Internal Length Research Articles

Geometric theory of mechanical screening in two-dimensional solids.

Holes in mechanical metamaterials, quasilocalized plastic events in amorphous solids, and bound dislocations in a hexatic matter are different mechanisms of generic stress relaxation in solids. Regardless of the specific mechanism, these and other local stress relaxation modes are quadrupolar in nature, forming the foundation for stress screening in solids, similar to polarization fields in electrostatic media. We propose a geometric theory for stress screening in generalized solids based on this observation. The theory includes a hierarchy of screening modes, each characterized by internal length scales, and is partially analogous to theories of electrostatic screening such as dielectrics and Debye-Hückel theory. Additionally, our formalism suggests that the hexatic phase, traditionally defined by structural properties, can also be defined by mechanical properties and may exist in amorphous materials.

Mixed mode I/II fracture behavior of CSTBD sandstone specimen under different loading angles

Cracks are often exposed to mixed mode I/II loading because of the random direction of crack relative to the external load, causing rock failure in rock engineering. However, many mixed mode I/II fracture properties of rock with different loading angles (β) are still poorly understood. To investigate the influence of β on mixed mode I/II fracture behavior, cracked straight through Brazilian disc tests were performed on sandstone under various β at the loading rate of 0.2 kN/s. The results show that the effect of β on the peak load and crack propagation velocity is slight. The mode I stress intensity factor (SIF) decreases from positive to negative with increasing β, while mode II SIF first increases and then decreases as β exceeds approximately 30°. The mixed mode I/II fracture toughness increases linearly with increasing β. When β exceeds 60°, the crack initiation location shifts from the center of the semi-circular notch tip to the surface of preset crack. For non-tip cracking, the crack propagates toward the loading point in a direction approximately perpendicular to the preset crack. The threshold β for it decreases with increasing internal friction coefficient and relative crack length. Moreover, the mixed-mode fracture surface becomes smoother as the mode I component contribution drops. There are remarkable differences in the prediction of fracture initiation angles and SIFs of various rock types applying the generalized maximum tangential stress criterion. The findings of this study could help understand the mixed mode I/II fracture observed in rock engineering.

Relaxed Incremental Formulations for Damage at Finite Strains Including Strain Softening

AbstractRelaxation is a promosing technique to overcome mesh‐dependency in computational damage mechanics originating from the non‐convexity of an underlying incremental variational formulation. This technique does not require an internal length scale parameter. However, in case of damage formulations, for many years the decrease of stresses with an increase of strains, referred to as strain‐softening, could not be modeled in the relaxed regime. This contribution discusses several possibilities of relaxation that lead to suitable models for stress‐ and strain‐softening.

Frontiers in homogenization methods towards generalized continua for architected materials

Architected materials have witnessed tremendous research activity in the last two decades, due to their ability to achieve unprecedented effective properties. The up-scaling of the mechanical response towards an effective continuum is challenging, due to the possible emergence of a homogenized macroscopic behavior that may substantially differ from that of the base material at the micro level. The need for enriched continuum models beyond Cauchy’s theory arises due to the existence of a new class of artificial materials exhibiting static and dynamic attributes typically not encountered in natural materials, deserving the name meta-materials. In such a situation with a failure of the strict scale separation hypothesis, the enrichment of the Cauchy continuum proceeds by adding new intrinsic parameters and internal length scales representative of the micro-structural impact at the level of the effective medium. Generalized continua including either new kinematic variables or higher gradients of the displacement have proven their ability to provide a faithful description of the response of materials with micro-structural effects; their derivation from a micro-mechanical approach starting directly from the scale of the microstructure raises specific difficulties, in contrast to the phenomenological standpoint, which however lacks a predictive capacity. A survey of the scientific issues raised by higher-order homogenization methods towards generalized continua is provided in this contribution. The main scientific issues and challenges raised by higher-order homogenization methods are reviewed in a critical manner, and solutions to these scientific issues are proposed in light of the more recent contributions. Specific attention is devoted to the elaboration of enriched models for architected materials in both static and dynamic regimes.

Coupled hydromechanical analysis of sparsely fractured crystalline rocks

This paper addresses the issue of evolution and coalescence of localized damage zones in sparsely fractured crystalline rocks. The approach incorporates a constitutive law with embedded discontinuity, which is phrased in terms of both the hydraulic and mechanical response. The formulation takes into account the hydromechanical interaction in regions intercepted by discontinuities. An internal length scale parameter is employed in the definition of equivalent hydraulic conductivity and the tangential stiffness operators, and the onset of newly developed macrocracks is detected by the bifurcation analysis. An enhanced mixed u-p finite element formulation is derived which considers the effect of progressive evolution of the fracture aperture in the weak statements of balance equations. Fully implicit temporal discretization is employed, and the finite element formulation is stabilized by invoking the Polynomial-Pressure-Projection (PPP) technique. A coupled FE analysis is conducted examining the response of Luc du Bonnet granite, with pre-existing fracture network, subjected to plane strain compression that triggers the crack propagation and coalescence. The approach is first verified on some benchmark problems that involve the presence of a dominant fracture. The results of simulations are compared with those obtained using a very fine mesh incorporating interface elements. Later, a series of coupled analyses are carried out examining the hydromechanical response in the presence of multiple fractures.

Synthesis and characterization of synthetic lubricants based on dibasic acid

AbstractSynthetic esters have long been used in a variety of applications due to their excellent thermal stability, excellent cleanliness, natural lubricity, and polarity. In the present work, we aimed to prepare some synthetic base oils through preparation of different dibasic esters by esterification of dicarboxylic acids (adipic acid and azelaic acid) with different linear alcohols (hexanol, octanol, and decanol) and branched alcohol (2‐ethyl hexanol) at 120°C. The reaction yield ranges between 85% and 94%. Fourier‐transform infrared spectroscopy (FT‐IR) and proton nuclear magnetic resonance (1H‐NMR) spectroscopy were used to analyze the structures of the produced compounds. Using thermo gravimetric analysis (TGA), the heat stability of the produced esters was determined, and it was found that the prepared esters have high thermal stability. The degradation of the prepared esters takes place in the range between 300 and 600°C. The rheological behaviour of prepared esters shows Newtonian behaviours, which means that Newtonian fluids obey viscosity Newton's law. The viscosity is independent of the shear rate. The results showed that the lubricity properties, based on their pour point, flash point, and oxidation stability of the esters, were significantly affected by the linear and branched alcohols used. There is a slight increase in kinematic viscosity and viscosity index values with decreasing the internal chain length of the dibasic acid. The esters which were based on adipic acid such as C1 exhibited maximum values of VI: 187 compared to those which were based on azelaic acid such as F1 with VI: 182. Viscosity and viscosity index increases with increasing the number of carbon atoms of the used mono‐ol alcohols. Using branched alcohols gave almost the same viscosity results compared to using linear alcohol with the same number of carbons. Almost all prepared esters give pour point results ≤ −30°C.

An efficient adaptive length scale insensitive phase-field model for three-dimensional fracture of solids using trilinear multi-node elements

As a diffused fracture theory, phase-field models can seamlessly simulate complex crack patterns such as extending, branching, and merging. Despite the success of phase-field models, there are two issues in previous methods of three-dimensional (3-D) fracture. Firstly, the nonlinear governing equations lead to the huge computational costs, which hinder the application of phase-field models in 3-D problems. Secondly, these models, which are mostly developed based on a simple quadratic degradation function, provide numerical solutions that are sensitive to a length scale. Hence, this work addresses an efficient adaptive phase-field model with the aid of trilinear multi-node elements. The order of the elements remains constant with the increase of the number of nodes. As the mesh size and length scale significantly influence the numerical precision, a robust adaptive criterion is established in which the element refinement is controlled by both internal length scale and phase-field. According to the criterion, an expected mesh density in the failure domain can be obtained even for nonuniform initial mesh. Besides, being able to extend the phase-field regularized cohesive zone model, the adaptive model provides length scale insensitive responses for both crack path and peak load. The failure of brittle and quasi-brittle materials in three-dimensional conditions, including simple and mixed-mode fracture, can be simulated by the proposed model. Several benchmark examples are analyzed to show the efficiency and accuracy of the trilinear element-based adaptive phase-field model (TAPFM), and the results are compared with the standard phase-field model as well as experimental data.

Free vibration analysis of pre-stretched hyperelastic micromorphic continua with arbitrary shapes

A novel numerical strategy is developed in this article to study the free vibrations of hyperelastic micromorphic continua under bending load. In the proposed approach, the variational differential quadrature (VDQ) method and the idea of position transformation are used. The 3D micromorphic hyperelasticity is first formulated via vector-matrix relations which can be readily utilized in the coding process of numerical methods. The present numerical approach is able to address problems with irregular domains. To this end, the domain of elements is transformed into a regular one by the technique of mapping of position field based on the finite element shape functions. Being locking-free, simple implementation, computational efficiency and fast convergence rate are other features of the present variational approach. Three numerical examples including rectangular, sector and circular plates under bending load are considered whose free vibration behavior is analyzed. It is shown that the method is capable of predicting the relation between natural frequencies of micromorphic hyperelastic continua with load factor in an efficient way. The effects of internal length scale, scale-transition parameter and mode transition on the results are investigated.

Multi-criteria selection of the diameter and length of the lift of gas-lift well

The problem of the optimal selection of the lift internal diameter and length used in gas-lift oil production method, with the optimal operating mode of the lift, is formulated as a two-criteria optimization problem with criteria and restrictions obtained on the basis of the equation of motion of a steady flow of fluid into the well. To solve the stated objective, the general problem of a multi-criteria problem with any finite number of criteria, constraints, and input independent variables is considered. A genetic algorithm for solving such a problem has been developed. Keywords: well; gas-lift; oil production; optimal operating; algorithm.

A comparative analysis of continuum plasticity, viscoplasticity and phase-field models for earthquake sequence modeling

This paper discusses continuum models for simulating earthquake sequences on faults governed by rate-and-state dependent friction. Through detailed numerical analysis of a conventional strike-slip fault, new observations regarding the use of various continuum earthquake models are presented. We update a recently proposed plasticity-based model using a consistently linearized formulation, show its agreement with discrete fault models for fault thicknesses of hundreds of meters, and demonstrate mesh objectivity for slip-related variables. To obtain a fully regularized fault width description with an internal length scale, we study the performance and mesh convergence of a plasticity-based model complemented by a Kelvin viscosity term and the phase-field approach to cohesive fracture. The Kelvin viscoplasticity-based model can introduce an internal length scale and a mesh-objective response. However, on grid sizes down to meters, this only holds for very high Kelvin viscosities that inhibit seismic slip rates, which renders this approach impractical for simulating earthquake sequences. On the other hand, our phase-field implementation for earthquake sequences provides a numerically robust framework that agrees with a discrete reference solution, is mesh objective, and reaches seismic slip rates. The model, unsurprisingly, requires highly refined grids around the fault zones to reproduce results close to a discrete model. Following this line, the effect of an internal length scale parameter on the phase-field predictions and mesh convergence are discussed.

A Meshfree Approach Based on Moving Kriging Interpolation for Numerical Solution of Coupled Reaction-Diffusion Problems

In this paper, a meshfree approach based on moving kriging interpolation is presented for numerical solution of coupled reaction-diffusion problems. The proposed approach is developed based upon local collocation using moving Kriging shape function. It is truly meshless and having the Kronecker delta property for accurate imposition of boundary conditions. In the proposed model, the weight function is used with correlation parameter treated as the model internal length factor. This produces a local moving kriging method with improved accuracy together with an ease to choose the weight function factor. The method can hence be used in an efficient manner without cumbersome effort for choosing its parameter. The meshless approach is presented for the first time for numerical solution of reaction-diffusion systems. Problems of Turing system and pattern formation in several 2D domains are solved in this study. The efficacy and accuracy of the proposed method for the reaction-diffusion systems in different problem domains are presented in comparison to available exact solution and other numerical methods. It is found that the present method is accurate and effective as a computational procedure for solving reaction-diffusion problems.

Determination of optimal padeye position of modified suction caissons in the sand under monotonic inclined loading

Modified suction caissons (MSCs) acting as floating platform foundations will be subjected to inclined loading induced by wind waves and currents, thus, it is of significance to investigate its inclined bearing capacity. Model tests and numerical simulations were conducted on the MSCs embedded in the sand to study the monotonic inclined bearing behavior. Test results show that the inclined capacity firstly increases, and then, decreases as the padeye position moves downward along the caisson wall. The optimum padeye position corresponding to maximum inclined bearing capacity locates at the 0.5 times the internal compartment length below the lid when the loading angle equals 20°. The contribution of passive suction to the capacity may be ignorable when the loading angle is relatively small, and the passive earth pressure is predominant. It was also found that the increase of the external structure length can improve the inclined bearing capacity compared with the increase of external structure width. The relationship between the ultimate bearing capacity and the padeye position follows a Gaussian function. Considering the external structure dimensions, loading angle, and padeye position, a normalized formula is proposed to determine the ultimate bearing capacity. Finally, the optimal padeye position will be found for the floating platform.

Assessment of amplitude factors of asymptotic expansion at crack tip in flexoelectric solid under mode I and II loadings

The direct flexoelectric effect is a consequence of the polarization of the material, which is proportional to the strain gradients. The strain gradients are prominent near material defects, especially at the crack tips, where the flexoelectric effect redistributes the stress field and consequently influences the crack propagation. The flexoelectricity is a size dependent effect, i.e. it depends on an internal material length parameter as the additional material characteristic. This fact makes the equilibrium, constitutive, and boundary equations complicated as well as the asymptotic solution at the crack tip contrary to the asymptotic solution in the linear elastic fracture mechanics. The matched asymptotic expansion method is applied to derive the expressions for the amplitude factors that appear in the flexoelectric asymptotic solution for the crack as functions of the stress intensity factors in the loadings of mode I or mode II. The application of the matched asymptotic expansion method is conditioned by the knowledge of the so-called boundary layer, which is evaluated from the energetic criteria at the crack tip.

Stylized versus voxel phantoms: quantification of internal organ chord length distances

Dosimetric calculations, whether for radiation protection or nuclear medicine applications, are greatly influenced by the use of computational models of humans, called anthropomorphic phantoms. As anatomical models of phantoms have evolved and expanded, thus has the need for quantifying differences among each of these representations that yield variations in organ dose coefficients, whether from external radiation sources or internal emitters. This work represents an extension of previous efforts to quantify the differences in organ positioning within the body between a stylized and voxel phantom series. Where prior work focused on the organ depth distribution vis-à-vis the surface of the phantom models, the work described here quantifies the intra-organ and inter-organ distributions through calculation of the mean chord lengths. The revised Oak Ridge National Laboratory stylized phantom series and the University of Florida/National Cancer Institute voxel phantom series including a newborn, 1-, 5-, 10- and 15 year old, and adult phantoms were compared. Organ distances in the stylized phantoms were computed using a ray-tracing technique available through Monte Carlo radiation transport simulations in MCNP6. Organ distances in the voxel phantom were found using phantom matrix manipulation. Quantification of differences in organ chord lengths between the phantom series displayed that the organs of the stylized phantom series are typically situated farther away from one another than within the voxel phantom series. The impact of this work was to characterize the intra-organ and inter-organ distributions to explain the variations in updated internal dose coefficient quantities (i.e. specific absorbed fractions) while providing relevant data defining the spatial and volumetric organ distributions in the phantoms for use in subsequent internal dosimetric computations, with prospective relevance to patient-specific individualized dosimetry, as well as informing machine learning definition of organs using these reference models.

The strain gradient elasticity via nonlocal considerations

Strain gradient elasticity and nonlocal elasticity are two enhanced elastic theories intensively used over the last fifty years to explain static and dynamic phenomena that classical elasticity fails to do. The nonlocal elastic theory has a clear differentiation from the classical case by considering stresses at a point of the continuum as an integral of classical stresses defined in the treated elastic body. On the other hand, strain gradient elasticity is characterized as a non-classical theory because considers both potential and kinetic energy densities as not only functions of strains and velocities, respectively but also functions of their gradients. Although the two considerations seem to be completely different from each other, it is a common belief that strain gradient elasticity has a lot in common with nonlocal elasticity. The goal of the present work is to derive all the strain gradient elastic theories appearing so far in the literature via the nonlocal definitions of the potential and kinetic energy densities and Hamilton’s principle. Such a consideration manifests the nonlocal nature of the SGE theories, correlates their internal length scale parameters with the nonlocal horizon of each material point, and proposes nonlocality in both elastic and inertia behavior of the continuum. For the sake of simplicity and brevity, only one-dimensional wave propagation phenomena in infinite domain are considered. However, since the Hamilton’s principle is employed, the definition of the corresponding boundary conditions for finite domains is accomplished simultaneously with the definition of the equation of motion.

One-Dimensional Failure Modes for Bodies with Non-convex Plastic Energies

In this paper, a complete picture of the different plastic failure modes that can be predicted by the strain gradient plasticity model proposed in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) is drawn. The evolution problem of the elasto-plastic strain is formulated in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) as an incremental minimization problem acting on an energy functional which includes a local plastic term and a non-local gradient contribution. Here, an approximate analytical solution of the evolution problem is determined in the one-dimensional case of a tensile bar. Different solutions are found describing specific plastic strain processes, and correlations between the different evolution modes and the convexity/concavity properties of the plastic energy density are established.The variety of solutions demonstrates the large versatility of the model in describing many failure mechanisms, ranging from brittle to ductile. Indeed, for a convex plastic energy, the plastic strain diffuses in the body, while, for a concave plastic energy, it localizes in regions whose amplitude depends on the internal length parameter included into the non-local energy term, and, depending on the convexity properties of the first derivative of the plastic energy, the localization band expands or contracts. Complex failure processes combining different modes can be reproduced by assuming plastic energy functionals with specific convex and concave branches.The quasi-brittle failure of geomaterials in simple tension tests was reproduced by assuming a convex-concave plastic energy, and the accuracy of the analytical predictions was checked by comparing them with the numerical results of finite element simulations.

Efficient gradient enhancements for plasticity with ductile damage in the logarithmic strain space

We contrast different gradient-enhancements for plasticity-damage material models in the logarithmic strain space and compare them to reference models based on multiplicative plasticity. The models being compared include plasticity - gradient-damage, where the gradient-enhancement is applied on the local damage variable, and gradient-plasticity - damage with a gradient-enhanced plastic hardening variable. Thereby, gradient-plasticity proved to be able to simultaneously regularise plastic and ductile (plasticity-driven) damage localisation as confirmed by numerical localisation analyses. This appears to be especially advantageous for logarithmic strain space plasticity-damage, because of the observed plastic localisation even for the case of plasticity with hardening. Even though gradient-plasticity appears to be numerically more demanding, two numerical examples indicate a good robustness and mesh objectivity in the softening regime. Moreover, the internal length for plasticity is able to adjust the damage zone width, similarly to gradient-damage, however ensuring a priori that damage takes place exclusively inside the plastic zone.

Adaptive mesh refinement for the phase field method: A FEniCS implementation

This paper reports insightful implementation details of the global adaptive refinement procedure for the phase-field method recently published in Freddi and Mingazzi (2022). Phase field approaches reproduce cracks within solids in a smeared manner. The small transition zone between broken and unbroken material, whose width is controlled by an internal scale length parameter, permits to precisely replicate complex sharp crack topologies only if an extremely fine mesh is adopted. Starting from a coarse mesh, the proposed refinement process utilizes an energetic criterion to selectively refine the elements on which cracks may propagate. In fully broken areas, where the phase field is no longer evolving, a specific refinement is adopted to capture the high displacement gradient. The implementation is performed within the open-source finite element software FEniCS (ver. 19.1.0) which provides a framework for automated solutions of partial differential equations. The fundamental aspects of the code are described starting from the functional definition to the various steps of the refinement technique. A representative example is illustrated to supply further information on the code functionality. The code can be downloaded from https://github.com/LorenzoMingazzi/AGu-AGal and be used to easily apply the proposed refinement strategy to different problems or as a starting point for more sophisticated formulations.

Geometrically necessary dislocations and related kinematic hardening in gradient grained materials: A nonlocal crystal plasticity study

Gradient grained metals whose microstructure is characterized by a spatially graded grain size distribution show a better strength-ductility combination than their homogeneous counterparts. Kinematic hardening associated with geometrically necessary dislocations (GNDs) is considered to be a dominant strengthening mechanism in gradient grained metals. However, the precise kinematics of GND accumulation and the nature of the back stress fields remain unclear, restricting the understanding of their deformation mechanisms. In this work, a nonlocal crystal plasticity model which explicitly accounts for the interaction between dislocations and grain boundaries is developed. The nonlocal feature is achieved by introducing a flux term to account for the spatial redistribution of dislocations due to their motion. In addition, back stress produced by the spatial variation of GND density introduces an explicit internal length scale into the model. The nonlocal nature of the model on the slip system level enables the direct investigation of strain gradient effects caused by internal deformation heterogeneities. Furthermore, the interaction between dislocations and grain boundaries leads to the formation of pileups near grain boundaries, which is key to studying the grain size effects in polycrystals. Finite element implementation of the model for polycrystals with different grain sizes quantitatively captures the grain size effect. Simulation results of gradient grained materials and their homogeneous counterparts demonstrate that smaller grains lead to higher GND density and enhanced back stress. Small grains significantly contribute to the GND-induced isotropic hardening and GND-induced kinematic hardening in gradient grained metals. This investigation helps to understand the underlying strengthening mechanisms of gradient grained metals, and the model can be readily applied to other kinds of heterogeneous materials.

Distal Embolization in Relation to Radiological Thrombus Characteristics, Treatment Details, and Functional Outcome.

Distal embolization (DE) is a common complication of endovascular treatment (EVT). We investigated the association of radiological thrombus characteristics and treatment details with DE. Patients with thin-slice (≤2.5 mm) baseline noncontrast computed tomography and computed tomography angiography from the ESCAPE-NA1 trial (Efficacy and Safety of Nerinetide for the Treatment of Acute Ischemic Stroke) were included. Thrombus annotation was performed manually on coregistered scans by experienced readers. We assessed thrombus location, distance from internal carotid artery terminus, length, perviousness, absolute attenuation, and hyperdense artery sign. In addition, we evaluated balloon guide catheter use during EVT, first-line EVT approach, the number of thrombectomy passes, and prior intravenous thrombolysis administration. DE was defined as the occurrence of emboli distal to the target artery or in new territories during EVT. The association between thrombus characteristics, treatment details, and DE was evaluated using descriptive statistics and multivariable mixed-effects logistic regression, resulting in adjusted odds ratios (aOR) with 95% CI. Interaction between IVT and radiological thrombus characteristics was assessed by adding interaction terms in separate models. In total, 496 out of 1105 (44.9%) ESCAPE-NA1 patients were included. DE was detected in 251 out of 496 patients (50.6%). Patients with DE had longer thrombi (median, 28.5 [interquartile range, 20.8-42.3] mm versus 24.4 [interquartile range, 17.1-32.4] mm; P<0.01). There were no statistically significant differences in the other thrombus characteristics. Factors associated with DE were thrombus length (aOR, 1.02 [95% CI, 1.01-1.04]), balloon guide catheter use (aOR, 0.49 [95% CI, 0.29-0.85]), and number of passes (aOR, 1.24 [95% CI, 1.04-1.47]). In patients with hyperdense artery sign, IVT was associated with reduced odds of DE (aOR, 0.55 [95% CI, 0.31-0.97]), P for interaction=0.04. DE was associated with longer thrombi, no balloon guide catheter use, and more EVT passes. IVT was associated with a reduced risk of DE in patients with hyperdense artery sign. These findings may support treatment decisions on IVT and EVT approaches.