Internal Length Scale Parameter Research Articles

A phase-field framework for modeling multiple cohesive fracture behaviors in laminated composite materials

This work proposes a phase-field (PF) framework to characterize the multiple fracture behaviors and quasi-brittle failure mechanisms in fiber reinforced laminated composites. The multi-phase field model considering the distinct failure mechanisms of composites is first derived based on thermodynamic principles. A hybrid PF formulation is proposed through the reasonable definition of the energy functional for each field, and a PF regularized cohesive zone model (CZM) combined with Hashin failure criteria is achieved for the typical failure modes in composites. The proposed PF-CZM is endowed with a linear softening behavior and is proven to be independent of the internal length scale parameter. The multiple fracture modeling of composite laminates is achieved by coupling the PF model for intralaminar failure with the cohesive interface element for interlaminar one. The model’s validation is demonstrated by several representative numerical examples, and results show that both the intralaminar fiber/matrix fractures and the induced interlaminar delamination, as well as their complex interactions, are well captured by the proposed model.

Simulation of hydraulic fracture propagation based on implicit gradient non-local model

Fracture extension simulation is an important tool for studying hydraulic fracture characteristics. To solve the problem of mesh sensitivity of strain and fluid pressure during hydraulic crack propagation, the equivalent strain in the damage evolution equation was non-localized, and implicit gradient was performed. A hydraulic-mechanical coupling model based on implicit gradient non-local damage was established. Numerical simulation program was further developed. Comparison of simulation results with experimental data on ‘L’-brittle material verified the validity of the program. The fracture propagation process of hydraulic fracturing in a perforated well was simulated. On this basis, the influence of internal length scale parameters and mesh size of the non-local model on the calculation results was analyzed. The influence of mesh size on hydraulic fracture morphology and damage variables is reduced by introducing an implicit gradient nonlocal model, which effectively alleviates the localization effect in hydraulic-mechanical coupling damage simulation. This new model achieves the nonlocalization of the fluid field by introducing the relationship among porosity-permeability and deformation and damage, which avoids the mesh sensitivity of equivalent strain, damage, fluid pressure, and improves the numerical stability. It provides a new method for simulating hydraulic fracture propagation and studying the failure behavior of porous media materials.

Nonlocal elasticity of Klein‐Gordon type with internal length and time scales: Constitutive modelling and dispersion relations

AbstractIn this work, a nonlocal elasticity theory with nonlocality in space and time is presented by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel. Based on the proposed theory, we consider the isotropic nonlocal elasticity of Klein‐Gordon type including a characteristic internal time scale parameter in addition to the characteristic internal length scale parameter. The dispersion relations for homogeneous isotropic media in the framework of nonlocal elasticity of Klein‐Gordon type are analytically determined. The obtained results reveal the ability of the presented nonlocal elasticity model to predict, for the first time in the framework of nonlocal elasticity, in addition to the acoustic modes (low‐frequency modes), optic modes (high‐frequency modes) as well as frequency band‐gaps. The phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) are determined showing that all four modes exhibit normal dispersion with positive group velocity. The presented model allows for physically realistic dispersive wave propagation.

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.

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.

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.

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.

A comprehensive characterization of fracture in unit cell open foams generated from Triply Periodic Minimal Surfaces

The main scope of the proposed study is to assess the occurrence of fracture in Triply Periodic Minimal Surfaces (TPMS) foams subjected to compressive loading. TPMS, developed by the mathematics community, may be exploited as a backbone for developing a new class of foams with open porosity for a wide range of engineering and biomedical applications. Therefore, a comprehensive analysis of their fracture response is fundamental and is herein attempted. To this aim, a 3D phase field model is herein proposed and applied to TPMS foam structures under compression, with the goal to predict critical points for crack nucleation, potential crack paths, and the stiffness and maximum force of the unit cell, which can be related to the apparent Young’s modulus and apparent strength of a macro-scale composite made of such TPMS unit cells. A careful mesh sensitivity analysis was conducted on the specimens, to provide guidelines on how to identify the optimal finite element discretization consistent with the internal length scale parameter of the phase field approach to fracture. The major predicted mechanical properties for five different TPMS open foams, and for different levels of porosity, are summarized in Ashby plots. The predicted trends are in agreement with previous results on TPMS taken from the literature and show that TPMS can outperform standard Aluminium open foams.

Nonlocal elasticity of Klein–Gordon type: Fundamentals and wave propagation

A nonlocal elasticity theory with nonlocality in space and time is developed by considering nonlocal constitutive equations with a dynamical scalar nonlocal kernel function. The proposed theory is specified to isotropic nonlocal elasticity of Klein–Gordon type, which is an extension of nonlocal elasticity of Helmholtz type, and it possesses one characteristic internal length scale parameter as well as one characteristic internal time scale parameter in addition to the elastic material parameters. Dynamical nonlocal kernels are analytically derived as retarded Green functions of the Klein–Gordon operator. The dispersion relations of the considered isotropic nonlocal elasticity model of Klein–Gordon type are analytically determined. The obtained results reveal the advantage of the proposed nonlocal elasticity model, that is, its ability to predict for the first time in the framework of nonlocal elasticity in addition to the acoustic modes (low-frequency modes), optic modes (high-frequency modes) as well as frequency band-gaps between the acoustic and optic modes. A qualitative examination of the phase and group velocities for all four modes (acoustic and optic branches of longitudinal and transverse waves) shows that the proposed model allows for physically realistic dispersive wave propagation.

Impacts of Length Scale Parameter on Material Dependent Thermoelastic Damping in Micro/nanoplates Applying Modified Coupled Stress Theory

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and need tobe alleviated in resonators inorder to enhance its performance parameters by improving its thermoelastic dampinglimited qualityfactor, QTED. The maximum energy dissipation is also interrelated with critical length (???????? ) of theplates and by optimizing the dimensions the peaking of energy dissipation can be diminished. As the size of thedevices is scaled down, classical continuum theories are not able to explain the size effect related mechanicalbehavior at micron or submicron levels and as a result non-classical continuum theories are pioneered with theinception of internal length scale parameters. In this paper, analysis of isotropic rectangular micro-plates based onKirchhoff model applying Modified Coupled Stress Theory is used toanalyzethe size-dependent thermoelasticdamping and its impact on quality factor and critical dimensions.Hamilton principle is adapted to derive thegoverning equations of motion and the coupled heat conduction equation is employed to formulate the thermoelasticdamping limited quality factor of the plates. Five different structural materials (PolySi, Diamond,Si, GaAs andSiC)are used for optimizing QTED which depends on the materialperformance index parameters. ThermoelasticDamping Index [TDI] and thermal diffusion length, lT. According to this work, the maximum QTED is attained forPolySi with the lowest TDI and Lcmax is obtained for SiC which is having the lowest lT. The impact of lengthscaleparameters (l), vibration modes, boundary conditions (Clamped–Clamped and Simply Supported), and operatingtemperatures on QTED and Lcare also investigated. It is concluded that QTED is further maximized by selecting lowtemperatures and higher internal length scale parameters (l).The prior knowledge of QTED and Lchelp the designers tocome out with high performance low loss resonators.

Thermoelastic Damping Limited Quality Factor Enhancement and Energy Dissipation Analysis of Rectangular Plate Resonators Using Nonclassical Elasticity Theory

Among the different energy dissipation mechanisms, thermoelastic damping plays a vital role and needs to be alleviated in vibrating resonators to mitigate parameters by improving the thermoelastic damping limited quality factor, QTED. The maximum energy dissipation is also interrelated with the critical dimension h c of the plates, and by optimizing the dimensions, the peaking of energy dissipation can be diminished. As the size of the devices is scaled down, classical continuum theories become incompetent to explain the size-effect related mechanical nature at the micron and submicron levels, and, as a result, nonclassical continuum theories have been pioneered with the inception of internal length scale parameters. In this work, an analysis of isotropic rectangular microplates based on the Kirchhoff model and a higher order theory like Modified Couple Stress Theory is utilized to study size-dependent thermoelastic damping and its impact on the quality factor and critical dimensions. The Hamilton principle is adapted to derive the governing equations of motion, and the coupled heat conduction equation is employed to formulate the thermoelastic damping limited quality factor of the plates. Five different structural materials (PolySi, diamond, Si, GaAs, and SiC) are used for optimizing QTED and hc, which depends on two material performance index parameters: the thermoelastic damping index (TDI) and the material thermal diffusion length, l T . According to this work, the maximum QTED is attained for PolySi with the lowest TDI, and hcmax is obtained for Si with the maximum l T . The impacts of the dimensionless length-scale parameters (l/h), vibration modes, and boundary conditions (clamped-clamped and simply supported) on QTED and hc are also investigated. From the current analysis, QTED can be further enhanced by selecting higher vibration modes and clamped-clamped boundary conditions. QTED can be maximized by fixing the internal length scale parameter (l) and making the thickness of the beam equal to l. The analytical study is numerically simulated by using MATLAB 2015 software. Prior knowledge of QTED and hc will help designers to produce high-performance and low-loss resonators for the futuristic technological applications.

On Hydromechanical Interaction during Propagation of Localized Damage in Rocks

This paper provides a mathematical description of hydromechanical coupling associated with propagation of localized damage. The framework incorporates an embedded discontinuity approach and addresses the assessment of both hydraulic and mechanical properties in the region intercepted by a fracture. Within this approach, an internal length scale parameter is explicitly employed in the definition of equivalent permeability as well as the tangential stiffness operators. The effect of the progressive evolution of damage on the hydro-mechanical coupling is examined and an evolution law is derived governing the variation of equivalent permeability with the continuing deformation. The framework is verified by a numerical study involving 3D simulation of an axial splitting test carried out on a saturated sample under displacement and fluid pressure-controlled conditions. The finite element analysis incorporates the Polynomial-Pressure-Projection (PPP) stabilization technique and a fully implicit time integration scheme.

Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory

The main subject of this study is the dynamic and static pull-in analyses of rectangular nanoplates made of functionally graded materials based on the nonlocal strain gradient theory compared with the strain gradient theory, Eringen’s differential method, and classical theory considering disparate boundary conditions (BCs). To this end, the governing equation is derived based on Kirchhoff’s plate theory and then, the Galerkin method is implemented to arrive at an ordinary differential equation of second-order in the time domain. The homotopy analysis method is used as an analytical solution methodology to solve the nonlinear ordinary differential equation which is under different nonlinear forces such as intermolecular, electrostatic and hydrostatic one. Likewise, an efficient technique is exerted for omitting secular terms. Accordingly, the dynamic pull-in voltage is obtained for different non-classical continuum theories, material gradient indices, nonlocal parameters and internal length scale parameters. The dynamic and static pull-in instabilities are also compared in the same situation. Additionally, the effects of the nonlocal parameter and internal length scale parameter on the non-dimensional frequency against the initial amplitude, electrostatic force and hydrostatic actuation are explored in disparate boundary conditions. Communicated by Eleonora Tubaldi.

Postbuckling analysis of hydrostatic pressurized FGM microsized shells including strain gradient and stress-driven nonlocal effects

Herein, with the aid of the newly proposed theory of nonlocal strain gradient elasticity, the size-dependent nonlinear buckling and postbuckling behavior of microsized shells made of functionally graded material (FGM) and subjected to hydrostatic pressure is examined. As a consequence, the both nonlocality and strain gradient micro-size dependency are incorporated to an exponential shear deformation shell theory to construct a more comprehensive size-dependent shell model with a refined distribution of shear deformation. The Mori–Tanaka homogenization scheme is utilized to estimate the effective material properties of FGM nanoshells. After deduction of the non-classical governing differential equations via boundary layer theory of shell buckling, a perturbation-based solving process is employed to extract explicit expressions for nonlocal strain gradient stability paths of hydrostatic pressurized FGM microsized shells. It is observed that the nonlocality size effect causes to decrease the critical hydrostatic pressure and associated end-shortening of microsized shells, while the strain gradient size dependency leads to increase them. In addition, it is found that the influence of the internal strain gradient length scale parameter on the nonlinear instability characteristics of hydrostatic pressurized FGM microsized shells is a bit more than that of the nonlocal one.

Effect of horizon shape in peridynamics

As a new continuum mechanics formulation, peridynamics has a non-local character by having an internal length scale parameter called horizon. Although the effect of the size of the horizon has been studied earlier, the shape of the horizon can also be influential. In this study, the effect of horizon shape is investigated for both ordinary state-based and non-ordinary state-based peridynamics. Three different horizon shapes are considered including circle, irregular and square. Both static and dynamic analyses are studied by considering plate under tension and vibration of a plate problems. For both static and dynamic conditions, square shape could not capture accurate vertical displacements for ordinary-state based peridynamics. On the other hand, results obtained for all three horizon shapes agree very well with finite element analysis results for non-ordinary state-based peridynamics.

A modulus gradient model for inhomogeneous materials with isotropic linear elastic constituents

The one-dimensional modulus gradient (E-grad) model proposed in Gülaşık et al. (2018) is extended to more general three-dimensional inhomogeneous materials with isotropic linear elastic constituents. In addition to the constitutive equations and balance relations, length scale dependent differential relations for the material parameters of isotropic linear elasticity are provided. The finite element formulation for axisymmetric problems is derived and a model problem of a soft cylindrical rod with a stiff spherical inclusion is solved. It is seen that discontinuities and/or very sharp changes in the modulus, displacement, strain and stress fields that exist in local formulations are smoothed with the proposed gradient model. Furthermore, a graded interphase region around the stiff inclusion is obtained by the model as reported in the literature of polymer nanocomposites. The thickness of the graded interphase region is interpreted as a characteristic length scale of the polymer nanocomposite. It is also shown that increasing the internal length scale parameter increases the stiffness of the model. The proposed model is compared with a micromechanical model from literature and experiments conducted with polyimide/silica nanocomposites. The results obtained by the proposed approach capture the experimentally measured values of the nanocomposite modulus. Finally, the model is extended to obtain anisotropic macroscopic response by choosing different length scale parameters in different directions.

Fracture energy in phase field models

The phase field approach to brittle fracture is based on smeared energetic representation of sharp fracture into surface. The passage between damaged and undamaged zones is influenced by an internal length scale parameter. In the present paper the approximation of fracture energy in phase field models is studied. Firstly, the diffusion equation of the phase field is numerically investigated. It is demonstrated through simple paradigmatic 2d and 3d cases that the fracture energy during crack initiation and propagation phenomena, such as crack branching and bifurcation, is strictly correlated with the internal length parameter. Moreover, it is shown that for finite value of the internal length parameter the dissipated energy does not depend only on the crack extension but on the geometrical configuration of fracture differently from the Griffith sharp approach. In particular, it is demonstrated that 3d cracks with same area may be characterized by different values of approximated fracture energy.

Electromechanical analysis of bilayer piezoelectric sensors due to flexoelectricity and strain gradient elasticity

The flexoelectric effect of materials, which is the coupling between strain gradient and electric polarization, is most noticeable for the micro/nano electromechanical systems. In the present paper, the size-dependent electromechanical properties of the bilayer piezoelectric sensor are studied and analyzed considering the strain gradient elastic and flexoelectric effects. The governing equation and the corresponding generalized mechanical boundary conditions of the bilayer cantilever sensor are derived utilizing the variational method of flexoelectric materials based on the electric Gibbs free energy. And a new piezo-flexoelectric coupling parameter is proposed and the relationship between the induced electric potential (voltage) and the rotation angles of the ends is obtained. The analytical expressions of deflection and induced electric potential are given when the bilayer piezoelectric sensor is subject to a uniform force. The numerical results show that the normalized deflection, normalized stiffness and induced electric potential are dependent on the structural size, material parameters and internal material length scale parameters. The piezoelectric effect will play a leading role in the induced electric potential when the sensor thickness is larger than a critical value. With decreasing sensor thickness, the flexoelectric and strain gradient elastic effects will dominate the induced electric potential. Moreover, an intrinsic size depending on the material properties is identified for the maximum induced electric potential. The thickness and polarization direction of the piezoelectric layer also have a great influence on the induced electric potential of the sensor systems.

Gradient elasticity and dispersive wave propagation: Model motivation and length scale identification procedures in concrete and composite laminates

Nano-scale experimental findings reveal that wave propagation in heterogeneous materials is dispersive. In order to capture such dispersive behavior, in this paper gradient elasticity theory is resorted to. A popular gradient elasticity model arising from Mindlin's theory incorporates two internal length scale parameters, which correspond to one micro-stiffness and one micro-inertia term. As an extension of Mindlin's model, an expanded three-length-scale gradient elasticity formulation with one additional micro-inertia term is used to improve the description of microstructural effects in dynamics. A non-local lattice model is introduced here to give the above micro-stiffness and micro-inertia terms a physical interpretation based on geometrical and mechanical properties of the microstructure. The purpose of this paper is to assess the effectiveness of such a three-length-scale formulation in predicting wave dispersion against experimental and micro-mechanical data from the literature. The dispersive wave propagation through laminated composites with periodic microstructure is investigated first. Length scale identification is carried out based on higher-order homogenization to link the constitutive coefficients of the gradient theory directly to microstructural properties of the layered composite. Secondly, experimental dispersion curves for phonons propagating in aluminum and bismuth crystals are scrutinized, thus highlighting the motivation for including multiple micro-inertia terms. Finally, ultrasonic wave dispersion experimentally observed in concrete specimens with various sand contents and water/cement ratios is analyzed, along with length scale quantification procedures. It is found that the proposed three-length-scale gradient formulation is versatile and effective in capturing a range of wave dispersion characteristics arising from experiments. Advantages over alternative formulations of gradient elasticity from the literature are discussed throughout the paper.

Asymptotic solution for interface crack between two materials governed by dipolar gradient elasticity

An asymptotic solution for interface crack between mismatched materials that obey a special form of linear isotropic strain gradient elasticity under conditions of plane strain is developed. It is shown that the asymptotic solution depends in a complicated manner both on a mismatch of elastic properties and a mismatch of the internal length scale parameter. Numerical analysis shows that the mismatch of elastic moduli and the gradient coefficient c respectively lift the degeneracy of the exponent p that is characteristic for a crack in homogeneous material. The total energy release rate Gint due to a crack extension along the interface is derived generalizing the classical virtual crack closure method. The reciprocal work contour integral method for the evaluation of amplitude factors in the asymptotic expansion is extended to interface crack in the linear isotropic strain gradient elasticity.