Potential Energy Release Rate Research Articles

In situ characterisation of the strain fields of intragranular slip bands in ferrite by high-resolution electron backscatter diffraction

High angular resolution electron backscatter diffraction has been used to quantify the local elastic field at the tip of mechanically loaded intragranular slip bands observed in situ in the ferrite grains of an age-hardened duplex stainless steel (Zeron 100). The surface elastic strain field was integrated to calculate in-plane and out-of-plane surface displacements. This allowed the elastic fields to be parameterised in a finite element analysis, which used the displacement field as the boundary conditions, to obtain the potential strain energy release rate (J-integral) and three-dimensional stress intensity factors (KI,II,III).This new analysis method is demonstrated by examining the elastic fields around the tip of an incipient slip band, an array of slip bands and the loading of a slip band. Direct measurement of the stress tensor in the grain identified the active slip systems with the highest Schmid factor. The stress intensity factors ahead of the slip band, measured under load, were directly affected by the magnitude of loading and the inclination angle of the slip band to the observed surface.

Timescales of successful and failed subduction: insights from numerical modelling

SUMMARY The relatively short duration of the early stages of subduction results in a poor geological record, limiting our understanding of this critical stage. Here, we utilize a 2-D numerical model of incipient subduction, that is the stage after a plate margin has formed with a slab tip that extends to a shallow depth and address the conditions under which subduction continues or fails. We assess energy budgets during the evolution from incipient subduction to either a failed or successful state, showing how the growth of potential energy and slab pull, is resisted by the viscous dissipation within the lithosphere and the mantle. The role of rheology is also investigated, as deformation mechanisms operating in the crust and mantle facilitate subduction. In all models, the onset of subduction is characterized by high lithospheric viscous dissipation and low convergence velocities, whilst successful subduction sees the mantle become the main area of viscous dissipation. In contrast, failed subduction is defined by the lithospheric viscous dissipation exceeding the lithospheric potential energy release rate and velocities tend towards zero. We show that development of a subduction zone depends on the convergence rate, required to overcome thermal diffusion and to localize deformation along the margin. The results propose a minimum convergence rate of ∼0.5 cm yr−1 is required to reach a successful state, with 100 km of convergence over 20 Myr, emphasizing the critical role of the incipient stage.

A novel method for measuring AFO stiffness using motion capture

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

Poling angle effect on two mode-III semi-permeable collinear cracks in a piezoelectric strip: Strip-saturation model

A mathematical strip-saturation model is proposed to analyze the influence of changing poling direction on two hairline electrically semi-permeable collinear cracks embedded in a piezoelectric strip. The remote boundaries of the piezoelectric strip are subjected to anti-plane mechanical and in-plane electrical loads. Fourier series method is applied to transform the model into triple series equations which are then reduced to singular integral equations using the integral equation technique. The closed-form analytical expressions are derived for saturation zones length, crack-sliding displacement (CSD), crack opening potential drop (COPD), intensity factors (IFs), and energy release rate (ERR). Influence of the poling angle on CSD, COPD, and ERR are shown graphically for various cracked piezoelectric ceramics. In addition, the numerical study is also presented for analyzing the effect of inter-crack distance and prescribed electrical and mechanical loadings on the fracture parameters. All graphical results are analyzed and compared.

Design Rules for TSV Membranes

The thermo-mechanical robustness of the bottom edge of an open through silicon via (TSV) is investigated by simulating the most critical step during the TSV process flow. In this step, stress arises due to the different thermal expansion coefficients of the materials forming the TSV. By employing the finite element method, the potential energy release rate of a crack at the TSV edge is calculated. The potential energy release rate, which quantifies the crack driving force, reaches its maximum when the diameter of the metal layers is ${\sim }1.25\times $ and ${\sim }1.5\times $ of the TSV diameter. Furthermore, when the metal diameter is three times the TSV, the crack driving force is reduced by ~50% for both aluminum and copper layers forming the TSV membrane. It was found that the crack tends to propagate perpendicularly to the TSV conductive layer, consistently with experimental observations. Meanwhile, the influence of the TSV membrane on the TSV conductive layer is negligible at distances further than 10% of the TSV height. Results show also that the reduction of 25% of the metal amount of the thickest layer lowers the crack driving force by 40% in comparison to the continuous layer.

On the total electro-mechanical potential energy and energy release rate at the interface crack tips in an initially stressed sandwich plate-strip with piezoelectric face and elastic core layers

An investigation into the values of the total electro-mechanical potential energy and energy release rate (ERR) at the interface crack tips in an initially stressed sandwich plate-strip is carried out for the opening crack mode. It is assumed that the face layers of the considered sandwich plate are made from piezoelectric materials but that the core is made of elastic material, and that this sandwich plate has initial stresses caused by the uniformly distributed normal forces acting on the planes of the two parallel grounded ends of the plate. The plate has two parallel interface cracks symmetrically located with respect to the middle planes and additional uniformly distributed normal opening mechanical forces acting on the cracks’ edges. Moreover, the face planes of the face layers and the cracks’ edges are unelectroded with vanishing normal electric displacement and are free of electrical charge. The investigations are made by utilizing the three-dimensional linearized theory of electro-elasticity in a plane-strain state within the framework of the piecewise homogeneous body model. The solution to the corresponding boundary value problems are found numerically by employing the finite element method. Numerical results, which are presented and analyzed, illustrate the influences of the geometrical parameters and the initial stresses, as well as the coupling effects between the electro-mechanical fields, on the values of the total electro-mechanical potential energy and of the ERR.

Strip-electro-mechanical yield model for transversely situated two semi-permeable collinear cracks in piezoelectric strip

A strip-electro-mechanical model is proposed for two semi-permeable collinear cracks, symmetrically situated and transversely oriented in a poled piezoelectric strip. The remote boundary of the strip is subjected to mode-III stress and in-plane electric-displacement. Fourier series method together with integral equation method is employed to obtain the analytic solution. Expressions are derived for crack-sliding displacement, crack opening potential drop, intensity factors and energy release rate. Also, explicit expressions for developed slide-yield and saturation zones are derived. An illustrative case study is presented for poled PZT-5H, BaTiO3, PZT-6B and PZT-7A piezoelectric ceramics to study the capability of the strip-electro-mechanical yield model for the crack arrest. Results obtained are graphically presented, discussed and concluded.

Crack deflection by the transformable particles dispersed in composites

The crack deflection in transformable particle-reinforced composites is studied in the present paper. The contribution of phase transformation on the crack tip J k -integral (k = 1, 2) is explicitly determined by the material configurational theory. For the crack deflection angle from its original crack path induced by the phase transformation it can be shown that the crack initiates in the direction along which the potential energy release rate in terms of the crack tip J k -integral possesses a stationary (maximum) value. The influence of one individual particle near the crack tip on the crack deflection is studied by accounting for both dilatant and shear transformation components. Furthermore, an FEM method is developed to model the stress-induced phase transformation on the basis of a macroscopic phenomenological constitutive model where multiple particles are taken to be non-uniformly distributed in a matrix.Numerical simulations are performed to observe the crack deflection by a cluster of particles. The results show a significant non-symmetric stress distribution locally at the crack tip, causing the crack to deflect. It is found that regions in the material with a higher volume fraction of transformable particles tend to deflect the crack growth more.

Experimental assessment of toughness in ceramic matrix composites using the J-integral with digital image correlation part I: methodology and validation

The complex nature of damage in ceramic matrix composites (CMCs) renders conventional methods of measuring quantitative fracture properties impractical. This paper is the first of two-part series that assesses the feasibility of using the J-integral with full-field deformation data from digital image correlation (DIC) to characterize toughness in continuous fiber SiC/SiC CMC laminates. It provides a resource for best practices when incorporating experimentally measured, full-field deformation data into J-integral evaluations of toughness. The techniques discussed are important for researchers attempting to measure fracture properties in advanced materials with damage mechanisms that have yet to be well characterized. Two methods for evaluating potential energy release rates J are presented: (1) numerical integration over a line contour and (2) Gaussian integration over an area contour. Accuracy and path independency for both methods were verified using analytically derived deformation fields for a center-cracked, infinite plate of isotropic material under equi-biaxial tension. Inherent noise in the deformation data and necessary contour truncation at the crack surfaces reduced accuracy and introduced path dependency. However, this was mitigated by careful noise filtering of the deformation data prior to the evaluation of J. Applying the line and area integrals to DIC data from tapered, double-cantilever beam, acrylic compact tension specimens resulted in the findings that (1) both integrals measured fracture toughness within a range of published values; (2) both integrals captured the constant stress intensity factor behavior that is characteristic of the tapered beam geometry; and (3) the area integral measurements were consistently larger than line integral measurements. The area integral is more accurate than the line integral as it samples more data points, reducing its sensitivity to experimental noise. Although variability in experimental data can be minimized, it cannot be eliminated. Measurement error is inevitable; thus, the line and area integrals must be used with caution when characterizing quantitative properties including fracture toughness.

Experimental assessment of toughness in ceramic matrix composites using the J-integral with digital image correlation part II: application to ceramic matrix composites

This paper is the second of a two-part series assessing the feasibility of using the J-integral with full-field deformation data from digital image correlation (DIC) to characterize toughness in continuous fiber, SiCf/SiCc ceramic matrix composites (CMCs). In part I, line and area integral adaptations of the J-integral were validated with analytical and experimental deformation fields. It was found that variability in experimental data and contour truncation at the crack wake introduced error and path-dependency, but this could be mitigated through data filtering. In this paper, the line and area integral methods were evaluated for cross-ply and longitudinal CMC tensile specimens. It is demonstrated that these approaches cannot be used for quantitative characterization, but can be successfully used to examine qualitative trends. For both architectures, area integral measurements of the potential energy release rate, J, were larger (and arguably more accurate) than line integral measurements. Resistance curves were characteristic of tough materials, but the stress intensity magnitudes were larger in the longitudinal laminates than the cross-ply (for equivalent crack extension). Experimentally derived bridging laws showed that fiber bridging tractions were largest in the cross-ply laminates, which was attributed to a lower volume fraction of longitudinal fibers. Quantitative, single-value measurements of fracture toughness could not be established since (1) noise in the DIC data masked strain signals indicative of first matrix cracking, and (2) widespread matrix cracking rendered the J-integral invalid (regardless of size and position, the boundaries of the line and area contours inevitably intercepted microcracks). Rather, gross estimates of toughness and stress intensity factors were made under the assumption that the limitations in the spatial resolution of the DIC data essentially transforms regions with fine microcracking along the contour into continuous regions of elastic deformation. When evaluated with the J-integral, the smoothed deformation fields manifest as resistance curves indicative of a tough material. Thus, the J-integral was valuable for qualitative assessments of toughening mechanisms in the CMCs.

Quantifying the energy dissipation of overriding plate deformation in three‐dimensional subduction models

AbstractIn a subduction system the force and the energy required to deform the overriding plate are generally thought to come from the negative buoyancy of the subducted slab and its potential energy, respectively. Such deformation might involve extension and back‐arc basin formation or shortening and mountain building. How much of the slab's potential energy is consumed during overriding plate deformation remains unknown. In this work, we present dynamic three‐dimensional laboratory experiments of progressive subduction with an overriding plate to quantify the force (FOPD) that drives overriding plate deformation and the associated energy dissipation rate (ΦOPD), and we compare them with the negative buoyancy (FBU) of the subducted slab and its total potential energy release rate (ΦBU), respectively. We varied the viscosity ratio between the plates and the sublithospheric upper mantle with ηSP/ηUM = 157–560 and the thickness of the overriding plate with TOP = 0.5–2.5 cm (scaling to 25–125 km in nature). The results show that FOPD/FBU has average values of 0.5–2.0%, with a maximum of 5.3%, and ΦOPD/ΦBU has average values of 0.05–0.30%, with a maximum of 0.41%. The results indicate that only a small portion of the negative buoyancy of the slab and its potential energy are used to deform the overriding plate. Our models also suggest that the force required to deform the overriding plate is of comparable magnitude as the ridge push force. Furthermore, we show that in subduction models with an overriding plate bending dissipation at the subduction zone hinge remains low (3–15% during steady state subduction).

Sensitivity Analysis of Linear Elastic Cracked Structures Using Generalized Finite Element Method

In this work, a sensitivity analysis of linear elastic cracked structures using two–scale Generalized Finite Element Method (GFEM) is presented. The method is based on computation of material derivatives, mutual potential energies, and direct differentiation. In a computational setting, the discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires the multiplication of the displacement vectors and stiffness sensitivity matrices. By judiciously choosing the velocity field, the method only requires displacement response in a sub-domain close to the crack tip, thus making the method computationally efficient. The method thus requires an exact computation of displacement response in a sub-domain close to the crack tip. To this end, in this study we have used a two-scale GFEM for sensitivity analysis. GFEM is based on the enrichment of the classical finite element approximation. These enrichment functions incorporate the discontinuity response in the domain. Three numerical examples which comprise mode-I and mixed mode deformations are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method.

A mathematical strip-induction-saturation model for an off-centric semipermeable crack in a piezoelectromagnetic strip

A mode-III problem of strip-induction-saturation model is considered for a transversely isotropic piezoelectromagnetic strip weakened by a semipermeable non-centric crack. Three cases are considered: when the developed induction zone is larger, smaller and equal to the saturation zone. Fourier integral transform is used to reduce the problem into Fredholm integral equations of the second kind and solve numerically. Closed-form analytic expressions are derived for developed zones, field intensity factors, crack sliding displacement, crack opening induction drop, crack opening potential drop, local and global energy release rates. An illustrative numerical example is considered for a poled cracked narrow BaTiO3–CoFe2O3 strip to show the effect of prescribed loading combinations and developed zone length on the aforementioned fracture parameters. Results obtained are presented graphically, analyzed and concluded.

A study on influence of poling direction on piezoelectric plate weakened by two collinear semi-permeable cracks

A study on the influence of change in poling direction is carried out for mechanical and electric strip-yield model for a transversely isotropic piezoelectric plate cut along two equal collinear semi-permeable straight hairline cracks. Two cases are considered, when the developed saturation zone is bigger or smaller than mechanical yield zone. The Stroh formalism and complex variable technique are employed to obtain the solution. The effect of change in poling direction is studied on crack opening displacement, crack opening potential drop and energy release rate. Also, a study is carried out on the suitability of crack closure for different piezoelectric ceramics. The obtained results are presented graphically and discussed.

Evolution of Subduction Zone Curvature and its Dependence on the Trench Velocity and the Slab to Upper Mantle Viscosity Ratio

Three‐dimensional laboratory models of upper mantle subduction are presented investigating the effect of the trench velocity (vt) and the slab to upper mantle viscosity ratio (ηSP/ηUM) on trench curvature and slab curvature. One set of experiments varies ηSP/ηUM from 66 to 1375. Another set of experiments modifies vt through applying different velocities at the trailing plate. The results show that the radius of trench curvature (RTC) progressively decreases with continuous trench retreat due to quasi‐toroidal mantle return flow from the subslab region around the lateral slab edges and toward the mantle wedge region, but starts to increase when trench retreat changes to trench advance due to quasi‐toroidal return flow in the opposite direction. Furthermore, RTC increases with increasing ηSP/ηUM (i.e., progressively stronger slabs are progressively less curved) following a cubic root function. The force required to curve the slab and trench varies with progressive subduction, but is only 0.1–2.5% of the total negative buoyancy force of the slab, while the viscous dissipation rate due to progressive slab curvature is only 0.01–0.60% of the potential energy release rate. Comparison of trench curvature in the models (scaled slab width w = 750 km) with that of the Scotia subduction zone (w ≈ 800 km) indicates that for the Scotia subduction zone the effective ηSP/ηUM is of the order 1–2 × 102. Finally, the elastic sphere indentation model for arc curvature is revisited, demonstrating that four main predictions are not met by observations, implying that the model should be rejected.

Analysis of Potential Energy Release Rate of Composite Laminate Based on Timoshenko Beam Theory

The Timoshenko beam theory is used to model each part of cracked beam and to calculate the potential energy release rate. Calculations are given for the double cantilever beam specimen, which is simulated as two separate beams connected elastically along the uncracked interface.

Integral Representation of Energy Release Rate in Graded Materials

It is well known that, for homogeneous materials, the path-independent J contour integral is the (potential) energy release rate. For general nonhomogeneous, or graded materials, such a contour integral as the energy release rate does not exist. This work presents a rigorous derivation of the extended J integral for general graded materials from the potential energy variation with crack extension. Effects of crack shielding and amplification due to a graded interlayer in an elastic-plastic material system are discussed in terms of this integral.

Investigation of the M -integral in crack-damaged piezoelectric ceramics

The physical interpretation of the M-integral is investigated in the analysis of crack-damaged piezoelectric problems. The relation between the M-integral and the change of the total electric enthalpy (CTEE), i.e., M = 2CTEE, is derived with a theoretical derivation procedure for two-dimensional piezoelectric problems. It is shown that the M-integral may provide a more natural description of electric enthalpy release due to the formation of the pre-existing microcracks associated with the damaged body, rather than the description of the total potential energy release rate as interpreted for conventional brittle solids. For crack-damaged piezoelectric ceramics, numerical calculation of the M-integral is discussed. Based on the pseudo-traction electric displacement method, M = 2CTEE has also been proved by the numerical results.

VectorJ-integral analysis of crack initiation at the edge of complete sliding contact

In this paper, the crack initiation at contact surface of solids is investigated on the basis of the concept of potential energy release rate. The expressions for path-independent integral vectorJi(i=1, 2) are derived and applied to the consideration of the process of crack initiation. The relationship is then established between the value of the path-independent integral vectorJiand the potential energy release rate for crack initiation in an arbitrary orientation. This allows the prediction of crack initiation angle on the basis of the maximum energy release rate criterion. The surface crack initiation angle in fretting fatigue is determined analytically as a function of the friction coefficient of the edge contact. This theoretical result is compared with the existing experimental results reported in the literature and a good agreement is found. The formulation provides a novel basis for numerical modelling of the complex process of fretting fatigue.

A general mixed-mode brittle fracture criterion for cracked materials

In order to predict the fracture direction and fracture loadings of cracked materials under the general mixed-mode state, this paper presents a new general mixed-mode brittle fracture criterion based on the concept of maximum potential energy release rate (MPERR). This criterion can be easily degraded to the pure-mode fracture criterion, and can also be reduced to the commonly accepted fracture criteria for the mixed-mode I/II state. In order to validate the proposed criterion, we have carried out the experiments with aluminium alloy specimens under various mixed-mode loading conditions. The experimental results agree well with the predictions of the proposed criterion.