Critical Stress Intensity Factor Research Articles

Evaluation of fracture toughness properties of polymer concrete composite using deep learning approach

AbstractUsing artificial intelligence‐based methods in predicting material properties, in addition to high accuracy, saves time and money. This paper models and predicts the fracture toughness properties of polymer concrete (PC) composites using the deep learning method. After preparing a database consisting of 209 experimental data from 19 relevant studies, the effect of seven important variables on critical stress intensity factor (KIc) and crack tip opening displacement (CTOD) is considered. Then, the deep neural network (DNN) model is developed and trained using the prepared database. The accuracy of the DNN model is examined by implementing four statistical criteria, MSE, R2, RMSE, and MAE. Finally, the sensitivity of the KIc and CTOD to each input variable is evaluated using a partial dependence plot (PDP) analysis. While aggregate size, nanofiller content, and a/R ratio have the most positive effect on KIc, aggregates and nanofiller content have the most positive influence on CTOD.

Effect of crack length on fracture behavior of particleboard

The effect of crack length on the fracture behavior of particleboard was investigated using the single-edge-notched bending (SENB) test method under mode I loading. The initial slope (kinit), critical stress intensity factor (KIC), specific fracture energy (Gf), and brittleness number were calculated for five different crack length/specimen width (a/W) ratios varying from 0.1 to 0.9 at intervals of 0.2. The results show that the fracture properties were significantly higher for specimens with an a/W ratio of 0.1 than for the others. However, for the critical stress intensity factor and specific fracture energy, there were no significant differences among the a/W ratios of 0.3, 0.5, and 0.7 where the crack tip was placed in the core layer of the particleboard. In general, as the a/W ratio decreased, the stiffness of the material increased, and the specimens with an a/W ratio of 0.1 showed brittle behavior. However, there was no statistically significant difference between a/W ratios of 0.5 and 0.7.

Investigation on the mixed mode fracture toughness of freshwater ice using the semi-circular bend method

The fracture toughness of ice is of great importance to understand the fracture behaviour of ice impact and blasting. In this study, the mixed mode (I + II) fracture toughness and mode II fracture toughness of granular freshwater ice were first investigated using the semi-circular bend method. A series of fracture tests were carried out to investigate the effects of the grain size (0.8 mm to 4.5 mm) and notch inclination angle (0° to 75°) on the fracture toughness under mixed mode loading conditions. The experimental results indicate that the mode II and mixed mode fracture toughness values slowly decrease in general as the grain size increases. The fracture initiation angles and critical stress intensity factors of the SCB ice specimens were predicted using the generalized maximum tangential stress (GMTS) criteria and conventional maximum tangential Stress (MTS). The GMTS criterion provides a better understanding for mixed mode fracture toughness of ice specimens with various grain sizes, particularly for mode II dominated loading conditions. The grain size of ice directly affects the size of the fracture process zone, which in turn affects the variation law of mixed mode fracture toughness. The strain field and displacement field were initially analysed based on digital image correlation. The results indicate that the variation in the mixed mode crack was complex, eventually exhibiting an asymmetrical distribution of the displacement field.

L-Arginine as Bio-Based Curing Agent for Epoxy Resins: Temperature-Dependence of Mechanical Properties.

The precise characterization of new bio-based thermosets is imperative for the correct assessment of their potential as matrix material in fiber-reinforced polymer composites. Therefore, the mechanical properties of diglycidyl ether of bisphenol a (DGEBA) cured with l-arginine were investigated to determine whether the bio-based thermoset possesses the required mechanical properties for application as a matrix material. The cured thermoset is called Argopox. The mixture of amino acid and epoxy resin was prepared via three-roll milling and cured in the presence of an urea-based accelerator. The tensile, compression, flexural and toughness properties of Argopox were characterized at T=−40 ∘C, 22 ∘C and 80 ∘C to determine the temperature-dependence of the thermoset’s mechanical properties in its service temperature range. The glass transition temperature Tg was analyzed via dynamic mechanical analysis (DMA) and is approximately 119 ∘C. The tensile, compression and flexural strength at 22 ∘C are about 56 MPa, 98 MPa and 85 MPa, respectively. The critical stress intensity factor KIC and fracture energy GIC at 22 ∘C are roughly 1.1 MPam0.5 and 510 Jm−, respectively. Consequently, Argopox possesses mechanical properties that reach performance levels similar to that of materials which are already used as matrix for fiber reinforced composites.

Interaction between crack and grain boundary in magnesium

In polycrystalline materials, grain boundaries (GBs) play an important role in resisting crack growth. In this study, the interaction of a crack on basal plane with GBs in magnesium was investigated by molecular statics simulations. Three types of GBs with different mis-orientation angles were considered, i.e., [21¯1¯0] asymmetric tilt GB, (011¯0) twist GB and [0001] asymmetric tilt GB. Simulation results show that, for [21¯1¯0] asymmetric tilt GBs, the crack always penetrates through the GBs but the crack growth path in neighboring grain changes from (0001) basal plane to (01¯11) pyramidal plane as the tilt mis-orientation angle increases. For (011¯0) twist GBs, the crack propagation mechanism changes from trans-granular fracture into inter-granular fracture with the increasing twist mis-orientation angle. For [0001] asymmetric tilt GBs, trans-granular fracture happens at small and high tilt mis-orientation angles, while inter-granular fracture occurs at intermedium mis-orientation angles. The critical stress intensity factor (SIF) for crack propagation across or along GB is correlated with the corresponding GB energy. Trans-granular fracture occurs when the GB energy is low, and the critical SIF also increases with the increasing GB energy. For high energy GBs, inter-granular fracture occurs and the critical SIF changes hardly with the increasing GB energy. The current work provides insights into understanding the behavior of cracks and ductility enhancement in magnesium.

Microstructures and properties of a high strength, toughness, and corrosion resistance Al-Zn-Mg-Cu alloy under an over-aging state

• Highest fracture toughness and critical stress intensity factor in Al-Zn-Mg-Cu alloy. • The excellent mechanical properties were attributed to the novel heat treatment. • Deformed structures without recrystallization retained after aging. • Pre-aging treatment introduced G.P. zones, leading to fine age hardening precipitates. • The size of the GBPs and the width of PFZs were significantly reduced. This work studied an Al-Zn-Mg-Cu alloy with excellent comprehensive properties using a novel heat treatment. The result shows that the fracture toughness ( K IC ) value of 65.2 ± 2.0 MPa∙m 1/2 and critical stress intensity factor ( K ISCC ) of 37.9 ± 1.7 MPa∙m 1/2 can be achieved. Each of which is the highest value reported. The fatigue strength with this heat treatment exceeded 180 MPa at the stress ratio of −1 ( σ -1 ), which was much higher than the other Al-Zn-Mg-Cu alloys with similar tensile strength. This superior combination of mechanical properties and fracture toughness for the studied alloy is greatly attributed to a significant fraction of deformed grain structures and dense age hardening precipitates evolved from existing G.P. zones.

Effect of reclaimed asphalt pavement (RAP) on fracture properties of stone matrix asphalt (SMA) at low temperature

In this research, the fracture properties and low temperature cracking of stone matrix asphalt)SMA(mixture containing reclaimed asphalt pavement)RAP(were evaluated. To achieve appropriate gradation of SMA with a nominal maximum aggregate size (NMAS) of 12.5 mm, 30 % dense graded RAP was used. Evaluation of fracture properties was done based on stress and energy approaches. To evaluate the fracture properties of the SMA mixture containing 30 % RAP and compare it with the non-RAP mixture, semicircular bending (SCB) test was performed at −10 °C. Air void in the mixture (Va), loading rate, and combined mode parameter (Me) was considered as research factors. The results showed that the critical stress intensity factor (KIC) of mixtures containing 30 % RAP was higher than the non-RAP mixture. In both types of mixtures, with increasing Me, KIC first decreased and then increased. In pure mode I (Me = 1) with increasing air void, KIC decreased. Regardless of the effect of RAP, in all fracture loading modes with increasing loading rate and air void, the fracture energy (Gf) decreased. Therefore, the presence of RAP did not affect the energy absorption of the mixture at low temperatures. The presence of RAP and coarse aggregate skeleton structure causes the brittle fracture of the mixture at low temperatures. The effect of RAP on the crack path only in samples with low air voids and high loading rate caused the crack path through coarse aggregates with smaller size and mastic to show a more share compared to non-RAP samples.

Stress corrosion characteristics of electron beam welded titanium alloys joints in NaCl solution

The Ti6321 titanium alloy has stress corrosion sensitivity in 3.5% NaCl solution, and the sample with cracks will fracture in a short time. The aim of this paper is to reveal the key factors affecting the stress corrosion resistance and clarify the stress corrosion mechanism of different regions of the joint by studying the crack propagation and stress corrosion critical stress intensity factor of the different regions of titanium alloy welded joint. Stress corrosion cracking, fracture and crack propagation path of electron beam welded titanium alloy joint were analyzed, characteristics of microstructure, texture and electrochemical were studied. The results indicated that the stress corrosion intensity factor KISCC of the weld metal is >76.50 MPa • m 1 / 2 , and the maximum KISCC value of heat affected zone is 51.62 MPa • m 1/2 , so KISCC of weld metal is better than heat affected zone. The crack growth rate of weld metal is slightly higher than that of heat affected zone. The fractured surface of weld metal has no obvious stress corrosion characterization, while heat affected zone has. The propagation path of weld metal is more zigzag than that of heat affected zone. The results reveal that microstructure, texture and electrochemical properties are the main reasons for stress corrosion. • The stress corrosion of titanium alloy electron beam welded joint in different regions has been studied. • The stress corrosion resistance of weld metal is significantly higher than heat affected zone, which is related to composition, microstructure and microtexture. • The stress corrosion crack mechanism of titanium alloy welded joint is the combined action of crack propagation and potential corrosion under stress.

Mixed-mode dynamic fracture behavior of a rubber toughened epoxy adhesive under stress wave loading

This work seeks to measure mixed-mode fracture parameters for a dynamically growing crack in a core shell rubber toughened epoxy adhesive (EPONTM 828 with PARALOID™ EXL-2691J particles). Digital image correlation technique coupled with ultrahigh-speed photography and a hybrid finite element-based post-processing approach are used to extract fracture parameters. The dynamic loading is accomplished by using a striker to induce stress waves in an incident bar in contact with a free-standing semi-circular specimen. A companion finite element analysis is performed for investigating suitable experimental parameters and configurations for implementing the approach and determining specific experimental parameters (crack length, crack angle, and impact velocity) necessary to achieve the desired experimental conditions (mode mixity at crack initiation and strain rate). The critical fracture parameters are used to generate mixed mode fracture envelope and fracture toughness-mode mixity relationship for the material. Further investigation is carried out to evaluate various fracture criteria for predicting initial crack kink angles. The measured dynamic fracture quantities are also compared with the quasi-static counterparts to assess strain rate-dependence of the adhesive. The results reveal a decrease in critical stress intensity factors of approximately 10 % and 30 % for the mode I and mode II states, respectively, with increasing strain rate. Post-initiation, the dynamic crack propagates at a near-constant mode I stress intensity factor of ∼1.4 MPa-√m. Under dynamic conditions, the critical, effective stress intensity factors are relatively constant with respect to mode mixity. This is in contrast to quasi-static results where the effective values increased with respect to mode mixity. Experimentally measured crack kink angles agree well with those predicted by the maximum energy release rate criteria and the maximum tangential stress criteria, particularly under mode I dominant conditions.

LEFM is agnostic to geometrical nonlinearities arising at atomistic crack tips

Various fields such as mechanical engineering, materials science, etc., have seen a widespread use of linear elastic fracture mechanics (LEFM) at the continuum scale. LEFM is also routinely applied to the atomic scale. However, its applicability at this scale remains less well studied, with most studies focusing on non-linear elastic effects. Using a harmonic “snapping spring” nearest-neighbor potential which provides the closest match to LEFM on a discrete lattice, we show that the discrete nature of an atomic lattice leads to deviations from the LEFM displacement field during energy minimization. We propose that these deviations can be ascribed to geometrical nonlinearities since the material does not have a nonlinear elastic response prior to bond breaking. We demonstrate that crack advance and the critical stress intensity factor in an incremental loading scenario is governed by the collectively loaded region, and can not be determined analytically from the properties (max. elongation, max. sustained force, etc.) of the stressed crack tip bond alone.

Experimental Study on Size Effect and Fracture Properties of Polypropylene Fiber Reinforced Lightweight Aggregate Concrete

In this experimental research, the effects of polypropylene fibers on size effect and fracture properties of lightweight aggregate concrete were studied. Two methods, including size effect method and work of fracture method, were used to investigate and analyze the size effect and fracture properties on different sizes of notched beams. The polypropylene fiber contents were 0.5%, 0.75%, and 1% by volume fraction. The obtained results revealed that increasing the polypropylene fibers in lightweight aggregate concrete relatively decreased the dependence of strength on the size effect parameter, while the addition of polypropylene fibers exhibited significant influence on decreasing the size dependency of ductility and fracture energy in lightweight aggregate concrete. Moreover, the increase of polypropylene fibers improved the total fracture energy (GF), initial fracture energy (Gf), characteristic length (lch), length of fracture process zone (cf), and critical stress intensity factor (KIc) of lightweight aggregate concrete in both size effect method and work of fracture method. This increase was more significant in work of fracture method because of considering the post-peak behavior. The size effect method was suitable and accurate for plain lightweight aggregate concrete. The GF ⁄ Gf ratio increased from 2.88 in plain lightweight aggregate concrete to 12.26 in polypropylene fiber reinforced lightweight aggregate concrete.

Experimental study on anisotropic fracture characteristics of coal using notched semi-circular bend specimen

To investigate the anisotropic fracture characteristics of coal, physical tests are carried out on semi-circular bend specimens with different pre-crack inclinations and bedding orientations. Regarding the coal as a transversely isotropic body, assigning its elastic properties to the finite element models, the crack geometry factors are calibrated, and stress distributions near the pre-crack tip are analyzed through numerical simulations. Then, the modes I and II critical stress intensity factors are determined based on fracture load. Experimental results show that bedding orientation and pre-crack inclination of coal specimens jointly affect their fracture mode and resistance, and mode I fracture dominates the fracture onset. For straight and inclined pre-cracks, the elliptic function and Fourier series can approximately describe the relationship between effective critical stress intensity factor and bedding angle, respectively. Besides, descriptions of typical propagation paths and their corresponding failure mechanisms are reported in detail. Further, the morphology of fracture surfaces is drawn based on scanning data; afterward, four equidistant cross-sectional profiles in different specimens are extracted and exhibited. Finally, an empirical equation is utilized to calculate the joint roughness coefficients of profiles. Comparing the differences among these coefficients is beneficial to quantitatively characterize anisotropic crack growth in coal.

Alumina-based ceramics reinforced with calcium hexaaluminate

In this study, alumina ceramics containing calcium hexaaluminate are investigated. The specimens are obtained by axial pressing of granulated powders and pressureless sintering. Fracture toughness is tested using the indentation method. Microhardness is determined by the Vickers method. XRD and microstructural studies using a scanning electron microscope are carried out. It was found that the use of Ca(OH)2 leads to the formation of CaAl12O19 in sintered alumina ceramics. As the platelet content increases, the average length of the platelets increases from 2 to 3 µm. The width of the platelets is about 0.4 µm. Increasing the CaAl12O19 content decreases the size of Al2O3 grains. The average grain size of the alumina ceramics is 1.65 ± 0.02 µm. For the material containing 6 wt.% CaAl12O19 the average grain size is 0.95 ± 0.05 μm. An increase in the critical stress intensity factor for the formation of 6 wt.% CaAl12O19 in a material compared to alumina ceramics without additives has been established. The relative density of such a material is 95.3 ± 0.5 %, microhardness is 1800 ± 50 HV, fracture toughness is 5.2 ± 0.4 MPa ∙ m1 / 2. The increase in fracture toughness of composites containing CaAl12O19 platelets is due to the fracture of platelets, crack bridging and crack deflection.

Fracture Performance of Cementitious Composites Based on Quaternary Blended Cements

This study presents test results and in-depth discussion regarding the measurement of the fracture mechanics parameters of new concrete composites based on quaternary blended cements (QBC). A composition of the two most commonly used mineral additives, i.e., fly ash (FA) and silica fume (SF), in combination with nanosilica (nS), has been proposed as a partial replacement for ordinary Portland cement (OPC) binder. Four series of concrete were made, one of which was the reference concrete (REF) and the remaining three were QBC. During the research, the main mechanical parameters of compressive strength (fcm) and splitting tensile strength (fctm), as well as fracture mechanics parameters and the critical stress intensity factor , along with critical crack-tip opening displacements (CTODc) were investigated. Based on the tests, it was found that the total addition of siliceous materials, i.e., SF + nS without FA, increases the strength and fracture parameters of concrete by approximately 40%. On the other hand, supplementing the composition of the binder with SF and nS with 5% of FA additive causes an increase in all mechanical parameters by approximately 10%, whereas an increase by another 10% in the FA content in the concrete mix causes a significant decrease in all the analyzed factors by 10%, compared to the composite with the addition of silica modifiers only.

A Physically Based Model Predicting the Degradation of Hydrogen on Crack Growth Critical Stress Intensity Factor of Metals

A simple, physically based model is developed to quantitatively predict the degradation of hydrogen on the crack growth critical stress intensity factor (CSIF) of metals. The model is formulated by combining a microscopically shielded Griffith criterion (MSGC) model for plasticity-induced cleavage fracture and thermodynamics decohesion (TDD) theory for hydrogen-enhanced interface decohesion. The hydrogen-influenced CSIF is described as a function of the intrinsic CSIF (hydrogen-free), initial hydrogen concentration (solubility), hydrogen trap binding energy and crack tip stress. All parameters in the model can be determined with a physical basis and the model is successfully validated by comparison with published experimental data.

Synthesis and Optical Characteristics of Gd0.96Eu0.01Sm0.01Tb0.01Er0.01Nb0.9Ta0.1O4 Ceramic Solid Solutions Prepared under Different Temperature Conditions

Fine powders of mixed gadolinium tantalum niobates doped with Eu, Sm, Tb, and Er were synthesized. Ceramic samples of polycomponent solid solutions of Gd0.96Eu0.01Sm0.01Tb0.01Er0.01Nb0.9Ta0.1O4 were obtained from synthesized powders using conventional sintering technology. The phase composition and phase structure characteristics of the Gd0.96Eu0.01Sm0.01Tb0.01Er0.01Nb0.9Ta0.1O4 ceramic phases were determined by XRD. The effect of ceramic sintering temperature on the physical characteristics of Gd0.96Eu0.01Sm0.01Tb0.01Er0.01Nb0.9Ta0.1O4 solid solutions is shown. The morphological features of the microstructure of the Gd0.96Eu0.01Sm0.01Tb0.01Er0.01Nb0.9Ta0.1O4 ceramics were studied in relation to its mechanical characteristics. At the same time, the strength characteristics (Young’s modulus, microhardness) and the critical stress intensity factor for mode I KIC were evaluated for the first time for the synthesized compounds. Photoluminescence and cathodoluminescence were studied in the visible region. The study confirms the potential application of Gd0.96Eu0.01Sm0.01Tb0.01Er0.01Nb0.9Ta0.1O4 ceramic solid solutions as scintillators and radioluminescent light sources.

Fatigue crack propagation and fracture toughness of cortical bone are radiation dose-dependent.

Cortical bone allograft sterilized with a standard γ-radiation dose of 25-35kGy has demonstrated reduced static and cyclic fracture resistance compared with unirradiated bone. To mitigate radiation damage, we recently observed a dose-dependent response of high-cycle fatigue behavior of human cortical bone from 0 to 25 kGy, with lower doses exhibiting logarithmically longer fatigue lives. The objectives of this study were as follows: (1) to determine whether fracture toughness, work-to-fracture, and fatigue crack propagation resistance of human cortical bone are also radiation dose-dependent, and (2) to determine the associations of radiation dose and a Raman biomarker for collagen disorder with fracture properties. Compact tension specimens were machined from two donor femoral pairs and allocated to four treatment groups: 0 (unirradiated control), 10, 17.5, and 25 kGy. Fracture toughness specimens were monotonically loaded to failure and the critical stress intensity factor (KC ) was determined. Work-to-fracture was calculated from the load versus displacement integral up to fracture. Fatigue crack propagation specimens were cyclically loaded under constant room-temperature irrigation and fatigue crack growth rate (da/dN) and cyclic stress intensity (∆K) were calculated. Fracture toughness, work-to-fracture, and fatigue crack propagation resistance decreased 18%, 33%, and 15-fold from 0 to 25 kGy, respectively (p < 0.05). Radiation dose was more predictive of fracture properties than collagen disorder. These findings support that quasi-static and fatigue fracture properties of cortical bone are radiation dose-dependent within this dose range. The structural alterations arising from irradiation that cause these losses in fracture resistance remain to be elucidated.

Improvement of the Mechanical Characteristics, Hydrogen Crack Resistance and Durability of Turbine Rotor Steels Welded Joints

This article is devoted to the following issues: calculating the values of temperatures obtained by simulating welding heating and the subsequent implementation of the welding process at the given mode parameters made it possible to obtain a welded joint of the rotor with an improved initial structure and increased mechanical properties, hydrogen resistance and durability by up to 10–15%; simulating welding heating in the areas of fusion, the overheating and normalization of the HAZ and the formation of austenite grains; specified welding heating creates the conditions for the formation of new products of austenite decomposition in the form of sorbitol in the area of the incomplete recrystallization of the HAZ. In air and gaseous hydrogen, the destruction of the combined joints took place on the weld metal, as well as on the fusion areas, the overheating and the incomplete recrystallization of the HAZ of 20H3NMFA steel as the base metal. Structural materials have a relatively low strength and high fracture toughness in air. This is manifested in a significant reduction in the elongation (δ), the area (ψ) and critical stress intensity factor (KIc) of welded joints and the endurance limit of cylindrical smooth rotor steel specimens, which were cut from transverse templates. Welded joints in the whole range of load amplitudes are sensitive to the action of hydrogen.

Applying the new experimental midpoint concept on strain energy density for fracture assessment of composite materials

A mixed-mode I/II fracture criterion for predicting the fracture response of composite materials is proposed. This criterion is derived based on a comprehensive study and the consideration of the physics of fracture onset. The fracture phenomenon that causes the various damage mechanisms at the vicinity of the crack tip is examined. It is elucidated that the stress distribution at the crack tip ought to be defined using the reinforcement isotropic solid (RIS) stress state. This new criterion, which is called Improved Strain Energy Density with Mid-point (ISEDM), includes the effect of the fracture process zone (FPZ) and T-stress, which remarkably affect the mixed-mode fracture process, particularly, when mode II is dominant. Substituting the strain energy density in a pure mode I with the strain energy density in the midpoint of mixed-mode I/II is the creative idea employed in proposing this criterion. Because in pure mode I the effects of FPZ are minimal, change of fitting point on fracture limit curve (FLC) from KIc to midpoint critical stress intensity factors (CSIF’s) makes it possible to consider the effects of FPZ more accurately. In the ISEDM criterion, fracture behavior depends on mechanical properties and CSIF’s of the midpoint. Crack initiation angle is considered along minimum strain energy density and this angle is derived in the midpoint of mixed-mode I/II. RIS theory is used as an applicable theory for modeling orthotropic materials in this paper and causes valid and reliable fracture behavior to be extracted. In addition, changing fracture point from pure mode I to the mid-point of experimental data causes the effects of FPZ to be considered without estimating the toughening mechanisms in this zone, and the fracture behavior is extracted with higher accuracy. FLC’s in comparison with available experimental data prove that the ISEDM criterion anticipates the fracture behavior of orthotropic materials well. Finding KIIc based on the analytical method is a valuable achievement. In this article, KIIc can be predicted with appropriate accuracy, only by CSIF’s of mid-point and using ISEDM criterion.

Study of tensile crack growth in rock-like materials under punch shear test

In this paper, the specification of tensile crack growth under punch shear test was studied. Also, direct and indirect tensile strength of gypsum and concrete (rock-like materials) were determined and compared in this research. The direct tensile strength was measured by a new apparatus using the cubic specimens with 20 cm * 20 cm * 5 cm in dimensions containing four edge notches. The indirect Brazilian tensile strength was measured with the Brazilian disc samples. The notch lengths were similar in one sample. Three different lengths were chosen for notches, i.e., 10 cm, 12 cm, 14 cm and 16 cm. Ligament lengths (horizontal distance between two notches) was 2 cm. These samples were tested under a punch shear test. The numerical simulation of these tests was also performed with the Particle Flow Code (PFC) and the stresses within the specimen and the critical stress intensity factors were estimated with the explicit finite difference code (FLAC2D). Ligament lengths were 2 cm, 3 cm and 4 cm. In this work, the results of indirect tensile strength were higher than those measured directly in the laboratory. When the joint length was 10 cm, the tensile force was concentrated around the notch tips while it was concentrated at the ligament by increasing the notch length. In the fixed value of ligament length, the tensile force concentration was distributed uniformly in the ligament by enhancing the joint length. In the fixed value of joint length, the tensile force concentration was distributed uniformly in the ligament by increasing the ligament length. It was found that, as the length of the joints increase the value of tensile was decreased. On the other hand, the tensile strength was increased by decreasing the ligament length.