Stable growth of a crack interacting with a circular inclusion

The problem of failure of a plate containing a circular inclusion and a crack is studied. The crack is oriented along a diameter of the inclusion and the plate is subjected to a remote uniaxial stress perpendicular to the crack axis. The process of slow stable crack growth from initiation to termination is studied by the strain energy density theory. The crack growth is simulated by predicting finite increments of crack extension when material elements near the crack tip absorb a critical amount of strain energy density level, . Unstable crack growth occurs when the strain energy density factor S reaches a critical value where r_c is the critical size of the final crack increment prior to instability. The stress at crack initiation and the critical stress and crack length at failure are determined. The influence of the mechanical properties of the plate and the inclusion, the relative position of the inclusion and the crack and the crack length on the characteristic quantities of stable crack growth is analyzed. The dependence of the stable crack growth process on the loading rate is also investigated. Results are displayed in graphical form.     

A pseudo-elastic material damage model, part II: Application to the scaling of specimen size, material behavior and loading rate

A failure criterion is presented which relates the strain energy density of the material to both yielding and fracture. Cumulative material damage throughout a structural component may be monitored and the relative influence of yielding and stable crack growth assessed. The criterion is demonstrated, using finite element analysis, for center cracked panel specimens differing by material toughness values. From crack growth increment predictions using the uniaxial stress-strain behavior of the material, the criterion predicts the critical value of the strain energy density factor S_c governing crack instability.     

Non-self-similar crack growth in elastic-plastic finite thickness plate

This work is concerned with non-self-similar crack growth in medium strength metal plates while the loading step, plate thickness and material properties are altered. The three-dimensional elastic-plastic finite element stress analysis is combined with the strain energy density criterion for modeling the material damage process from crack initiation to final global instability including the intervening stage of slow crack growth. Both inelastic deformation and crack growth are accounted for each increment of loading such that the redistribution of stresses and strains are made for each new crack profile. Numerical results are obtained for the center cracked plate configuration under uniform extension with twenty-seven (27) different combinations of specimen thickness, loading step and material type. The fracture toughness S_c being related to K_1c for three different materials are predicted analytically from the corresponding uniaxial tensile test data. Effective strain energy density factor and half crack length are defined so that the results can be compared with their two-dimensional counterparts. Crack growth resistance curves (R-curves) are constructed by plotting as a function of . The condition is found to prevail during slow crack growth. Translation and/or rotation of the lines can yield results other than those calculated and serve a useful purpose for scaling component size and test time. The minimum thickness requirement for the ASTM valid K_1c test is also discussed in connection with predictions based on the strain energy density criterion. The corresponding K_1c for smaller specimens that exhibit moderate ductility and nonlinearity can also be obtained analytically. In such cases, the influence of loading step can be significant and should not be neglected. Notwithstanding the shortcomings of the theory of plasticity, the qualitative features of non-self-similar crack growth are predicted by the strain energy density criterion. Any refinements on the analytical modeling of the material damage process would only affect the results qualitatively, a subject that is left for future investigation.     

Influence of load amplitude and uniaxial tensile properties on fatigue crack growth

The rate at which energy is accumulated within a unit volume of material in fatigue is assumed to depend not only on load-time history but also on the specimen size and geometry in addition to material type. A threshold level for the hysteresis strain energy density function accumulated in the material is used for predicting macrocrack growth. This is accomplished by application of the incremental theory of plasticity for each increment of crack growth. The accumulated hysteresis strain energy density factor ΔS to crack growth increment Δa ratio is found to be constant for fixed specimen size and loading, i.e., . Results are obtained for the cylindrical bar specimens with a penny-shaped defect at the center subjected to a constant amplitude and frequency loading. The resistance curves in the ΔS versus Δa plot are parallel lines as specimen size is altered. This information provides a rational means for predicting the influence of specimen size on fatigue lifetime.The results are also compared with those found for geometrically similar plate specimens with line cracks. Cylinder bar specimens of the same material are found to sustain more load cycles prior to catastrophic failure.     

Growth of crack caused by temperature gradients with change in surface insulation

This work is concerned with thermoelastic stress and failure analysis of a centrally cracked panel subjected to temperature gradients while the insulation on the crack surface is varied. The corresponding temperature and thermoelastic stress fields are obtained by application of the finite element method. According to the strain energy density criterion, the crack grows incrementally when the maximum of the minimum strain energy density function reaches a critical value for a given material. Crack growth resistance curves involving plots of the strain energy density factor S versus the half crack length a are developed for crack surfaces with varying degree of heat resistance. The resulting curves are straight lines satisfying the condition dS/da = const. and useful for determining combined influence of thermal loading and structural geometry that lead to global instability.     

Strain energy density criteria for dynamic fracture and dynamic crack branching

Dynamic extension of Sih's fracture criterion based on strain energy density factor, r_c (dW/dV), is used to analyze dynamic crack propagation and branching. Influence of the nonsingular components, which are known as the higher order terms (HOT) in the crack tip stress field, on the strain energy density distribution at a critical distance surrounding the crack tip moving at constant crack velocity is examined. This r_c (dW/dV) fracture criterion is then used to analyze available dynamic photoelastic results of crack branching and of engineering materials.     

A pseudo-elastic material damage model,part I: Strain energy density criterion

A failure criterion is presented which relates the strain energy density of the material to both yielding and fracture. Cumulative material damage throughout a structural component may be monitored and the relative influence of yielding and stable crack growth assessed. The criterion is demonstrated, using finite element analysis, for center cracked panel specimens differing by material toughness values. From crack growth increment predictions using the uniaxial stress-strain behavior of the material, the criterion predicts the critical value of the strain energy density factor S_c governing crack instability.     

Size effect of cylindrical specimens with fatigue cracks

The rate at which energy is accumulated within a unit volume of material in fatigue is assumed to depend not only on load-time history but also on the specimen size and geometry in addition to material type. A threshold level for the hysteresis strain energy density function accumulated in the material is used for predicting macrocrack growth. This is accomplished by application of the incremental theory of plasticity for each increment of crack growth. The accumulated hysteresis strain energy density factor ΔS to crack growth increment Δa ratio is found to be constant for fixed specimen size and loading, i.e., $\text{ΔS}\text{Δa}\text{=}\text{const}$. Results are obtained for the cylindrical bar specimens with a penny-shaped defect at the center subjected to a constant amplitude and frequency loading. The resistance curves in the ΔS versus Δa plot are parallel lines as specimen size is altered. This information provides a rational means for predicting the influence of specimen size on fatigue lifetime.The results are also compared with those found for geometrically similar plate specimens with line cracks. Cylinder bar specimens of the same material are found to sustain more load cycles prior to catastrophic failure.     

Comparison of energy release rate and minimum strain energy density for potential failure of composite with bi-material interface

An analytical method is developed to describe the fields of stress and displacement in a bi-material strip specimen with an edge interfacial crack. All of the basic governing equations, boundary conditions on crack surfaces and conditions of continuity along the interface are satisfied by the eigenfunction expansion method. The other boundary conditions are satisfied by the generalized variational principle. The stress intensity factors are calculated for determining the energy release rate and minimum strain energy density factor S_min that is used the strain energy density criterion. Problems with oscillatory singularity and contact zone are discussed. Not only the effects of bi-material modulus ratio, thickness ratio, Poisson's ratio and crack length to S_min, but also the influences of bi-material modulus ratio, thickness ratio to phase angle are presented. Among these parameters, particular situations where S_min become jeopardously high and lead to failure are discussed.     

Crack growth resistance in non-perfect plasticity: Isotropic versus kinematic hardening

The Strain Energy Density Theory is applied for analyzing energy dissipation and crack growth in the three-point bending specimen when the material behavior follows a multilinear strain-hardening stress-strain relationship. The problem is solved through the application of incremental theory of plasticity and finite element method.The rate of change of the strain energy density factor S with crack length a is verified to be governed by the relation . Results are obtained for isotropic and kinematic hardening. Moreover, the effects of loading step and specimen size are pointed out.     

Mixed mode fatigue crack growth and the strain energy density factor

The strain energy density factor S was first proposed by Sih for the prediction of the critical of the load and failure direction under monotonic, mixed mode loading condition. It seems a natural extension to apply the same concept to fatigue crack propagation. However, a close examination of the existing theory indicates that the Strain Energy Density Factor cannot logically account for the phenomena of the R-ratio effect and crack arrest. Thus, modification is necessary before the concept can be applied successfully for the prediction of mixed mode fatigue crack propagation.Based on the concept of hysteresis energy dissipation, an effective strain energy density factor range, ΔS_p,eff, is proposed for the correlation of fatigue crack growth data. ΔS_p,eff is consistent with the concept of crack closure. Experimental investigation indicates that it could predict the crack growth rates and trajectories.     

Thickness reduction rate of plastically compressed metal plate by hardened rollers

The rate at which a solid deforms permanently depends on the load history, geometry and material properties. When a metal plate is compressed between two hardened rollers, its thickness reduces continuously if the material elements are deformed beyond their elastic limits. Those near the region of contact will experience more distortion as compared with those interior to the plate. This effect is analyzed incrementally in time by the theory of plasticity coupled with the strain energy density criterion. Failure is examined by assuming that the location of crack initiation coincides with the maximum of the minimum strain energy density function, (dW/dV)_min^max, when reaching its critical value. This is found to occur at the center of the plate depending on the rate of deformation. An increase in plate thickness reduction without failure can be achieved by taking smaller loading steps. Displayed graphically are numerical results for five different load histories that provide useful insights into the rate dependent process of metal forming.     

Thermal/mechanical interaction of subcritical crack growth in tensile specimen

The mutually interacting thermal/mechanical effects are accounted for by retaining the rate of change of volume with surface dV/dA, in the surface/volume energy density theory. An exchange between surface and volume energy would thus prevail in accordance with the rates at which the loads are transmitted throughout the system. Obtained are results for a center-cracked specimen subjected to monotonically rising tensile load at the rate of . Eight load steps are taken with an even increment increase of 69 MPa starting from 276 MPa. The temperature is found to oscillate about the ambient condition as the crack grew incrementally in a stable fashion. What differed considerably from the plasticity theory are the local stress and strain distribution and the crack growth characteristics. This is particularly pronounced in regions close to the crack tip where the local strain rates and strain rate history change from element to element, an affect that is not accounted for in plasticity.     

Mode-I crack projection procedure using the strain energy density criterion

A practice used in linear elastic fracture mechanics is the projection of a crack onto a plane normal to the principal tensile stress axes for computing the stress intensity factor K_I. The minimum strain-energy criterion is applied for different crack configurations with the introduction of a safety factor S_i which is the ratio of the strain energy density factor of the projected crack and that of the original crack. Numerous crack configurations are investigated to illustrate the degree of conservativeness of the crack projection procedure.     

Mixed mode fatigue crack growth with a sudden change in loading direction

With a sudden change in the maximum load level, there will be a corresponding change in the crack driving force regardless of whether the load is applied monotonically or cyclically. The effective strain energy density factor range ΔS_p,eff has been used to correlate mixed mode fatigue crack propagation where the crack growth direction is not known as an a priori. Examined in this work is a sudden change of load direction on fatigue crack growth while the load level remains unchanged. Yielding is assumed to be localized near the crack tip such that the crack growth behavior can be described adequately by the elastic stress field. Under the conditions investigated, minimal change on crack growth rates is observed. No firm conclusion could be drawn on deviation of crack path for the case considered.     

A damage model for crack initiation and growth

A damage accumulation model is presented for the study of the problem of crack initiation and stable growth in an elastic-plastic material. A centre-cracked specimen subjected to a uniform stress perpendicular to the crack plane is considered. A coupled stress and failure analysis is performed by using a finite element computer program based on J₂-plasticity theory in conjunction with the strain energy density theory. After initial yielding, each material element follows a different equivalent uniaxial stress-strain behavior depending on the amount of energy dissipation by permanent deformation. A host of uniaxial stress-strain curves constituting parts of the same stress-strain curve were assigned to material elements for each increment of loading. The path-dependent nature of the onset of crack initiation and growth was revealed. The proposed model predicts faster crack growth rates than those obtained on the basis of a single uniaxial stress-strain curve and is closer to experimental observation.     

Influence of indenter geometry on half-plane with edge crack subjected to fretting condition

An edge crack is analyzed to study fretting failure. A flat punch with rounded corners and a half-plane are regarded as an indenter and a substrate, respectively. Plane strain condition is considered. Contact shear traction in the case of partial slip is evaluated numerically. It is assumed that an initial crack is extended to the point of minimum strain energy density in the half-plane from the trailing edge of contact. Dislocation density function method is used to evaluate K_I and K_II. The variations of K_I and K_II during crack growth are examined in the case of indentation by a punch with different ratio of the flat region (l) to the punch width (L). Sih's minimum strain energy density theory [1] is also applied to predict the propagation direction of the initial crack. The direction evaluated is similar to that found in the experiment. Stress intensity factor ranges (ΔK_I and ΔK_II) are examined during cyclic shear on the contact. For the design of contacting bodies, a suggestible geometry of punch for alleviating cracking failure is studied.     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

Fracture instability of reinforced panel with cracks emanating from hole

Initiation of failure by yielding and/or fracture depends on the magnitude of the distortion and dilatation of material elements. According to the strain energy density theory (SED), failure is assumed to initiate at the site of the local maximum of maxima [(dW/dV)^max_max]_L by yielding and the maximum of minima [(dW/dV)^max_min]_L by fracture. The fracture is assumed to start from point L where [(dW/dV)^max_min]_L appears and tends toward G where the global maximum of dW/dV minima appears, denoted by [(dW/dV)^max_min]_G. The distance l between L and G along the anticipated crack trajectory is an indication of failure instability of the system by fracture. If l is sufficiently large and [(dW/dV)^max_min]_L exceeds the threshold, fracture initiation could lead to global failure. The local and global failure instability of a composite structural component is studied by application of the strain energy density theory. The depicted configuration is that of a panel with a circular hole reinforced by two side strips made of different material. The case of two symmetric cracks emanating from the hole and normal to the applied uniaxial tensile stress is also analyzed. Results are displayed graphically to illustrate the geometry and dissimilar material properties influence the fracture instability behavior of the two examples.     

Fatigue crack growth rate in ferrite-martensite dual-phase steel

An empirical study is made on the fatigue crack growth rate in ferrite-martensite dual-phase (FMDP) steel. Particular attention is given to the effect of ferrite content in the range of 24.2% to 41.5% where good fatigue resistance was found at 33.8%. Variations in ferrite content did not affect the crack growth rate da/dN when plotted against the effective stress intensity factor range ΔK_eff which was assumed to follow a linear relation with the crack tip stress intensity factor range ΔK. A high ΔK_eff corresponds to uniformly distributed small size ferrite and martensite. No other appreciable correlation could be ralated to the microstructure morphology of the FMDP steel. The closure stress intensity factor K_cl, however, is affected by the ferrite content with K_cl/Kmax reaching a maximum value of 0.7. In general, crack growth followed the interphase between the martensite and ferrite.Dividing the fatigue crack growth process into Stage I and II where the former would be highly sensitive to changes in ΔK and the latter would increase with ΔK depending on the R = σ_min/σ_max ratio. The same data when correlated with the strain energy density factor range ΔS showed negligible dependence on mean stress or R ratio for Stage I crack growth. A parameter α involving the ratio of ultimate stress to yield stress, percent reduction of area and R is introduced for Stage II crack growth so that the da/dN data for different R would collapse onto a single curve with a narrow scatter band when plotted against αΔS.