Continuum Mechanics Theory Research Articles

A Pre-peak Elastoplastic Damage Model of Gosford Sandstone Based on Acoustic Emission and Ultrasonic Wave Measurement

The determination of internal material damage is always an arduous challenge. Non-destructive monitoring methods show great potential in quantitatively determining the internal material properties, whereas most of the studies relying on external observations remain in a qualitative stage. They either violate the basic thermodynamic assumptions or are difficult to guide engineering practice. In this paper, following the theory of continuum mechanics, an elastoplastic damage model based on non-destructive monitoring methods (i.e., acoustic emission and ultrasonic wave velocity measurement) has been developed. To capture the continuous and precise damage evolution inside rock mass, P wave velocity obtained by ultrasonic wave measurement was utilised and then considered as an input for the proposed elastoplastic damage model. Triaxial loading test results on six Gosford sandstone samples were analysed first to characterise critical stresses along the stress–strain loading curves, such as crack closure stress, stable crack propagation stress and unstable crack propagation stress. The drop of ultrasonic wave velocity can be seen as an indicator to represent the damage evolution inside rock material. Damage initiation is also closely related to the confining stress and dilation induced volumetric expansion. The test results also suggested that the Drucker–Prager criterion is sufficient to describe the plastic yielding surface and the following material hardening. A non-associated plastic flow assumption was adopted, considering the essence of microcrack shearing in rock failure and the effect of hydrostatic pressure on plastic deformation. A modified Drucker–Prager plastic potential was also introduced to track the orientation of plastic increment with material hardening. A scalar damage variable was derived from ultrasonic wave measurement results to indirectly represent the deterioration of rock properties (modulus). The proposed model was used to match lab test results with high consistency, and the main features of rock behaviour in triaxial loading tests were successfully captured by the model. Finally, the damage evolution of rock samples was analysed, which indicates that damage is dependent on its conjugate force, namely damage energy release rate Y. This study proves that P wave velocity can be an effective approach to measure and forecast the internal damage evolution inside rock mass, which has broad prospects for engineering applications.

Elasticity-associated rebinding rate of molecular bonds between soft elastic media

A quantitative understanding of how cells interact with their extracellular matrix via molecular bonds is fundamental for many important processes in cell biology and engineering. In these interactions, the deformability of cells and matrix are usually comparable with that of the bonds, making their rebinding events globally coupled with the deformation states of whole systems. Unfortunately, this important principle is not realized or adopted in most conventional theoretical models for analyzing cellular adhesions. In this study, we considered a new theoretical model of a cluster of ligand-receptor bonds between two soft elastic bodies, in which the rebinding rates of ligands to receptors are described, by considering the deformation of the overall system under the influence of bond distributions. On the basis of theory of continuum and statistical mechanics, we obtained an elasticity-associated rebinding rate of open bonds in a closed analytical form that highly depends on the binding states and distributions of all other bonds as well as on the overall deformation energy stored in the elastic bodies and all closed bonds. On the basis of this elasticity-associated rebinding rate and by performing Monte Carlo simulations, we uncovered new mechanisms underlying the adhesion stability of molecular-bond clusters associated with deformable elastic bodies. Moreover, we revealed that the rebinding processes of molecular bonds is not only dependent on interfacial separation but is related to overall energy. This newly proposed rebinding rate may substantially improve our understanding of how cells adapt to their microenvironments by adjusting their mechanical properties through cytoskeleton remodeling.

Size dependent torsional electro-mechanical analysis of flexoelectric micro/nanotubes

The free vibration and static torsion of a coupling electromechanical flexoelectric micro/nanotube have been investigated, in the present study. Using the non-classical theory of continuum mechanics which is based on a strain gradient, the coupling governing equations of torsional flexoelectric micro/nanotubes have been developed. To derive the coupling governing equations the variational method has been used and the formulation with the classical and non-classical boundary conditions is generally derived for torsional flexoelectric micro/nanotubes. For the first time, the formulation of micro/nanotube torsional flexoelectric coupling with independent polarization effect is presented in this paper which can be used for specific applications. At last, the static analysis and free vibrations results are presented for specific mechanical and electrical boundary conditions. The parameter of mechanical and electrical size effect changes has a great effect on the results. However, the effect of the mechanical and electrical size effect parameter on direct flexoelectric torsion is also opposite, but the effect of mechanical and electrical size effect parameter on reverse flexoelectric torsion is similar. Polarization is also introduced as a new variable in the equations and its effect is investigated as an independent variable of the electric field in the results, in this paper. As a significant parameter, the results show that polarization plays an important role to simulate the torsional electromechanical structures.

Advances in Permanent Deformation Modeling of Asphalt Concrete-A Review.

Permanent deformation is one of the dominant asphalt concrete damages. Significant progress has been made to realistically predict the damage. In the last decade, the mechanistic approach has been the focus of research, and the fundamental theories of viscoelasticity, viscoplasticity, continuum mechanics, and micromechanics are applied to develop the material laws (constitutive equations). This paper reviews the advancement of permanent deformation models including analogical, microstructural, and continuum-based methods. Pavement analysis using the nonlinear damage approach (PANDA) is the most comprehensive and theoretically sound approach that is available in the literature. The model coupled different damages and other phenomena (such as cracking, moisture, and phenomena such as healing, aging, etc.). The anisotropic microstructure approach can be incorporated into the PANDA approach for a more realistic prediction. Moreover, the interaction of fatigue and permanent deformation is the gap that is lacking in the literature. The mechanistic approaches have the capacity to couple these damages for unified asphalt concrete damage prediction.

A method for determining elastic constants and boundary conditions of three-dimensional hyperelastic materials

Although there have been many studies on the identification of elastic constants and boundary conditions, it is still challenging to simultaneously estimate the unknown elastic constants and boundary conditions of a three-dimensional hyperelastic material under locally observed boundary conditions with noise. In this paper, a novel inverse method based on two different objective functions is proposed. The first objective function is constructed by a “dual fields” construction method, which utilizes redundant Cauchy boundary conditions (displacement boundary conditions and force boundary conditions) on the observable boundary. The second objective function is based on the gap between measured and calculated displacements of the observable boundary. The inverse analysis is conducted by an adjoint method and a linear approximation method based on the theories of continuum mechanics and finite element method to obtain the Jacobian matrices of the objective functions with respect to different variables. An alternating iterative algorithm based on Gauss–Newton method is proposed to minimize the objective functions. The proposed inverse algorithm is tested using nearly incompressible Neo-Hookean and nearly incompressible Mooney–Rivlin hyperelastic model in numerical experiments and physical experiments with a silicone-based tissue-mimicking material. The results of both numerical and physical experiments show the feasibility and accuracy of the proposed method.

Transient Continuum Mechanics and Chemomechanics

Abstract A transient chemomechanical coupling formulation for solid continuum is presented. The second-order rate and the characterized time are introduced to include the transient effect through Taylor expansion. The transient Reynold’s transport theorem is derived with the new products or material elimination considered. Based on conservation laws and the second law of thermodynamic, we state a consistent Helmholtz-energy-based framework. The transient field equations take mechanical and chemical contributions and microscopic time into account. Either microscopic time or chemical reactions leads to the unsymmetry of the stress tensor. The relationship of Helmholtz energy and constitutive properties, the evolution equations, and the entropy are consistent with the classical continuum thermodynamics and the constitutive theory in continuum mechanics. Further, the transient equations of thermal conduction and diffusion with finite velocity are naturally derived rather than postulated, and a comparison with the existing theories is discussed.

Regularization of shear banding and prediction of size effects in manufacturing operations: A micromorphic plasticity explicit scheme

Good quality manufacturing operation simulations are essential to obtain reliable numerical predictions of the processes. In many cases, it is possible to observe that the deformation localizes in narrow areas, and since the primary deformation mode is under shear, these areas are called shear bands. In classical continuum mechanics models, the deformation localization may lead to spurious mesh dependency if the material locally experiences thermal or plastic strain softening. One option to regularize such a non-physical behavior is to resort to non-local continuum mechanics theories. This paper adopts a scalar micromorphic approach, which includes a characteristic length scale in the constitutive framework to enforce the plastic strain gradient theory to regularize the solution. Since many manufacturing process simulations are often assessed through finite element methods with an explicit solver to facilitate convergence, we present an original model formulation and procedure for the implementation of the micromorphic continuum in an explicit finite element code. The approach is illustrated in the case of the VPS explicit solver from ESI GROUP. According to the original formulation, we propose an easy way to implement a scalar micromorphic approach by taking advantage of an analogy with the thermal balance equation. The numerical implementation is verified against the analytical solution of a semi-infinite glide problem. Finally, the correctness of the method is addressed by successfully predicting size effects both in a cutting and a bending tests.

Dynamics Analysis of Penetration against Pebble-Concrete Target.

The classical continuum mechanics theory cannot sufficiently describe the effect of pebbles on projectile, which leads to a large calculation error. In this paper, an orthogonal curvilinear coordinate system is constructed, which effectively describes and perfects the normal cavity expansion theory. A couple stress theory based on the normal cavity expansion is proposed in which not only the tangential movements but also the rotations of the concrete medium are considered. According to the high-speed impact of pebble concrete, combined with dynamic equations and the FE simulation, the theoretical and simulation results of pebble particles scale on warhead resistance are compared. It is shown that, the larger the scale of pebble particles, the stronger the effect of rotation on the resistant force applied on the warhead.

Investigation of microscale fracture mechanisms in glass–ceramics using peridynamics simulations

AbstractGlass–ceramics (GCs), obtained by controlled crystallization of a specially formulated precursor glass, are interesting materials that show great promise in obtaining superior properties compared to those of the precursor glass. Controlled crystallization enables creation of a microstructure with multiple phases which impacts macroscale properties in interesting ways. The present work develops microstructure‐scale computational models using the theory of peridynamics to investigate the increase in fracture toughness of GCs compared to traditional glass. Computational modeling is a promising tool to probe microstructural mechanics, but such studies in the literature are scarce. In this work, the theory of peridynamics, a non‐local theory of continuum mechanics, is applied to simulate crack propagation through microstructural realizations of a model lithium‐disilicate glass–ceramic. The crystalline and glassy phases within the microstructure are explicitly considered, with the size and shape of crystals inspired by experimental data. Multiple toughening mechanisms are revealed, which are functions of crystallinity and morphology, and the impact on fracture toughness is demonstrated. Crack path tortuosity is studied, and it is found that an optimum level of crack path tortuosity can be obtained in the range of 0.6–0.8 crystallinity. Numerical results are shown to agree well with previously published experimental and modeling results.

A hybrid meshfree discretization to improve the numerical performance of peridynamic models

Efficient and accurate calculation of spatial integrals is of major interest in the numerical implementation of peridynamics (PD). The standard way to perform this calculation is a particle-based approach that discretizes the strong form of the PD governing equation. This approach has rapidly been adopted by the PD community since it offers some advantages. It is computationally cheaper than other available schemes, can conveniently handle material separation, and effectively deals with nonlinear PD models. Nevertheless, PD models are still computationally very expensive compared with those based on the classical continuum mechanics theory, particularly for large-scale problems in three dimensions. This results from the nonlocal nature of the PD theory which leads to interactions of each node of a discretized body with multiple surrounding nodes. Here, we propose a new approach to significantly boost the numerical efficiency of PD models. We propose a discretization scheme that employs a simple collocation procedure and is truly meshfree; i.e., it does not depend on any background integration cells. In contrast to the standard scheme, the proposed scheme requires a much smaller set of neighboring nodes (keeping the same physical length scale) to achieve a specific accuracy and is thus computationally more efficient. Our new scheme is applicable to the case of linear PD models and within neighborhoods where the solution can be approximated by smooth basis functions. Therefore, to fully exploit the advantages of both the standard and the proposed schemes, a hybrid discretization is presented that combines both approaches within an adaptive framework. The high performance of the developed framework is illustrated by several numerical examples, including brittle fracture and corrosion problems in two and three dimensions.

A Review on the Mechanical Behavior of Size-Dependent Beams and Plates using the Nonlocal Strain-Gradient Model

Nowadays, the mechanical characteristics of micro-/nano-structures in the various types of engineering disciplines are considered as remarkable criteria which may restrict the performance of small-scale structures in the reality for a certain application. This paper deals with a comprehensive review pertinent to using the nonlocal strain-gradient continuum mechanics model of size-dependent micro-/nano-beams/-plates. According to the non-classical features of materials, using size-dependent continuum mechanics theories is mandatory to investigate accurately the mechanical characteristics of the micro-/nano-structures. Recently, the number of researches related to the analysis of micro-/nano-structures with various geometry including beams as well as plates is considerable. In this regard, the mechanical behavior of these structures induced by different loadings such as vibration, wave propagation, and buckling behavior associated with the nonlocal strain-gradient continuum mechanics model is presented in this review work. Proposing the most valuable literature pertinent to the nonlocal strain-gradient continuum mechanics theory of micro-/nano-beams/plates is the main objective of this detailed survey.

A New Technique for Curved Rod Bending Tests Based on Digital Image Correlation

BackgroundThe study of the deformation of curved rods subjected to bending and its associated stress state is a complex task that has not been treated in depth in the literature, which makes difficult to obtain constitutive models or Finite Element Models (FEM) in which it is necessary to know all the components of the stress and strain tensors.ObjectivesThis study focuses on a new calculation methodology to obtain stress and strain tensors of curved rods under bending.MethodsThe stress and strain tensors have been determined based on the theory of continuum mechanics and differential geometry of curves (moving bases), in a general methodology and valid for large strains, curved geometries and variable cross-sections along the specimen. This has been applied to the human rib and, in addition, a new experimental method for bending of curved specimens based on Digital Image Correlation (DIC) is presented.ResultsBoth the test method and the proposed calculations applied to the human rib show results according to expectations, allowing to know the rib curvature changes along the test, the stresses and strains along the rib and the components of both stress and strain in all directions, in order to build the stress and strain tensors. In addition, the results of stress, strain and young’s modulus correspond to those of previous literature in tensile testing of human rib cortical bone.ConclusionsThe proposed calculations allow the construction of the strain and stress tensors of a curved specimen subjected to bending, which is of great importance for the development of constitutive models. Moreover, since with this method it is possible to calculate both tensors along the entire length of the specimen and in all directions, it is possible to apply this method in finite element models. Finally, the new test methodology allows to know the stress and strain in curved specimens such as the human rib, from bending tests.

The Performance of Five Extended Bond-Based Peridynamic Models on Crack Simulation

Peridynamics (PDs) is a nonlocal continuum mechanics theory used to solve fracture and failure problems. The classical bond-based peridynamics (BBPDs) involves one parameter to describe the microelastic bond stretch response, which makes the Poisson’s ratio restricted. In previous studies, kinds of extended bond-based peridynamic (EBBPD) models were proposed to deal with the limitation of Poisson’s ratio in BBPDs. In this paper, four benchmarks with different load conditions are designed to compare the performance of five EBBPD models and to investigate the effects of model parameters on crack propagation. Results show that compared with BBPDs, the crack patterns of the EBBPD models have the main form of BBPDs; on the other hand, the introduction of tangential stiffness makes the form of crack tend to shear failure, and the increase of tangential stiffness will enhance this effect. The crack patterns transition of wing cracks from tensile failure to mixed tensile–shear failure when the tangential stiffness increases. For the models with rotational stiffness, the rotational stiffness affects the width and propagation direction of the secondary crack and makes the trends to shear failure. Besides, the rotational stiffness affects the forms of crack coalescence and divides the results of these five models into three types.

Scale Effects on Thermoelastic Coupling Wave Propagation of Micro-Beam Resonator Using Nonlocal Stain Gradient and Generalized Thermoelasticity

Coupled thermoelastic analysis in the study of microscale solid mechanics has attracted considerable attention along with the miniaturization and functionalization of devices. Higher-order continuum mechanics theories have considered scale effects; however, the theories to predict thermoelastic performances of structures in micro-scale are still incomplete. The purpose of this paper is to study scale and thermal-induced effects by considering stress nonlocality and strain gradient elasticity in micro-structure resonators. The thermoelastic constitutive relations are established based on the nonlocal strain gradient (NSG) theory. The wave propagation problem of a Timoshenko micro-beam in conjunction with Green and Naghdi’s generalized thermoelastic (G–N) theory is investigated. The governing equations are derived and analytically solved by the eigenvalue method. Finally, the dispersion relation of a frequency spectrum is determined and the influences of scale parameters and coupled thermoelastic effect are discussed; in addition, the analysis results are compared with those under different size-dependent theories accordingly.

ОБҐРУНТУВАННЯ ТРАНСПОРТНИХ ПАРАМЕТРІВ ТА МОДЕЛЮВАННЯ ПРОЦЕСУ РУЙНУВАННЯ ПРУЖНО-ПЛАСТИЧНОГО СЕРЕДОВИЩА

Purpose. The substantiation of the mathematical model of mechanical deformation of the elastic-plastic medium and the modeling of the transport process of the expansion assemblies. Methodology. Mathematical and theoretical studies were based on the fundamental theory of continuum mechanics and general positions of engineering mechanics. Analytical and graphical analysis of mathematical models carried out on a PC in a specialized software complex. Results. In the current conditions of economic development of the country, considerable attention should be paid to the development and modernization of certain sectors of the economy. Significant amounts of work, which are accompanied by the development of soil of different properties, which can be defined as an elastic-plastic material. These studies will solve a number of problems that are acute not only in agriculture but also in construction, in the open pit mining, reclamation, one of which is an imperfect process of loosening the soil during its cultivation. Therefore, the urgent problem is to establish rational parameters of the transport surface of the unit for loosening the elastic-plastic material. The developed mathematical model allows determining the emerging stress, which in turn determines the boundary of the destruction of the elastic-plastic material. In the design of lining assemblies, it is advisable to have a radius of the outlet section R = 0.18 m or more. Originality. The mathematical models for the process of destruction of elastic-plastic material and forecasting of optimal transport parameters for designing of expansion assemblies are obtained. Practical value. The developed mathematical model will provide an improvement in the process of loosening elastic-plastic material, and engineering calculations during the design of the rutter can prevent unwarranted increase in resistance, with displaced materials on the cutting surface. References 10, figures 8.

Active elasticity drives the formation of periodic beading in damaged axons.

In several pathological conditions, such as coronavirus infections, multiple sclerosis, Alzheimer's and Parkinson's diseases, the physiological shape of axons is altered and a periodic sequence of bulges appears. Experimental evidences suggest that such morphological changes are caused by the disruption of the microtubules composing the cytoskeleton of the axon. In this paper, we develop a mathematical model of damaged axons based on the theory of continuum mechanics and nonlinear elasticity. The axon is described as a cylinder composed of an inner passive part, called axoplasm, and an outer active cortex, composed mainly of F-actin and able to contract thanks to myosin-II motors. Through a linear stability analysis we show that, as the shear modulus of the axoplasm diminishes due to the disruption of the cytoskeleton, the active contraction of the cortex makes the cylindrical configuration unstable to axisymmetric perturbations, leading to a beading pattern. Finally, the nonlinear evolution of the bifurcated branches is investigated through finite element simulations.

Thermo-Mechanically Coupled Peridynamic Modelling of Fracture Formation in Solid Oxide Fuel Cell Anode

Fracture formation can be observed in brittle yttria-stabilized zirconia (YSZ) in Ni-YSZ composite anode during high-temperature operation of solid oxide fuel cells (SOFCs). It is quite difficult to predict the fracture formation in solid materials accurately based on the classical continuum mechanics theory, because of the singularity problems in partial differential equations. In this work, an ordinary state-based-peridynamic (OSB-PD) model is proposed to predict the fracture formation in SOFC anode based on real microstructures reconstructed using focused ion beam-scanning electron microscopy (FIB-SEM) dual-beam technique. The key factors determining anode microstructure strength is analyzed based on the simulation results upon the comparison among different anode microstructures. The thermo-mechanical coupling effect is investigated by considering the influence of both external mechanical loads and temperature distribution.

Thermo-Mechanically Coupled Peridynamic Modelling of Fracture Formation in Solid Oxide Fuel Cell Anode

Fracture formation can be observed in brittle yttria-stabilized zirconia in Ni-YSZ composite anode during high-temperature operation of solid oxide fuel cells. It is difficult to predict the fracture formation in brittle solid materials accurately based on the classical continuum mechanics theory because of the singularity problems in partial differential equations. An ordinary state-based-peridynamic model is proposed to predict the fracture formation under external loads in SOFC anode in micro-scale using focused ion beam-scanning electron microscopy dual-beam technique in this work. The thermo-mechanical coupling effect is also investigated by simulating the formation and propagation of macro-scale crack in single cell under thermal shock.

Myths and misconceptions related to unsaturated soil mechanics

There have been three main pillars associated with the development of an applied engineering science for both saturated and unsaturated soil mechanics; namely, i) the synthesis of continuum mechanics theories of physical behavior, ii) the laboratory measurement of relevant soil properties, and iii) analyses that illustrate the solution of practical example problems. Geotechnical engineers have, however, been relatively slow in adopting unsaturated soil mechanics into geotechnical engineering practice. There have been several so-called “myths or misconceptions” that appear to have hindered the application of unsaturated soil mechanics. This paper attempts to describe and dispel what are deemed to be misconceptions related to the more general implementation of unsaturated soil mechanics into engineering practice. The so-called “myths” come from the acceptance of false information related to unsaturated soil behavior and a hesitancy to embrace changes to existing empirical protocols. Several misconceptions are identified in the paper that are related to: i) complexity of unsaturated soil mechanics theories, ii) inability to readily measure soil suctions in-situ, iii) the nonlinearity of unsaturated soil property functions, iv) permanency of soil suctions above the water table, v) difficulties associated with assessing ground surface moisture flux conditions, and vi) difficulties associated with numerical modeling that involves solving nonlinear partial differential equations. Each of the above-mentioned items are dealt with as myths or misconceptions in the sense of being impediments to the application of unsaturated soil mechanics in geotechnical engineering practice.

Research Progress of Initial Mechanism on Debris Flow and Related Discrimination Methods: A Review

The initial of debris flow can be classified into two types based on their triggering positions, that is, debris flow from slope and debris flow from gully or channel. For the former, great progress has been achieved on the mechanisms of soil failure and liquefaction. The framework established by a series of theories or laws, such as the Mohr–Coulomb criteria, the unsaturated soil mechanics, and the critical state of soil mass, has been used widely in industry and research. However, the details and discrimination basis for the transformation process from landslide into debris flow still need to be further clarified. Relatively, debris flow from gully or channel is more complex due to its various mass sources and the diversity of processes. Nevertheless, through a great number of case studies and experimental statistics, people have gradually recognized the influential rule and critical condition of factors from landform, hydrology, and other aspects on debris flow initiation. Furthermore, based on the theories of granular flow, continuum mechanics, and rheological law, some typical event-based scenarios can also be reproduced by different single-/two-phase depth integral/average numerical models. However, some key knowledge on mechanism and application level is still insufficient, such as the erosion and entrainment mechanism of materials from different sources, the boundary tractions and materials exchange, as well as the selection of prediction indicators. Three current discriminated methodologies for debris flow initiation, that is, the safety factor method, the rainfall indicator method, and the comprehensive assessment method, were summarized in this article. Considering the technical limitation of each methodology, it is believed that the establishment or improvement of a unified, stable, and open-access database system for event registration and query, as well as the development of large-scale and high-precision rainfall monitoring, is still regarded as the important aspect of debris flow prevention in the future. In addition, as an economic and efficiency means for obtaining information on potential threats and real-time hazard messages, the multielement method for debris flow is recommended as a long-term reference.