Crack growth resistance behavior of a functionally graded material: computational studies

This paper describes crack growth resistance simulation in a ceramic/metal functionally graded material (FGM) using a cohesive zone ahead of the crack front. The plasticity in the background (bulk) material follows J₂ flow theory with the flow properties determined by a volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model). A phenomenological, cohesive zone model with six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) describes the constitutive response of the cohesive zone. Crack growth occurs when the complete separation of the cohesive surfaces takes place. The crack growth resistance of the FGM is characterized by a rising J-integral with crack extension (averaged over the specimen thickness) computed using a domain integral (DI) formulation. The 3-D analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. The paper describes applications of the cohesive zone model and the DI method to compute the J resistance curves for both single-edge notch bend, SE(B), and single-edge notch tension, SE(T), specimens having properties of a TiB/Ti FGM. The numerical results show that the TiB/Ti FGM exhibits significant crack growth resistance behavior when the crack grows from the ceramic-rich region into the metal-rich region. Under these conditions, the J-integral is generally higher than the cohesive energy density at the crack tip even when the background material response remains linearly elastic, which contrasts with the case for homogeneous materials wherein the J-integral equals the cohesive energy density for a quasi-statically growing crack.     

A method for dynamic fracture toughness determination using short beams

This paper deals with dynamic fracture toughness testing of small beam specimens. The need for testing such specimens is often dictated by the characteristic dimensions of the end product. We present a new methodology which combines experimentally determined loads and fracture time, together with a numerical model of the specimen. Calculations are kept to a minimum by virtue of the linearity of the problem. The evolution of the stress intensity factor (SIF) is obtained by convolving the applied load with the calculated specimen response to unit impulse force. The fracture toughness is defined as the value of the SIF at fracture time. The numerical model is first tested by comparing numerical and analytical solutions (Kishimoto et al., 1990) of the impact loaded beam. One point impact experiments were carried out on of commercial tungsten base heavy alloy specimens. The robustness of the method is demonstrated by comparing directly measured stress intensity factors with the results of the hybrid experimental-numerical calculation. The method is simple to implement, computationally inexpensive, and allows testing of large sample sizes, without restriction on the specimen geometry and type of loading.     

Interface cracking between functionally graded coatings and a substrate under antiplane shear

Coating technology plays a significant role in a number of applications such as high temperatures, corrosion, oxidation, wear, and interface. In this paper, we investigate the interface cracking between ceramic and/or functionally graded coatings (FGM coatings) and a substrate under antiplane shear. Four coating models are considered, namely single layered homogeneous coating, double layered piece-wise homogeneous coating, single layered FGM coating and double layered coating with an FGM bottom coat. Mode III stress intensity factors (SIFs) are calculated for the different coating models. In the case of μ_L > μ₀ where μ₀ is the shear modulus of the substrate and μ_L the shear modulus of the material at the surface of the coating, it is found that the single layered FGM coating reduces SIF slightly, whereas the coating system with a top homogeneous layer and a thin FGM bottom layer reduces SIF significantly. In the case of μ_L < μ₀ the SIF is found to be larger for the FGM coatings than for the homogeneous coatings. The FGM coating, however, may still be superior to homogeneous coatings in this case as FGM coatings usually provide better bonding strength between the coating and substrate. Finally, the applicability of the SIF concept in the fracture of FGM coatings is discussed. Large modulus gradients in thin coatings may seriously restrict the application of SIFs as the SIF-dominant zone may fall into the crack tip nonlinear deformation and damage zone. The same argument exists for some interphase models in interface crack solutions.     

Study of mode I crack dynamic propagation behaviour and rock dynamic fracture toughness by using SCT specimens

To study crack dynamic propagation behaviour and rock dynamic fracture toughness, a single cleavage triangle (SCT) specimen was proposed in this paper. By using these specimens and a drop‐weight test system, impact experiments were conducted, and the crack propagation velocity and the fracture time were measured by using crack propagation gauges. To examine the effectiveness of the SCT specimen and to predict the test results, finite difference numerical models were established by using AUTODYN code, and the simulation results showed that the crack propagation path agrees with the test results, and crack arrest phenomena could happen. Meanwhile, by using these numerical models, the crack dynamic propagation mechanism was investigated. Finite element code ABAQUS was applied in the calculation of crack dynamic stress intensity factors (SIFs) based on specimen dimension and the loading curves measured, and the curves of crack dynamic SIFs versus time were obtained. The fracture toughness (including initiation toughness and propagation toughness) was determined according to the fracture time and crack speeds measured by crack propagation gauges. The results show that the SCT specimen is applicable to the study of crack dynamic propagation behaviour and fracture toughness, and in the process of crack propagation, the propagation toughness decreases with crack propagation velocity, and the crack arrest phenomena could happen. The critical SIF of an arrest crack (or arrest toughness) was higher than the crack propagation toughness but was lower than the initiation toughness.     

Cohesive modeling of dynamic fracture in functionally graded materials

The dynamic fracture of functionally graded materials (FGMs) is modeled using an explicit cohesive volumetric finite element scheme that incorporates spatially varying constitutive and failure properties. The cohesive element response is described by a rate-independent bilinear cohesive failure model between the cohesive traction acting along the cohesive zone and the associated crack opening displacement. A detailed convergence analysis is conducted to quantify the effect of the material gradient on the ability of the numerical scheme to capture elastodynamic wave propagation. To validate the numerical scheme, we simulate dynamic fracture experiments performed on model FGM compact tension specimens made of a polyester resin with varying amounts of plasticizer. The cohesive finite element scheme is then used in a parametric study of mode I dynamic failure of a Ti/TiB FGM, with special emphasis on the effect of the material gradient on the initiation, propagation and arrest of the crack.     

XFEM simulation of cracks, holes and inclusions in functionally graded materials

The present work aims at the numerical simulation of inhomogeneities/discontinuities (cracks, holes and inclusions) in functionally graded materials (FGMs) using extended finite element method (XFEM). A FGM with unidirectional gradation in material properties is modeled under plane strain condition. The domain contains a major crack either at the center or at the edge of the domain along with multiple minor discontinuities/flaws such as minor cracks and/or voids/inclusions distributed all over the domain. The effect of the variation in stress intensity factor (SIF) of the major crack due to the presence of the minor cracks and voids/inclusions is studied in detail. The simulations show that the presence of minor discontinuities significantly affects the values of SIFs.     

Effects of uniform interior pressure distribution on vibration of FGM cylindrical shell with rings support based on first-order theory subjected to ten boundary conditions

This paper presents the study on natural frequency characteristics of a thin-walled functionally graded material (FGM) cylindrical shell with rings support under symmetric uniform interior pressure distribution. The FGM properties are graded along the thickness direction of the shell. The FGM shell equations with rings support and interior pressure are established based on first-order shear deformation theory. The governing equations of motion were employed, using energy functional and by applying the Ritz method. Ten boundary conditions represented by end conditions of the FGM shell are the following: simply supported-simply supported, clamped-clamped, free-free, clamped-free, clamped-simply supported, free-simply supported, sliding-sliding, sliding-simply supported, sliding-free and sliding-clamped. This problem was solved with computer programming using MAPLE package for numerical investigation. Comparison of the results is carried out to verify the validity of the proposed procedure with published works. The influence of interior pressure, ring support position and number of rings support, and effect of the ten boundary conditions on natural frequency characteristics are studied. The results presented can be used as an important benchmark for researchers to validate their numerical methods when studying natural frequencies of shells with ring and pressure.     

Cohesive fracture modeling of elastic-plastic crack growth in functionally graded materials

This work investigates elastic-plastic crack growth in ceramic/metal functionally graded materials (FGMs). The study employs a phenomenological, cohesive zone model proposed by the authors and simulates crack growth by the gradual degradation of cohesive surfaces ahead of the crack front. The cohesive zone model uses six material-dependent parameters (the cohesive energy densities and the peak cohesive tractions of the ceramic and metal phases, respectively, and two cohesive gradation parameters) to describe the constitutive response of the material in the cohesive zone. A volume fraction based, elastic-plastic model (extension of the original Tamura-Tomota-Ozawa model) describes the elastic-plastic response of the bulk background material. The numerical analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates solid elements with graded elastic and plastic properties and interface-cohesive elements coupled with the functionally graded cohesive zone model. Numerical values of volume fractions for the constituents specified at nodes of the finite element model set the spatial gradation of material properties with isoparametric interpolations inside interface elements and background solid elements to define pointwise material property values. The paper describes applications of the cohesive zone model and the computational scheme to analyze crack growth in a single-edge notch bend, SE(B), specimen made of a TiB/Ti FGM. Cohesive parameters are calibrated using the experimentally measured load versus average crack extension (across the thickness) responses of both Ti metal and TiB/Ti FGM SE(B) specimens. The numerical results show that with the calibrated cohesive gradation parameters for the TiB/Ti system, the load to cause crack extension in the FGM is much smaller than that for the metal. However, the crack initiation load for the TiB/Ti FGM with reduced cohesive gradation parameters (which may be achieved under different manufacturing conditions) could compare to that for the metal. Crack growth responses vary strongly with values of the exponent describing the volume fraction profile for the metal. The investigation also shows significant crack tunneling in the Ti metal SE(B) specimen. For the TiB/Ti FGM system, however, crack tunneling is pronounced only for a metal-rich specimen with relatively smaller cohesive gradation parameter for the metal.     

Experimental and numerical investigations of low velocity impact on functionally graded circular plates

This paper addresses low-velocity impact behaviour of functionally graded clamped circular plates. An experimental work was carried out to investigate the impact behaviour of FG circular plates which is composed of ceramic (SiC) and metal (Al) phases varying through the plate thickness by using a drop-weight impact test system. The influence of the compositional gradient exponent and impactor velocity on the contact forces and absorbed energies was concentrated on the tests. The explicit finite element method, in which a volume fraction based elastic–plastic model (the TTO model) was implemented for the functionally graded materials, was used to simulate their drop-weight impact tests. Effective material properties at any point inside FGM plates were determined using Mori–Tanaka scheme. The experimental and numerical results indicated that the compositional gradient exponent and impactor velocity more effective on the elasto-plastic response of the FG circular plates to a low-velocity impact loading. The comparison at the theoretical and experimental results showed that the use of the TTO model in modelling the elasto-plastic behaviour of FG circular plates results in increasing deviations between the numerical and experimental contact forces for ceramic-rich compositions whereas it becomes more successful for metal-rich compositions.     

Face/core debond fatigue crack growth characterization using the sandwich mixed mode bending specimen

Face/core fatigue crack growth in foam-cored sandwich composites is examined using the mixed mode bending (MMB) test method. The mixed mode loading at the debond crack tip is controlled by changing the load application point in the MMB test fixture. Sandwich specimens were manufactured using H45 and H100 PVC foam cores and E-glass/polyester face sheets. All specimens were pre-cracked in order to define a sharp crack front. The static debond fracture toughness for each material configuration was measured at different mode-mixity phase angles. Fatigue tests were performed at 80% of the static critical load, at load ratios of R = 0.1 and 0.2. The crack length was determined during fatigue testing using the analytical compliance expression and verified by visual measurements. Fatigue crack growth results revealed higher crack growth rates for mode I dominated loading. For specimens with H45 core, the crack grew just below the face/core interface on the core side for all mode-mixities, whereas for specimens with H100 core, the crack propagated in the core or in the face laminate depending on the mode-mixity at the debond crack tip.     

SiCP/Al功能梯度装甲板抗侵彻性能的试验与数值模拟

采用粉末冶金方法制备碳化硅陶瓷颗粒(SiC_P)增强金属铝基复合材料板(MMCs), 并采用热压扩散法制备功能梯度装甲板(FGM)。利用高速冲击空气炮系统, 对纯铝靶板和两种不同铺层结构的功能梯度装甲靶板进行侵彻试验, 并利用LS-DYNA软件对侵彻试验过程进行数值模拟分析, 同时考察等厚、 等面密度下SiC颗粒分布对抗侵彻性能的影响。研究结果表明, 功能梯度板的抗侵彻性能比纯铝板好, 而两种不同铺层结构功能梯度板的抗侵彻性能相差不大。数值计算结果与现有试验结果取得了较好的一致, 说明了数值模拟的有效性。从数值计算结果可以看出, 层状功能梯度板比等厚、 等面密度均质复合材料靶板的抗侵彻能力好, 并可近似地认为等厚、 等面密度下多层功能梯度板的抗侵彻性能对颗粒分布不敏感。     

Ductile–brittle fatigue and fracture behaviour of aluminium/PMMA bimaterial 3PB specimens

Fracture toughness and fatigue crack growth tests and numerical simulations on 3PB specimens were carried out to study the behaviour of a crack lying perpendicular to the interface in a ductile/brittle bimaterial. Polymethylmethacrylate acrylic (PMMA) and aluminium alloy 2024 T531 were joined together using epoxy resin. A precrack was introduced into the ductile material and tests were carried out to obtain fracture toughness and fatigue properties. The body force method and elastic–plastic finite-element analyses were used to simulate the experimental stress intensity K_I and cracking behaviour under monotonic and cyclic loads. It was found that the bimaterial fatigue crack growth rate is higher than that for monolithic aluminium 2024 but lower than the rate for a monolithic PMMA. This agreed with the trend for the fracture toughness values and was consistent with the numerical method results. The initial Mode I stable ductile cracking in the aluminium appears to ‘jump’ the interface and continues under mixed fracture Mode (I and II) in the PMMA material up to the final failure. A consistency between the simulation methods has indicated that the bimaterial fatigue crack growth is dominantly elastic with a small plastic zone near the crack tip.     

Determination of dynamic fracture toughness for PMMA

A dynamic FEM (finite element method) and a strain gage method are applied to analyze the dynamic fracture toughness and SIF (stress intensity factor) for PMMA (polymethyl methacrylate). The analyses are carried out for plates with an edge crack subjected to one-point bending in a plane of the plate. A simple procedure that the present author has proposed is applied to the problem of using a triangular element of assumed constant strain on finite element analysis. The numerical simulation by FEM provides values for the applied forces as measured with the strain gages. Also, a crack initiation time is measured with the strain gage mounted around the crack tip. The dynamic fracture toughness is determined by adapting the crack initiation time to the simulation curve of the dynamic SIF calculated by the FEM. In this study, the usefulness of the method to determine the dynamic fracture toughness is investigated by comparing predictions with the experimental results for dynamic stresses and SIFs.     

Sickle-shaped crack in a round bar under complex Mode I loading

A sickle‐shaped surface crack in a round bar under complex Mode I loading is considered. First, the stress‐intensity factor (SIF) along the front of the flaw is numerically determined for five elementary Mode I stress distributions (constant, linear, quadratic, cubic and quartic) directly applied on the crack faces. The finite element method and linear elastic fracture mechanics concepts are employed. Then, a numerical procedure to calculate approximate values of SIF for a complex Mode I stress distribution on the crack faces is proposed based on both the power series expansion of the function describing such a stress distribution and the superposition principle. In order to validate the results obtained through the above procedure, a comparison with numerical data available in the literature is made.     

Inverse problem of material distribution for desired fracture characteristics in a thick-walled functionally graded material cylinder with two diametrically-opposed edge cracks

This study concerns the inverse problem of evaluating the optimum material distribution for desired fracture characteristics in a thick-walled functionally graded material (FGM) cylinder containing two diametrically-opposed edge cracks emanating from the inner surface of the cylinder. The thermal eigenstrain developed in the cylinder material due to nonuniform coefficient of thermal expansion as a result of cooling from sintering temperature is taken into account. Based on a generalized method of evaluating stress intensity factors developed in a previous study, an inverse method is developed to optimize material distribution intending to realize prescribed apparent fracture toughness in the FGM cylinder. To present some numerical results, a TiC/Al₂O₃ FGM cylinder is considered and the inverse problems are solved to evaluate material distributions for two examples of prescribed apparent fracture toughness. The effect of cylinder wall thickness on the material distribution and comparison of material distributions corresponding to a single and two cracks are also discussed. The numerical results reveal that the apparent fracture toughness of FGM cylinders can be controlled by choosing the material distributions properly.     

Influence of material properties of SIFs determined by frozen stress

In a prior study utilizing two materials for which thermal and mechanical properties of both materials in bimaterial specimens were matched except for modulus, it was found that modulus mismatch had a negligible effect on the stress intensity factor (SIF) level and distribution for cracks in bond lines of stress frozen photoelastic bimaterial specimens. However, bond line residual stresses increased the SIF values. Since it is almost impossible to obtain commercial stress freezing materials with matching critical temperatures (T_c) with different moduli, the present study of cracks parallel to and within bond lines utilized such materials with T_c mismatch and compared results with parallel studies with matched T_c values in order to evaluate T_c mismatch effects. Results show that T_c mismatch creates mixed mode conditions in cracks parallel to the bond line with significant increases in the mode I SIF (K₁) for cracks very near the bond line but shows little effect for cracks within or away from the bond line. The absence of a modulus mismatch effect is again confirmed as well.     

Free vibration analysis of pre-twisted rotating FGM beams

The natural frequencies of vibration of a rotating pre-twisted functionally graded cantilever beam are investigated. Rotating cantilever beam with pre-twist made of a functionally gradient material (FGM) consisting of metal and ceramic is considered for the study. The material properties of the FGM beam symmetrically vary continuously in thickness direction from core at mid section to the outer surfaces according to a power-law form. Equations of motion for free vibration are derived using Lagrange’s equation and the natural frequencies are determined using Rayleigh–Ritz method. The effect of parameters such as the pre-twist angle, power law index, hub radius and rotational speed on the natural frequencies of rotating functionally graded pre-twisted cantilever beams are examined through numerical studies and comparison is made with the numerical results obtained using other methods reported in literature. The effect of coupling between chordwise and flapwise bending modes on the natural frequencies has also been investigated.     

Experimentelle und numerische Ermittlung dynamischer Risswiderstandskurven im Kerbschlagbiegeversuch

Experimental and numerical determination of crack resistance curves in the notched‐bar impact test The assessment of the reliability of components requires the knowledge of crack resistance curves, which are often not available due to lack of specimen material. More likely is the availability of typical material parameters such as the yield strength, tensile strength, uniform elongation, elongation at rupture as well as upper shelf impact energy and the lateral elongation of notched‐bar impact test specimens. The material model of Gurson describes ductile crack growth due to the nucleation, growth and coalescence of voids in the material. Although dependent on the material and temperature, the material parameters of the Gurson model are independent of the specimen geometry and rate of loading. This latter fact allows one to use the values of these parameters determined on statically‐loaded fracture mechanics specimens to model specimens with other geometries and subjected to different loading conditions, in particular to model impact loaded Charpy‐V specimens. A method is proposed to construct crack resistance curves based on available data of tension tests and on quasi‐static yield curves. Dynamic yield curves are determined using proven procedures as based on the analysis of the dislocation activation energy. The ductile damage parameters are then obtained via simulation of tests on notched tensile specimens and notched‐bar impact tests as well as the fitting to the upper shelf impact energy. In this way, the ductile damage parameters are determined, which in turn enable the determination of the required J‐resistance curves via simulation of ductile crack growth in fracture mechanics specimens. Thus, the application of the classical J‐integral concept gets possible. Furthermore, the independence of the identified material parameters from the geometry of the specimen then allows the use of the Gurson model to analyse the safety of structural components with cracks directly.     

A model for interface cracks in layered orthotropic solids: Convergence of modal decomposition using the interaction integral method

The mode‐partitioning problem for bimaterial interfaces is still not resolved by the classical fracture mechanics approach in a satisfactory manner. Stress oscillations and overlapping crack faces are a direct consequence of the rigorous solution of the elastic boundary value problem, if the constitutive law changes discontinuously across the interface. Conversely, continuously varying material properties, also referred to as functionally graded materials (FGM), avoid these physically not admissible drawbacks. In this case the crack tip fields are of the same nature as those known from homogeneous materials. Therefore, the well‐established stress intensity factor concept can be used without any changes. Following this motivation an FGM‐interface model for delaminated composite beam structures was developed and its characteristics with respect to the modal decomposition of the crack tip fields were investigated. The considered beam structures consisted of two orthotropic layers, each of a different material. The spatial variation of the material properties in the interface region was modeled by a tanh ‐function introducing one transition parameter that controlled the FGM‐gradient. Four load cases were analyzed for each structural configuration: either a unit normal force or a unit bending moment was imposed on each end of the split beam. Thus, any load case can be simply reconstructed from the presented results by means of superposition. The stress intensity factors for modes I and II were then evaluated using an interaction integral method along with the finite element method. The corresponding results are given depending on the mesh density of the interface region, the integration domain and the transition parameter. In this manner, the influence of the transition parameter on the mode ratio and on the convergence behavior of the modal decomposition scheme with respect to its integration domain was identified. Finally, the ability of the FGM‐interface model to represent bimaterial interfaces while still maintaining the advantages of crack analysis in homogeneous materials was highlighted. Copyright © 2008 John Wiley & Sons, Ltd.     

具有理想与非理想界面的双材料四点弯曲试件的界面断裂韧性

本文作者用边界元法研究了具有理想与非理想界面的双材料四点弯曲试件的界面断裂韧性.分析了理想界面裂纹尖端的复应力强度因子以及非理想界面裂纹前沿的应变能密度与试件的几何尺寸、双材料的性质等的关系.计算结果表明,当裂纹的长度小于试件内支点的跨度时,上述两个物理量在很大范围内不随裂纹长度而变化.这种稳态的特性为两者的临界值的实验测定提供了方便.