Symmetric Three-point Bending Research Articles

Combined ANN-FEM approach for spatial–temporal structural response prediction: Method and experimental validation

The prediction of the mechanical behavior and possible failure mechanisms of engineering systems is essential in the design process. It is a challenging task in view of numerous influencing parameters, various unavoidable uncertainties and time-consuming numerical analyses based on complex finite element models. Therefore, a combined hierarchical ANN-FEM approach is proposed in this contribution. By means of this approach, the advantages of both simulation methods are used to predict uncertain spatial–temporal structural responses with less effort and comparable accuracy. A non-linear stability analysis of a three-dimensional carbon fiber reinforced plastic (CFRP) structure is conducted to illustrate the efficiency of the combined ANN-FEM approach. The individual components space, time and uncertainty are investigated in detail regarding their influence on the ANN architecture and complexity. The numerical predictions of the buckling behavior and the material failure have been compared then to experimental values measured in a symmetric three-point bending test. A sensitivity analysis is finally conducted which clarifies the strong dependence of the outcomes for the ultimate limit state on the fiber volume content, the structural thicknesses and the stiffness in fiber direction.

Combined diffused material interface and hybrid phase-field model for brittle fracture in heterogeneous composites

In this article, we propose a novel approach for modeling brittle fracture in heterogeneous composites using a combined diffused material interface method and a hybrid phase-field model (PFM). We have used the diffused material interface method to derive analytical expressions for the regularized material properties across the material interfaces that are treated as an input to the PFM for the simulation of brittle fractures in composites. The use of diffused material interface method regularizes the material properties in such a way that one can avoid the discontinuity in stress across the material interface and thus bypass the requirement of explicit tracking of material interfaces for finite element implementation of the hybrid PFM. We have implemented the proposed combined model using a recently developed open-source finite element toolbox, Gridap in Julia. We have considered tests on a single circular fiber inclusion under tension, a bi-material rectangular plate under tension, single square inclusion under tensile loading, a notched composite beam with a central notch under symmetric three-point bending, and a rectangular plate with multiple fibers under tensile loading to show how the proposed combined model can be used to simulate brittle fracture in composites.

A criterion for fatigue failure mode evaluation of U-rib to deck joints

In this study, a numerical analysis model was established, on basis of the traction structural stress method, for fatigue failure mode prediction of orthotropic bridge deck (OBD) under symmetric three-point bending conditions. The structural stress-based criterion was validated by comparison with a collection of 64 test results. Fatigue failure modes are found to hinge on a dimensionless normalized constraint distance l/lcr that is related to specimen geometric dimensions and supporting arrangements. OBD specimens in symmetric three-point bending fatigue tests fail at toe-U-rib and external toe-deck as the actual constraint distance l is lower and higher than the critical constraint distance lcr where fatigue failure mode changed. The prediction power of the evaluation criterion is not affected by the fatigue load patch width in fatigue tests, U-rib profile shape, or weld penetrations.

U-notch fracture in additively manufactured ABS specimens under symmetric three-point bending

This paper aims to investigate the fracture behavior of U-notched semi-circular bend (USCB) specimens fabricated from the Acrylonitrile Butadiene Styrene (ABS) using the fused deposition modelling (FDM) method under opening-mode loading. The notched specimens are manufactured with different notch tip radii and various unidirectional layup configurations for which the fracture loads are experimentally acquired by conducting fracture tests under symmetric three-point bending. Some additional tests are also performed on cracked and plain tensile specimens to obtain the fracture toughness and tensile properties of the ABS material. With the goal of estimating the fracture loads theoretically, the recently proposed virtual isotropic material concept (VIMC) is linked to three different brittle fracture models, namely the point-stress (PS), mean-stress (MS), and averaged strain energy density (ASED) criteria. Comparing with the experimental results, it is shown that the VIMC-MS criterion is generally the best model among the three ones, while VIMC-PS criterion gives fair results and VIMC-ASED criterion is excellent only for larger notch tip radii. The crack growth path of the mentioned samples as well as their type of failure is also evaluated. It is revealed that the unidirectional USCB samples with 0, 30, and 45 degrees layup configurations experience inter/cross-laminar fracture, while those with 60 and 90 degrees layups experience inter-laminar fracture.

An open-source implementation of a phase-field model for brittle fracture using Gridap in Julia

This article proposes an open-source implementation of a phase-field model for brittle fracture using a recently developed finite-element toolbox, Gridap in Julia. This work exploits the advantages of both the phase-field model and Gridap toolbox for simulating fracture in brittle materials. On one hand, the use of the phase-field model, which is a continuum approach and uses a diffuse representation of sharp cracks, enables the proposed implementation to overcome such well-known drawbacks of the discrete approach for predicting complex crack paths as the need for re-meshing, enrichment of finite-element shape functions, and an explicit tracking of the crack surfaces. On the other hand, the use of Gridap makes the proposed implementation very compact and user-friendly that requires low memory usage, and provides a high degree of flexibility to the users in defining weak forms of partial differential equations. Tests on a single-edge notched plate under tension, an L-shaped panel, a notched plate with a hole, a notched beam under symmetric three-point bending and a notched beam with three holes under asymmetric three-point bending are considered to demonstrate how the proposed Gridap-based phase-field Julia code can be used to simulate fracture in brittle materials.

A phase field model with the mixed-mode driving force of power-law relation

A new phase field model is proposed through introducing the mixed-mode critical energy release rate (ERR) following the power-law relation as the crack driving force. Compared with the classical phase field models involving only mode-I critical ERR GIC, the present phase field model takes mode-II critical ERR GIIC into account and introduces two power-law factors. As a result, a pure mode-II crack propagation can be observed in the present simulation of a single material under shear loading when the mode-I critical ERR is significantly larger than the mode-II one. A phase field simulation on a symmetric three-point bending semi-circular Brazilian disc specimen of polymethylmethacrylate shows that a better agreement with experimental results can be achieved through adjusting the power-law factors. Then, a shear test of a single edge notched plate is simulated and the results show that: (i) The peak load decreases gradually as either power-law factor increases. (ii) The mixity factor (GIIC/GIC) affects not only the cracking mode but also the brittle or tough extent. With the increase of the mixity factor, the initial crack deflection angle converges to a constant value of 70.8±0.5°, which is close to 70.5° predicted by the classical maximum tangential stress criterion. (iii) The energy split methods have an obvious effect on the crack propagation behaviors. For the volumetric-deviatoric split method, a pure mode-II crack propagation is observed when the mixity factor approaches zero and a pure mode-I crack propagation occurs when the mixity factor is no less than 4.0. For the spectral decomposition split method, a mode-I crack propagation always occurs regardless of the mixity factor. To summarize, with the change of the power-law factors and critical ERRs, diverse cracking modes can be predicted by the present phase field model.

Comparison of Testing Method Effects on Cracking Resistance of Asphalt Concrete Mixtures

As an inherent characteristic of materials, the fracture toughness is an important parameter to study the cracking behavior of asphalt concrete mixtures. Although material compositions and environmental conditions have a significant effect on the fracture toughness, for a certain material and testing environment, the test condition including the specimen configuration and loading type may also affect the obtained fracture toughness. In this paper, the effect of specimen configuration and applied loading type on the measured pure mode-I fracture toughness (KIc) is investigated. In order to achieve this purpose, using a typical asphalt mixture, four different test specimens including Semi-Circular Bend (SCB), Edge Notch Disc Bend (ENDB), Single Edge Notch Beam (SENB) and Edge Notch Diametral Compression (ENDC) disc are tested under pure mode I. The mentioned specimens have different shapes (i.e., full disc, semi-disc and rectangular beam) and are loaded either with symmetric three-point bending or diametral compressive force. The tests were performed at two low temperatures (−5 °C and −25 °C) and it was observed that the critical mode-I fracture toughness (KIc) was changed slightly (up to 10%) by changing the shape of the test specimen (i.e., disc and beam). This reveals that the fracture toughness is not significantly dependent on the shape of the test specimen. However, the type of applied loading has a significant influence on the determined mode I fracture toughness such that the fracture toughness determined by the disc shape specimen loaded by diametral compression (i.e., ENDC) is about 25% less than the KIc value with the same geometry but loaded with the three-point bending (i.e., ENDB) specimen. In addition, the fracture toughness values of all tested samples were increased linearly by decreasing the test temperature such that the fracture toughness ratio (KIc (@−25 °C)/KIc (@−5 °C)) was nearly constant for the ENDB, ENDC, SCB and SENB samples.

Fracture Behavior of Two Biopolymers Containing Notches: Effects of Notch Tip Plasticity

This paper analyzes the notch effect on the fracture behavior of two biomaterials (a brittle bone cement and a ductile dental material) under mode I loading. U-notched Brazilian disk (UNBD) specimens of both materials were tested under remote compression, determining the corresponding fracture loads and load-displacement curves. Additionally, cracked rectangular and semicircular bend (SCB) specimens were tested under symmetric three-point bending in order to determine the fracture toughness of the two materials. Then, fracture loads were derived theoretically by applying the maximum tangential stress (MTS) and the mean stress (MS) criteria. Due to the brittle linear elastic behavior of the bone cement material, the MTS and MS criteria were directly applied to this material; however, given the significant nonlinear behavior of the dental material, the two fracture criteria were combined with the Equivalent Material Concept (EMC) for the fracture analyses of the dental material specimens. The results reveal a very good accuracy of both the MTS and the MS criteria for the fracture analysis of bone cement notched specimens. In the case of the dental material, very good results are also obtained when combining the MTS and the MS criteria with the EMC. The proposed approach can be useful for the fracture analysis of a wide range of biopolymers, from brittle to ductile behavior.

Sub-sized short bend beam configuration for the study of mixed-mode fracture

In this paper an angled edge cracked short beam specimen subjected to symmetric three-point bend loading was designed and examined for conducting mixed-mode I + II fracture toughness experiments. This specimen is suitable for being used in those experimental crack growth studies in which a very limited amount of raw material is available for the preparation of the test specimen. By conducting several finite element analyses, it was shown that the short bend beam specimen is able to produce full modes I and II mixities if the affecting parameters such as the crack length ratio, loading span ratio and crack inclination angle are set in suitable ranges. The practical ability of the short bend beam specimen was investigated for obtaining the mixed-mode I + II fracture toughness of bovine bone and the corresponding values of KIc and KIIc were determined for different mode mixities. While a significant discrepancy was observed between the experimental results and predictions of the conventional singular based fracture theories, it was demonstrated that the two-parameter (K &T) based fracture models such as GMTS and EMTSN criteria can provide significantly improved predictions for the fracture behavior of tested short bend beam specimen made of bone material.

On the use of edge cracked short bend beam specimen for PMMA fracture toughness testing under mixed-mode I/II

In this paper an inclined edge cracked short beam specimen subjected to symmetric three-point bend loading was designed and examined for conducting mixed-mode I/II fracture toughness experiments. The aspect ratio (i.e. length to width ratio) and the loading span distance are considered much lower than the other conventional cracked bend beam samples. Crack tip parameters such as stress intensity factors and T-stress were computed numerically for this specimen by several finite element analyses and it was demonstrated that the specimen is able to produce full combinations of mode I and II including pure mode II. The practical capability of the short bend beam specimen was studied experimentally by conducting a set of mixed-mode fracture tests on PolymethylMethacrylate (PMMA) as a well-known model brittle material. The critical stress intensity factors, the direction of fracture kinking and the path of fracture trajectory were investigated both experimentally and theoretically using two stress and strain-based fracture criteria. The fracture toughness of tested PMMA was decreased by moving towards mode II case due to the effect of T-stress on the fracture mechanism of the short bend beam specimen.

System transverse in-plane shear stiffness of pultruded GFRP bridge decks

The static transverse behavior of two pultruded GFRP deck systems with trapezoidal (DS) and triangular (AS) cell cross-sectional geometry was experimentally investigated in order to study their transverse in-plane shear stiffness. Symmetric three-point bending experiments up to failure were performed on 200-mm-wide beams. Their stiffness, strength and failure modes were compared. Different load transfer mechanisms were found in the DS (frame-dominated) and AS (truss-governed) systems depending on the cell geometry. The DS beams exhibited a lower apparent bending stiffness (24–30 times less) and degree of composite action between the flanges (14–17 times less) than the AS beams. These dissimilarities were attributed to the lower transverse in-plane shear stiffness provided by the trapezoidal core than by the triangular core. The low bound values for both system in-plane shear moduli were estimated from the experimental deflection results. The system in-plane shear modulus of the DS beams represented approximately 2–3% of that of the AS beams.

Fracture behaviour of epoxy biocomposite reinforced with short coconut fibres (Cocos nucifera) and walnut particles (Juglans regiaL.)

Mixed-mode fracture characteristics of epoxy-based biocomposite reinforced with 20 wt% walnut shell particle and 10 wt% coconut fibres are investigated. The biocomposite is fabricated using the squeeze casting method. The positive aspect of hybrid combination of fibre and particle reinforcement is advocated by comparing mode I, mode II and mixed-mode I/II fracture surfaces under a scanning electron microscope. An edge-cracked semicircular arc specimen subjected to symmetric three-point bend (TPB) loading is suggested for fracture toughness testing of biocomposite material. A series of fracture tests are conducted on hybrid biocomposite using the proposed semicircular bend (SCAB) specimen geometry, TPB and four-point bend (FPB) specimens. The average mode I and II fracture toughness obtained from semicircular arc bend (SCAB) specimen are 1.319 MPa and 1.219 MPa [Formula: see text], respectively. The average values of modes I and II toughness obtained from TPB and FPB specimens are 1.267 and 0.754 MPa [Formula: see text]. The overall ratio of mode I and mode II fracture toughness is found to be 0.762, which is very close to the predicted value of 0.87 by maximum tangential stress criterion. The results obtained from new SCAB geometry are validated with the results obtained from TPB and FPB specimen geometries by statistical significance test. Very good agreement is found between the experimental results obtained from TPB, FPB and the proposed SCAB specimens.

Elastic and anelastic properties of α- and b-quartz single crystals

The anisotropic complex Young’s modulus of synthetic quartz single crystals is investigated, using a symmetric three-point bending experimental setup in a Dynamic Mechanical Analysis (DMA) testing system. The storage modulus and the dissipation modulus are examined at ambient pressure between room temperature and 620°C, across the α - β phase transition (573°C). The moduli are determined from measured stress, strain and phase lag data of oriented single crystals at five to ten distinct frequencies between 0.5 and 20 Hz. Quartz shows a strong frequency and temperature dependence of its complex mechanical properties, especially in the vicinity of the α - β phase transition. Around 573°C, the storage modulus steadily increases with frequency, while the dissipation modulus reaches a maximum between 1 and 2 Hz. The observed elastic and anelastic behaviour can be well described by the Poynting-Thomson model, yielding low- and high-frequency limits of the Young’s modulus. This accounts for differences among published data obtained with different experimental techniques, where inconsistent Young’s moduli are reported for similar temperature conditions, especially close to the α - β phase transition. Softening of the Young’s modulus of α -quartz with increasing temperature is accompanied by a decrease of the dynamic viscosity over several orders of magnitude and can be described by two Arrhenian processes. Using published shear moduli and the derived dynamic viscosity, mean relaxation times are derived, which decline from > 1700 s to c -axis and from > 700 s to 0.5 s perpendicular to the c -axis of the crystal, when increasing the temperature from 59°C to 571°C. The ratio of storage to dissipation modulus corresponds to the seismic quality factor Q , a measure of the attenuation of seismic waves. It exhibits a strong frequency and temperature dependence and reaches a minimum at the α - β phase transition. Longitudinal sound wave velocities show significant dispersion of >6 % parallel and >14 % perpendicular to the c -axis of the crystal at 571°C, between 1 and 20 Hz. The characteristics of both attenuation and dispersion of seismic waves close to the α – β phase transition could be used as an in situ temperature probe for quartz-rich crustal rocks.

Experimental Fracture Toughness Study for some Modified Asphalt Mixtures

The use of modifiers or additives in the asphalt mixtures is a suitable and common method for improving their mechanical properties especially under high temperature service conditions. However, for cold climates which the pavement of roads usually experience subzero temperatures, the overall failure mechanism of asphalt layers may be occurred mainly due to elastic brittle fracture and growth of initiated cracks inside the pavements. Fracture toughness is the most important parameter for characterizing the crack growth and failure of cracked materials and structures such as the asphalt pavements. Hence, the main aim of this research is to study the effect of different additives including Poly phosphoric acid (PPA), Styrene butadiene styrene (SBS), Anti striping agent (ANTI), Crumb rubber (CR) and FT-paraffin wax (Sasobit) on the low temperature mode I fracture resistance of asphalt mixtures. A series of asphalt samples with different percentages of the mentioned additives were manufactured in the shape of semi circular specimens containing vertical edge cracks. The test samples were then loaded monotonically using a symmetric three-point bend fixture at a constant subzero temperature of-15°C. The value of mode I fracture toughness (KIc) were determined by recording the critical fracture loads of tested specimens. It is shown that all the investigated additives increase the low temperature fracture toughness of the asphalt mixture and the maximum increase in the value of KIc occurs when the sasobit and CR additives are used.

Ultimate Bending Strength Evaluation of U-Notched Ductile Steel Samples Under Large-Scale Yielding Conditions

The load-bearing capacity of several U-notched specimens, reported in literature, made of very ductile steel and loaded under symmetric three-point bending were theoretically estimated by using the Equivalent Material Concept (EMC), proposed originally by the author, combined with two well-known brittle fracture models namely the mean stress (MS) and the point stress (PS) criteria. The results revealed that the MS-EMC model with an accuracy of more than 90% was more efficient than the PS-EMC model with about 72% accuracy. By using the MS-EMC model, one can predict well the onset of mode I cracking in U-notched ductile components under large-scale yielding conditions without requiring conducting elastic-plastic analyzes or using ductile fracture criteria.

Damage accumulation in specimens with edge crack in the prefracture region under nonstationary few-cycle loading

We present the results of experimental studies of strain and fracture of the aluminum alloy D16-T and titanium alloy VT5-1 under nonstationary few-cycle loading by a symmetric three-point bending and consider several loading conditions under which the load is gradually increased or decreased. The parameter determining the damage accumulation process is found. Special attention is paid to the distinction between the damage accumulation mechanisms in titanium and aluminum alloys. It is shown that, as a rule, the damage accumulation is of nonlinear character and significantly depends on the type of the stepwise variation in the load (increase or decrease). In experiments, we also succeeded in relating the characteristic point on the “deflection-force” graph in the case of single loading to a similar point on the graph constructed for a few-cycle loading, which permits predicting the structure survivability in a few-cycle loading from a single loading.

Sampling Moiré Method for Accurate Small Deformation Distribution Measurement

In this paper, a novel accurate deformation distribution measurement technique by using sampling moire method is proposed. The basic principle and an experimental result of a steel beam in symmetric three-point bending are reported. In this method, the measurement area of a target is attached with an adhesive tape of a known pitch grating firstly. An ordinary CCD camera is installed on a fixed point to record the image during deformation. The captured image is analyzed by performing easy image processing, i.e., thinning-out and linear interpolation, to obtain the multiple phase-shifted moire patterns. Then, the phase distribution of the moire pattern can be calculated using phase-shifting method. Finally, the deformation distribution is calculated by the grating pitch times the phase difference of before deformation and after deformation. The experimental results in symmetric three-point bending test show that the displacement of the steel beam at loading point agree well with those obtained by an accurate displacement sensor. The average error of displacement measurement is less than 4 μm when 2 mm grating pitch is used, and it corresponds to 1/500 of the grating pitch accuracy. This indicates that noncontact deformation distribution measurement is possible by simple and easy procedure with high accuracy, high speed, and low cost for the structural evaluation of infrastructures.

A numerical study of inclusion–matrix debonding in the presence of a nearby crack

Debonding of an inclusion from the surrounding matrix in the presence of a nearby crack tip is studied numerically. Finite element models of symmetric three-point bend configurations are implemented in conjunction with a stiffness degradation method and the maximum tensile stress criteria to investigate the influence of debonding on crack tip parameters. The geometry considered is a single edge cracked beam having a symmetrically positioned stiff or soft cylindrical inclusion ahead of the crack tip. The numerical model is first validated by interferometric measurements on an edge cracked epoxy specimen with a glass inclusion. The measured quantities namely, the crack mouth opening displacement, crack mouth compliance, energy release rate, dominant strain, are all successfully captured by the adopted numerical methodology, before and after the inclusion debonds from the matrix. Subsequently, the effects of parameters such as the distance between the crack tip and inclusion center ( L), the inclusion diameter ( d), and the Young’s modulus ratio between the inclusion and the matrix, are studied using the model. A stiff inclusion of constant ( d/ L) ratio debonds from the matrix at higher load levels when the inclusion interface is farther away from the crack tip. The increase in the crack driving force due to debonding is the highest when the inclusion proximity parameter ρ is approximately 0.4 and it decreases when ρ is increased or decreased relative to this value. However, when d/ L ratio is varied, higher crack driving forces due to debonding are observed for larger size inclusions due to a greater loss of crack tip shielding and reinforcement. The influence of the modulus ratio ( E i/ E m) due to debonding is most prominent in the range 0–1 for fixed d/ L and ρ values. Additionally, a stiffer inclusion tends to debond from the matrix at lower loads for a constant interfacial strength.

Evaluation of interfacial crack growth in bimaterial metallic joints loaded by symmetric three-point bending

Bimetallic three-point bend specimens (TPB) were produced from explosive clad alloy aluminium LY12/pure aluminium Al by electron beam welding of homogeneous Al and homogeneous LY12 to the respective surfaces. Cracks were introduced at the interface in the specimens by milling and spark-erosion. Similar bimetallic specimens of explosive clad LY12/steel were also prepared. Finite element analyses were undertaken to evaluate the J-integral and the deformation ahead of the bimetallic interfacial crack tip field of the TPB specimens loaded normally. The calculations showed that, even for symmetric TPB loading of the specimens, a sharpening–blunting deformation existed on the profile of the interfacial crack tip for bimetals with a mismatch in mechanical properties. J-resistance curves for two kinds of bimetal interface were obtained from symmetric TPB tests. The results demonstrated that the resistance to crack propagation along the bimetallic interfaces was always lower than those of the respective counterpart metals.