Three-Dimensional Free Vibration Analyses of Preloaded Cracked Plates of Functionally Graded Materials via the MLS-Ritz Method

In this study, the moving least squares (MLS)-Ritz method, which involves combining the Ritz method with admissible functions established using the MLS approach, was used to predict the vibration frequencies of cracked functionally graded material (FGM) plates under static loading on the basis of the three-dimensional elasticity theory. Sets of crack functions are proposed to enrich a set of polynomial functions for constructing admissible functions that represent displacement and slope discontinuities across a crack and appropriate stress singularity behaviors near a crack front. These crack functions enhance the Ritz method in terms of its ability to identify a crack in a plate. Convergence studies of frequencies and comparisons with published results were conducted to demonstrate the correctness and accuracy of the proposed solutions. The proposed approach was also employed for accurately determining the frequencies of cantilevered and simply supported side-cracked rectangular FGM plates and cantilevered internally cracked skewed rhombic FGM plates under uniaxial normal traction. Moreover, the effects of the volume fractions of the FGM constituents, crack configurations, and traction magnitudes on the vibration frequencies of cracked FGM plates were investigated.

The effect of material anisotropy on the mechanics of a thin-film/substrate system under mechanical and thermal loads

In this study, the stress analysis for an orthotropic thin film bonded to an orthotropic elastic substrate is addressed using both the analytical and finite element methods. The analytical method employs the integrodifferential formulation with the aid of membrane assumption. Utilizing the finite element method, the effect of orientation of the material principal directions are studied. The loading scenarios include a temperature gradient imposed on the film and a far-field uniaxial tension on the substrate. The results of current study indicate that the ratio of the film to the substrate stiffness plays a leading role in the film stress distribution. For the mechanical loading applied to the substrate, a soft thin film attached to a relatively stiffer substrate is preferred. The film can tolerate the induced thermal stresses as it is bonded to a softer host structure. The rotation angle of material orthotropy directions significantly affects the stress singularity near the film edges up to a certain extent.

Elastic solution of a polyhedral particle with a polynomial eigenstrain and particle discretization

The paper extends the recent work (JAM, 88, 061002, 2021) of the Eshelby's tensors for polynomial eigenstrains from a two dimensional (2D) to three dimensional (3D) domain, which provides the solution to the elastic field with continuously distributed eigenstrain on a polyhedral inclusion approximated by the Taylor series of polynomials. Similarly, the polynomial eigenstrain is expanded at the centroid of the polyhedral inclusion with uniform, linear and quadratic order terms, which provides tailorable accuracy of the elastic solutions of polyhedral inhomogeneity by using Eshelby's equivalent inclusion method. However, for both 2D and 3D cases, the stress distribution in the inhomogeneity exhibits a certain discrepancy from the finite element results at the neighborhood of the vertices due to the singularity of Eshelby's tensors, which makes it inaccurate to use the Taylor series of polynomials at the centroid to catch the eigenstrain at the vertices. This paper formulates the domain discretization with tetrahedral elements to accurately solve for eigenstrain distribution and predict the stress field. With the eigenstrain determined at each node, the elastic field can be predicted with the closed-form domain integral of Green's function. The parametric analysis shows the performance difference between the polynomial eigenstrain by the Taylor expansion at the centroid and the 𝐶0 continuous eigenstrain by particle discretization. Because the stress singularity is evaluated by the analytical form of the Eshelby's tensor, the elastic analysis is robust, stable and efficient.

Couette Resistance Due to a Sliding Plate Over a Plate with Stripes of Non-Homogeneous Slip

Couette flow with non-homogeneous partial-slip stripes on one plate is studied. Drag and flow rate are found by an efficient eigenfunction expansion and point match method. Longitudinal motion (parallel to the stripes) experiences lower drag than transverse motion. As the gap width between the two plates approaches zero, the drag increases to a finite value if the stripes have partial slip, as comparison to the infinite value for no slip. Analysis of the region near the junction of a perfect stick-slip boundary shows a weak stress singularity while there is no singularity for partial slip junctions.

Automatic Adaptive Recovery Stress ES-FEM for Lower-Bound Limit Load Analysis of Structures

The paper proposes a novel automatic adaptive recovery stress edge-smoothed finite element method (ES-FEM) that determines the maximum load capacity of inelastic structures at plastic collapse. This approach performs solely a series of elastic ES-FEM analyses with systematic modulus variations (considering the influences of stress singularity) to converge the collapse load solutions. The smoothed [Formula: see text]-continuous recovery stress field ensures the satisfaction of static admissible stress and yield conformity conditions underpinning lower-bound limit analysis theorems. A modified modulus error function within the newest node bisection algorithm enables automatic mesh refining and coarsening constructions, and fast convergence to the lower-bound collapse limit.

Effect of mesoscale phase contrast on fatigue-delaying behavior of self-healing hydrogels

We investigate the fatigue resistance of chemically cross-linked polyampholyte hydrogels with a hierarchical structure due to phase separation and find that the details of the structure, as characterized by SAXS, control the mechanisms of crack propagation. When gels exhibit a strong phase contrast and a low cross-linking level, the stress singularity around the crack tip is gradually eliminated with increasing fatigue cycles and this suppresses crack growth, beneficial for high fatigue resistance. On the contrary, the stress concentration persists in weakly phase-separated gels, resulting in low fatigue resistance. A material parameter, λtran, is identified, correlated to the onset of non-affine deformation of the mesophase structure in a hydrogel without crack, which governs the slow-to-fast transition in fatigue crack growth. The detailed role played by the mesoscale structure on fatigue resistance provides design principles for developing self-healing, tough, and fatigue-resistant soft materials.

Fracture mechanics of rate-and-state faults and fluid injection induced slip

Propagation of a slip transient on a fault with rate- and state-dependent friction resembles a fracture whose near tip region is characterized by large departure of the slip velocity and fault strength from the steady-state sliding. We develop a near tip solution to describe this unsteady dynamics, and obtain the fracture energy G c , dissipated in overcoming strength-excursion away from steady state, as a function of the rupture velocity v r . This opens a possibility to model slip transients on rate-and-state faults as singular cracks characterized by approximately steady-state frictional resistance in the fracture bulk, and by a stress singularity with the intensity defined in terms of G c ( v r ) at the crack tip. In pursuing this route, we develop and use an analytical equation of motion to study 1-D slip driven by a combination of uniform background stress and a localized perturbation of the fault strength with the net Coulomb force Δ T . In the context of fluid injection, Δ T is a proxy for the injection volume V inj . We then show that, for ongoing fluid injection, the propagation speed of a transient induced on a frictionally stable fault is bounded by a large-time limiting value proportional to the injection rate dV inj /d t , while, for stopped injection, the maximum slip run-out distance is proportional to V inj , total 2 . This article is part of the theme issue ‘Fracture dynamics of solid materials: from particles to the globe’.

Analysis of Linear Elastic Fracture Mechanics for Cracked Functionally Graded Composite Plate by Enhanced Local Enriched Consecutive-interpolation Elements

This paper reports the application of consecutive-interpolation procedure into four-node quadrilateral elements for analysis of two-dimensional cracked solids made of functionally graded composite plate. Compared to standard finite element method, the recent developed consecutive-interpolation has been shown to possess many desirable features, such as higher accuracy and smooth nodal gradients it still satisfies the Kronecker-delta property and keeps the total number of degrees of freedom unchanged. The discontinuity in displacement fields along the crack faces and stress singularity around the crack tips are mathematically modeled using enrichment functions. The Heaviside function is employed to describe displacement jump, while four branch functions being developed from asymptotic stress fields are taken as basis functions to capture singularities. The interesting characteristic of functionall graded composite plate is the spatial variation of material properties which are intentionally designed to be served for particular purposes. Such variation has to be taken into account during the computation of Stress Intensity Factors (SIFs). Performance of the proposed approach is demonstrated and verified through various numerical examples, in which SIFs are compared with reference solutions derived from other methods available in literatures.

On the scale dependence in the dynamics of rupture

<p>Potential energy stored during the inter-seismic period by tectonic loading around faults can be released through earthquakes as radiated energy, heat and rupture energy. The latter is of first importance, since it controls both the nucleation and the propagation of the seismic rupture. On one side, the rupture energy estimated for natural earthquakes (also called Breakdown work) ranges between 1 J/m<sup>2</sup> and tens of MJ/m<sup>2</sup> for the largest events, and shows a clear slip dependence. On the other side, recent experimental studies highlighted that at the scale of the laboratory, rupture energy is a material property (energy required to break the fault interface), limited by an upper bound value corresponding to the rupture energy of the intact material (1 to 10 kJ/m<sup>2</sup>), independently of the size of the event, i.e. of the seismic slip.</p><p>To reconcile these contradictory observations, we performed stick-slip experiments, as an analog for earthquakes, in a bi-axial shear configuration. We analyzed the fault weakening during frictional rupture by accessing to the on-fault (1 mm away) stress-slip curve through strain-gauge array. We first estimated rupture energy by comparing the experimental strain with the theoretical predictions from both Linear Elastic Fracture Mechanics (LEFM) and the Cohesive Zone Model (CZM). Secondly, we compared these values to the breakdown work obtained from the integration of the stress-slip curve. Our results showed that, at the scale of our experiments, fault weakening is divided into two stages; the first one, corresponding to an energy of few J/m<sup>2</sup>, coherent with the estimated rupture energy (by LEFM and CZM), and a long-tailed weakening corresponding to a larger energy not observable at the rupture tip.</p><p>Using a theoretical analysis and numerical simulations, we demonstrated that only the first weakening stage controls the nucleation and the dynamics of the rupture tip. The breakdown work induced by the long-tailed weakening can enhance slip during rupture propagation and can allow the rupture to overcome stress heterogeneity along the fault. Additionally, we showed that at a large scale of observation the dynamics of the rupture tip can become controlled by the breakdown work induced by the long-tailed weakening, leading to a larger stress singularity at the rupture tip which becomes less sensitive to stress perturbations. We suggest that while the onset of frictional motions is related to fracture, natural earthquakes propagation is driven by frictional weakening with increasing slip, explaining the large values of estimated breakdown work for natural earthquakes, as well as the scale dependence in the dynamics of rupture.</p>