Abstracts From Recent Papers


Boundary Layer Framework Considering Material Gradation Effects
Shim, D., Paulino, G. and Dodds, R.
Engineering Fracture Mechanics, Vol. 73, pp. 593-615, 2006

This paper describes the development and application of a novel modified boundary layer (MBL) model for graded nonhomogeneous materials, e.g. functionally graded materials (FGMs). The proposed model is based on a middle-crack tension, M(T), specimen with traction boundary conditions applied to the top and lateral edges of the model. Finite element analyses are performed using WARP3D, a fracture mechanics research finite element code, which incorporates elements with graded elastic and plastic properties. Elastic crack-tip fields obtained from the proposed MBL model show excellent agreement with those obtained from full models of the cracked component for homogeneous and graded nonhomogeneous materials. The K-T dominance of FGMs is investigated by comparing the actual stress fields with the asymptotic stress fields (the Williams' solution). The examples investigated in the present study consider a crack parallel to the material gradient. Results of the present study provide insight into the K-T dominance of FGMs and also show the range of applicability of the proposed MBL model. The MBL model is applied to analyze the elastic-plastic crack-tip response of a Ti/TiB FGM SE(T) specimen. The numerical results demonstrate that the proposed MBL model captures the effect of T-stress on plastic zone size and shape, constraint effects, etc. for such configurations.

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Effect of Material Gradation on K-Dominance of Fracture Specimens
Shim, D., Paulino, G. and Dodds, R.
Engineering Fracture Mechanics, Vol. 73, pp. 643-648, 2006

This work describes the effect of material gradation (parallel to the crack plane) on stress intensity factors and K-dominance, i.e. the dominance of the singular region, of fracture specimens; SE(T), SE(B) and C(T). The extent of K-dominance is investigated by comparing the actual stress field with the Williams' asymptotic stress field. Linear-elastic finite element analyses are performed using graded elements which incorporate graded material properties at the element level. Material gradation and crack geometry are systematically varied to perform the parametric study. Results reveal that the effect of material gradation on KI is most pronounced when a short crack is located on the stiffer side of the fracture specimen. For a given specimen and crack geometry, the extent of K-dominance yields a curve with a peak point at a certain material gradation. Results of the present study provide valuable insight into the K-dominance of FGMs.

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J Resistance Behavior in Functionally Graded Materials Using Cohesive Zone and Modified Boundary Layer Models
Shim, D., Paulino, G. and Dodds, R.
International Journal of Fracture

To be published...

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Computation of Mixed-Mode Stress Intensity Factors for Cracks in Three-Dimensional Functionally Graded Solids
Walters, M., Paulino, G. and Dodds, R.
ASCE Journal of Engineering Mechanics, Vol. 132, No. 1, pp. 1-15, 2006

This work applies a two-state interaction integral to obtain stress intensity factors along cracks in three-dimensional functionally graded materials. The procedures are applicable to planar cracks with curved fronts under mechanical loading, including crack-face tractions. Interaction-integral terms necessary to capture the effects of material nonhomogeneity are identical in form to terms that arise due to crack-front curvature. A discussion reviews the origin and effects of these terms, and an approximate interaction-integral expression that omits terms arising due to curvature is used in this work to compute stress intensity factors. The selection of terms is driven by requirements imposed by material nonhomogeneity in conjunction with appropriate mesh discretization along the crack front. Aspects of the numerical implementation with (isoparametric) graded finite elements are addressed, and examples demonstrate the accuracy of the proposed method.

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Effects of Ductile Void Growth on Cleavage Fracture in Ferritic Steels
Petti, J. and Dodds, R.
International Journal of Solids & Structures, Vol. 42, No. 13, pp. 3655-3676, 2005

Over the mid-to-upper region of the ductile-to-brittle transition region, transgranular cleavage and ductile tearing define two competing failure mechanisms in ferritic steel. At metallurgical scales (≤ 50μm), formation and growth of the voids driving ductile crack extension likely alter the local stress fields acting on the smaller inclusions that trigger cleavage fracture. Here we study the effects of void growth on cleavage fracture by modeling discrete cylindrical voids lying on the crack plane ahead of the crack tip within a small-scale yielding (SSY) boundary layer model. These discrete voids have a spacing, D, within a highly refined crack-front region. This enables identification of both single void growth and multiple void growth mechanisms that depend primarily on the initial void porosity, f0. The crack grows in this model by release of nodal reactions (enforcing zero displacement) along the ligament (symmetry plane) between the blunted crack tip and closest void when the void obtains a specified critical porosity. This process grows the crack in discrete increments of size equal to the length of an intervoid ligament. Continued external loading leads to subsequent void growth and crack extensions through additional node releases. The external loads at the point of each crack extension define the crack growth resistance (JR) curves. This enables comparison with conventional JR-Δa curves obtained using computational cell (Gurson-Tvergaard) analyses. The Weibull stress model is then employed to quantify the stress concentration effects on the probability of cleavage fracture. We describe a non-dimensional function, h(Jbar), to represent stress concentration effects on the Weibull stress in a convenient framework (Jbar=J/(D*σ0) denotes a non-dimensional loading for SSY analyses). These h-functions also reflect the increase in volume of material sampled as the crack grows from the original tip to the first void, the second void, etc. The h-functions vary with material flow properties, initial porosity (f0), critical porosity (fc), Weibull modulus (m), and T-stress (Tσ) or constraint level.

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Interaction-Integral Procedures for 3-D Curved Cracks Including Surface Tractions
Walters, M., Paulino, G. and Dodds, R.
Engineering Fracture Mechanics, Vol. 72, pp. 1635-1663, 2005

This study examines a two-state interaction integral for the direct computation of mixed-mode stress intensity factors along curved cracks under remote mechanical loads and applied crack-face tractions. We investigate the accuracy of stress intensity factors computed along planar, curved cracks in homogeneous materials using a simplified interaction integral that omits terms to reflect specifically the effects of local crack-front curvature. We examine the significance of the crack-face traction term in the interaction integral, and demonstrate the benefit of a simple, exact numerical integration procedure to evaluate the integral for one class of three-dimensional elements. The work also discusses two approaches to compute auxiliary, interaction integral quantities along cracks discretized by linear and curved elements. Comparisons of numerical results with analytical solutions for stress intensity factors verify the accuracy of the proposed interaction integral procedures.

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Loading Rate Effects on Parameters of the Weibull Stress Model for Ferritic Steels
Gao, X. and Dodds, R.
Engineering Fracture Mechanics, Vol. 72, pp. 2416-2425, 2005

This study investigates the effects of loading rate on parameters of the Weibull stress model for prediction of cleavage fracture in a low strength, strongly rate-sensitive A515-70 pressure vessel steel. Based on measured, dynamic fracture toughness data from deep- and shallow-cracked SE(B) specimens, the calibrated Weibull modulus (m) at K*I=22.5 MPa√m/s shows little difference from the value calibrated previously using static toughness data. This newly obtained result supports the hypothesis in an earlier study [Gao X, Dodds RH, Tregoning RL, Joyce JA. Weibull stress model for cleavage fracture under high-rate loading. Fatigue Fract Engng Mater Struct 2001;24:551-64] that the Weibull modulus likely remains rate independent for this material over the range of low-to-moderate loading rates. Additional experimental and computational results for higher rates show that a constant m-value remains applicable up to the maximum loading rate imposed in the testing program (K*I ≈ 2200 MPa√m/s). Rate dependencies of the scale parameter (σu) and the threshold parameter (σw-min) are computed using the calibrated m, and the results indicate that σu decreases and σw-min increases with higher loading rates. The predicted cumulative probability for cleavage fracture exhibits a strong sensitivity to small changes in σu. Consequently, σu must be calibrated using dynamic fracture toughness data at each loading rate of interest in an application or selected to make the Weibull stress model predict a dynamic master curve of macroscopic toughness for the material.

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Mode Mixity for Circular Hollow Section X-Joints with Weld Toe Cracks
Qian, X., Dodds, R. and Choo, Y.S.
ASME Journal of Offshore Technology & Artic Engineering, Vol. 127, pp. 269-279, 2005

This paper describes the mode mixity of stress-intensity factors for surface cracks at weld toes located at the saddle point in circular hollow section X joints. The remote loading applies a uniform tensile stress at the end of the brace along its axis. The three dimensional finite element models employ mesh tieing between a topologically continuous, global mesh and a separate, local crack-front mesh. Analyses of a simple plate model that approximates key features of toe cracks at the brace-chord intersection verify the negligible effects of the recommended mesh-tieing scheme on stress intensity factors. The linear-elastic analyses compute the mixed-mode stress intensity factors along the crack front using an interaction-integral approach. The mixed-mode stress intensity factors indicate that the crack front experiences predominantly mode I loading, with KIII→0 near the deepest point on the front φ=π/2. The total crack driving force, described by the J integral, reaches a maximum value at the deepest point of the crack for the crack aspect ratio a/c=0.25 considered here. The mode-mixity angle, ψ=tan-1(KII/KI), at φ=π/2 is compared for a range of practical X-joint configurations and crack-depth ratios. The present study demonstrates that the mode-mixity angle increases with increasing brace-to-chord diameter ratio β and decreasing chord radius to wall thickness ratio γ. Values of the nondimensional stress intensity factors FI=KI/σbarbr√(πa), and FII=KII/σbarbr√(πa), , however, show an opposite trend, with higher crack driving forces for small β and large γ ratios. The variations in the brace-to-chord wall thickness ratio τ and the crack depth ratio (a/t0) do not generate significant effects on the mode mixity.

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Three Dimensional Model for Overload Effects on Fatigue Crack Closure in the Small-Scale Yielding Regime
RoyChowdhury, S. and Dodds, R.
Fatigue & Fracture of Engineering Materials and Structures, Vol. 28, pp. 891-907, 2005

This paper describes the effects of a single overload event, within otherwise constant amplitude cycles, on the plasticity-induced closure process for mode I fatigue crack growth in the small-scale yielding (SSY) regime. The 3-D finite element (FE) analyses extend the initially straight, through-thickness crack front by a fixed amount in each load cycle, using a node release procedure. Crack closure during reversed loading occurs when nodes behind the growing crack impinge on a frictionless, rigid plane. A bilinear, purely kinematic hardening model describes the constitutive response of the elastic-plastic material. Extensive crack growth in the analyses, both before and after the overload, allows the crack to grow out of the initial and the post-overload transient phases, respectively. The work presented here shows that the large plastic deformation in the overload cycle reduces the crack driving force through enhanced closure. Further, the residual plastic deformations due to the overload cause a disconnected pattern of closure in the wake long after the crack front passes through the overload plastic zone. The computational studies demonstrate that the 3-D scaling relationship for crack opening loads established in our earlier work for constant amplitude cycling (with and without a T-stress) also holds before, during and after the overload event. For a specified ratio of overload-to-constant amplitude loading (ROL=KOLmax/Kmax), the normalized opening load (Kop/Kmax) at each location along the crack front remains unchanged when the constant amplitude peak load (Kmax), thickness (B) and material flow stress (σ0) all vary to maintain a fixed value of Kbar=Kmax/(σ0√B). The paper concludes with a comparison of the post-overload response predicted by the 3-D analyses and by the conventional Wheeler model.

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Effects of Weld-Strength Mismatch on Elastic-Plastic Fracture in Circular Hollow Section X-Joints
Qian, X., Dodds, R. and Choo, Y.S.
Engineering Structures, Vol. 27, pp. 1419-1434, 2005

This study describes the elastic-plastic driving force in surface cracks located at weld toes near the saddle point of circular hollow section X-joints, with strength mismatch between the chord material and welds. The remote loading at the brace end imposes displacements acting along the brace axis. The 3-D finite element models couple a global, topologically continuous mesh and a separate, local crack-front model through mesh-tieing. The numerical solver computes the elastic-plastic crack driving force (J-value) through a domain-integral approach. Comparisons of the elastic-plastic J-values evaluated from a continuous model and from a mesh-tied model for a simple plate configuration, which represents key features of the brace-chord intersection near the saddle point, verify the accuracy of J-values computed from mesh-tied models containing both homogeneous and mismatched material properties. The numerical analyses employ stress-strain curves for representative high-strength steels now used in offshore construction. The yield strength of the welds follows σyw=mσyc, where m denotes the mismatch ratio and σyc is the chord yield stress. The strain hardening property of the welds remains the same as that of the chord material. Unlike historical research on weld mismatch effects for simple fracture specimens, the surface crack in the tubular X-joint resides in the base metal (chord) adjacent to the weld toe rather than in the welds. The computed J-values demonstrate that the crack driving force increases with increased weld strength—and thus a higher potential for initiation of ductile tearing. The numerical results show that a large elastic-plastic crack driving force exists for joints with a high chord radius to wall thickness ratio (γ) or with a small brace to chord diameter ratio (β). For joints with β>0.8, the model with uniform material properties σyw=σyb=σyc exhibits the largest crack driving force among the different mismatch ratios.

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Temperature Dependence of Weibull Stress Parameters: Studies Using the Euro-material
Wasiluk, B., Petti, J. and Dodds, R.
Engineering Fracture Mechanics, Vol. 73, pp. 1046-1069, 2006.

This work demonstrates the temperature invariance of the Weibull stress modulus, m, for a 22Ni-MoCr37 pressure vessel steel through calibrations at two extreme temperatures of the ductile-to-brittle transition. This temperature invariance reflects the characterization of microcrack size distribution in the material described by the modulus. The calibrations performed here also demonstrate the clear dependence of the Weibull scale parameter, σu, on temperature. The increase of σu with temperature reflects the increase in microscale toughness of ferritic steels. The calibration procedure employs a three-parameter Weibull stress model which includes the effects of a minimum (threshold) toughness, Kmin. The calibrations suggest that Kmin increases gradually with temperature. Finally, an engineering procedure is presented to enable practical applications of the Weibull stress model for defect assessments. This procedure combines the demonstrated temperature invariance of m, a recently developed method for predicting the variation of σu with temperature using the Master Curve, and calibration of the Weibull stress parameters at one temperature. The (calibrated) temperature invariant m and the estimated σu as a function of temperature are used to predict the cumulative probability of fracture for several large datasets without direct calibration.

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Mode-Mixity for Tubular K-Joints with Weld Toe Cracks Under Different Boundary Conditions
Qian, X., Dodds, R. and Choo, Y.S.
Accepted for publication November 2005, Engineering Fracture Mechanics

This paper examines the mode-mixity of stress-intensity factors for surface cracks located near weld toes of the crown point at the toe of the tension brace in circular hollow section K-joints under different chord/brace end constraints. The remote loading applies a uniform stress at the end of the brace along its axis. The 3-D finite element models employ mesh-tieing between a topologically continuous, global mesh and a separate, local crack-front mesh. The linear-elastic analyses compute the mixed-mode stress-intensity factors along the crack front using an interaction-integral approach. The analyses for three different crack locations along the brace-chord intersection demonstrate that the critical location for the surface flaw lies at the crown point of the brace toe. The numerical investigation of six different boundary conditions indicate that the unbalanced loading conditions cause significant bending stresses in the chord and generate a considerably larger mode I stress-intensity factor than found for the balanced loading case. The mixed-mode stress-intensity factors indicate that the crack front experiences predominantly mode I loading, with KIII→0 near the deepest point on the front (φ=π/2). The non-dimensional, mode I crack driving force, described by FI=KI/σbarbr√(πa), reaches a maximum value at the deepest point of the crack for the crack aspect ratio a/c=0.25 considered here. The mode-mixity angle, ψ=tan-1(KII/KI), at φ=π/2 is compared for a range of practical K-joint configurations. The present study demonstrates that the mode-mixity angle ψ becomes significantly larger in the balanced loading conditions than in the unbalanced loading condition. Values of the non-dimensional stress-intensity factors (FI=KI/σbarbr√(πa), and FII=KII/σbarbr√(πa)), however, decrease for the balanced loading condition. Variations in the brace-to-chord diameter ratio (β) and chord radius to wall thickness ratio (γ) generate significant changes in the mode mixity. Thin-walled joints (γ=20) generally experience a larger crack driving force with a higher mode-mixity angle.

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Modeling of Three-Dimensional Effects on Fatigue Crack Closure Processes
Dodds, R. and Roychowdhury, S.
Journal of ASTM International, Vol. 2, No. 5, pp. 2-19, 2005 (2003 Jerry L. Swedlow Lecture)

In ductile metals, plasticity-induced closure of fatigue cracks often retards significantly measured crack growth rates in the Paris regime and contributes strongly to the observed R-ratio effect in experimental data. This work describes a similarity scaling relationship based on the 3D small-scale yielding framework wherein the thickness, B, defines the only geometric length-scale of the model. Dimensional analysis suggests a scaling relationship for the crack opening loads relative to the maximum cyclic loads (Kop/Kmax) governed by the non-dimensional load parameter, Kmax/(s0√B), i.e., a measure of the in-plane plastic zone size normalized by the thickness. Both Kop and Kmax refer to remotely applied values of the mode I stress-intensity factor. Large-scale, 3D finite element analyses described here demonstrate that Kop/Kmax values vary strongly across the crack front in thin sheets but remain unchanged when Kmax, B, and s0 vary to maintain Kmax/(s0√B) = constant. The paper also includes results to demonstrate that the scaling relationship holds for non-zero values of the T-stress (which affect the Kop/Kmax values) and for an overload interspersed in the otherwise constant amplitude cycles. The present results focus on R=Kmin/Kmax=0 loading, although the scaling relationship has been demonstrated to hold for other R>0 loadings as well. The new similarity scaling relationship makes possible more realistic estimates of crack closure loads for a very wide range of practical conditions from just a few analyses of the type described here.

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Constraint Comparisons for Common Fracture Specimens: C(T)s and SE(B)s
Petti, J. and Dodds, R.
Engineering Fracture Mechanics, Vol. 71, pp. 2677-2683, 2004

New testing standards (e.g., ASTM E1921) remain under continuing development to measure the fracture toughness of ferritic steels over the ductile-to-brittle transition. The procedures assume that relatively small, deep-notch test specimens maintain near small-scale yielding conditions at fracture, which simplifies greatly the interpretation of measured values. However, 3-D finite element analyses suggest that the geometry and small size of common fracture specimens leads frequently to constraint loss, e.g., the decay of small-scale yielding conditions, at only moderate levels of deformation. The Weibull stress micromechanical model, or "local approach," is employed here to quantify these constraint effects. Previous research along these same lines quantifies constraint loss in common fracture specimens relative to strict plane-strain, small-scale yielding conditions with a zero T-stress. Here we present a more practical approach for application within experimental testing programs by comparing directly the two most commonly tested fracture specimens, the single-edge notched bend, SE(B), and the compact tension, C(T), specimens. Developers of testing standards may thus choose a "reference" specimen then correct values measured with other specimens to the adopted reference configuration.

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Calibration of the Weibull Stress Scale Parameter σu, Using the Master Curve
Petti, J. and Dodds, R.
Engineering Fracture Mechanics, Vol. 72, pp. 91-120, 2005

This work proposes that the Weibull stress scale parameter, σu, increases with temperature to reflect the increasing microscale toughness of ferritic steels caused by local events that include plastic shielding of microcracks, microcrack blunting, and microcrack arrest. The Weibull modulus, m, then characterizes the temperature invariant, random distribution of microcrack sizes in the material. Direct calibration of σu values at temperatures over the DBT region requires extensive sets of fracture toughness values. A more practical approach developed here utilizes the so-called Master Curve standardized in ASTM Test Method E1921-02 to provide the needed temperature vs. toughness dependence for a material using a minimum number of fracture tests conducted at one temperature. The calibration procedure then selects σu values that force the Weibull stress model to predict the Master Curve temperature dependence of KJc values for the material. At temperatures in mid-to-upper transition, the process becomes more complex as fracture test specimens undergo gradual constraint loss and the idealized conditions of high-constraint, small-scale yielding assumed in E1921-02 gradually degenerate. The paper develops the σu calibration process to incorporate these effects in addition to consideration of threshold toughness effects and the testing of fracture specimens with varying crack-front lengths. Initial illustrations of the calibration process for simpler conditions, i.e. 1T crack-front lengths, use the temperature dependent flow properties and a range of toughness levels for an A533B pressure vessel steel. Then using the extensive fracture toughness data sets for an A508 pressure vessel steel generated recently by Faleskog et al. [Engng. Fract. Mech., in press], the paper concludes with calibrations of both m and σu over the DBT region and assessments of the Master Curve calibration approach developed here.

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Coupling of Micro and Macro-Scale Models to Predict Cleavage Fracture
Petti, J. and Dodds, R.
Engineering Fracture Mechanics, Vol. 71, pp. 1079-2103, 2004

This study couples the ASTM E1921 procedure to characterize the ductile-to-brittle toughness of ferritic steels in terms of KJc (or Jc) values with the Weibull stress model, i.e., the "local approach" for fracture at the microscale. The E1921 procedures assume that uniform, small-scale yielding (SSY) conditions exist at fracture along the full crack-front, which supports the use of a simple thickness scaling relationship to adjust experimental toughness values to an equivalent 1T size. For smaller specimens tested at temperatures in the mid-to-upper transition, plasticity induced constraint loss (crack-front triaxiality) frequently invalidates the simple scaling relationship. The non-dimensional functions, g(M=bσ0/J), derived from application of the Weibull stress approach for a specific specimen and material, describes the evolution of constraint loss effects on the fracture toughness relative to a plane-strain, SSY reference condition. The g-functions vary with parameters of the Weibull stress model, material flow properties, and specimen geometry, but not with the absolute specimen size. By combining the g-functions and a Weibull stress-based expression for the cumulative probability, a new procedure is proposed that adjusts (scales) measured toughness values simultaneously for both thickness and constraint loss directly within the existing E1921 framework. Monte Carlo simulations are used with the new approach to estimate the effects of constraint loss on the E1921 reference temperature, T0, for a range of specimen types, sizes and material properties.The paper concludes with an application of the new g-function approach in the E1921 framework for fracture tests performed on an A36 structural steel to correct the data sets for constraint loss and to estimate constraint loss effects on the T0 value.

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Crack Growth Resistance Behavior of a Functionally Graded Material: Computational Studies
Jin, Z.H. and Dodds, R.
Engineering Fracture Mechanics, Vol. 71, pp. 1651-1672, 2004

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.

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Effect of T-Stress on Fatigue Crack Closure in 3-D Small-Scale Yielding
RoyChowdhury, S. and Dodds, R.
International Journal for Solids and Structures, Vol. 41, pp. 2581-2606, 2004

This paper investigates the effects of in-plane constraint on 3-D fatigue crack closure in the small-scale yielding regime. The finite element analyses grow a sharp, straight-through crack in a modified boundary layer model under mode I, constant amplitude cyclic loading with prescribed but independent peak values of stress intensity factor, Kmax, and the T-stress, Tmax. A purely kinematic hardening law with constant modulus represents the material constitutive behavior. The computational results demonstrate that a two parameter characterization of crack tip fields in terms of Kmax/(σ0√B) and Tmax/σ0, where σ0, denotes the yield stress of the material, correlates successfully the normalized opening load Kop/Kmax across variations of thickness (B), constraint level and material flow properties. Both negative and positive T-stress reduce the through-thickness variation in local opening load levels along the crack front. A negative T-stress increases Kop/Kmax values, particularly at low peak loads where the plastic zone size remains a fraction of the thickness; a positive T-stress has limited effect on Kop/Kmax values. The fringe plots of individual plastic strain components reveal (a) in the absence of T-stress (Tmax/σ0=0), plastic contraction in the thickness direction compensates primarily for permanent stretching in the direction normal to the crack plane required for closure, (b) for negative T-stress (Tmax/σ0<0), plastic contraction in the in-plane transverse direction contributes the larger share of material flowing into the normal direction, and (c) for positive T-stress (Tmax/σ0>0), both in-plane directions experience permanent stretching and the thickness direction alone undergoes plastic contraction.

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Stress Intensity Factors for Surface Cracks in Functionally Graded Materials Under Mode-I Thermo-Mechanical Loading
Walters, M., Paulino, G. and Dodds, R.
International Journal for Solids and Structures, Vol. 41, pp. 1081-1118, 2004

This paper describes the development and application of a general domain integral method to obtain J-values along crack fronts in three-dimensional configurations of isotropic, functionally graded materials (FGMs). The present work considers mode-I, linear-elastic response of cracked specimens subjected to thermomechanical loading, although the domain integral formulation accommodates elastic-plastic behavior in FGMs. Finite element solutions and domain integral J-values for a two-dimensional edge crack show good agreement with available analytical solutions for both tension loading and temperature gradients. A displacement correlation technique provides pointwise stress-intensity values along semi-elliptical surface cracks in FGMs for comparison with values derived from the proposed domain integral. Numerical implementation and mesh refinement issues to maintain path independent J-values are explored. The paper concludes with a parametric study that provides a set of stress-intensity factors for semi-elliptical surface cracks covering a practical range of crack sizes, aspect ratios and material property gradations under tension, bending and spatially-varying temperature loads.

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