Linear Elastic Systems Research Articles

Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

Abstract We extend the Calderón–Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations ℒ A s ⁢ u = f {\mathcal{L}^{s}_{A}u=f} , where the operator ℒ A s {\mathcal{L}^{s}_{A}} is formally given by ℒ A s ⁢ u = ∫ ℝ n A ⁢ ( x , y ) | x - y | n + 2 ⁢ s ⁢ ( x - y ) ⊗ ( x - y ) | x - y | 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ 𝑑 y . \mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-% y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For 0 < s < 1 {0<s<1} and A : ℝ n × ℝ n → ℝ {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{R}} taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier–Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if A ⁢ ( ⋅ , y ) {A(\,\cdot\,,y)} is uniformly Holder continuous and inf x ∈ ℝ n ⁡ A ⁢ ( x , x ) > 0 {\inf_{x\in\mathbb{R}^{n}}A(x,x)>0} , then for f ∈ L loc p {f\in L^{p}_{\rm loc}} , for p ≥ 2 {p\geq 2} , the solution vector u ∈ H loc 2 ⁢ s - δ , p {u\in H^{2s-\delta,p}_{\rm loc}} for some δ ∈ ( 0 , s ) {\delta\in(0,s)} .

Improved State-Space Approach Based on Lumped Mass Matrix for Transient Analysis of Large-Scale Locally Nonlinear Structures

Due to the assumption of acceleration variation in traditional step-by-step integration methods such as Newmark, sufficiently small time steps are required to ensure numerical stability and accuracy in dynamic systems. In contrast, the state-space approach, based on piecewise interpolation of discrete load functions, does not rely on predetermined acceleration assumptions and has demonstrated high efficiency in terms of stability and accuracy. The original state-space method requires the calculation of the inverse of the structural mass in the transition matrix. However, when a lumped mass matrix is used, this computation renders the entire mass matrix singular, resulting in an invalid solution expression. To address this issue, this study proposes an improved state-space approach for the transient analysis of large-scale structural systems with local nonlinearities. In this approach, a nonlinear force corrector is introduced as an external force term applied to the linear elastic system to account for the nonlinear behavior of locally yielding components. Consequently, the original nonlinear dynamic system can be transformed into an equivalent linear elastic transient system. Furthermore, based on the lumped mass matrix, a first-order ordinary differential state-space equation for such an equivalent linear elastic transient system is derived. Simulation results from three transient system examples show that the state-space approach outperforms the Newmark method in terms of accuracy and stability for dynamic systems characterized by high frequency and low damping. The prediction results show that the state-space approach appears to be insignificantly affected by the choice of the consistent or lumped mass matrix. The numerical results show that the root-mean-square errors between the consistent and lumped matrices in the top displacement time histories of a 15-storey plane frame under various seismic intensities are all less than 1%, and in the base reaction time histories responses the discrepancies are only about 0.5%, indicating that the use of lumped mass matrices is quite reliable. When many nodes or degrees of freedom have no assigned mass, the dimensionality of the state-space equation can be significantly reduced using the lumped mass approach. Therefore, the simulation of large-scale systems can be simplified by employing the improved state-space approach with lumped mass matrices, yielding results nearly identical to those obtained using traditional methods. In conclusion, the improved state-space approach has great potential for the simulation of transient behavior in large-scale systems with local nonlinearities.

Discrete fracture‐matrix model of poroelasticity

AbstractThe paper presents the derivation and analysis of a poroelasticity model in a domain with fracture represented by a codimension‐one manifold. The system of saturated flow and linear elasticity both in the rock matrix and in the fracture coupled through an appropriate interface condition is obtained from a continuum description by integration and semi‐discretization in the normal direction of the fracture. The existence and uniqueness of a weak solution are proved with the help of an iterative splitting, whose convergence rate is analyzed. The analysis is complemented by a numerical example.

Probability distributions for dynamic and extreme responses of linear elastic structures under quasi-stationary harmonizable loads

The non-stationary load models based on the evolutionary power spectral density (EPSD) may lead to ambiguous structural responses. Quasi-stationary harmonizable processes with non-negative Wigner-Ville spectra are suitable for modeling non-stationary loads and analyzing their induced structural responses. In this study, random environmental loads are modeled as quasi-stationary harmonizable processes. The Loève spectrum of a harmonizable load process contains several random physical parameters. An explicit approach to calculate the probability distributions for the dynamic and extreme responses of a linear elastic structure subjected to a quasi-stationary harmonizable load is proposed. Conditioned on the specific values of the load spectral parameters, the harmonizable load process is assumed to be Gaussian. The conditional joint probability density function (PDF) of structural dynamic responses at any finite time instants and the conditional cumulative distribution function (CDF) of the structural extreme response are provided. By multiplying these two conditional probability distributions with the joint PDF of the load spectral parameters, and then integrating these two products over the parameter sample space, the joint PDF of structural dynamic responses at any finite time instants and the CDF of the structural extreme response can be calculated. The efficacy of the proposed approach is numerically validated using two linear elastic systems, which are subjected to non-stationary and non-Gaussian wind and seismic loads, respectively. The merit of the harmonizable load process model is highlighted through a comparative analysis with the EPSD load model.

Reiterated homogenization of elliptic systems with Neumann boundary conditions

This paper is devoted to the quantitative estimates in reiterated homogenization of second‐order elliptic systems of linear elasticity, where the coefficients oscillate on multiple separated scales. We establish the uniform boundary Lipschitz estimate and the uniform estimate for the problem. The convergence rate in the space is also obtained as a by‐product.

Asymptotic behavior of the linear elasticity system with varying and unbounded coefficients in a thin beam

We study the asymptotic behavior of the solutions of the linear elasticity system in a thin beam of small thickness ε>0. The elasticity tensor also varies with ε, and it is assumed to be uniformly elliptic but non-uniformly bounded. Namely, we just impose that its norm in L∞ is an infinitesimal of 1/ε and its norm in L1 is bounded. We obtain an homogenized problem corresponding to a linear system in one dimension. It gives an approximation of the solution of the problem in the thin beam which consists in the sum of a Bernouilli-Navier's deformation plus a torsion term. This limit system provides a general asymptotic model for the behavior of an elastic beam composed by the mixture, at a mesoscopic level, of several materials, and therefore, which does not satisfy any homogeneity and/or isotropy conditions.

Structural pounding of three adjacent multi degree of freedom system under subsurface blast and seismic action by considering and ignoring the effect of soil structure interaction

A lack of spacing between the adjacent structures is mostly observed in metropolitan cities due to more land cost, which has led to more serious problem called as structural pounding in many recent earthquakes. Large impact forces and short acceleration pulses are mostly noticed during the structural pounding, and which is hazardous to structural health of the buildings involved in this phenomena. In the present paper three adjacent multi-degree-of-freedom linear elastic systems subjected to seismic and subsurface blast actions are analyzed by considering and ignoring the effect of soil structure interactions. Three dimensional buildings are analyzed and designed for deciding the column and beam sizes in SAP 2000 NL software considering all possible load combinations of dead load, live load and earthquake load as per IS 1893:2016 Part I. The subsurface indirect method is used for studying the effect of soil structure interaction. The SSI spring stiffness’s and damping constants are calculated as per FEMA 356. These buildings are converted into the lumped mass model as per the principles of dynamics of structures. The entire analysis of three adjacent lumped mass models with insufficient separation is carried out in time domain form using MATLAB programming. The outputs are measured in the forms of top storey displacement time history plots, top floor level pounding force time history plots and base shear force time history plots. From the outputs, it is noticed that the SSI amplifies the displacement response of building than that of fixed base system. SSI also overrates the pounding response of structure. Blast force response is much smoothens than seismic response of structure.

Modal analysis-based calculation of periodic nonlinear responses of harmonically forced piecewise linear elastic systems

This paper introduces a new solution method based on modal analysis to calculate periodic nonlinear response to harmonic forcing. The method can be used for piecewise linear (PL) elastic multi-degree-of-freedom (MDOF) systems with two linear states, provided that the unloaded equilibrium position of both states coincides, and the system is in both states once during a period of forcing. The method formulates a linear system of equations, which yields the initial conditions of the periodic response as a function of the time spent in each state and the forcing phase. Error functions are constructed to secure the switching between the states of the system at the given time instances and the nonlinear equations are solved for the candidates of periodic responses with the scanning of the parameter space. Finally, the vibrations with unwanted switches are filtered out to obtain the actual periodic responses. An example with support vibration shows that the proposed method is capable to find disconnected branches of the periodic responses to a harmonic excitation. As the dimension of the scanning space is independent of the discretization, the method scales almost linearly with the increase of the number of degrees-of-freedom.

Controllability of the linear elasticity as a first-order system using a stabilized space-time mixed formulation

<p style='text-indent:20px;'>The aim of this paper is to study the boundary controllability of the linear elasticity system as a first-order system in both space and time. Using the observability inequality known for the usual second-order elasticity system, we deduce an equivalent observability inequality for the associated first-order system. Then, the control of minimal <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm can be found as the solution to a space-time mixed formulation. This first-order framework is particularly interesting from a numerical perspective since it is possible to solve the space-time mixed formulation using only piecewise linear <inline-formula><tex-math id="M2">\begin{document}$ C^0 $\end{document}</tex-math></inline-formula>-finite elements. Numerical simulations illustrate the theoretical results.</p>

Asymptotic analysis of stretching modes for a folded plate

<abstract><p>In this paper, we show that the spectral problem associated to stretching modes in a thin folded plate can be derived from the three-dimensional eigenvalue problem of linear elasticity through a rigourous convergence analysis as the thickness of the plate goes to zero. We show, using a nonstandard asymptotic analysis technique, that each stretching frequency of an elastic thin folded plate is the limit of a family of high frequencies of the three-dimensional linearized elasticity system in the folded plate, as the thickness approaches zero.</p></abstract>

Singularities of the stress concentration in the presence of 𝐶^{1,𝛼}-inclusions with core-shell geometry

In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lamé systems of linear elasticity, may exhibit the singularities with respect to the distance ε \varepsilon between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with C 1 , α C^{1,\alpha } boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance ε \varepsilon between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates.

Pre-strains and buckling in mechanosensitivity of contractile cells and focal adhesions: A tensegrity model

We demonstrate that several key aspects of the contractile activity of a cell interacting with the substrate can be captured by means of a non linear elastic tensegrity mechanical system made of a tensile element in parallel with a buckling-prone component, and exchanging forces with the surroundings through an extracellular matrix-focal adhesion complex. Mechanosensitivity of the focal adhesion plaque is triggered by pre-strain-driven buckling of the system induced either by pre-contraction or pre-polymerization of the constituents. The impact of pre-polymerization on the mechanical force and the implications of using linear and nonlinear elasticity for the focal adhesion plaque are assessed.

Energy-Based Approach: Analysis of a Laterally Loaded Pile in Multi-Layered Non-Linear Elastic Soil Strata

Several studies have been reported in published literature on analytical solutions for a laterally loaded pile installed in a homogeneous single soil layer. However, piles are rarely installed in an ideal homogeneous single soil layer. The present study describes a new continuum-based analysis or energy-based approach for predicting the pile displacement responses subjected to static lateral loads and moments considering the soil non-linearity. This analytical analysis treats the pile as an elastic Euler–Bernoulli beam and the soil as a three-dimensional (3D) continuum in which the non-linear elastic properties are described by a modulus degradation relationship. The principle of virtual work was applied to the energy equation of a pile–soil system in order to obtain the governing differential equation for the pile and soil displacements. An iterative procedure was adopted to solve the equations numerically using a finite difference method (FDM). The pile displacement response was obtained using the software MATLAB R2021a, and the results from the energy-based method were compared with those obtained from the field test data as well as the finite element analysis (FEA) based on the software ANSYS Workbench 2021R1. The present study investigated the effect of explicit incorporation of soil properties and layering through a parametric study in order to understand the importance of predicting appropriate pile displacement responses in a linear elastic soil system. The responses indicated that the effect of soil layers and their thicknesses, pile properties and the variation in soil moduli have a direct impact on the displacements of piles subjected to lateral loading. Hence, a proper emphasis has to be given to account for the soil non-linearity. Considering the effect of soil non-linearity, it is observed that the results obtained from the energy-based method agreed well with the field measured values and those obtained from the FEA. The results indicated a difference of approximately less than 7% between the proposed method and the FEA. The approach presented in this study can be further extended to piles embedded in multi-layered soil strata subjected to the combined action of axial loads, lateral loads and moments. Furthermore, the same approach can be extended to study the response of the soil to group piles.

Energy-Based Approach: Analysis of a Vertically Loaded Pile in Multi-Layered Non-Linear Soil Strata

Numerous studies have been reported in the published literature on analytical solutions for a vertically loaded pile installed in a homogeneous single soil layer. However, piles are rarely installed in an ideal homogeneous single soil layer. This study presents an analytical model based on the energy-based approach to obtain displacements in an axially loaded pile embedded in multi-layered soil considering soil non-linearity. The developed analytical model incorporating Euler-Bernoulli beam theory proved to be an effective way in estimating the load-displacement responses of piles embedded in multi-layered non-linear elastic soil strata. The differential equations are solved analytically and numerically using the variational principle of mechanics. A parametric study investigated the effect of explicit incorporation of soil properties and layering in order to understand the importance of predicting appropriate pile displacement responses in linear elastic soil system. It is clear from the results that the analyses which consider the soil as a single homogeneous layer will not be able to produce an accurate estimation of the pile stiffnesses. Therefore, it is highly important to account for the effect of soil layering and the non-linear response. The pile displacement response is obtained using the software MATLAB R2019a and the results from the energy-based method are compared with those obtained from the field test data as well as the Finite Element Analysis (FEA) based on the software ANSYS 2019R3. The non-linear elastic constitutive relationship which described the variation of secant shear modulus with strain through a power law has shown reasonably accurate predictions when compared to the published field test data and the FEA. The developed mathematical framework is also more computationally efficient than the three-dimensional (3D) FEA.

Maximum dynamic response of linear elastic SDOF systems based on an evolutionary spectral model for thunderstorm outflows

The study aims to estimate the maximum dynamic response of linear elastic SDOF systems subjected to thunderstorm outflows. Starting from a recently developed Evolutionary Power Spectral Density (EPSD) model for the wind velocity, the dynamic response is decomposed into a time-varying mean and a non-stationary random fluctuation. The EPSD and the Non-Geometrical Spectral Moments (NGSMs) of the random fluctuation are derived both accounting and neglecting the transient dynamics due to the modulating function of the load. The mean value of the maximum nonstationary fluctuating component of the response is estimated based on the definition of an equivalent stationary process following an approach proposed in the literature. In order to mitigate the overestimations of the maximum dynamic response due to the Poisson approximation, analogously to the formulation developed by Der Kiureghian for withe noise excitation, an equivalent expected frequency is introduced for thunderstorm excitation. Finally, the maximum dynamic response to thunderstorms is estimated as the sum of the maximum mean and fluctuating parts and a numerical validation of the results against real recorded thunderstorms is provided, highlighting the reliability of adding up the mean and fluctuating contributions and the advantages and limits of neglecting the transient dynamics.

Dynamic recoil in metamaterials with nonlinear interactions

Biological systems can generate motions with high acceleration and velocity via dynamic recoil. Traditionally, recoil dynamics in unstructured linear elastic systems are based on inertial and elastic properties, with the recoil velocity determined by the materials' speed of sound and the strain at recoil. Metamaterials with designed elastic structures and nonlinear interactions (e.g., magnetic interactions) have recently demonstrated the capability to control impulsive motions and manage high-rate energy transfer events. However, a predictive model for understanding the kinematics and recoil dynamics has not been developed. Here, we study the dynamic recoil that couples with a general nonlinear power-law interaction within a metamaterial framework. Starting with a 1D discrete model, we demonstrate how the nonlinear force interactions and the elastic structures in metamaterials control the dynamic performance. Under a weakly nonlinear condition, the governing equation of the dynamic recoil is described by the KdV (Korteweg-de Vries) equation and under the condition of strong nonlinearity by a more general strongly nonlinear wave equation. Our model predicts a “sonic vacuum” response in the recoiling metamaterials, where a near-zero speed of sound is induced by strongly nonlinear interactions and zero initial prestress. We propose a sequential wave motion during the dynamic recoil, where a linear wave propagates at a large stretch and switches to a nonlinear one below a critical stretch. We further discover that the nonlinear wave speed has a non-monotonic dependence upon the strength of the power-law interaction. We find good agreements in the recoil velocity by comparing our analytical model to experiments of the recoiling metamaterials with magnetic interactions. Our work demonstrates how power-law interactions within metamaterials can program dynamic recoil dynamics, setting the stage for new materials designs for energy management and conversion in high-rate applications.

Periodically alternated elastic support induced topological phase transition in phononic crystal beam systems

Determining strong robustness in flexural wave propagation in beam systems has always been a promising achievement with various real-life applications such as energy harvesting and waveguiding. In this manuscript, the bending characteristics of a phononic crystal beam on periodically alternated linear elastic support systems with topologically protected wave propagation are proposed. Based on the Timoshenko beam theory, general solutions of this periodic beam-foundation topological system are obtained. Subsequently, by applying the Bloch theory, the dispersion relation is obtained using the transfer matrix method. Comparing with finite element numerical solutions, excellent agreement is observed with only a minor difference due to the assumptions in the Timoshenko beam theory. Generally, any breaking of spatial symmetry in beam systems can be achieved by changing geometry of beam cross-sections or tuning the material properties of substructures. We herein propose a new approach to break the spatial symmetry, i.e., tuning the elastic stiffnesses of the periodically alternated elastic supports. It is found that beam bending on an infinite continuously distributed elastic foundation leads to a band-crossing point. After tuning the elastic support stiffness in a periodic manner, the band-crossing point is broken and a new bandgap appears due to the breaking of structural symmetry. Further, a mode shape inversion and Zak phase transition are captured. Based on a unit cell analysis, we examine the topologically protected interface modes in an array constructed with periodically arranged unit cells with two corresponding phase states. Furthermore, several numerical examples are used to demonstrate the defect- and disorder-immune properties in this one-dimensional topological system. This new proposed general mechanism can be extended to establish topological phase transitions in other mechanical and dynamic systems.

The topological ligament in shape optimization: a connection with thin tubular inhomogeneities

In this article, we propose a formal method for evaluating the asymptotic behavior of a shape functional when a thin tubular is added between two distant regions of the boundary of a domain. In the contexts of the conductivity equation and the linear elasticity system, we relate this issue to a perhaps more classical problem of thin tubular inhomogeneities: we analyze the solutions to versions of the physical partial differential equations which are posed inside a fixed background medium, and whose material coefficients are altered inside a tube with vanishing thickness. Our main contribution from the theoretical point of view is to propose a heuristic energy argument to calculate the limiting behavior of these solutions with a minimum amount of effort. We retrieve known formulas when they are available, and we manage to treat situations which are, to the best of our knowledge, not reported in the literature (including the setting of the 3d linear elasticity system). From the numerical point of view, we propose three different applications of the formal ligament approach derived from these expansions. At first, it is an original way to account for variations of a domain, and it thereby provides a new type of sensitivity for a shape functional, to be used concurrently with more classical shape and topological derivatives in optimal design frameworks. Besides, it suggests new, interesting algorithms for the design of the scaffold structure sustaining a shape during its fabrication by a 3d printing technique, and for the design of truss-like structures. Several numerical examples are presented in two and three space dimensions to appraise the efficiency of these methods.

On the linearized system of elasticity in the half-space

<abstract><p>The purpose of this paper is twofold. The first goal is to provide a simple and constructive proof of Korn inequalities in half-space with weighted norms. The proof leads to explicit values of the constants. The second objective is to use these inequalities to show that the linear elasticity system in half-space admits a coercive variational formulation. This formulation corresponds to the physical case in which the solution is evanescent at infinity.</p></abstract>

On singularities of solution of the elasticity system in a bounded domain with angular corner points

This paper aims to give a mathematically rigorous description of the corner singularities of the weak solutions for the plane linearized elasticity system in a bounded planar domain with angular corner points on the boundary. The qualitative properties of the solution including its regularity depend crucially on these corner points or such types of boundary conditions. In particular, the resulting expansion of the solutions of the underlying problem involves singular vector functions, inlines, depending on a certain parameter ξμ. We derive the transcendental equations for all 10 possible cases of combinations of the boundary conditions generated by the basic four ones in classical elasticity proposing in the two natural directions of the boundary, that is, tangential and normal direction, respectively, which depends on ξμ. So a MATLAB program is developed whereby ξμ can be computed, and figures showing their distributions are presented. The leading singular exponents are computed for these combinations of the boundary conditions, wherein critical angles ωcritical are listed such that for interior angles ω < ωcritical the H2‐regularity of solution can be guaranteed. Moreover, the characterization of stress singularities in terms of the inner angle of a corner point is studied, and the regularity results are given.