piezoelectric materials with composite materials can achieve state perception and deformation control of the structure. Based on the molecular dynamics method, the continuum finite element method and the cohesive force model, the serial multi-scale research on the bonded composite piezoelectric intelligent structure was carried out from the micro, meso and macro scales. Molecular models of adhesive, macro fiber composite (MFC) surface and composite surface were established. The effects of temperature and pressure on the surface interaction energy of adhesive/MFC and adhesive/composite materials were analyzed. The traction-separation (T-S) relationship of cohesive region was obtained by virtual stretching method. Combined with the microscopic observation of the mesoscopic adhesive layer, the representative volume element (RVE) model of the mesoscopic adhesive layer considering the bubble defect was established, and the cohesive relationship curve of the adhesive layer was obtained. A double cantilever beam (DCB) macro finite element model was established to simulate the crack propagation behavior of the adhesive layer, and compared with the experimental results.

A novel experimental technique to determine the tack development during automated placement of uncured thermoset carbon/epoxy tows

In-situ bond strength toughness (IBST), commonly referred to as tow tackiness, is a first order property affecting the adherence quality and defect formation (e.g., wrinkles, folds) during automated tow placement (ATP) processing of uncured thermoset polymer matrix composite (PMC) tows. Tow-tow IBST develops over characteristic millisecond timescales (tc≤50ms) due to the rapid tow placement velocity of ∼1 m/s. In this paper, an experimental technique is presented to determine millisecond timescale tow-tow mode-I IBST in terms of fracture mechanics-based traction-separation relationship. The test specimen consists of two tows bonded to rigid platens with a pre-crack between them to induce crack initiation. Experiments are performed using IM7-G/8552 carbon/epoxy prepregs for both long and short timescales at a constant debonding rate of 5 mm/s, contact pressure of 0.23 MPa, and bonding temperature of 40 °C. In addition, high-speed two-dimensional digital image correlation (2D-DIC) is used to image and measure the tow-tow interfacial deformation during debonding. The peak traction and apparent energy release rate are found to significantly decrease (about 60 %) at the short millisecond timescale contact hold times compared to longer (i.e., second) timescale.

Rate-dependent augmented finite element method for fast arbitrary crack growth

Experimental observations indicate that fast crack growth often exhibits a rate dependency, which is caused by the viscosity of the material and the dissipation of energy resulting from the branching of microcracks. To address this phenomenon in simulation modeling, this study introduces a rate-dependent augmented finite element method designed for analyzing fast arbitrary crack growth. The method utilizes the maximum principal stress criterion to simulate arbitrary crack growth and predict the path of crack growth. The elements where crack growth is expected are divided into subelements, and they are re-assembled by incorporating rate-dependent cohesive elements between them to construct augmented finite elements. The cohesive elements are based on a rate-dependent trapezoid traction–separation relationship, where the critical traction and fracture toughness are modeled as functions of the separation rate and crack growth speed. The augmented finite elements only contain external nodal degrees of freedom (DoFs), while internal nodal DoFs are condensed. They can be utilized as conventional elements. A numerical algorithm based on the proposed method was developed in this article, and it was implemented in ABAQUS via user element subroutine. The effectiveness of this method and algorithm is validated through an example of dynamic crack growth in a double cantilever beam. The results demonstrate agreement with experimental and computational findings reported in the literature. Additionally, the method was applied to other structures, yielding reasonable outcomes. Analysis of these examples confirms the method’s utility in predicting fast arbitrary crack growth under dynamic load conditions.

Using Machine Learning and Finite Element Analysis to Extract Traction-Separation Relations at Bonding Wire Interfaces of Insulated Gate Bipolar Transistor Modules.

For insulated gate bipolar transistor (IGBT) modules using wire bonding as the interconnection method, the main failure mechanism is cracking of the bonded interface. Studying the mechanical properties of the bonded interface is crucial for assessing the reliability of IGBT modules. In this paper, first, shear tests are conducted on the bonded interface to test the bonded interface's strength. Then, finite element-cohesive zone modeling (FE-CZM) is established to describe the mechanical behavior of the bonded interface. A novel machine learning (ML) architecture integrating a convolutional neural network (CNN) and a long short-term memory (LSTM) network is used to identify the shape and parameters of the traction separation law (TSL) of the FE-CZM model accurately and efficiently. The CNN-LSTM architecture not only has excellent feature extraction and sequence-data-processing abilities but can also effectively address the long-term dependency problem. A total of 1800 sets of datasets are obtained based on numerical computations, and the CNN-LSTM architecture is trained with load-displacement (F-δ) curves as input parameters and TSL shapes and parameters as output parameters. The results show that the error rate of the model for TSL shape prediction is only 0.186%. The performance metric's mean absolute percentage error (MAPE) is less than 3.5044% for all the predictions of the TSL parameters. Compared with separate CNN and LSTM architectures, the proposed CNN-LSTM-architecture approach exhibits obvious advantages in recognizing TSL shapes and parameters. A combination of the FE-CZM and ML methods in this paper provides a promising and effective solution for identifying the mechanical parameters of the bonded interfaces of IGBT modules.

A new mechanism based cohesive zone model for Mode I delamination coupled with fiber bridging of composite laminates

Based on identification of the two distinguishing delamination mechanisms within the two delamination zones associated with Mode I fracture toughness testing of composite laminates using the well-known ASTM standard double cantilever specimen (DCB), a new mechanism based cohesive zone model (MB-CZM) is proposed in this work. Overcoming the limitations with the widely used superposed cohesive zone models, the proposed MB-CZM develops two traction-separation relations to individually represent the two distinctive delamination mechanisms. One for the quasi-brittle linear elastic behavior of composite material and another for the nonlinear characteristics of fiber bridging which is commonly simplified with tri-linear to multi-linear approximation in the previous cohesive zone models (CZMs). Energy decomposition is carried out based on different damage and toughening mechanisms associated with delamination initiation and propagation. The proposed new MB-CZM is implemented in the finite element analysis via two UMAT subroutines and used in the numerical simulations. The good agreement of the simulation results with the experimental results provides the verification and demonstration of the capabilities of the proposed MB-CZM.

Cohesive Zone Modelling and the Fracture Process of Cement-Based Composites

Computational modelling of quasi-brittle fracture in cement-based composites needs to cover both i) the damage caused by micro-fractured zones, referring to some nonlocal strain-stress relations, respecting quite different behaviour of such composites in tension and compression, and ii) the initiation and propagation of macroscopic cracks, exploiting the cohesive zone model, handled by some modification of the finite element technique, together with the discretization in time. A fundamental issue for such model is the introduction of a traction-separation (stress-displacement) relationship. This contribution pays particular attention to the design, identification and estimation of material parameters for the traction separation law suitable for predicting the deformation behaviour of samples of materials and structures.

A numerical modelling framework for simulating rate-dependent delamination along polyurea/steel interfaces under Mode–I loading conditions

Polyurea coated structures have drawn significant attention from researchers across the globe for various applications such as waterproof roofing, vibration control, and protection from impact and blast threats. Despite widespread studies on the rate-dependent nature of polyurea and its bulk deformation response, there are very few works which study the delamination in polyurea coated structures and the associated rate sensitivity. The present work is aimed at two objectives: (i) to develop a suitable rate-dependent traction–separation model that fits an existing experimental dataset from literature related to rate-dependent delamination of polyurea/steel interface and, (ii) to develop a simple, robust numerical modelling framework to simulate the rate-dependent delamination process in a double–cantilever beam experiment found in the same literature source. Fulfilling the first objective resulted in identifying the important parameters in a traction–separation relation and their rate-dependency. A non-zero traction at zero crack opening displacement is one of the striking features of the experimental data and the proposed model takes that into account. As part of the second objective, a numerical modelling involving separate constitutive equations for the bulk and interface regions of the polyurea adhesive has been proposed. The interface is modelled as a rate-dependent linear elastic plastic material while the bulk is considered as hyperelastic material using two parameter Mooney–Rivlin strain energy function. This model was implemented in LS-DYNA. The proposed approach could reproduce the fracture mechanisms of the dynamic delamination process reported in the literature and helped in understanding the interaction between the bulk and the interface during the delamination at different strain rates. This approach could be a potential alternative to the cohesive zone method, easing the complexity of simulations involving dynamic delamination in commercial packages.

Fracture Energy Measurement in Different Concrete Grades

Fracture energy is regarded as an intrinsic (material) properties that dominates crack mechanisms and associated crack growth in concrete damage under applied stress. In recent times, significant advancements in computing technology have driven the adoption of finite element analysis (FEA) methodologies that necessitate the integration of constitutive models, includingthe traction-separation relationship derived from cutting-edge fracture mechanics. A physically-based model requires fracture energy values; therefore, a properly measured fracture energy value is essential to exhibit better structure response within FEA models. There are large arrays of parameters involved during the concrete mixture, such as beam size effect, aggregate size, and concrete grade, that affect the flexural resistance of the concrete. The fracture and failure in concrete ahead of the crack tip are represented by fracture energy values where micro-damage events such as interfacial failure, fiber-bridging, and matrix cracking occurred.This study aims to determinethe fracture energy of concrete specimens with combination of notch depth aoat mid-span, design concrete strength as specified in the testing series. Independent compression strength, fcand measured load-displacement profiles underathree-point bendingtest were used to determine fracture energy by incorporating three available fracture energy expressions such as Bazant, Hillerborg,and CEB-FIP models.

Multiscale investigation on bitumen-aggregate interfacial debonding using molecular dynamics and finite element method

Adhesive failure between bitumen and aggregates is the primary cause of cracking in asphalt mixtures and pavement moisture damage. The resulting pavement distress poses a significant challenge to the service durability of the road infrastructure. Previous studies have validated that aggregate mineralogy has a significant influence on the adhesion properties between bitumen and aggregates; however, there is still a lack of a mechanistic understanding of the influence of the mineral constitution and distribution on the interfacial debonding behavior. This study aims to fill this knowledge gap by proposing a multiscale approach based on molecular dynamics (MD) simulations and the finite element method (FEM). First, the MD method was employed to extract the traction-separation curve of the interface between bitumen and minerals at a molecular scale. The obtained traction-separation relationship was then utilized as the input parameter for the zero-thickness cohesive element in the microscale bitumen-aggregate finite element model. The combined simulation results indicate that quartz with bitumen has the best cracking resistance at the molecular scale, followed by albite, mica, pyroxene, and anorthite. Among the three selected aggregates, the greywacke aggregate exhibited the best cracking resistance at the microscale level. Moreover, the distribution microstructure of the minerals was found to affect the cracking path of the bitumen-aggregate interface, and the bitumen-mica interface elements at the junction area with quartz have a late debonding. The proposed research method spearheads the prediction of multiscale bitumen-aggregate interfacial behavior by considering the aggregate mineralogical heterogeneity.

A rate-dependent cohesive zone model for simulating fast crack evolution and growth

This paper proposes a rate-dependent cohesive zone model for simulating the evolution and growth of cracks under dynamic loading. This model takes into consideration the effects of cohesive separation rate and crack growth speed on the cohesive traction-separation relationship to reflect the rate-dependent characteristics of both material viscosity and energy dissipation of micro-crack branches. The evolution algorithm of the cohesive law is presented, wherein separation rate and crack growth speed are evaluated using the backward difference method explicitly in incremental computation, leading to the determination of rate-dependent critical traction and fracture toughness. The proposed model was implemented in ABAQUS software via UEL subroutine and utilized in an example of interfacial crack growth simulation, with satisfactory agreement found among the present results, experimental data, and numerical results from the literature. The effectiveness of the rate-dependent model and numerical algorithm in simulating high speed crack growth is validated through further numerical tests.

Experimental investigation on translaminar fracture behavior of cross-laminated bamboo

Cross-laminated bamboo (CLB) is a bamboo-based construction product with bamboo's advantages, which are green, environmentally protective, and sustainable. However, bamboo is easy to crack. Structural elements in bamboo constructions are assembled onsite, making splitting cracks easily form at joints and bolt connections. Therefore, it is essential to assess cracked structure safety with fracture mechanics. However, rare studies are addressed on fracture properties of bidirectional laminated bamboo, especially the translaminar fracture resistance, which represents the ability to arrest a catastrophic crack in structures. This paper studies the translaminar fracture properties of CLB composite using three-point bending beam tests. Load-displacement and load-CMOD curve characteristics, fracture process, and the failure modes of bidirectional laminated bamboo are analyzed. The R-curve expressed by the energy release rate is calculated by three methods. Results demonstrate that the R-curves obtained by the three different methods show similar results. The methods to determine R-curves employed in this study are simple and can describe the translaminar fracture toughness of laminated bamboo. The methods are validated by comparing the traction-separation relations calculated from the R-curves to those measured by double-edge notched tests (DENT). A comparison of the bidirectional laminated bamboo's energy release rate to the matrix-dominated energy release rate of unidirectional laminated bamboo reveals that bidirectional fiber arrangement has a remarkable improvement on the toughening of laminated bamboo at and after crack initiation. Moreover, the lay-up effects of CLB are analyzed by relating the energy release rate results to the failure pattern of fractured specimens. It is found that fiber pullout/bridging caused by 90°direction crack propagation around fiber bundles is the dominant toughening mechanism of CLB. The lay-up sequence of 0° and 90° plies also has an effect on 90°direction crack propagation; therefore, it will influence the toughness of CLB.

Utilizing XFEM model to predict the flexural strength of woven fabric Kenaf FRP plate strengthened on plain concrete beam

The evolution of advancing computing technology has encouraged the use of finite element analysis to require constitutive models, mostly adopted extensive experimental datasets. Nevertheless, fewer material properties are required within the bilinear traction-separation relationship incorporated with the XFEM technique and require fewer computation efforts due to the energetic approach simulation. This study used ABAQUS CAE 2021 to predict the flexural strength of plain concrete beams strengthened with Kenaf Fibre Reinforced Polymer (KFRP) plates under a four-point bending test, later validated with experimental datasets. The varying parameters, including woven fabric architectures, lengths, widths, and plate thickness, were considered. The results demonstrated that the consistency proposed modelling technique in this study well-captured failure modes and crack development as experimental observations. All models demonstrated an increase in peak load, and a comparison of load-deflection profiles was made between the numerical FEA modelling with the experimental works. The peak load discrepancies between the numerical outputs and the experimental datasets were found to be less than 12% in all testing series. Despite promising findings, stress analysis on peel and shear stresses associated with failure modes exhibited was performed. It was found that KFRP ruptures occurred due to peel stress peak at plate mid-span, while shear mode failure demonstrates concentrated peel stress (to lesser extent shear stress) at KFRP edge tip. Hence, a more unified approach is promoted to require only two material properties (preferably independently measured values). This approach enables a designer to choose the optimum FRPs size using the current modelling tool, substantially reducing laborious and expensive experimental setup.