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 notched cracked plates characterized by the strain energy density theory

The fracture instability of a mechanical system is analyzed by the strain energy density theory. The local relative minima of the strain energy density function dW/dV referred to local coordinate systems at each point of the body are distinguished from the global minimum of dW/dV, G, which is referred to a fixed global coordinate system. Failure by fracture starts from the maximum of the local minima of dW/dV, L, and passes from point G. The distance l between L and G along the fracture trajectory is introduced as a length parameter to characterized the fracture instability of the system. Numerical results are obtained and discussed for a cracked plate with two symmetrical notches subjected to a monotonically rising tensile stress perpendicular to the crack axis.     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.     

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.     

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.     

Characterization of material inhomogeneity by stationary values of strain energy density

The strain energy density theory is applied to analyze the fracture instability of a mechanical system whose behavior is governed by the interaction of geometry, load and material inhomogeneity. This is accomplished by obtaining the location of the global and local relative minima of the strain energy density function dW/dV denoted, respectively, by [(dW/dV)_min]_g and [(dW/dV)_min]_¢l. The former refers to a fixed global coordinate system for the entire solid while the latter corresponds to local coordinate systems referred to each material point. An unique length parameter “ℓ” representing the distance between [(dW/dV)_min^max]_g and [(dW/dV)_min^max]_ℓ can thus be found and serves as a measure of the degree of system instability tending toward failure by fracture.Numerical results are obtained and displayed graphically for the case of a solid containing an inclusion of dissimilar material. The changes that take place in material inhomogeneity, loading type and physical dimensions of the solid and inclusion are reflected through ℓ. The method suggests the compatibility of ℓ for each member of a multi-component structure in order to avoid premature failure of a single member.     

Piezomagnetic and piezoelectric poling effects on mode I and II crack initiation behavior of magnetoelectroelastic materials

The ferrite and ferroelectric phase of magnetoelectroelastic (MEE) material can be selected and processed to control the macroscopic behavior of electron devices using continuum mechanics models. Once macro- and/or microdefects appear, the highly intensified magnetic and electric energy localization could alter the response significantly to change the design performance. Alignment of poling directions of piezomagnetic and piezoelectric materials can add to the complexity of the MEE material behavior to which this study will be concerned with.Appropriate balance of distortional and dilatational energy density is no longer obvious when a material possesses anisotropy and/or nonhomogeneity. An excess of the former could result in unwanted geometric change while the latter may lead to unexpected fracture initiation. Such information can be evaluated quantitatively from the stationary values of the energy density function dW/dV. The maxima and minima have been known to coincide, respectively, with possible locations of permanent shape change and crack initiation regardless of material and loading type. The direction of poling with respect to a line crack and the material microstructure described by the constitutive coefficients will be specified explicitly with reference to the applied magnetic field, electric field and mechanical stress, both normal and shear. The crack initiation load and direction could be predicted by finding the direction for which the volume change is the largest. In contrast to intuition, change in poling directions can influence the cracking behavior of MEE dramatically. This will be demonstrated by the numerical results for the BaTiO₃–CoFe₂O₄ composite having different volume fractions where BaTiO₃ and CoFe₂O₄ are, respectively, the inclusion and matrix.To be emphasized is that mode I and II crack behavior will not have the same definition as that in classical fracture mechanics where load and crack extension symmetry would coincide. A striking result is found for a mode II crack. By keeping the magnetic poling fixed, a reversal of electric poling changed the crack initiation angle from θ₀=+80° to θ₀=−80° using the line extending ahead of the crack as the reference. This effect is also sensitive to the distance from the crack tip. Displayed and discussed are results for r/a=10⁻⁴ and 10⁻¹. Because the theory of magnetoelectroelasticity used in the analysis is based on the assumption of equilibrium where the influence of material microstructure is homogenized, the local space and temporal effects must be interpreted accordingly. Among them are the maximum values of (dW/dV)_max and (dW/dV)_min which refer to as possible sites of yielding and fracture. Since time and size are homogenized, it is implicitly understood that there is more time for yielding as compared to fracture being a more sudden process. This renders a higher dW/dV in contrast to that for fracture. Put it differently, a lower dW/dV with a shorter time for release could be more detrimental.     

Mechanics and physics of energy density theory

As a departure from the classical continuum mechanics approach, irreversible material behavior is uniquely identified with the exchange of surface and volume energy density through the rate of change of volume with surface area. This provides an one-to-one correspondence between the uniaxial and multiaxial stress or energy state. Equivalent uniaxial stress and strain response can thus be determined for each material element that undergoes damage by permanent deformation and/or fracture. Discussed in detail are the applications of surface energy or volume energy with continuum mechanics theories for analyzing failure in contrast to the strain energy density that analyzes stress and failure simultaneously. In particular, the results for the progressive damage of a slowly moving crack are presented and compared with those obtained from the theory of plasticity. It is shown that the neglected of dilatational energy in plasticity results in inaccurate prediction of the state of affairs near the crack tip.Since the stresses and strains in the strain energy density theory are determined separately, nonlinear and finite strains may be easily included into dV/dA and incorporated into the formulation. This opens the door to a class of nonlinear problems that can be solved directly even for finite and large strains including energy dissipation.     

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.     

The strain energy density failure criterion applied to notched elastic solids

Application of the strain energy density failure criterion is made to plane notch problems, where the crack now becomes a special case of a more generalized approach to failure. The specific case considered is that of the plane elliptical cavity under remote tension and compression. Both failure loads and fracture trajectories are discussed. It is shown that an additional characteristic dimension provides satisfactory agreement of the theory with available data. Finally, known characteristics of fracture trajectories from a notch tip are shown to be predicted for unstable fracture conditions.     

Crack growth instability studied by the strain energy density theory

The strain energy density theory has successfully been used to address the problem of material damage and structural failure in problems of engineering interest. The theory makes use of the strain energy density function, dW/dV, and focuses attention in its stationary values. The directions of crack growth and yielding are determined from the minimum and maximum values of dW/dV, respectively, along the circumference of a circle centered at the point of failure initiation. Failure by crack growth or yielding takes place when these values of dW/dV become equal to their critical values which are material constants. In the present work the basic principles of the strain energy density theory were reviewed. Furthermore, this theory was used to study three problems of structural failure, namely the problem of slow stable growth of an inclined crack in a plate subjected to uniaxial tension, the problem of fracture instability of a plate with a central crack and two notches, and the problem of unstable crack growth in a circular disc subjected to two equal and opposite forces. The results of stress analysis were combined with the strain energy density theory to obtain the whole history of crack growth from initiation to instability. A length parameter was introduced to define the fracture instability of a mechanical system. Fracture trajectories were obtained for fast unstable crack propagation.     

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.     

Failure initiation sites and instability of structural components with localized energy density

When the surface or interior of a solid undergoes curvature and/or material change, there results localized fluctuation of the energy density field depending on the type of loading. These fluctuations are related to changes in the distortion and dilatation of material elements that could lead to failure by yielding and/or fracture should their magnitude become sufficiently large. According to the strain energy density criterion, failure is assumed to initiate at site of local maximum of minimum strain energy density denoted by [(dW/dV)_min^max]_L and tends toward the global maximum of [(dW/dV)_min^max]_G. The distance l between these two stationary values of dW/dV at L and G provides an indication of failure instability. That is, large l corresponds to more wide spread failure while the opposite holds for small l.Specimens with three different geometries are analyzed; they consist of round shoulder, hole and edge notch. The loads are either bending or tension. As the severity of notch or hole curvature is varied, predicted failure path also altered from boundary to boundary or an interior point of the specimen. The narrowest section turns out to be most vulnerable. If the hole is filled with a material of higher modulus, it acts as a reinforcer such that failure site would be shifted away from the interface. In general, there prevails a trade off between l and [(dW/dV)_min^max]_L. The undesirable combination would be for l and [(dW/dV)_min^max]_L to increase simultaneously. Failure initiation and global instability would then likely occur in tandem. This corresponds to the bending of a specimen with round shoulders. A variety of other conditions are analyzed with results displayed graphically so that the ways with which load, geometry and material inhomogeneity affect the failure behavior of structural components with notches and holes could be better understood.     

Isoparametric shear spring element applied to crack patching and instability

In this work the isoparametric shear spring element is applied to the stress and energy analysis of a center-crack panel reinforced by a rectangular patch. In this model, only transverse shears are assumed to prevail in the adhesive layer. The stresses and crack-tip stress intensity factors are obtained for reinforcement on both sides and one side of the panel, and are found to be in agreement with those obtained by previous authors using the triangular shear spring element.Crack stability that tends to vary with patch thickness is determined from the local and global maximum of the minimum strain energy density function denoted, respectively, as [(dW/dV)_min^max]_L at point L and [(dW/dV)_min^max]_G at point G. The distance l between L and G gives the prospective path of subcritical crack growth and its magnitude provides a measure of the degree of crack stability. A patched panel with small l tends to be more stable than that with large l. By increasing the patch thickness beyond a certain value, l can be contained within the patch such that failure, if initiated, will be highly localized. Such a behavior is exhibited. Numerical results are provided to support the foregoing conclusion.     

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.     

Strain energy density prediction of crack initiation and direction in cracked T-beams and pipes

In this paper, the S-theory is applied to determine crack initiation and direction for cracked T-beams and circumferentially cracked pipes. It makes use of a parameter called strain energy density factor, S, which is a function of the stress intensity factors. The strain energy density theory provides a more general treatment of fracture mechanics problems by virtue of its ability in describing the multiscale feature of material damage and in dealing with mixed mode crack propagation problem. A simple method for obtaining approximate stress intensity factors is also applied. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Some basic loading conditions in beams and pipes are studied.     

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.     

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.     

Fracture of piezoelectric materials: energy density criterion

In this paper, the concept of energy density factor S for piezoelectric materials is presented. In addition to the mechanical energy the electrical energy is included as well. The direction of crack initiation is assumed to occur when S_min reaches a critical value S_cr that can be used as an intrinsic materials parameter and is independent of the crack geometry and loading. The result agrees with empirical evidence qualitatively and explains rationally the effect of applied electric field on fracture strength: positive electric fields decrease the apparent fracture toughness of piezoelectric materials while negative electric fields increase it.