Linear Elasticity Research Articles

Acquiring elastic properties of thin composite structure from vibrational testing data

Abstract The paper is devoted to a problem of acquiring elastic properties of a composite material from the vibration testing data with a simplified experimental acquisition scheme. The specimen is considered to abide by the linear elasticity laws and subject to viscoelastic damping. The boundary value problem for transverse movement of such a specimen in the frequency domain is formulated and solved with finite-element method. The correction method is suggested for the finite element matrices to account for the mass of the accelerometer. The problem of acquiring the elastic parameters is then formulated as a nonlinear least-square optimization problem. The usage of the automatic differentiation technique for stable and efficient computation of the gradient and hessian allows to use well-studied first and second order local optimization methods. We also explore the possibility of generating initial guesses for local minimization by heuristic global methods. The results of the numerical experiments on simulated data are analyzed in order to provide insights for the following real life experiments.

Discovering interpretable physical models using symbolic regression and discrete exterior calculus

Computational modeling is a key resource to gather insight into physical systems in modern scientific research and engineering. While access to large amount of data has fueled the use of machine learning to recover physical models from experiments and increase the accuracy of physical simulations, purely data-driven models have limited generalization and interpretability. To overcome these limitations, we propose a framework that combines symbolic regression (SR) and discrete exterior calculus (DEC) for the automated discovery of physical models starting from experimental data. Since these models consist of mathematical expressions, they are interpretable and amenable to analysis, and the use of a natural, general-purpose discrete mathematical language for physics favors generalization with limited input data. Importantly, DEC provides building blocks for the discrete analog of field theories, which are beyond the state-of-the-art applications of SR to physical problems. Further, we show that DEC allows to implement a strongly-typed SR procedure that guarantees the mathematical consistency of the recovered models and reduces the search space of symbolic expressions. Finally, we prove the effectiveness of our methodology by re-discovering three models of continuum physics from synthetic experimental data: Poisson equation, the Euler’s elastica and the equations of linear elasticity. Thanks to their general-purpose nature, the methods developed in this paper may be applied to diverse contexts of physical modeling.

The biomechanical effects of different membrane layer structures and material constitutive modeling on patient-specific cerebral aneurysms.

The prevention, control and treatment of cerebral aneurysm (CA) has become a common concern of human society, and by simulating the biomechanical environment of CA using finite element analysis (FEA), the risk of aneurysm rupture can be predicted and evaluated. The target models of the current study are mainly idealized single-layer linear elastic cerebral aneurysm models, which do not take into account the effects of the vessel wall structure, material constitution, and structure of the real CA model on the mechanical parameters. This study proposes a reconstruction method for patient-specific trilaminar CA structural modeling. Using two-way fluid-structure interaction (FSI), we comparatively analyzed the effects of the differences between linear and hyperelastic materials and three-layer and single-layer membrane structures on various hemodynamic parameters of the CA model. It was found that the numerical effects of the different CA membrane structures and material constitution on the stresses and wall deformations were obvious, but does not affect the change in its distribution pattern and had little effect on the blood flow patterns. For the same material constitution, the stress of the three-layer membrane structure were more than 10.1% larger than that of the single-layer membrane structure. For the same membrane structure, the stress of the hyperelastic material were more than 5.4% larger than that of the linear elastic material, and the displacement of the hyperelastic material is smaller than that of the linear elastic material by about 20%. And the maximum value of stress occurred in the media, and the maximum displacement occurred in the intima. In addition, the upper region of the tumor is the maximum rupture risk region for CA, and the neck of the tumor and the bifurcation of the artery are also the sub-rupture risk regions to focus on. This study can provide data support for the selection of model materials for CA simulation and analysis, as well as a theoretical basis for clinical studies and subsequent research methods.

Comprehensive validation of three-dimensional finite element modelling of wheel-rail high-frequency interaction via the V-Track test rig

Wheel-rail high-frequency interaction is closely related to the formation of railway short-wave defects. Finite element (FE) method has been widely used to simulate wheel-rail dynamic systems, but its validity in modelling high-frequency interaction has not been fully demonstrated in three dimensions (3D). This work aims at comprehensively validating the 3D FE modelling of wheel-rail high-frequency interaction using a downscale V-Track test rig. First, the FE model of the V-Track is developed that comprehensively includes the 3D track elasticity. The simulated track dynamic behaviours are validated against hammer tests, and the major vibration modes are analyzed employing modal analysis. Afterwards, the simulate wheel-rail dynamic responses are comprehensively compared with measurement results up to 10 kHz. Their characteristic frequencies are identified and correlated to the eigenmodes of the vehicle-track system. The results indicate that the proposed 3D FE model is capable of comprehensively and accurately simulating the 3D track dynamics and wheel-rail dynamic interaction of the V-Track up to 10 kHz. Rail vibrations dominate the wheel-rail dynamic contact within 10 kHz, while the wheel vibrations play an increasingly important role at higher frequencies and become decisive near the wheel eigenmode frequencies. The V-Track overall achieves dynamic similarity to the real vehicle-track system.

Analysis of axisymmetric hollow cylinder under surface loading using variational principle

In this work, a variational principle–based approach has been adopted to analyze one of the classical linear elasticity problem of the axisymmetric cylinder under surface loading. The use of variational principle results in a set of governing partial differential equations with associated boundary conditions. The equations have been solved using the separation of variable approach and the Frobenius method. A general solution has been derived and used to solve two test cases. The proposed solution is capable of meeting all the boundary conditions. The solution has been validated by comparing it with a finite element–based numerical solution and considering a special limiting condition of a solid cylinder, for which results are available in the literature. Further various studies have been carried out to understand the robustness and limitation of the presented solution.

The Spectral Decomposition of the Continuous and Discrete Linear Elasticity Operators with Sliding Boundary Conditions

The Spectral Decomposition of the Continuous and Discrete Linear Elasticity Operators with Sliding Boundary Conditions

Quantitative stability estimate for the inverse coefficients problem in linear elasticity

In this article we consider the inverse problem of reconstructing piece-wise Lamé coefficients from boundary measurements. We reformulate the inverse problem into a minimization one using a Kohn-Vogelius type functional. We study the stability of the parameters when the jump of the discontinuity is perturbed. Using tools of shape calculus, we give a quantitative stability result for local optimal solution. Dans cet article, nous considérons le problème inverse de reconstruction des coefficients de Lamé constants par morceaux à partir de mesures au bord. Nous reformulons le problème inverse en un problème de minimisation utilisant une fonctionnelle de type Kohn-Vogelius. Nous étudions la stabilité des paramètres lorsque le saut de la discontinuité est perturbé. En utilisant les outils du calcul de forme, nous donnons un résultat de stabilité quantitative pour une la solution optimale locale.

Multiscale simulation of polymer curing of composites combined mean-field homogenisation methods at large strains

This work introduces a coupled thermo-chemo-mechanical framework for simulating the polymerisation process of composites, accounting for large strains. The framework encompasses the microscale, mesoscale and, macroscale. The microscale represents a step-growth polymerisation mechanism, which is modelled as a system of multiple growing spheres. These spheres are further decomposed into concentric layers of solid, resin, and curing agent material. At the mesoscale, the composite is divided into inclusions and the polymer matrix. The macroscale is considered as a homogeneous material. In order to describe the behaviour of the heterogeneous scales, mean-field homogenisation methods are employed. These methods have been developed before for linear elastic materials. However, considering finite deformations, a special treatment is necessary to accurately account for the effects of such deformations. Effective properties at all scales are obtained using the Mori–Tanaka, Voigt, Reuss, and Voigt–Reuss–Hill average mean-field homogenisation method. The results demonstrate how the different homogenisation methods influence the curing process and, consequently, the macroscopic response of the composite material.

A quasi-meshfree method for constructing boundary-aware reproducing bases on geometrically complex domains using manifold geodesics

In many applications, physical domains are geometrically complex making it challenging to perform coarse-scale approximation. A defeaturing process is often used to simplify the domain in preparation for approximation and analysis at the coarse scale. Herein, a methodology is presented for constructing a coarse-scale reproducing basis on geometrically complex domains given an initial fine-scale mesh of the fully featured domain. The initial fine-scale mesh can be of poor quality and extremely refined. The construction of the basis functions begins with a coarse-scale covering of the domain and generation of weighting functions with local support. Manifold geodesics are used to define distances within the local support for general applicability to non-convex domains. Conventional moving least squares is used to construct the coarse-scale reproducing basis. Applications in quasi-interpolation and linear elasticity are presented.

Full weak Galerkin finite element discretizations for poroelasticity problems in the primal formulation

This paper develops novel finite element solvers for linear poroelasticity problems on quadrilateral meshes. These solvers are based on the primal formulations of linear elasticity and Darcy flow. Specifically, the fluid pressure and solid displacement are approximated by scalar- or vector-valued polynomials of degree k≥0 separately in element interiors and on edges. The discrete weak gradients of these shape functions are established in the broken (vector- or matrix-version) Arbogast–Correa spaces for approximations of the classical gradients in the variational forms. These weak Galerkin spatial discretizations are combined with the implicit Euler or Crank–Nicolson temporal discretizations to develop locking-free numerical solvers that have optimal order (k+1) convergence rates in pressure, velocity, displacement, stress, and dilation. Rigorous analysis is presented and illustrated by numerical experiments on popular test cases.

Water Effects on Elastic S-Wave Propagation and Attenuation Across Single Clay-Rich Rock Fractures: Insights from Ultrasonic Measurements

Though previous studies have shown that the presence of water strongly influences wave responses of rock fractures, water effects on wave behaviours across clay-rich rock fractures have been ambiguous until now. In the present study, we conducted considerable ultrasonic measurements on single rock fractures filled with kaolinite-dominant gouges at varying water saturation degrees to investigate the water effects on elastic S-wave propagation and attenuation across clay-rich rock fractures. The experimental results reveal that the S-wave velocity across single clay-rich rock fractures slightly increases and decreases with the progressively increasing water saturation degree. An increase in water saturation leads to a concave trend of the spectral amplitudes, while it moderately affects the central frequency of transmitted S-waves. In addition, the seismic quality factor across single clay-rich rock fractures follows an exponential growth trend with the water saturation, suggesting the exponentially negative relation between S-wave attenuation and the water saturation. We attribute the water saturation-dependent S-wave attributes across single clay-rich rock fractures to the combined effects of the local flow and the degradation of grain contacts. Compared to P-wave, S-wave exhibits less sensitivity to varying water saturation of clay-rich rock fracture. Upon increasing water saturation, S-waves display similar velocity and central frequency trends with P-waves. The tendencies of spectral amplitude and seismic quality factor for S-waves are approximately opposite to those for P-waves as the water saturation degree increases. We interpret these discrepancies by the fact that S-wave attributes across single water-saturated clay-rich rock fractures mainly depend on the properties of the skeletal frame, while the characteristics of the particles, pore fluid, and skeleton frame dominate P-wave behaviours. The outcomes of the current work facilitate our understanding of the fluid effects on the interaction of waves with clay-rich rock discontinuities.

First-principles investigation of physical properties of transition metal-based ternary TM5Si3C (TM = Nb, Mo, Ta, W) silicides of nowotny phase

In this work, the structural, electronic, mechanical, elastic anisotropy, thermodynamic, and optical properties of TM5Si3C silicides have been examined using first-principles calculations. All studied silicides fulfill the structural stability criteria. Computed mechanical properties reveal the ductile nature of Mo5Si3C, Nb5Si3C, and W5Si3C silicides, while Ta5Si3C silicide is brittle in character. Elastic anisotropy plots, including 2D and 3D, show highly anisotropic behaviour. The ascertained electronic and dielectric properties substantiate the conductive nature of all these silicides. All silicides in the visible region have an average reflectivity of 46 %, making them ideal for coatings to reduce solar heating. Debye temperature, melting point, thermal conductivity, and Gr u¨ neisen constant has also been investigated in current work. Thermodynamic properties have been computed at different pressure and temperature values including bulk modulus (B), volume (V), thermal expansion coefficient (α), Debye temperature (θD), and lattice heat capacity (Cv). The probed results of TM5Si3C ternary silicides demonstrate their viability for applications at elevated temperatures and as well as solar reflectors. Our findings may provide a gateway for exploring new ternary silicides and can act as an efficient, cost-effective guide for experimentalists.

Nonlinear Performance of Steel Tube Tower in Ultra-High Voltage Transmission Lines under Wind Loads

As complex, statically indeterminate structures, transmission towers are subject to complex forces and are usually simplified into truss structures that only consider the effects of axial force. When the load and deformation of a tower are small, it is reasonable to carry out analysis according to the linear elasticity theory. However, the height of an ultra-high voltage (UHV) transmission tower is significantly large, meaning that the calculation result according to the current elastic analysis method often has a large deviation from the actual stress of the structure. With the influence of the bending moment at the end of the member, a numerical model is established considering the influence of geometric nonlinearity and material nonlinearity in this paper. The stress distribution characteristics and development law of UHV transmission towers in linear and nonlinear stress states are analyzed and studied. The real tower test and elastoplastic ultimate bearing capacity analysis show that the elastoplastic analysis is closer to the actual tower. The UHV steel pipe tower designed according to the linear elasticity and small deformation theory has a large safety margin under the design load, resulting in a significant waste of materials. Under the action of heavy load, the tower exhibits strong nonlinearity, and the influence of geometric and material nonlinear factors should be fully considered when designing the structural components in UHV transmission towers.

A Binary Medium Constitutive Model for Frozen Solidified Saline Soil in Cold Regions and Its Fractal Characteristics Analysis

In addressing the issue of strength degradation in saline soil foundations under the salt-freeze coupling effects, a binary medium constitutive model suitable for un-solidified and solidified frozen saline soil is proposed considering both bonding and friction effects. To verify the validity of the constitutive model, freezing triaxial tests are carried out under different negative temperatures, confining pressures, and water contents. The pore structure and fractal characteristics of saline soil are analyzed using mercury intrusion porosimetry (MIP) and the fractal dimension D qualitatively and quantitatively, which shed light on the strength enhancement mechanism during the solidification of frozen saline soils. The results show that the constitutive model for frozen solidified saline soil based on binary medium theory aptly captures the stress–strain relationship before and after the solidification of frozen saline soil. The stress–strain relationship of frozen saline soil before and after solidification can be delineated into linear elasticity, elastoplasticity, and strain-hardening or -softening phases. Each of these phases can be coherently interpreted through the binary medium constitutive model. The un-solidified and solidified frozen both show pronounced fractal characteristics in fractal analysis. Notably, the fractal dimension D of the solidified saline soil exhibits a significant increase compared to that of un-solidified ones. In Regions I and III, the values of D for solidified saline soil are lower than those for untreated saline soil, which is attributed to the filling effect of hydration products and un-hydrated solidifying agent particles. In Region II, the fractal dimensions DMII and DNII of the solidified saline soil exhibit a “non-physical state”, which is mainly caused by the formation of a significant number of inkpot-type pores due to the binding of soil particles by hydration products.

Signorini problem as a variational limit of obstacle problems in nonlinear elasticity

<abstract><p>An energy functional for the obstacle problem in linear elasticity is obtained as a variational limit of nonlinear elastic energy functionals describing a material body subject to pure traction load under a unilateral constraint representing the rigid obstacle. There exist loads pushing the body against the obstacle, but unfit for the geometry of the whole system body-obstacle, so that the corresponding variational limit turns out to be different from the classical Signorini problem in linear elasticity. However, if the force field acting on the body fulfils an appropriate geometric admissibility condition, we can show coincidence of minima. The analysis developed here provides a rigorous variational justification of the Signorini problem in linear elasticity, together with an accurate analysis of the unilateral constraint.</p></abstract>

Lecture on mineral physics 2: linear elasticity

Lecture on mineral physics 2: linear elasticity

Analiza elastično-plastičnog naprezanja skupljajućih debelih FGM cilindara

This paper addresses the stress analysis of shrink-fitted thick-walled cylinders made of functionally graded material (FGM). Analytical solutions are provided for the elastic-plastic behavior of shrink-fitted axisymmetric thick-walled FGM cylinders based on the linear plane elasticity theory and plasticity laws. Due to the material's functional gradation, mechanical characteristics like Young's modulus, yield stress, and plastic parameters are controlled by a power function along the wall thickness. Considering a plane strain case, elastic-perfectly plastic model and Von Mises yield criterion; theoretical solutions are obtained for both elastic and plastic phases. The radial, hoop, axial, and equivalent Von Mises stresses are obtained for up to 25%, 50%, and 75% of cylinder plasticization for the elastic and plastic zones. Stresses are given in terms of interference and geometrical and mechanical parameters of the assembly. In addition, the interferences to start plastic deformation and the full plastic collapse of the cylinder are discussed. Moreover, the effect of the gradient index on the contact pressure and both the start of yield and plastic collapse pressures are conducted. The analytical results found in the current research are compared with the numerical solutions carried out by Ansys Workbench. Both results show a good agreement.

The relevance of analytical formulations predicting stiffener tripping

In search for the governing failure mode of externally pressurized ring-stiffened cylinders the focus is, due to its loading and slender geometry, drawn to elastic buckling. However, the hydrostatic pressure loading and imperfections cause nonlinear behaviour, and excessive deformation results in plastic failure, denoted by collapse. This implies that a true bifurcation point is often hardly observed, and this raises the question whether it is worthwhile to spend much effort to determine the elastic buckling pressures other than to ensure they are well above the collapse pressure? This paper focuses on the elastic buckling of ring frames, generally referred to as frame tripping. Some latest attempts in improving the accuracy of the analytical formulations will be questioned. Firstly, this study will compare non-linear buckling with linear-elastic material behaviour to elastic buckling by means of finite element analysis. Secondly, the non-linear elastic-plastic effects will be included too. Doing so, the effect of the geometrical non-linearity and material elastic-plastic non-linearity will be visible separately and will give an indication of the relevance of frame tripping results predicted by analytical elastic buckling formulations. Literature offers a variety on analytical formulations, each with an underlying idea to remove undesired assumptions that impair results. Even recently comprehensive formulations are published to establish an accurate value of the elastic tripping pressures. This paper shows whether those methods are still relevant in modern pressure hull design or not.

Bayesian experimental design for linear elasticity

Bayesian experimental design for linear elasticity

A Force-Matched Approach to Large-Strain Nonlinearity in Elasticity Imaging for Breast Lesion Characterization.

Ultrasound elasticity imaging is a class of ultrasound techniques with applications that include the detection of malignancy in breast lesions. Although elasticity imaging traditionally assumes linear elasticity, the large strain elastic response of soft tissue is known to be nonlinear. This study evaluates the nonlinear response of breast lesions for the characterization of malignancy using force measurement and force-controlled compression during ultrasound imaging. 54 patients were recruited for this study. A custom force-instrumented compression device was used to apply a controlled force during ultrasound imaging. Motion tracking derived strain was averaged over lesion or background ROIs and matched with compression force. The resulting force-matched strain was used for subsequent analysis and curve fitting. Greater median differences between malignant and benign lesions were observed at higher compressional forces (p-value < 0.05 for compressional forces of 2-6N). Of three candidate functions, a power law function produced the best fit to the force-matched strain. A statistically significant difference in the scaling parameter of the power function between malignant and benign lesions was observed (p-value = 0.025). We observed a greater separation in average lesion strain between malignant and benign lesions at large compression forces and demonstrated the characterization of this nonlinear effect using a power law model. Using this model, we were able to differentiate between malignant and benign breast lesions. With further development, the proposed method to utilize the nonlinear elastic response of breast tissue has the potential for improving non-invasive lesion characterization for potential malignancy.