Linear Isotropic Elasticity Research Articles

Plane Asymptotic Interface Crack Solutions in Gradient Elasticity Theory

The goal of the contribution is to develop an asymptotic interface crack-tip solution under conditions of plane strain for a bi-material that obeys a special form of linear isotropic gradient elasticity. Several fracture mechanics problems have been solved in the past within the framework of strain gradient elasticity which is capable to capture additional length/size parameters. However to our best knowledge no solution concerning an interface crack is available in the literature.

On the accuracy and fitting of transversely isotropic material models

Fiber reinforced structures are central to the form and function of biological tissues. Hyperelastic, transversely isotropic material models are used widely in the modeling and simulation of such tissues. Many of the most widely used models involve strain energy functions that include one or both pseudo-invariants (I4 or I5) to incorporate energy stored in the fibers. In a previous study we showed that both of these invariants must be included in the strain energy function if the material model is to reduce correctly to the well-known framework of transversely isotropic linear elasticity in the limit of small deformations. Even with such a model, fitting of parameters is a challenge. Here, by evaluating the relative roles of I4 and I5 in the responses to simple loadings, we identify loading scenarios in which previous models accounting for only one of these invariants can be expected to provide accurate estimation of material response, and identify mechanical tests that have special utility for fitting of transversely isotropic constitutive models. Results provide guidance for fitting of transversely isotropic constitutive models and for interpretation of the predictions of these models.

Exact solution of Eshelby–Christensen problem in gradient elasticity for composites with spherical inclusions

The gradient elasticity theory is employed to solve exactly the problem of Eshelby–Christensen for filled composites with spherical inclusions across length scales. Relying on the fundamental symmetry considerations and using Lagrange’s variational formalism, we derive the governing relations of linear isotropic gradient elasticity. We demonstrate that to avoid spurious solutions, one should necessarily impose some additional symmetry restrictions on the operational strain gradient elastic constants that can be considered as a new correctness condition. By enforcing the “strain gradient” symmetry condition, we offer the variant of the correct applied one-parametric gradient theory of interfacial layer model. To solve the Eshelby–Christensen problem, we employ the generalized Eshelby’s integral representations for the gradient elasticity models that allow to formulate the closing equations in a self-consistent three-phase method, and we also use the generalized Papkovich–Neuber representation to determine the general form of the gradient solution and the structure of the scale effects. As a result, we obtained for the first time an exact solution of Eshelby–Christensen problem for composites reinforced with spherical inclusions in framework of the gradient interfacial layer model. There are known analogs of fundamental results for gradient models related to closed solution for composites with spherical inclusions obtained by R.M. Christensen and K.H. Lo in 1976. The obtained analytical solution of Eshelby–Christensen problem for correct gradient theory is used to determine the stress–strain state and the effective properties of dispersed composites. The analysis of the effect of scale factors is given; the error associated with the use of gradient theories that do not obey the proposed condition of correctness is estimated.

Structure and stability of threading edge and screw dislocations in bulk GaN

Computational studies of a-type threading edge dislocations parallel to the [0001] direction in bulk GaN reveal three possible structures of their cores that are characterized as the 4-atom ring, the 8-atom ring and the 5/7-atom ring configurations. Similar calculations made on the c-type threading screw dislocation suggest the existence of a full core, although the precise structure is still a matter of debate. Most studies associate the position of the dislocation with only high-symmetry points in the lattice and do not discuss the core stability in non-equiatomic configurations. Here, we investigate how the core structures of a-type edge and c-type screw threading dislocations depend on the position in the (0001) plane at which the dislocation is inserted perpendicular to this plane. Furthermore, by successively removing the high-energy atoms from the cores of these dislocations, we track the stability of each core in non-equiatomic configurations and identify the core structures of minimum energy. The edge dislocation is shown to prefer the 5/7-atom ring structure, while the core of the screw dislocation transforms into a new Ga-rich configuration with the associated energy drop of nearly 0.4eV per one Angström of the dislocation length. For all types of dislocation cores found in this paper, we employ isotropic linear elasticity to estimate their core radii and core energies.

Uniqueness and stability of Lamé parameters in elastography

This paper concerns an hybrid inverse problem involving elastic measurements called Transient Elastography (TE) which enables detection and characterization of tissue abnormalities. In this paper we assume that the displacements are modeled by linear isotropic elasticity system and the tissue displacement has been obtained by the rst step in hybrid methods. We reconstruct Lamé parameters of this system from knowledge of the tissue displacement. More precisely, we show that for a suciently large number of solutions of the elasticity system and for an open set of the well-chosen boundary conditions, Lamé parameters can be uniquely and stably reconstructed. e set of well-chosen boundary conditions is characterized in terms of appropriate complex geometrical optics solutions.

Reconstruction of constitutive parameters in isotropic linear elasticity from noisy full-field measurements

Within the framework of linear elasticity we assume the availability of internal full-field measurements of the continuum deformations of a non-homogeneous isotropic solid. The aim is the quantitative reconstruction of the associated moduli. A simple gradient system for the sought constitutive parameters is derived algebraically from the momentum equation, whose coefficients are expressed in terms of the measured displacement fields and their spatial derivatives. Direct integration of this system is discussed to finally demonstrate the inexpediency of such an approach when dealing with noisy data. Upon using polluted measurements, an alternative variational formulation is deployed to invert for the physical parameters. Analysis of this latter inversion procedure provides existence and uniqueness results while the reconstruction stability with respect to the measurements is investigated. As the inversion procedure requires differentiating the measurements twice, a numerical differentiation scheme based on an ad hoc regularization then allows an optimally stable reconstruction of the sought moduli. Numerical results are included to illustrate and assess the performance of the overall approach.

An Invariant Method of Fundamental Solutions for the Cauchy Problem in Two-Dimensional Isotropic Linear Elasticity

In this paper, we propose a numerical algorithm based on the method of fundamental solutions (MFS) for the Cauchy problem in two-dimensional linear elasticity. Through the use of the double-layer potential function, we give the invariance property for a problem with two different descriptions. In order to adapt this invariance property, we give an invariant MFS to satisfy this invariance property, i.e., formulate the MFS with an added constant and an additional constraint. The method is combining the Newton method and classical Tikhonov regularization with Morozov discrepancy principle to solve the inverse Cauchy problem. Some examples are given for numerical verification on the efficiency of the proposed method. The numerical convergence, accuracy, and stability of the method with respect to the the number of source points and the distance between the pseudo-boundary and the real boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are also analysed with some examples.

ANALYSIS OF 3D PLANAR CRACKS WITH CONSIDERATION OF SURFACE STRESS EFFECTS

An efficient numerical procedure for modeling planar cracks in three-dimensional, linear elastic, infinite media accounting for the surface stress effect is presented in this paper. The concept of surface stresses, which has been widely employed in the modeling of nano-scale problems, is considered in the present study to derive a suitable mathematical model capable of simulating nano-sized cracks. An infinitesimally thin layer of material on the crack surface is modeled by a surface with zero-thickness and perfectly adhered to the bulk material, with its behavior governed by the Gurtin-Murdoch constitutive law. In the formulation, the classical theory of isotropic linear elasticity is utilized to establish the governing equation of the bulk material in terms of completely regularized boundary integral equations for the displacements and tractions on the crack surface. For the zero-thickness layer, the final governing equation incorporating the surface stress effect is obtained in a weak form following the standard weighted residual technique. Solutions of the fully coupled system of equations are then obtained by the FEM-SGBEM coupling numerical procedure. Owing to the weakly singular feature of all involved boundary integral equations, standard C0 interpolation functions are used everywhere in the approximation of crack-face data and only special quadrature for evaluating nearly singular and weakly singular integrals is required. Once the implemented numerical scheme is validated with available benchmark solutions, it is applied to investigate the nano-scale influence of nano-sized cracks. Results from an extensive parametric study reveal that, the presence of surface stresses not only increases the near-surface material stiffness but also introduces size-dependent behavior of predicted solutions and the reduction of stresses in the region ahead of the crack front

Cross slip of dislocation loops in GaN under shear

AbstractThis work explores possible cross‐slip mechanisms of gliding type <a > = a /3[1 –2 1 0] dislocation loops in wurtzite gallium nitride (GaN) as a function of slip plane. A modified form of the dislocation dynamics code ParaDiS was employed using isotropic linear elasticity and dislocation mobilities estimated in part from molecular dynamics (MD) simulations. Under an externally applied uniform stress, the occurrence of cross slip events is highly dependent on the initial dislocation slip plane. The basal plane is the preferred active plane, owing to the greater mobility of <a > type segments on that plane, over the other planes considered including the prismatic (–1 0 1 0) and two equivalent pyramidal planes (–1 0 1 1) and (1 0 –1 1). For an applied stress state, cross slip processes are more readily seen from the prismatic‐to‐basal planes or the pyramidal‐to‐basal planes, and neither is found to occur in reverse. Cross slip by climb is not presently considered. In all cases, cross‐slip events occur after the loop expands until a greater number of screw‐oriented segments are able to access the cross slip plane and the resolved stresses on the plane become sufficiently large. In comparison to dislocations found in GaN previously, the calculations suggest that some threading dislocations along the [0001] direction that have edge character may have been formed from loops whose screw segments slip and escape on basal planes leaving only the edge segments.

Thermodynamic aspects of Cottrell atmospheres

Using open system elastic constants, we can convert any of the many existing solved small-strain problems of stress fields of dislocation configurations to what the stresses and solute distribution equilibrium would be in alloys. We confine ourselves to small strain linear isotropic elasticity in which the solute atoms are centres of pure dilation and compute the inhomogeneous compositional distributions that are called Cottrell atmospheres, as well as the energy changes due to the segregation. We correct the misconception in the literature that the solute distribution does not affect the stress field of the dislocation and show that the atmosphere around an edge dislocation acts like an edge dislocation with a Burgers vector of opposite sign.

Reconstruction of elasticity: a stochastic model-based approach in ultrasound elastography

BackgroundThe convectional strain-based algorithm has been widely utilized in clinical practice. It can only provide the information of relative information of tissue stiffness. However, the exact information of tissue stiffness should be valuable for clinical diagnosis and treatment.MethodsIn this study we propose a reconstruction strategy to recover the mechanical properties of the tissue. After the discrepancies between the biomechanical model and data are modeled as the process noise, and the biomechanical model constraint is transformed into a state space representation the reconstruction of elasticity can be accomplished through one filtering identification process, which is to recursively estimate the material properties and kinematic functions from ultrasound data according to the minimum mean square error (MMSE) criteria. In the implementation of this model-based algorithm, the linear isotropic elasticity is adopted as the biomechanical constraint. The estimation of kinematic functions (i.e., the full displacement and velocity field), and the distribution of Young’s modulus are computed simultaneously through an extended Kalman filter (EKF).ResultsIn the following experiments the accuracy and robustness of this filtering framework is first evaluated on synthetic data in controlled conditions, and the performance of this framework is then evaluated in the real data collected from elastography phantom and patients using the ultrasound system. Quantitative analysis verifies that strain fields estimated by our filtering strategy are more closer to the ground truth. The distribution of Young’s modulus is also well estimated. Further, the effects of measurement noise and process noise have been investigated as well.ConclusionsThe advantage of this model-based algorithm over the conventional strain-based algorithm is its potential of providing the distribution of elasticity under a proper biomechanical model constraint. We address the model-data discrepancy and measurement noise by introducing process noise and measurement noise in our framework, and then the absolute values of Young’s modulus are estimated through the EFK in the MMSE sense. However, the initial conditions, and the mesh strategy will affect the performance, i.e., the convergence rate, and computational cost, etc.

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lame parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners.

Solvability of boundary singular integral operators of elasticity on surfaces with conic points

A system of boundary singular integral equations of the linear isotropic elasticity is studied in the case where the double layer potential is generated by the stress operator. These equations are considered on a bounded two-dimensional surface without boundary which is smooth outside a finite number of conical points. The solvability of the system is proved in various weighted spaces of differentiable functions. We obtain a representation of the inverse operator of the system in terms of the inverse operators of some boundary value problems. We also obtain pointwise estimates for the kernel of this operator and “quasilocal” estimates for solutions of the integral equations in question. Bibliography: 12 titles.

Characteristics of the Elastic Response of Isotropic Linear Micropolar Solids under Imposed Loads

This article proposes a simplified way to solve solid mechanic problems in micropolar elasticity using the solution found in the classic theory of elasticity as a starting point. In this study, an analysis of the linear isotropic micropolar elasticity is conducted based on the properties imposed on the micropolar medium by the constitutive and equilibrium equations. To ascertain how the micropolar medium responses deviate from Hooke’s theory of elasticity, different loading conditions were classified. Three cases have been found so far: the rotational couple response, the quasi-classic equilibrium of momentum response and the general case. This study is the first in a series planned to explore the use of commercial packages of finite element in order to solve micropolar elasticity problems.

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lame equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lame equations. The system of Lame equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lame equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.

Estimates for kernels of inverse operators of integral equations of elasticity on surfaces with conic points

Boundary integral equations of linear isotropic elasticity, with the double layer potential generated by the preudostress operator, are considered on surfaces with a finite number of conic points. Representations for solutions are obtained in terms of the inverse operators of the Dirichlet and Neumann problems in the interior and exterior of the surface. Pointwise estimates for kernels of inverse operators and their derivatives of any order are derived with the help of estimates for fundamental solutions of those boundary value problems. The Laplace operator is contained here as a special case. Bibliography: 15 titles.

Invertibility of boundary integral operators of elasticity on surfaces with conic points

The system of boundary integral equations of linear isotropic elasticity, with the double layer potential generated by the preudostress operator, is considered on surfaces with a finite number of conic points. The solvability of the system is established in various function spaces. Representation for the inverse operator of the system in question is obtained in terms of inverse operators of some boundary value problems. Pointwise estimates for the kernel of the inverse operator of the system and their derivatives of any order are derived together with “quasilocal” estimates for solutions of the integral equations. The Laplace operator is contained here as a special case. Bibliography: 6 titles.

Some remarks on the range of Poisson's ratio in isotropic linear elasticity

We study the range of Poisson's ratio ν in the theory of isotropic linear elasticity. Using the ratio of the Lamé constants χ = λ/μ, we defined the parametrized Poisson's ratio as ν = ν(χ) and found that it represents a standard hyperbola with center (χ, ν) = (−1/3, 1/2) and eccentricity . One of the focal points of the hyperbola represents a Born instability criterion (or the equivalent strong ellipticity condition) and the other determines the sign of ν. The hyperbola has a vertex point in the admissible range of ν, and, hence, it naturally divides the range into the two subranges: and . We also found that there are three equivalent formulas among the nine elastic-constant ratios χ i considered in this study. The geometric conjugate of Poisson's ratio, which is defined as ν* ≔ 1 − ν, has also been proposed.

Closed-form solution for a coated circular inclusion under uniaxial tension

A coated circular inclusion embedded in an infinite matrix is analyzed in the framework of two-dimensional isotropic linear elasticity. A closed-form solution is obtained for the case of far-field uniaxial tension using Muskelishvilis complex potential method. The solutions for the stress and strain distributions for all three regions, that is, matrix, coating, and inclusion, were obtained for various coating-to-matrix shear modulus ratios, while keeping the fiber and matrix shear moduli the same. Test cases for an inclusion without the coating and hollow inclusion were also studied. The energy release rate was evaluated using the path-independent M-integral, which is used to calculate the energy release rate for the self-similar expansion of defects surrounded by the closed contour of the integral. The results for the stress and strain concentrations along with the energy release rate due to this material inhomogeneity were analyzed to yield a better understanding of the mechanics of materials with circular inclusions. This can be helpful in designing intelligent composite structures with embedded optical fiber sensors.

A shear deformation micro-plate model based on the most general form of strain gradient elasticity

In this study, a first order shear deformation micro-plate model based on strain gradient elasticity theory is developed. The most general form of Mindlin's linear isotropic strain gradient elasticity is employed and a general micro-plate formulation is obtained. The governing equations of motion and all possible boundary conditions are determined using the variational method. For some specific values of the gradient-based material parameters, the general plate formulation can be reduced to those based on simple forms of the strain gradient elasticity theory. Accordingly, a simple form of the plate formulation is introduced. To illustrate the behavior of a micro-plate predicted by the new plate formulation, the static bending and free vibration problems of a simply supported micro plate are investigated. Numerical results reveal that the plate defection decreases and natural frequencies increase remarkably when the plate thickness becomes comparable to its material length scale parameter. These size effects decrease or even diminish as the thickness of the plate is far greater than the material length scale parameter. It was also observed that the effects of shear deformation are more sensible at micro scales.