Griffith Energy Research Articles

A generalized V-shaped peeling model on rigid curved substrates incorporating pre-stretch effects

Abstract In biological adhesion systems, it is often observed that organisms climb on curved surfaces by controlling the relaxation and contraction of their muscles. In this paper, we propose a general model for V-peeling on an arc-shaped substrate based on Griffith's energy release theory. The developed model can handle V-peeling problems on arc-shaped substrates with arbitrary initial configurations and considers the effects of pre-stretching in the bonded segment. When the substrate radius and initial peeling angle are relatively large, the results approach those obtained for flat substrates. However, due to the constraints of the substrate size and its active alteration of the peeling angle, a steady state, as seen on flat substrates, does not occur during the peeling process. Pre-stress has a certain inhibitory effect on the initiation of delamination, but once delamination begins, the presence of pre-stress promotes it. Meanwhile, same degree of pre-stretch has a more pronounced effect on enhancing the peel resistance of structures with smaller curvatures, and pre-stretch further amplifies the differences between arc-shaped and flat substrates. The paper also discusses the impact of variations in different parameters on energy. The conclusions drawn in this study have implications for understanding biological adhesion and designing multilayered structures.

Variational damage model: A new paradigm for fractures

The computational modeling of fracture in solids using damage mechanics faces challenges with complex crack topology. This can be addressed by using a variational framework to reformulate the damage mechanics. In this paper, we propose several mathematically elegant variational damage models (VDMs) for fracture mechanics without explicitly using damage variables. Based on the energy density ϕ, the fracture energy density is formulated as ϕ^=ϕ(1+ℓϕ/Gc) and the damage variable is expressed as s = ϕ/(ϕ + Gc/ℓ), which satisfy ϕ^∣ϕ→∞=Gc/ℓ and s∣ϕ→∞ = 1 as ϕ approaches infinity. These limits demonstrate that the new energy density converges to the Griffith energy release rate at full-damage state. The VDM profoundly modified the energy functional, implicitly incorporating the damage field. As a generalization of previous model, we propose a family of VDMs of varying orders. Additionally, we develop a multi-damage model to account for different types of energy densities, such as elastic thin plate and gradient elasticity. Using this functional, it is straightforward to deduce the governing equation for automatically evolving fractures. These formulations can be employed in conventional finite element method or other numerical methods with minimal modifications. Compared to the phase field method with the same mesh density, a sharper crack interface can be achieved. We demonstrate the capabilities of the proposed variational damage formulations using representative numerical examples.

A Griffith description of fracture for non-monotonic loading with application to fatigue

With the fundamental objective of establishing the universality of the Griffith energy competition to describe the growth of large cracks in solids not just under monotonic but under general loading conditions, this paper puts forth a generalization of the classical Griffith energy competition in nominally elastic brittle materials to arbitrary non-monotonic quasistatic loading conditions, which include monotonic and cyclic loadings as special cases. Centered around experimental observations, the idea consists in: (i) viewing the critical energy release rate Gcnot as a material constant but rather as a material function of both space X and time t, (ii) one that decreases in value as the loading progresses, this solely within a small region Ωℓ(t) around crack fronts, with the characteristic size ℓ of such a region being material specific, and (iii) with the decrease in value of Gc being dependent on the history of the elastic fields in Ωℓ(t). By construction, the proposed Griffith formulation is able to describe any Paris-law behavior of the growth of large cracks in nominally elastic brittle materials for the limiting case when the loading is cyclic. For the opposite limiting case when the loading is monotonic, the formulation reduces to the classical Griffith formulation. Additional properties of the proposed formulation are illustrated via a parametric analysis and direct comparisons with representative fatigue fracture experiments on a ceramic, mortar, and PMMA.

A frost heave pressure model for fractured rocks subjected to repeated freeze-thaw deterioration

Rock engineering in the Tibetan Plateau is usually suffered severe freeze-thaw (F-T) deterioration, triggering numerous geotechnical engineering disasters. As the essential triggering factor of F-T damage, the frost heave pressure (FHP) in rocks should be quantitatively described and predicted. In this study, a novel theoretical model is established and validated to estimate the critical entire evolution process of FHP in fractured rocks subjected to repeated F-T deterioration, in which the influence of fracture geometric configuration, rock mechanical properties and F-T conditions are comprehensively considered. Besides, given the obvious five-stage manner of FHP evolution, the combination effect of different dominant mechanisms in different stages is theoretically addressed. During freezing, after a silence stage induced by temperature above freezing point of water, volumetric expansion and frost-heave cracking separately controls the generation and reduction stages of FHP, and the solutions are obtained based on elastic mechanics and Griffith energy balance theory, respectively. During thawing, ice segregation dominates the second-arising stage of FHP that is quantitatively described via segregation potential and water migration velocity, and then the FHP enters the dissipation stage owing to complete melting of fracture ice. In addition, a decay coefficient is introduced to estimate the influence of F-T cycle number on FHP, and a piecewise FHP model is finally constructed for fractured rocks subjected to repeated F-T deterioration. To validate our new model, a total of 12 F-T cycle tests with real-time FHP monitoring are conducted on granite and sandstone specimens considering three typical influencing factors, i.e., fracture width, rock types and freezing temperatures. A rational consistency is identified between the theoretical and testing results in terms of the FHP evolution curves and the peak FHPs in rock fracture, demonstrating the generalizability and robustness of our proposed model.

Modulating adhesion strength in multi-ferroic composite materials: Insights from adhesive contact with arbitrary profile indenters

Adhesion control is a critical aspect of various applications, from industrial adhesion devices to the locomotion of insects on ceilings and walls. Multi-ferroic materials, which encompass mechanical, electrical, and magnetic properties, offer approaches for reversible adhesion control. This study presents a comprehensive theoretical framework for adhesive contact in multi-ferroic composite materials when subjected to axisymmetric rigid indenters with arbitrary profiles. We analytically derive the physical fields for various contact models, including Hertz contact, Johnson–Kendall–Roberts (JKR), and Maugis–Dugdale (MD) adhesive models. The obtained energy release rates indicate that the electric and magnetic potentials can modulate adhesion strength. The Griffith energy balance relation is employed to derive the indentation forces and penetration depths, which can be extended to a range of common indenters. Notably, the classical approximations fail when using small spherical indenters, but the study provides valid alternatives. The influence of amplitude and wavelength on contact behavior is explored, with greater effects observed for larger or smaller values. For cosine-shaped indenters, equivalence to flat-ended cylindrical punches is established under specific conditions. The study also reveals that the power indices for power-law-shaped indenters change the influence of electric and magnetic potentials on pull-off forces. These findings provide a theoretical fundament associated with the biomimetic and artificial adhesive systems and the modern testing techniques.

Develop A new mathematical approach to predict the crack propagation rate in a thin plate subjected to in-plane biaxial loading

The mechanical design of metals must deal with structures collapsing due to cracks occur in the material. This study focused on the edge crack at non-proportional cycle loading because it propagates quickly. This research attempted to determine two mathematical models that could predict the crack propagation rate for thin samples of aluminum alloy types 6061 and 5052 under constant tensile stresses and cyclic shear stress applied within the elastic limits of the material using “Griffith energy of dynamic fracture”. This was performed to evaluate if these models can predict crack growth rates and compare with the numerical results of a computer system in the ANSYS program R1 2021. The direction of the crack path was calculated and compared with the analytical program. The results of the two mathematical models in predicting the dynamic growth rate in the studied alloys gave low error rates to the numerical solution.

A unified treatment of axisymmetric adhesive contact for piezoelectric materials

Nanoindentation technique has been widely applied to characterize the electromechanical properties of piezoelectric materials, where the adhesion effect induced by different surface forces becomes very prominent. The indenter tip should have more general profile rather than just three common simple shapes (flat, cone and sphere) in practical testing. In this study, the classical Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (M-D) models are extended to investigate the adhesion behaviors of a piezoelectric half-space indented by the power-law shaped punches with different electric properties, whose shape index is denoted as n. The JKR-n and M-D-n adhesion models for both the electrically conducting and electrically insulating punches are set up by means of Griffith energy balance. The Derjaguin-Muller-Toporov-n (DMT-n) adhesion solutions are derived from the corresponding M-D-n adhesion models as the limiting cases. The generalized Tabor parameter and interfacial adhesion strength applicable to piezoelectric materials are defined for the first time, with the former being used to describe the transition behavior from JKR-n model to DMT-n model. The impacts of the electric loading and the shape index on adhesion behaviors are discussed in detail. It is found that the pull-off force increases with the electric potential, which reveals the adhesion strengthening effect caused by electric loading. The shape index has a prominent influence on the pull-off force, interfacial adhesion strength, contact radius at pull-off moment, indentation depth and contact radius at self-equilibrium state, and the transition behavior from JKR-n model to DMT-n model. The results derived from this work not only are helpful to understand the contact behaviors of piezoelectric materials at micro/nano scale, but also provide a theoretical basis for nanoindentation technique in evaluating material properties of piezoelectric mediums.

Regularity improvement for the minimizers of the two-dimensional Griffith energy

In this paper, we prove that the singular set of connected minimizers of the planar Griffith functional has Hausdorff dimension strictly less than one, together with the higher integrability of the symmetrized gradient.

Equilibrium Configurations for Nonhomogeneous Linearly Elastic Materials with Surface Discontinuities

We prove a compactness and semicontinuity result that applies to minimisation problems in nonhomogeneous linear elasticity under Dirichlet boundary conditions. This generalises a previous compactness theorem that we proved and employed to show existence of minimisers for the Dirichlet problem for the (homogeneous) Griffith energy.

Atomistic insight into the defect-induced tunable plasticity and electronic properties of tetragonal zirconia

Tetragonal zirconia (t-ZrO2) with exceptional properties is crucial in catalytic applications. Defects effectively enhance its electronic properties and plasticity, making t-ZrO2 a valuable material for high-efficiency industrial catalysts, and flexible electronic devices. In this study, we utilize first-principles calculations to compare the impact of vacancy defects and nitrogen doping on the structural and electronic properties of wide bandgap semiconductor t-ZrO2. Additionally, we analyze the tuned plasticity of t-ZrO2 by varying N-dopant and vacancy concentrations using molecular dynamics simulations and cyclic-nanoindentation tests. The estimation of energy release associated with plastic deformation is conducted using the Griffith energy balance model. Our study reveals significantly distinct electronic properties and plastic deformation maps in oxygen-deficient and N-doped t-ZrO2 compared to the perfect counterpart, where the defect nature dictates the band gap energy and plastic zone size. t-ZrO2−x exhibits a greater bandgap narrowing than t-ZrO2−xNx, resulting from increased atomic displacement, decreased free energy for plastic deformation, and enhanced plastic dissipated energy. Furthermore, we demonstrate that electronic properties and the plasticity of t-ZrO2−xNx, including bandgap energy and energy release rate during cyclic-nanoindentation, are unable to compete with those of the small-bandgap counterpart t-ZrO2−x. Herein, t-ZrO2−x, x = 0.2 exhibits the highest plasticity and the smallest bandgap of 1.28 eV, contrasting with the 5.6 eV bandgap of perfect t-ZrO2. Thereby, t-ZrO2−x displays pronounced band gap tightening and enriched flexibility, providing it a favorable semiconductor for photoelectrochemical energy conversion (PEC) applications.

A modified single edge V-notched beam method for evaluating surface fracture toughness of thermal barrier coatings

The surface fracture toughness is an important mechanical parameter for studying the failure behavior of air plasma sprayed (APS) thermal barrier coatings (TBCs). As APS TBCs are typical multilayer porous ceramic materials, the direct applications of the traditional single edge notched beam (SENB) method that ignores those typical structural characters may cause errors. To measure the surface fracture toughness more accurately, the effects of multilayer and porous characters on the fracture toughness of APS TBCs should be considered. In this paper, a modified single edge V-notched beam (MSEVNB) method with typical structural characters is developed. According to the finite element analysis (FEA), the geometry factor of the multilayer structure is recalculated. Owing to the narrower V-notches, a more accurate critical fracture stress is obtained. Based on the Griffith energy balance, the reduction of the crack surface caused by micro-defects is corrected. The MSEVNB method can measure the surface fracture toughness more accurately than the SENB method.

Nonlocal analysis of edge‐cracked nanobeams under Mode I and Mixed‐Mode (I + II) static loading

AbstractIn the present paper, the behavior of an edge‐cracked nanobeam under both Mode I and Mixed‐Mode (I + II) static bending loading is analyzed by using the stress‐driven nonlocal model (SDM), being the SDM applied for the first time in the case of Mixed‐Mode loading. More precisely, the cracked nanobeam is modeled by using a modification of the classical cracked‐beam theory, consisting in dividing the beam into two beam segments connected through a massless elastic rotational spring located at the cracked cross section. The bending stiffness of the cracked cross section is computed by exploiting: the Griffith energy criterion and the conventional linear elastic fracture mechanics. Cracks under both Mode I and Mixed‐Mode (I + II) loading are considered. Finally, an edge‐cracked cantilever nanobeam subjected to a transversal point load at the free end is analyzed. The static deflection is calculated and discussed for different values of relative crack depth, crack location, and crack orientation.

INCREASING MOISTURE CONTENT AS FAILURE RATE PROPAGATOR OF HOT MIX ASPHALT PAVEMENT USING FRACTURE MECHANICS

The design and construction of safe and durable roads is a necessity to promote national development. The challenges encountered with roads design and construction industries worldwide is the high financial implication for quality and durability. This study proposes a new approach for modelling the nonlinear elastic behavior of Hot Mix Asphalt using fracture mechanics (a principle concerned with the propagation of cracks in material elements). This is achieved by simulating crack propagation or crack growth in asphalt pavements using a bottom-up crack growth progression. However, time rate delamination of flexible pavement is of paramount importance. There is a need to determine a safer and sustainable means of predicting and determining the performance of roadways before, during and after its construction. This will result to cheaper road design and analysis, effective maintenance measure and reduced cost of maintenance. This study is on time rate delamination of flexible pavement, the effect of Moisture/Hydraulic Coefficient, temperature variation and stiffness ratio of the pavement. Consequently, adaptive meshing is used in the Axis-Symmetric model followed by a model for nonlinear viscoelastic behavior of Hot Mix Asphalt and simulating crack propagation in asphalt pavements using Griffith energy criterion. The findings indicate that there is a strong relationship between environmental condition (moisture content) and Asphalt Concrete resilient Modulus. Increasing temperature and moisture content results to reduction in pavement strength. Thou the failure is not visible at the onset of crack propagation (crack tip), but continual exposure to increasing temperatures as well as increasing moisture content will lead to failure of the pavement before the design life is attained. Furthermore, there is also a strong linear relationship between Griffith strain energy and the Pavement Resilient Modulus.

The Effect of Adhesion on Indentation Behavior of Various Smart Materials

The nanoindentation technique plays a significant role in characterizing the mechanical properties of materials at nanoscale, where the adhesion effect becomes very prominent due to the high surface-to-volume ratio. For this paper, the classical adhesion theories were generalized to study the contact behaviors of various piezoelectric materials indented by conical punches with different electric properties. With the use of the Hankel integral transform, dual integral equations, and superposing principle, the closed-form solutions of the physical fields for the Johnson-Kendall-Roberts (JKR) and Maugis-Dugdale (M-D) models were obtained, respectively. The contribution of the electrical energy to the energy release rate under the conducting punch was taken into consideration. The relationships between the contact radius, the indentation load, and the indentation depth were set up using the total energy method for the JKR model and the Griffith energy balance for the M-D model, respectively. Numerical results indicate that increasing the half cone angle of the conical punch enhances the adhesion effect, which can significantly affect the accuracy of the results of characterization in nanoindentation tests. It was found that the effect of electric potential on adhesion behaviors is sensitive to different material properties, which are not revealed in the existing studies of axisymmetric adhesive contact of piezoelectric materials and multiferroic composite materials. The load-displacement curves under conical punches with different half cone angles have very different slopes. These results indicate that the half cone angle has a prominent effect on the characterization of mechanical properties of piezoelectric solids in nanoindentation tests.

Adhesive behavior between dissimilar materials subjected to thermo-elastic loadings with normal-tangential coupling effect

The temperature difference always exists between contacting objects in various thermal bonding technologies, where the adhesion strength is found to be very sensitive to the temperature. In this paper, the anisothermal adhesive contact between an elastic cylinder subjected to the external force with an inclined angle and an elastic substrate is investigated based on the non-slipping JKR model, where the coupling effect between the normal and tangential tractions within the contact region is taken into consideration. The stated problem is reduced to a system of coupled singular integral equations, which can be solved with the analytical function theory. The full analytical solutions of the oscillatory stress fields are obtained for the first time by taking the series expansion of the integrand induced by the thermoelastic effect, which is of substantial significance. The relationship between the contact size and the external loadings is established by virtue of the Griffith energy balance criterion. The difference between the oscillatory and non-oscillatory solutions of stress fields is discussed in detail. Numerical results demonstrate that the oscillatory solution cannot be well approximated by the corresponding non-oscillatory solution for the cases of large Dundurs’ index and external compressive force. Furthermore, it is found that both the thermoelastic effect and the inclined angle of the applied force have significant influences on the adhesive contact behaviors. The presence of temperature difference between contacting objects can either enhance or weaken the adhesion effect, which is related to the inclined angle of the external force. The results obtained from this paper may provide a new insight into the influence mechanism of the temperature on adhesion behaviors.

Nano-samples give higher brittle strength by the Griffith energy principle

The purpose of this paper is to show that brittle test samples give a huge size effect that can take several different forms depending on the sample geometry, crack position and mode of force application. Sometimes crack equilibrium force depends on sample dimension d or d1/2 and sometimes the force is independent of area, for example in peel or lap joint cracking. This big size effect arises from the potential energy term in the conservation theory, not considered by Griffith but dominating certain cracks. These examples illustrate the fact that strength of a brittle material containing a crack is an unsatisfactory concept because the cracks absorb surface energy driven by volume energy terms or by potential energy terms or a mixture of the two, leading to a disconnection between applied cracking force and sample cross-section area. The flaw statistics argument mentioned by Griffith is unnecessary, though strength can be affected in certain circumstances by the presence of random flaws. An unusually large size effect is shown experimentally for thermal shock of ceramic tubes, in which the cracking force increases as the cube of diameter goes down. This thermal shock resistance of fine tubes has proved important for application of ceramic fuel cells but cannot be explained by fracture mechanics theory at present. The conclusion is that experimental results show the Griffith energy criterion for cracking is correct whereas the Galilean stress criterion fails. The concept 'strength of brittle materials' is therefore untenable for most crack testing geometries. This article is part of the theme issue 'Nanocracks in nature and industry'.

New theory explaining Griffith strength results on nano-cracked glass fibres.

The purpose of this paper is to propose a mechanism and theory that can explain the extraordinary increase in measured strength that Griffith observed for glass fibres containing nano-cracks. His 1921 theory (Griffith 1921 Phil. Trans. R. Soc. Lond. A 221, 163-198. (doi:10.1098/rsta.1921.0006)) predicted that the strength of a cracked sample should be independent of sample size, yet his results on stretched glass fibres gave strength increasing almost as the inverse of fibre diameter. He proposed a 'flaw statistics' argument in an attempt to explain these bizarre results, suggesting strength increased because large defects were less likely in the smaller volumes. But this 'flaw statistics' concept is unnecessary because the Griffith energy criterion of cracking must give a size effect, as demonstrated in many different crack-testing configurations. In general, the Griffith energy criterion for crack equilibrium predicts strength rising for smaller samples because such samples contain less volume energy to create new crack surface energy. The problem is that this 'size effect' idea has not until now been properly defined for the simple tension crack test. The new idea proposed is that many nano-cracks are likely to exist in an experimental glass sample, so these must also obey the thermodynamic analysis. A problem then arises because, as the main crack propagates, other cracks may close, but healing is not reversible in glass so thermodynamics does not apply completely to these secondary cracks. Crack healing is in the Griffith theory, which is perfectly reversible mathematically, though not explicitly stated. This article is part of the theme issue 'Nanocracks in nature and industry'.

Cassette-like peeling system for testing the adhesion of soft-to-rigid assemblies

A novel cassette-like peeling system is developed to address the limitations of current peeling standards when evaluating bonding quality of soft-to-rigid assemblies. The system transforms the translation of a specimen in the conventional peeling configuration to rotation via a cassette-like spool clamping the specimen. The peeled film is loaded by tension to drive the winding of the spool, thus achieving self-similar crack propagation and a stationary peeling front unrelated to the stiffness of the film. These features enable the system’s compatibility with most universal testers and in situ observation of crack tip morphology with optical instruments. Analysis to derive the intrinsic fracture energy when peeling a soft film is conducted based on Griffith energy balance, making use of which, a parametric study is performed to clarify the related mechanisms. We carry out a comprehensive validation of the cassette-like peeling system by performing a series of peeling tests using our in-house prototype and by comparing the results with those from the conventional system. Owing to its universality and ease-of-use, the proposed cassette-like peeling system can potentially be applied to the development of the next generation of peel test standards.

Asymmetric non-slipping adhesion behavior of layered piezoelectric structures

Layered piezoelectric structures have found wide applications in micro-electro-mechanical systems (MEMS) due to their intrinsic electro-mechanical coupling effect. The adhesion effect resulting from various surface forces (e.g., the electrostatic force, capillary force, and Van der Waals force) has become one dominant factor in determining the reliability and service life of the MEMS devices. In this study, the asymmetric non-slipping adhesion behavior of layered piezoelectric structures under the action of different external loadings is investigated. A system of coupled singular integral equations are obtained for the stated problem in the framework of the generalized JKR theory, where the coupling effect between the normal and tangential contact stresses is taken into consideration. The closed-form analytical solutions of the electric displacement and stress fields within the contact zone are derived. The relations between the contact size and applied loadings are set up with the Griffith energy balance criterion. The effect of the mechanical and electric loads and the mismatch strain on the adhesion behavior of layered piezoelectric structures is discussed in detail. It is found that applying the electric load can only strengthen the adhesion effect for different types of layered piezoelectric structures. Both the inclined angle of the mechanical force and the geometric parameter of the contacting objects have significant effects on the adhesion behavior. The results obtained from present paper should be very useful for understanding the adhesion failure mechanism of the MEMS devices composed of layered piezoelectric structures.

Adhesive behavior of transversely isotropic piezoelectric bimaterials

The adhesive contact behavior between two dissimilar transversely isotropic piezoelectric cylinders is investigated in this paper. The effect of adhesion on the contact behavior between piezoelectric bimaterials is considered within the framework of the non-slipping JKR model, which takes the coupling effect between the tangential and normal tractions inside the contact region into account. The explicit analytical solutions of stress and electric displacement fields within the contact region are obtained by solving the resulting coupling singular integral equations with the analytical function theory. The relationship between the contact size and external loadings are established by adopting the Griffith energy balance criterion at the edges of the contact zone. It is found that different types of piezoelectric bimaterials show the same adhesive contact behaviors, which is different from the case of a single layer piezoelectric material. The result obtained from the present paper should be helpful for the design of micro-electro-mechanical systems (MEMS) devices involving multi-layered smart structures.