Linear Isotropic Elasticity Research Articles

Effect of margin designs and loading conditions on the stress distribution of endocrowns: a finite element analysis

BackgroundMargin designs and loading conditions can impact the mechanical characteristics and survival of endocrowns. Analyzing the stress distribution of endocrowns with various margin designs and loading conditions can provide evidence for their clinical application.MethodsThree finite element analysis models were established based on the margin designs: endocrown with a butt-joint type margin (E0), endocrown with a 90° shoulder (E90), and endocrown with a 135° shoulder (E135). The E0 group involved lowering the occlusal surface and preparing the pulp chamber. The E90 group created a 90° shoulder on the margin of model E0, measuring 1.5 mm high and 1 mm wide. The E135 group featured a 135° shoulder. The solids of the models were in fixed contact with each other, and the materials of tooth tissue and restoration were uniform, continuous, isotropic linear elasticity. Nine static loads were applied, with a total load of 225 N, and the maximum von Mises stresses and stress distribution were calculated for teeth and endocrowns with different margin designs.ResultsCompared the stresses of different models under the same loading condition. In endocrowns, when the loading points were concentrated on the buccal side, the maximum von Mises stresses were E0 = E90 = E135, and when there was a lingual loading, they were E0 < E90 = E135. In enamel, the maximum von Mises stresses under all loading conditions were E0 > E90 > E135. In dentin, the maximum von Mises stresses of the three models were basically similar except for load2, load5 and load9. Compare the stresses of the same model under different loading conditions. In endocrowns, stresses were higher when lingual loading was present. In enamel and dentin, stresses were higher when loaded obliquely or unevenly. The stresses in the endocrowns were concentrated in the loading area. In enamel, stress concentration occurred at the cementoenamel junction. In particular, E90 and E135 also experienced stress concentration at the shoulder. In dentin, the stresses were mainly concentrated in the upper section of the tooth root.ConclusionStress distribution is similar among the three margin designs of endocrowns, but the shoulder-type designs, especially the 135° shoulder, exhibit reduced stress concentration.

Discrete dislocation dynamics simulations of [formula omitted]-type prismatic loops in zirconium

Neutron irradiation of zirconium alloys in light water nuclear reactors generates nano-scale defects in the form of vacancy and interstitial 〈a〉-type prismatic loops which lie in prismatic planes of the sample. The dynamics of idealised conservative square 〈a〉-type prismatic loops have been investigated for a range of loop lengths in the framework of linear isotropic elasticity. Three-dimensional dislocation dynamics (DD) simulations of a dislocation-loop interaction have been performed to investigate the dislocation-loop interaction mechanism. For this purpose, a mobility law developed for hexagonally close-packed materials has been implemented and described in detail. Analytical and numerical calculations have been performed to obtain expressions for the restoring force and angular stability of prismatic loops. These analyses have been used to inform a 2.5D discrete dislocation plasticity (DDP) model in order to emulate realistic prismatic loop physics and improve irradiation hardening simulations. From the 2.5D DDP prismatic loop analyses, it has been observed that the stable angle of smaller sized loops is less sensitive to external stresses compared to that of larger loops, which may have implications for the mechanisms of irradiation hardening. Furthermore, initial 2.5D single-slip simulations predict that prismatic loops cause significantly elevated flow stress that increases with increasing loop density in accord with experimental observations, and that the restraining effect of the out-of-plane loop segments (the restoring force) plays an important role in the strengthening caused by loops.

Physics-informed UNets for discovering hidden elasticity in heterogeneous materials

Soft biological tissues often have complex mechanical properties due to variation in structural components. In this paper, we develop a novel UNet-based neural network model for inversion in elasticity (El-UNet) to infer the spatial distributions of mechanical parameters from strain maps as input images, normal stress boundary conditions, and domain physics information. We show superior performance – both in terms of accuracy and computational cost – by El-UNet compared to fully-connected physics-informed neural networks in estimating unknown parameters and stress distributions for isotropic linear elasticity. We characterize different variations of El-UNet and propose a self-adaptive spatial loss weighting approach. To validate our inversion models, we performed various finite-element simulations of isotropic domains with heterogenous distributions of material parameters to generate synthetic data. El-UNet is faster and more accurate than the fully-connected physics-informed implementation in resolving the distribution of unknown fields. Among the tested models, the self-adaptive spatially weighted models had the most accurate reconstructions in equal computation times. The learned spatial weighting distribution visibly corresponded to regions that the unweighted models were resolving inaccurately. Our work demonstrates a computationally efficient inversion algorithm for elasticity imaging using convolutional neural networks and presents a potential fast framework for three-dimensional inverse elasticity problems that have proven unachievable through previously proposed methods.

Numerical method for quasi-static adhesive elastic contact subjected to tangential loading

This paper presents a three-dimensional (3D) numerical model for simulating the adhesive contact with linear isotropic elasticity considering the interaction between normal and tangential loads. The model is based on the Boussinesq-Cerruti integral equations for elasticity, the Maugis-Dugdale (MD) model for adhesion, and the friction/adhesion interaction theory in the McMeeking model; it uses the Coulomb's law of friction to identify the occurrence of local microslip. Efficient and accurate determination of the contact behavior, together with adhesive and stick/slip regions, uses the iterative conjugate gradient method (CGM) and the discrete convolution and fast Fourier transform (DC-FFT) algorithm. The model analyzes adhesive-contact pressure, shear traction, and subsurface stress fields. It is used to study the effects of increasing tangential loading on the variations of stick/slip zones and adhesive-energy dissipation, as well as the influence of sinusoidal roughness on surface adhesion behavior. Compared with most theoretical models, the proposed numerical model has no restrictions on surface geometry and roughness of contact bodies, and it should have a wider range of applications.

On the identification of Lamé parameters in linear isotropic elasticity via a weighted self-guided TV-regularization method

Abstract Recently in [V. Markaki, D. Kourounis and A. Charalambopoulos, A dual self-monitored reconstruction scheme on the TV \mathrm{TV} -regularized inverse conductivity problem, IMA J. Appl. Math. 86 2021, 3, 604–630], a novel reconstruction scheme has been developed for the solution of the inclusion problem in the inverse conductivity problem on the basis of a weighted self-guided regularization process generalizing the total variation approach. The present work extends this concept by implementing and investigating its applicability in the two-dimensional elasticity setting. To this end, we employ the framework of the reconstruction of linear and isotropic elastic structures described by their Lamé parameters. Numerical examples of increasingly challenging geometric complexities illustrate the enhanced accuracy and efficiency of the method.

A non-iterative methodology to reconstruct boundary shapes and conditions in isotropic linear elasticity based on the BEM

A non-iterative method is proposed to estimate unknown boundary geometry shapes and boundary conditions in 2D and 3D isotropic linear elasticity problems based on the boundary element method (BEM). A virtual domain with a virtual boundary is introduced to the inverse problem. For the boundary geometry shape identification problem, the displacements of the virtual boundary are obtained through the measurement point displacements and the singular value decomposition is applied to solve the ill-posed matrix. The BEM is used to solve the problem with the displacements of the virtual boundary. The unknown boundary geometry shapes can be estimated by searching the internal point isodisplacement curve. For the boundary condition reconstruction problem, the virtual boundary is overlapped with the real boundary. The displacements of the virtual boundary can also be acquired by the measurement point displacements. The unknown boundary tractions can be calculated by the BEM. Finally, the influences with different virtual domains, measurement points, measurement positions and measurement errors on the results are discussed. The results show that the proposed method is an accurate and efficient way to identify boundary geometry shapes and reconstruct boundary conditions.

Static Unified Inelastic Model: pre- and post-yield dislocation-mediated deformation

Modelling dislocation glide over the initial part of a stress–strain curve of metals received little attention up to now. However, dislocation glide is essential to ones understanding of the fundamental relationship between inelastic deformation and the evolution of the dislocation network structure. Therefore, we present a model of dislocation-driven deformation under static loading conditions.We reproduce repeated cyclic uniaxial tensile tests on Interstitial-Free and Low-Alloy steels. The elastic mechanical behaviour is described by isotropic linear elasticity, pre-yield anelastic mechanical behaviour by a dislocation bow-out model with dissipation, and the post-yield evolution of dislocation network structure by a statistical storage model. We hypothesise that when the local anelastic compliance is lower than the global plastic compliance, deformation is mechanically recoverable, and vice versa. This hypothesis is corroborated with the classical Taylor relation. We report the relation between stable and unstable dislocation glide using this prototypical modelling framework.We find four structural variables, that are based on dislocation physics, to describe the stress–strain curve: total dislocation density, average dislocation segment length, dislocation junction formation rate, and average dislocation junction length. Firstly, we quantify the dislocation network evolution during uniaxial monotonic loading, and verify work-hardening by dislocation junction formation and a Taylor-type equation for flow. Finally, we present a semi-empirical relation for the evolution of the dislocation network structure. Which allows us to: refine the physical interpretation of the Taylor relationship, and rationalise experimental observations on apparent modulus degradation by thermomechanical processing. Both these findings circumvent the limitations of current, physics-based hardening models.

A developed hybrid experimental–analytical method for thermal stress analysis of a deep U-notched plate

A general framework of hybridizing experimentally-recorded load-induced thermal information by means of thermoelastic stress analysis (TSA) of a loaded structure with the Michell solution of Airy stress function in isotropic linear elasticity is proposed for in-plane stresses determination. The capability of the proposed hybrid method in separating the TSA signals into individual stresses is demonstrated by stress-analyzing a deep U-notched aluminum plate without neither knowing the entire geometry and distant loading/boundary conditions nor using supplementary experimental information. Even though no experimental data were processed at, and adjacent to, the edges of the U-notched plate, the individual stresses are determined throughout the surface of the plate. Prior processing actual experimental TSA data, the accuracy and the stability of the proposed hybrid method through adding artificial noise scatters in the processed simulated data with the number of retained terms in the generalized Airy stress expressions are firstly investigated. The effects of the superimposed noise scatters as well as the number of employed data in the evaluated stresses are also assessed. Finally, the reliability of the determined hybrid-experimental stresses is supported through a comparison with finite-element predictions and strain gauge measurements. The ease of implementation, the accuracy and the precision of the evaluated stress, and the versatility to the type of employed data make the proposed hybrid method more attractive and superior than other hybrid methods.

Determination of the Spherical Couple-Stress in Polar Linear Isotropic Elasticity

It is well known that the conventional couple-stress theory leaves the spherical part of the couple-stress indeterminate. This indeterminacy problem is recently resolved for fibrous composites subjected to either small or large deformations and containing a single family of fibres resistant in bending (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595). However, the problem remains still unsolved in simpler cases where the implied preference material direction is not related to fibre bending resistance, and even in the simplest possible case where the polar material of interest is linearly elastic and isotropic. This communication aims (i) to show that a relevant virtual spin concept employed (Soldatos in Math. Mech. Solids, 2021, https://doi.org/10.1177/10812865211061595) is further applicable in the latter case of polar linear isotropic elasticity, (ii) to demonstrate the process in which that concept thus leads to determination of the spherical part of the couple-stress, (iii) to exemplify this process by providing a couple of simple illustrative examples, (iv) to specify and discuss the reason that the outlined method meets a hurdle in cases of linear anisotropic elasticity that is due to one or more preferential material directions, and, hence, (v) to further discuss the manner in which that newly identified difficulty is currently confronted, and may thus be handled successfully.

Direct connection between Navier and spherical harmonic kernels in elasticity

&lt;abstract&gt; &lt;p&gt;Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.&lt;/p&gt; &lt;/abstract&gt;

Rayleigh and Stoneley waves in linear elasticity

We construct microlocal solutions of Rayleigh and Stoneley waves in isotropic linear elasticity with the density and the Lamé parameters smooth up to a curved boundary or interface. We compute the direction of the microlocal polarization and show a retrograde elliptical motion of these two type of waves.

From two- to three-dimensional continuum-kinematics-inspired peridynamics: More than just another dimension

Continuum-kinematics-inspired Peridynamics (CPD) has been recently proposed as a geometrically exact formulation of peridynamics that is also thermodynamically and variationally consistent. Unlike the original formulation of peridynamics (PD), CPD can accurately capture the Poisson effect. For a three-dimensional analysis, CPD builds upon one-, two- and three-neighbor interactions. The isotropic three-dimensional CPD formulation of non-local elasticity therefore involves three material constants associated with length, area and volume. This manuscript aims to establish the relationships between the material parameters of CPD and isotropic linear elasticity for three-dimensional problems. In addition to addressing significant technical difficulties that arise when advancing from two- to three-dimensional problems, this contribution unravels several key features that are entirely absent in a two-dimensional analysis (Ekiz et al., 2022). It is shown that the three material parameters of CPD reduce to two independent parameters in the linearized framework, and can be expressed in terms of any pairs of isotropic linear elasticity constants, such as Lamé parameters. The analysis here provides a physical interpretation for the first Lamé constant, for the first time. Finally, we establish the admissible ranges for CPD material parameters.

Analytical strain localization analysis of isotropic and anisotropic damage models for quasi-brittle materials

Strain and damage localization are usually precursors of rupture. We present a three-dimensional method dedicated to quasi-brittle materials based on the works of Bigoni & Hueckel, and Jirásek and coworkers, aiming at simplifying the analysis of the localization properties of continuous damage models of a general form, and possibly anisotropic. The method reformulates the localization problem as a two-variable polynomial maximization problem, a strategy commonly used in softening plasticity models, but not so much in Continuum Damage mechanics. The quasi-brittle hypothesis is exploited to render the problem solvable in a fully analytical way, and a post-analysis criterion for the validity of the analysis is also exhibited. In this work, the method is fully established from a theoretical viewpoint, and examples illustrating its use are provided. Multiaxial calculations are performed for four continuous damage models (two isotropic and two anisotropic ones). The method applies to induced anisotropy and constitutive models representing isotropic linear elasticity before damage growth, and remains accurate when models display immediate softening after the elastic limit (and thus to multiaxial tensile cases). The analytical method is, however, entirely general and allows for the calculation of (i) the orientation of a potential localization plane, (ii) the mode angle of the weak discontinuity, and (iii) the validity domain of such a simplified analysis.

Boundary Integral Equations for Isotropic Linear Elasticity

Boundary Integral Equations for Isotropic Linear Elasticity

Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods

A new reduced-dimension adaptive generalized Dryja--Smith--Widlund (GDSW) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition, the reduced-dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced-dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented.

A novel approach to consider triaxial tensile stresses within the framework of a failure criterion

AbstractFor use in micromechanical simulations of continuous fiber reinforced polymers, a more general form of the paraboloid failure criterion by Stassi‐D'Alia for matrix failure was developed with explicit consideration of the hydrostatic tension strength. Regarding polymers, limits for hydrostatic tensile strength based on isotropic linear elasticity could be derived. The comparison of the newly developed extended paraboloid criterion with experimental data for yielding as well as for material separation (fracture) shows good agreement.

Density functional theory method for twisted geometries with application to torsional deformations in group-IV nanotubes

We present a real-space formulation and implementation of Kohn-Sham Density Functional Theory suited to twisted geometries, and apply it to the study of torsional deformations of X (X = C, Si, Ge, Sn) nanotubes. Our formulation is based on higher order finite difference discretization in helical coordinates, uses ab initio pseudopotentials, and naturally incorporates rotational (cyclic) and screw operation (i.e., helical) symmetries. We discuss several aspects of the computational method, including the form of the governing equations, d ils of the numerical implementation, as well as its convergence, accuracy and efficiency properties.The technique presented here is particularly well suited to the first principles simulation of quasi-one-dimensional structures and their deformations, and many systems of interest can be investigated using small simulation cells containing just a few atoms. We apply the method to systematically study the properties of single-wall zigzag and armchair group-IV nanotubes in the range of (approximately) 1 to 3 nm radius, as they undergo twisting. For the range of deformations considered, the mechanical behavior of the tubes is found to be largely consistent with isotropic linear elasticity, with the torsional stiffness, ktwist, varying as the cube of the nanotube radius. Furthermore, for a given tube radius, ktwist is seen to be highest for carbon nanotubes and the lowest for those of tin, while nanotubes of silicon and germanium are found to have intermediate values of this quantity close to each other. We also describe different aspects of the variation in electronic properties of the nanotubes as they are twisted. In particular, we find that akin to the well known behavior of armchair carbon nanotubes, armchair nanotubes of silicon, germanium and tin also exhibit bandgaps that vary periodically with imposed rate of twist, and that the periodicity of the variation scales in an inverse quadratic manner with the tube radius. These examples highlight the utility of the proposed method in the accurate and efficient computational characterization of important nanomaterials from first principles.

Relationships between the material parameters of continuum-kinematics-inspired peridynamics and isotropic linear elasticity for two-dimensional problems

Continuum-kinematics-inspired Peridynamics (CPD) has been recently proposed as a geometrically exact formulation of peridynamics that is also thermodynamically and variationally consistent. CPD can capture the Poisson effect exactly, unlike the original formulation of peridynamics (PD). Due to its geometrically exact nature, CPD does not suffer from zero-energy modes and displacement oscillations that may be observed in state-based PD formulations. For a two-dimensional analysis, CPD builds upon one-neighbor and two-neighbor interactions. The one-neighbor interactions of CPD are equivalent to the bond-based interactions of the original PD formalism. Two-neighbor interactions, however, are key in CPD since the basic notions of classical continuum kinematics, namely length and area, are preserved exactly. The isotropic two-dimensional CPD formulation of non-local elasticity therefore involves two material constants, namely C1 and C2, associated with length and area, respectively. This manuscript aims to establish relationships between the material parameters of CPD and isotropic linear elasticity for an affine deformation in a two-dimensional setting. It is shown that each of the CPD material parameters can be expressed in terms of any pairs of isotropic linear elasticity constants, such as Lamé parameters. Finally, we establish the admissible ranges for CPD material parameters.

A sequential linear complementarity problem for multisurface plasticity

In this paper, a general algorithm is presented for the integration of multisurface plasticity models. The algorithm combines the return mapping with the technique of the sequential linear complementarity problem (SLCP) to update stress. The set of active constraints is naturally identified by the classical Lemke’s algorithm. With the assumptions of isotropic linear elasticity, perfect plasticity, and the associated flow rule, all details are provided in matrix notations to facilitate computer implementation. The extension to hardening/softening multisurface plasticity models is also presented. The application of the algorithm is demonstrated via simulation of three types of geotechnical problems in 2D and 3D. Both linear and nonlinear multisurface plasticity models, e.g. Mohr-Coulomb and generalized Hoek-Brown yield criteria, are examined within the framework of the proposed algorithm. The numerical results are in good agreement with the analytic solutions. Moreover, the accuracy of the proposed stress integration procedure is investigated through 3D isoerror map. The convergence using the consistent tangent matrix at the global level is examined by a one-increment example consisting of one element.

Dislocation transmission across Σ3{112} incoherent twin boundary: a combined atomistic and phase-field study

Grain boundaries (GBs) in polycrystalline materials act as impediments to dislocation motion and result in strengthening. Understanding slip transmission through GBs, specifically twin boundaries, is essential to understand the plastic deformation behavior of polycrystalline fcc materials. In this study the interaction between a glide dislocation and Σ3{112} incoherent twin boundary (ITB) in copper is investigated using a combined atomistic and mesoscale approach. The material parameters and structure of the GB in the mesoscale phase field dislocation dynamics (PFDD) model are informed from Molecular Statics (MS) simulations. The structural unit of the ITB consists of an array of three partial dislocations. The interaction between a glide dislocation impinging on each of the GB partial dislocations is investigated using both PFDD and Molecular Dynamics (MD) with two boundary conditions. Transmission planes predicted by both PFDD and MD (NVT) are in agreement, and show that not all transmission events are direct. Critical transmission stresses predicted by PFDD are in the range of 276 MPa to 1380 MPa, while MD predictions are in the range from 100 MPa to 700 MPa. The PFDD and MD predictions of slip transmission are explained using dislocation theory based on isotropic linear elasticity.