Showing papers on "Linear elasticity published in 2019"
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TL;DR: In this paper, the authors introduce the notion of odd elasticity, a generalization of linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity.
Abstract: Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. In this Letter, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity. This generalized framework, which we dub odd elasticity, reveals that two additional moduli can exist in a two-dimensional isotropic solid with active bonds. Such an odd-elastic solid can be regarded as a distributed engine: work is locally extracted, or injected, during quasi-static cycles of deformation. Using continuum equations, coarse-grained microscopic models, and numerical simulations, we uncover phenomena ranging from activity-induced auxetic behavior to wave propagation powered by self-sustained active elastic cycles. Besides providing insights beyond existing hydrodynamic theories of active solids, odd elasticity suggests design principles for emergent autonomous materials.
139 citations
TL;DR: In this article, an analytical solution for the longitudinal bending stiffness of a segmental liner, typically used on tunnels built with a shield, is derived and verified by providing comparisons between its results and those from the Finite Element program ABAQUS.
101 citations
TL;DR: A novel approach for the construction of isogeometric numerical methods for elliptic PDEs on trimmed geometries, seen as a special class of more general V-reps, and validated both on two and three dimensional problems, for diffusion and linear elasticity.
55 citations
TL;DR: A novel and effective computational approach based on polygonal XFEM (named as PolyXFEM) for the analysis of two-dimensional (2D) linear elastic fracture mechanics problems, equipped with a new numerical integration technique that uses the concept of Cartesian transformation method (CTM) overpolygonal domains.
52 citations
TL;DR: In this article, the authors considered the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimated its magnitude, and they introduced singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lame system.
Abstract: In the region between close-to-touching hard inclusions, stress may be arbitrarily large as the inclusions get closer. This stress is represented by the gradient of a solution to the Lame system of linear elasticity. We consider the problem of characterizing the gradient blow-up of the solution in the narrow region between two inclusions and estimating its magnitude. We introduce singular functions which are constructed in terms of nuclei of strain and hence are solutions of the Lame system, and then show that the singular behavior of the gradient in the narrow region can be precisely captured by singular functions. As a consequence of the characterization, we are able to regain the existing upper bound on the blow-up rate of the gradient, namely, ɛ−1/2 where ɛ is the distance between two inclusions. We then show that it is in fact an optimal bound by showing that there are cases where ɛ−1/2 is also a lower bound. This work is the first to completely reveal the singular nature of the gradient blow-up and to obtain the optimal blow-up rate in the context of the Lame system with hard inclusions. The singular functions introduced in this paper play essential roles in overcoming the difficulties in applying the methods of previous works. The main tools of this paper are the layer potential techniques and the variational principle. The variational principle can be applied because the singular functions of this paper are solutions of the Lame system.
51 citations
TL;DR: The proposed approach guarantees that the additional parameters vanish if the material is purely homogeneous, in other words, it is fully compatible with conventional homogenization schemes based on spatial averaging techniques.
Abstract: Owing to additive manufacturing techniques, a structure at millimeter length scale (macroscale) can be produced by using a lattice substructure at micrometer length scale (microscale). Such a system is called a metamaterial at the macroscale as the mechanical characteristics deviate from the characteristics at the microscale. As a remedy, metamaterial is modeled by using additional parameters; we intend to determine them. A homogenization approach based on the asymptotic analysis establishes a connection between these different characteristics at micro- and macroscales. A linear elastic first order theory at the microscale is related to a linear elastic second order theory at the macroscale. Relation for parameters at the macroscale is derived by using the equivalence of energy at macro- and microscales within a so-called Representative Volume Element (RVE). Determination of parameters are succeeded by solving a boundary value problem with the Finite Element Method (FEM). The proposed approach guarantees that the additional parameters vanish if the material is purely homogeneous, in other words, it is fully compatible with conventional homogenization schemes based on spatial averaging techniques. Moreover, the proposed approach is reliable as it ensures that such resolved additional parameters are not sensitive to choices of RVE consisting in the repetition of smaller RVEs but depend upon the intrinsic size of the structure.
46 citations
TL;DR: In this paper, a detailed derivation of an incremental formulation of the phase field model for a hydraulic fracture in a poroelastic medium is presented, and the authors prove that the fracture remains small in the third direction in comparison to the first two principal directions.
Abstract: We study the propagation of hydraulic fractures using the fixed stress splitting method. The phase field approach is applied and we study the mechanics step involving displacement and phase field unknowns, with a given pressure. We present a detailed derivation of an incremental formulation of the phase field model for a hydraulic fracture in a poroelastic medium. The mathematical model represents a linear elasticity system with fading elastic moduli as the crack grows that is coupled with an elliptic variational inequality for the phase field variable. The convex constraint of the variational inequality assures the irreversibility and entropy compatibility of the crack formation. We establish existence of a minimizer of an energy functional of an incremental problem and convergence of a finite dimensional approximation. Moreover, we prove that the fracture remains small in the third direction in comparison to the first two principal directions. Computational results of benchmark problems are provided that demonstrate the effectiveness of this approach in treating fracture propagation. Another novelty is the treatment of the mechanics equation with mixed boundary conditions of Dirichlet and Neumann types. We finally notice that the corresponding pressure step was studied by the authors in Mikelic et al. (SIAM Multiscale Model Simul 13(1):367–398, 2015a).
44 citations
TL;DR: This is the first attempt to quantitatively understand structure, quiescent relaxation and shear elasticity, and nonlinear yielding of dense microgel suspensions using microscopic force based theoretical methods that include activated hopping processes.
Abstract: We present an integrated experimental and quantitative theoretical study of the mechanics of self-crosslinked, slightly charged, repulsive pNIPAM microgel suspensions over a very wide range of concentrations (c) that span the fluid, glassy and putative “soft jammed” regimes. In the glassy regime we measure a linear elastic dynamic shear modulus over 3 decades which follows an apparent power law concentration dependence G′ ∼ c5.64, a variation that appears distinct from prior studies of crosslinked ionic microgel suspensions. At very high concentrations there is a sharp crossover to a nearly linear growth of the modulus. To theoretically understand these observations, we formulate an approach to address all three regimes within a single conceptual Brownian dynamics framework. A minimalist single particle description is constructed that allows microgel size to vary with concentration due to steric de-swelling effects. Using a Hertzian repulsion interparticle potential and a suite of statistical mechanical theories, quantitative predictions under quiescent conditions of microgel collective structure, dynamic localization length, elastic modulus, and the structural relaxation time are made. Based on a constant inter-particle repulsion strength parameter which is determined by requiring the theory to reproduce the linear elastic shear modulus over the entire concentration regime, we demonstrate good agreement between theory and experiment. Testable predictions are then made. We also measured nonlinear rheological properties with a focus on the yield stress and strain. A theoretical analysis with no adjustable parameters predicts how the quiescent structural relaxation time changes under deformation, and how the yield stress and strain change as a function of concentration. Reasonable agreement with our observations is obtained. To the best of our knowledge, this is the first attempt to quantitatively understand structure, quiescent relaxation and shear elasticity, and nonlinear yielding of dense microgel suspensions using microscopic force based theoretical methods that include activated hopping processes. We expect our approach will be useful for other soft polymeric particle suspensions in the core–shell family.
43 citations
TL;DR: In this paper, a constitutive theory of isotropic linear elastic solids where Cauchy's stress tensor is not necessarily symmetric thus earning them the attribute "polar" was proposed.
Abstract: The paper initiates a constitutive theory of isotropic linear elastic solids where Cauchy’s stress tensor is not necessarily symmetric thus earning them the attribute “polar”. Mainly, expressions for the fourth-order elasticity tensors of isotropic polar solids in two and three space dimensions are derived. It is found that the constitutive tensors feature, in addition to the usual bulk and shear moduli, one extra independent elastic constant in 3D, and two extra constants in 2D. In the latter case, the new constants quantify to which degree stress and mirror symmetries are broken. Indeed, it turns out that in 2D, stress asymmetry enables isotropic yet chiral behaviors and the interplay between chirality and polarity is subsequently investigated. To motivate the theory, it is shown that 2D isotropic polar solids naturally arise in the application of the transformation method when the background medium is isotropic and when the underlying spatial transformation is conformal (i.e., angle-preserving). Accordingly, isotropic polar solids, in conjunction with conformal transformations, can simplify the design of invisibility cloaks and other wave-steering devices by circumventing the need for anisotropic behaviors. As a demonstration, a conformal carpet cloak is designed out of graded hexachiral lattices and numerically tested. The cloaking performance is shown to be satisfactory over a range of pressure-shear coupled dynamic loadings.
40 citations
TL;DR: In this article, the generalized finite difference method (GFDM) was used to stably and accurately solve 2D inverse Cauchy problems in linear elasticity by using the Navier equations.
40 citations
TL;DR: In this article, the homogenized behavior of three-dimensional periodic structures made of welded elastic bars was determined in the framework of static linear elasticity, and it has been shown that such structures can b
Abstract: We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars It has been shown that such structures can b
TL;DR: The present work derives and implements a finite element approach to simulate elastic wave propagation in a mechanically-modulated metamaterial and results on non-reciprocal wave propagate in a negative stiffness chain are shown.
Abstract: Acoustic and elastic metamaterials with time- and space-dependent effective material properties have recently received significant attention as a means to induce non-reciprocal wave propagation. Recent analytical models of spring-mass chains have shown that external application of a nonlinear mechanical deformation, when applied on time scales that are slow compared to the characteristic times of propagating linear elastic waves, may induce non-reciprocity via changes in the apparent elastic modulus for perturbations around that deformation. Unfortunately, it is rarely possible to derive analogous analytical models for continuous elastic metamaterials due to complex unit cell geometry. The present work derives and implements a finite element approach to simulate elastic wave propagation in a mechanically-modulated metamaterial. This approach is implemented on a metamaterial supercell to account for the modulation wavelength. The small-on-large approximation is utilized to separate the nonlinear mechanical deformation (the "large" wave) from superimposed linear elastic waves (the "small" waves), which are then analyzed via Bloch wave analysis with a Fourier expansion in the harmonics of the modulation frequency. Results on non-reciprocal wave propagation in a negative stiffness chain, a structure exhibiting large stiffness modulations due to the presence of mechanical instabilities, are then shown as a case example.
TL;DR: In this paper, an orthotropic model for in-plane linear elasticity is proposed in the micropolar peridynamic analysis framework, which is derived from the definition of a specific microelastic energy function for micropolastic nonlocal lattices.
TL;DR: It is concluded that lipid membrane deformations are to a very good approximation volume-preserving, with a Poisson ratio that is likely about 3% smaller than the common soft matter limit.
Abstract: We investigate the Poisson ratio ν of fluid lipid bilayers, i.e., the question how area strains compare to the changes in membrane thickness (or, equivalently, volume) that accompany them. We first examine existing experimental results on the area- and volume compressibility of lipid membranes. Analyzing them within the framework of linear elasticity theory for homogeneous thin fluid sheets leads us to conclude that lipid membrane deformations are to a very good approximation volume-preserving, with a Poisson ratio that is likely about 3% smaller than the common soft matter limit . These results are fully consistent with atomistic simulations of a DOPC membrane at varying amount of applied lateral stress, for which we instead deduce ν by directly comparing area- and volume strains. To assess the problematic assumption of transverse homogeneity, we also define a depth-resolved Poisson ratio ν(z) and determine it through a refined analysis of the same set of simulations. We find that throughout the membrane's thickness, ν(z) is close to the value derived assuming homogeneity, with only minor variations of borderline statistical significance.
TL;DR: In this paper, the authors developed full range true stress-strain curves of various steel materials for large deformations, which are then incorporated into advanced finite element models to simulate large deformation fields of these steel coupons observed in the tensile tests.
TL;DR: In this article, the effect of inter-fiber adhesion on the mechanical behavior of cross-linked ran-dom fiber networks in two dimensions is investigated. And the degree of shrinkage is determined as a function of the strength of adhesion and network parameters.
Abstract: We study the effect of inter-fiber adhesion on the mechanical behavior of cross-linked ran- dom fiber networks in two dimensions. To this end, we consider networks with connectiv- ity number, z , below, at, and above the isostaticity limit of the structure without adhesion, z c . Fibers store energy in the axial and bending deformation mode and the cross-links are of freely rotating type. Adhesive forces lead to fiber bundling and to a reduction of the total volume of the network. The degree of shrinkage is determined as a function of the strength of adhesion and network parameters. The mechanical response of these struc- tures is further studied in uniaxial tension and compression. The stress-strain curves of networks without inter-fiber adhesion exhibit an initial linear regime, followed by strain stiffening in tension and strain softening and strain localization in compression. In pres- ence of adhesion, the response becomes more complex. The initial linear regime persists, with the effective modulus decreasing and increasing with increasing adhesion in cases with z > z c and z z c subjected to tension strain-stiffen at rates that depend on the adhesion strength, but eventually enter a large strain/stress regime in which the response is independent of this parameter. Networks with z z c case, increasing the adhe- sion strength reduces the linear elastic modulus and significantly increases the range of the linear regime, delaying strain localization. This first investigation of the mechanics of cross-linked random networks with inter-fiber adhesion opens the door to the design of soft materials with novel properties.
TL;DR: In this article, an efficient and accurate framework for the post-buckling analysis of sandwich structures with elastic-plastic material behaviors is proposed, which is based on the Euler-Bernoulli beam theory and high-order functions.
TL;DR: In this paper, a scaled boundary finite element method is used for both static and dynamic analyses of cylindrical shells. But the scaling strategy is employed to ensure that the shell boundaries and the cross sections at the element inferfaces are accurately represented through the scaling process.
Abstract: A modeling technique based on the scaled boundary finite element method is developed for both static and dynamic analyses of cylindrical shells. A new scaling strategy is employed to ensure that the shell boundaries and the cross sections at the element inferfaces are accurately represented through the scaling process. The formulation starts directly from the three-dimensional linear elasticity theory for cylindrical shells. The principle of virtual work involving the inertial force is applied to derive the scaled boundary finite element equation. Only the in-plane dimensions of the structure are discretized with finite elements while the solution through the thickness is expressed analytically as a Pade expansion. A variable transformation procedure facilitates the development of the dynamic stiffness matrix, which leads to the static stiffness matrix and mass matrix naturally. A laminate model with arbitrary number of layers can readily be constructed. Numerical examples demonstrate the accuracy, applicability and efficiency of the two-layer model.
TL;DR: This work presents an approach to predict from the discrete particle configuration the type of deformation resulting for a finite-sized system of magnetic gels and elastomers, and finds that regular simple cubic configurations show elongation of the sphere along the magnetization if oriented along a face or space diagonal of the cubic unit cell.
Abstract: Magnetic gels and elastomers are promising candidates to construct reversibly excitable soft actuators, triggered from outside by magnetic fields. These magnetic fields induce or alter the magnetic interactions between discrete rigid particles embedded in a soft elastic polymeric matrix, leading to overall deformations. It is a major challenge in theory to correctly predict from the discrete particle configuration the type of deformation resulting for a finite-sized system. Considering an elastic sphere, we here present such an approach. The method is in principle exact, at least within the framework of linear elasticity theory and for large enough interparticle distances. Different particle arrangements are considered. We find, for instance, that regular simple cubic configurations show elongation of the sphere along the magnetization if oriented along a face or space diagonal of the cubic unit cell. Contrariwise, with the magnetization along the edge of the cubic unit cell, they contract. The opposite is true in this geometry for body- and face-centered configurations. Remarkably, for the latter configurations but the magnetization along a face or space diagonal of the unit cell, contraction was observed to revert to expansion with decreasing Poisson ratio of the elastic material. Randomized configurations were considered as well. They show a tendency of elongating the sphere along the magnetization, which is more pronounced for compressible systems. Our results can be tested against actual experiments for spherical samples. Moreover, our approach shall support the search of optimal particle distributions for a maximized effect of actuation.
TL;DR: In this paper, the relationship between macroscopic nonlinear mechanics and the microscopic crystal structural evolution of pre-oriented high-density polyethylene (HDPE) is investigated by in situ synchrotron radiation wide-angle X-ray diffraction (WAXD) measurement over a wide temperature range from −10 to 130 °C.
TL;DR: In this paper, a digital image correlation (DIC) is used to analyze an adhesively-bonded double-cantilever beam (DCB) and to determine the traction-separation law for a cohesive-zone model.
Abstract: Digital-image correlation (DIC) is used to analyze an adhesively-bonded double-cantilever beam (DCB), and to determine the traction-separation law for a cohesive-zone model. The issues involved with how to extract useful information from the digital data of DIC are addressed. In addition, DIC is used to explore how the cohesive zone evolves, and to determine how the elastic arms deform in response to the loading and to the adhesive. The results of these observations are compared to numerical and analytical models for the DCB geometry. In particular, the well-known concept of root rotation is demonstrated. It is shown that, by combining the effects of shear into an effective root rotation, it is possible to use a simple Euler-beam approximation to describe the compliance of the DCB. The experiments and analysis also illustrate the lesser-known concept that significant compression can occur beyond the tensile region in the cohesive zone ahead of a DCB crack tip. Therefore, for accurate numerical predictions, a cohesive-zone model must incorporate compressive deformation. The DIC results are further used to illustrate the concept of a cohesive-length scale. This is defined in terms of the work done against crack-tip tractions, the opening displacement, the stiffness of the arms, and a characteristic geometrical length. The cohesive-length scale is measured experimentally in this paper, and its magnitude is shown to indicate when linear elasticity can be used to describe the deformation of a DCB geometry. The cohesive-length scale is shown to correlate with both the root rotation and the length of a cohesive zone in a fashion that is very similar to what is predicted analytically by elastic-foundation models. Finally, it is demonstrated that, when used in a cohesive-zone model of the geometry, the experimentally determined traction-separation law gives excellent predictions for the evolution of the cohesive zone and for the deformation of the elastic beams. A very minor discrepancy is associated with in-plane tensile stresses that must develop within an adhesive layer in response to the deformation of the beams.
TL;DR: In this paper, the authors consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure, where the fluid motion is represented by the classical Navier-Stokes equations while the elastic displacement is described by the linearized elasticity equation.
Abstract: In this paper, we consider a system modeling the interaction between a viscous incompressible fluid and an elastic structure. The fluid motion is represented by the classical Navier-Stokes equations while the elastic displacement is described by the linearized elasticity equation. The elastic structure is immersed in the fluid and the whole system is confined into a general bounded smooth domain of R3. Our main result is the local in time existence and uniqueness of a strong solution of the corresponding system.
TL;DR: It is customary that the residual stress and deformation acquired by the modified model of CHILE conform to those ones assessed through adopting the viscoelastic model.
Abstract: A multi-physics coupling numerical model of the curing process is proposed for the thermosetting resin composites in this paper, and the modified "cure hardening instantaneously linear elastic (CHILE)" model and viscoelastic model are adopted to forecast residual stress and deformation during the curing process. The thermophysical properties of both models are evolved in line with temperature and degree of cure (DOC). Accordingly, the numerical simulation results are improved to be more accurate. Additionally, the elastic modulus of the materials is calibrated to be equal to the modulus of viscoelastic relaxation by a defined function of time in the CHILE model. Subsequently, this work effectuates the two proposed models in a three-dimensional composite laminate structure. Through comparing the two numerical outcomes, it is customary that the residual stress and deformation acquired by the modified model of CHILE conform to those ones assessed through adopting the viscoelastic model.
TL;DR: In this article, a mixed virtual element method was proposed for a pseudostress-displacement formulation of the linear elasticity problem with nonhomogeneous Dirichlet boundary conditions.
TL;DR: In this paper, analytical expressions for the six Cauchy stress components in untwisted, straight, thin-walled beams with rectangular and circular cross sections characterised by constant taper and subjected to three cross-section forces were derived.
Abstract: Tapered beams are widely employed in efficient flexure dominated structures. In this paper, analytical expressions are derived for the six Cauchy stress components in untwisted, straight, thin-walled beams with rectangular and circular cross sections characterised by constant taper and subjected to three cross-section forces. These expressions pertain to homogeneous, isotropic, linear elastic materials and small strains. In fact, taper not only alters stress magnitudes and distributions but also evokes stress components, which are zero in prismatic beams. A parametric study shows that increasing taper decreases the von Mises stress based fatigue life, suggesting that step-wise prismatic approximations entail non-conservative designs.
TL;DR: This work embraces a three scales asymptotic homogenization approach to investigate the effective behavior of hierarchical linear elastic composites reinforced by cylindrical, uniaxially aligned fibers and possessing a periodic structure at each hierarchical level of organization.
Abstract: In the present work we embrace a three scales asymptotic homogenization approach to investigate the effective behavior of hierarchical linear elastic composites reinforced by cylindrical, uniaxially aligned fibers and possessing a periodic structure at each hierarchical level of organization. We present our novel results assuming isotropy of the constituents and focusing on the effective out-of-plane shear modulus, which is computed exploiting the solution of the arising anti-plane problems. The latter are solved semi-analytically by means of complex variables and successfully benchmarked against the results obtained by finite elements. Our findings can pave the way for multiscale modeling of complex hierarchical materials (such as bone and tendons) at a negligible computational cost.
TL;DR: In this paper, a second-gradient elastic (SGE) material is identified as the homogeneous solid equivalent to a periodic planar lattice characterized by a hexagonal unit cell, which is made up of three different linear elastic bars ordered in a way that the hexagonal symmetry is preserved and hinged at each node, so that the lattice bars are subject to pure axial strain while bending is excluded.
TL;DR: In this paper, the free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated based on third-order shear deformation plate theory w.r.t.
Abstract: In this paper, free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated. The study is based on third-order shear deformation plate theory w...
TL;DR: In this article, the authors used dipolar approximation and linear elasticity theory for small deformations to determine the components of deformations and tensions of magnetorheological elastomers, showing that their elasticity state depends not only on magnetic field intensity but also on volume fraction.
TL;DR: In this paper, a mixed-mode failure model for crack initiation in nonlinear elastic materials has been proposed to determine critical loads of hyperelasticadhesive bonds in both shear and tension dominated configurations.