Zero-stress Conditions Research Articles

Exact solutions for clamped spherical and cylindrical panels via a unified formulation and boundary discontinuous method

Numerous works in both recent and historical literature have concentrated on formulating theories to perform static analysis on simply-supported shell structures. However, it is worth noting that obtaining analytical solutions for clamped boundary conditions presents a strong challenge. In this paper, closed-form solutions for clamped cross-ply laminated and sandwich shells are achieved by employing a robust and hybrid methodology not previously reported in the literature. The high versatility of the Carrera Unified Formulation (CUF), based on the Equivalent-Single-Layer (ESL) description, is utilized to implement several refined shell theories. The Principle of Virtual Displacements (PVD) is utilized to derive the strong form of the governing equations in terms of displacement variables. As the main novelty, these equations are solved by the Boundary Discontinuous Fourier-based method (BDM) which provides highly accurate analytical solutions. The validity and robustness of the proposed methodology are assessed through a detailed comparison with references available in the open literature, as well as with FEM 3D results obtained with commercial software. Furthermore, the stress recovery technique is exploited to fulfill zero-stress and interlaminar continuity (IC) conditions. The findings might be useful in training artificial intelligence (AI) models, which, for instance, could facilitate the development of digital twin structures.

Influence of thermo-mechanical loads on variable stiffness composite beams: a comprehensive study using semi-analytical and finite element approaches

This study investigates the stability and vibration characteristics of variable stiffness laminated composite (VSLC) beams subjected to thermo-mechanical loadings. A semi-analytical model is developed to determine the buckling and vibration characteristics of variable stiffness composite beams. The model employs a displacement-based Ritz approach to derive the matrix representation of the governing equations. Modified constitutive relations are derived to account for the Poisson effects that arise due to the development of zero-stress conditions in the width direction of beams. The material properties and temperature variations are assumed to be constant across the thickness of the beam. Moreover, finite element calculations are carried out using commercial software ABAQUS in conjunction with a MATLAB-generated input file, which is utilized to capture fiber angle variations within the beam. Numerical results are obtained to elucidate the effect of slenderness ratio, modulus of elasticity ratio, boundary conditions, number of layers, and ply-sequence on the free vibration and instability characteristics of VSLC beams subjected to mechanical loadings and thermal environments. This study underscores the crucial importance of taking these factors into account for precise and dependable design and analysis of such composite beams in engineering applications.

Anomalous buckling of odd elastic plates

Buckling of thin-walled structures such as plates and shells is a consequence of in-plane stress being released through out-of-plane displacements. Generally, thin-walled structures that are subjected to tension or zero stress conditions remain stable. In this article, the anomalous tension buckling and stress-free active buckling of odd elastic plates are reported, which are a novel instability caused by odd elastic effects. The latter can only occur in the form of left- or right-handed chiral deformation and it does not involve external loads or internal active stress in a critical state. We demonstrate that the chiral rotation angle deformation is responsible for the active buckling of the plates, because the energy required for instability can be obtained based on the odd elastic effect. These findings can serve as an interpretation in a novel way for the occurrence of surface morphologies with biological activities, as well as provide references for buckling designs and applications of active structures.

Internal Elastic Strains of AZ31B Plate during Unloading at Twinning-Active Region

Magnesium alloys, being the lightest structural metals, have garnered significant attention in various fields. The characterization of inelastic behavior has been extensively investigated by researchers due to its impact on structural component performance. However, the occurrence of twinning in the absence of any applied driving force during unloading has lacked reasonable explanations. Moreover, the influence of deformation mechanisms other than twinning on inelastic behavior remains unclear. In this study, uniaxial tension and compression tests were conducted on hot-rolled magnesium alloy plates, and neutron diffraction experiments were employed to characterize the evolution of macroscopic mechanical response and microscopic mechanisms. Additionally, a twinning and detwinning (TDT) model based on the elastic visco-plastic self-consistent (EVPSC) model has been proposed, incorporating back stress to describe the deformation behavior during stress relaxation. This approach provides a comprehensive understanding of the inelastic behavior of magnesium alloys from multiple perspectives and captures the influence of microscale mechanisms. A thorough understanding of the inelastic behavior of magnesium alloys and a reasonable explanation for the occurrence of twinning under zero-stress conditions offer valuable insights for the precise design of magnesium alloy structures.

An isogeometric frictionless beam‐to‐beam contact formulation for hyperelastic Cosserat rods with unconstrained directors

AbstractWe present an isogeometric finite element formulation of frictionless beam‐to‐beam contact, using a Cosserat beam formulation with unconstrained directors. The beam's cross‐sectional deformation can be efficiently represented by director vectors. For the first order beam kinematics, Poisson locking due to the inability to represent linear in‐plane strain field in the cross‐section is alleviated by an enhanced assumed strain (EAS) method [1]. We employ a Gauss point‐to‐surface contact formulation combined with an active set iteration and a penalty method. The beam's cross‐sectional boundary as well as the axis is parameterized by a NURBS (non‐uniform rational B‐spline) curve, so that the lateral boundary surface has at least C2‐continuity, which yields a continuous surface metric and curvature in the closest point projection [2]. Since we do not employ zero stress conditions, a three‐dimensional nonlinear constitutive law can be straightforwardly utilized. We present several numerical examples, where a compressible Neo‐Hookean material is particularly considered.

Vibration of variable stiffness composite laminate and hybrid composite laminate plates coupled to sloshing fluid

In the present article, Reddy shear deformation plate theory is employed to investigate the free vibration of vertical laminated composite plates coupled to sloshing liquid. Two different types of composite plate are considered; hybrid composite laminate (HCL) plate, which consists of two types of fiber, and variable stiffness composite laminate (VSCL) plate, which is made of curvilinear fibers. The applied Reddy shear deformation theory introduces a nonlinear through-thickness distribution for the transverse shear stress, and it satisfies the zero stress conditions at the bottom and top surfaces of the plate. The sloshing fluid, which is partly in contact with the plate, is modeled as ideal, and the bulging and sloshing modes related to that are calculated using the fluid velocity potential. The interaction between fluid and solid is modeled through the continuity equation and boundary conditions at the fluid–solid interface. To extract the vibrational characteristics of the structure coupled to fluid, a polynomial approximation based on the Rayleigh-Ritz method is employed. After validating the proposed methodology, a parametric study is conducted to show the effects of various parameters associated with the fluid and structure on the natural frequencies of the system.

Scattering of a scalene trapezoidal hill with a shallow cavity to SH waves

Abstract Both surface ground motion and cavity stress concentration have always been considered in the designs of earthquake engineering. In this paper, a theoretical approach is used to study the scattering problem of circular holes under a scalene trapezoidal hill. The wave displacement function was obtained by solving the Helmholtz equation that meets the zero-stress boundary conditions by the variable separation method and the image method. Based on the complex function, the multipolar coordinate method and the region-matching technique, algebraic equations were established at auxiliary boundaries and free boundary conditions in the complex domain. Auxiliary circles were used to solve the singularity of the reflex angle at the trapezoidal corner. Then, according to the sample statistics, instead of the Fourier expansion method, the least-squares method was used to solve the undetermined coefficient of the algebraic equations by discrete boundaries. Frequency responses for some parameters were calculated and discussed. The numerical results demonstrate that the continuity of the auxiliary boundaries and the accuracy of the zero-stress boundary are good; the displacement of the free surface and the stress of the circular hole are related to the shape of the trapezoid, the position of the circular hole, the direction of the incident wave and the frequency content of the excitation. Finally, time-domain responses were calculated by inverse fast Fourier transform based on the frequency domain theory, and the results have revealed the wave propagation mechanism in the complicated structure.

Scattering of SH waves by a non-symmetrical V-shaped canyon with a shallow cavity in half-space

Both surface motion and hole stress concentration have always been concerned in engineering design. In this paper, a theoretical approach is used to study the scattering problem of circular cavities under a non-symmetrical V-shaped canyon. The wave displacement function is obtained by solving the Helmholtz equation that meets the zero-stress boundary conditions by adopting the method of separation of variables and the symmetric method. Based on the complex function method, multi-polar coordinate method and region-matching technique, algebraic equations are established at relevant conditions, which include auxiliary and free boundary conditions. Then, according to sample statistics, a least-square method is used instead of the Fourier expansion method to solve the undetermined coefficients of the algebraic equations by discrete boundary. Numerical results show that the theoretical method is verified valid by the finite element method and the reduced model, and the displacement of the free surface and the stress of the circular cavity are related to the shape of the canyon, the position of the circular cavity, the direction of the incident wave and the frequency content of the excitation.

Cavity flow characteristics and applications to kidney stone removal

Abstract

Free Vibration Analysis of Laminated Functionally Graded Carbon Nanotube-Reinforced Composite Doubly Curved Shallow Shell Panels Using a New Four-Variable Refined Theory

In this paper, a new four-variable refined shell theory is developed for free vibration analysis of multi-layered functionally graded carbon nanotube-reinforced composite (FG-CNTRC) doubly curved shallow shell panels. The theory has only four unknowns and satisfies zero stress conditions at the free surfaces without correction factor. Five different types of carbon nanotube (CNTs) distribution through the thickness of each FG-CNT layer are considered. Governing equations of simply supported doubly curved FG-CNTRC panels are derived from Hamilton’s principle. The resultant eigenvalue system is solved to obtain the frequencies and mode shapes of the anti-symmetric cross-ply laminated panels by using the Navier solution. The numerical results in the comparison examples have proved the accuracy and efficiency of the developed model. Detailed parametric studies have been carried out to reveal the influences of CNTs volume fraction, CNTs distribution, CNTs orientation, dimension ratios and curvature on the free vibration responses of the doubly curved laminated FG-CNTRC panels.

Layer-effect on variable seismic ground motions of SV-wave scattered by a double-stratified circular-arc crossing by layer-interface

This study is to theoretically investigate and quantify the variability of the seismic surface motions of a double-stratified half space with a circular-arc canyon by the SV wave incidence. Compared with the SH-wave induced scattering by the canyon with the single-layer (ideally homogeneous) medium, the challenge is to simultaneously satisfy the boundary conditions at the ground surface and layer-interface and deal with the relatively complex wave transmission on layer-interface. Theoretical solution for the SV-wave induced scattering by a double-stratified half space with circular-arc canyon is derived explicitly. The present method is validated by reducing the proposed solution with that in the homogenous medium. Then, an example is considered to further verify the derived results by simultaneously satisfying the continuity and zero-stress boundary conditions. The effects of the layer medium, the incident frequencies and incident angles on the ground motion of the double-stratified medium are investigated parametrically and compared with that of the homogeneous case. Results show that the derived solution is theoretically reliable and the layer-effect on the variability of the canyon ground motions is obvious.

A mixed finite element model with enhanced zigzag kinematics for the non-linear analysis of multilayer plates

In the present work, a non-linear finite element model for the analysis of composite and sandwich plates is proposed. The underlying variational formulation is based on the kinematics of the refined zigzag theory (RZT) as well as on a modified version of the two-field Hellinger–Reissner (HR) functional, where the displacements and the transverse shear stresses are involved as independent field variables. In fact, a consistent expression for each shear stress function is obtained by integrating the local equilibrium equations written in terms of derivated strain components. Regardless of the number of material layers, the proposed four-node plate element, which is labeled HR–RZT, exhibits only seven nodal degrees of freedom. The additionally introduced variables are effectively eliminated on the element level. One key advantage of the HR–RZT formulation is the fact that interlayer-continuity of the transverse shear stresses is automatically obtained without resorting to any post-processing procedures. The same applies to the zero stress conditions at the outer surfaces of the laminate. Further, due to the enhanced kinematics, the layerwise distortion of the laminate’s cross-section can also be accurately captured. The performance of the novel element formulation is studied by several numerical applications, where the solutions of the two-dimensional HR–RZT elements are verified by comparison with fully three-dimensional finite element models.

Scattering and amplification of SV waves by a semi-cylindrical hill in a half-space by a wavefunction-based meshless method using mapping and point-matching strategies

The zero-stress boundary conditions along the semi-infinite surface of a half-space have always posed challenging issues for the classical wavefunction expansion method to deal with in-plane wave propagation and scattering problems. A mapping and a point matching method are both introduced in this study to overcome the issue to propose a new wavefunction series solution to the scattering of SV-waves by a semi-cylindrical hill. The semi-infinite space is divided into an open and an enclosed region with a virtual boundary. After mapping the infinite surface into a finite interval, a good match of the zero-stress conditions in the open region is achieved. The zero-stress conditions in the enclosed hill region are satisfied analytically using the orthogonality of complex exponential functions. Taking into account the stress and displacement continuity conditions on the virtual boundary, the series solution can be obtained by solving a matrix to get the unknown coefficients of the wave fields. The accuracy of the solution is verified by comparing with two existing numerical solutions. To be helpful for engineering applications, the acceleration time histories at different locations under a real earthquake input are found to be obtainable based on the transfer functions of the frequency domain solution.

In-plane soil-structure interaction excited by incident plane SV waves

In-plane soil-structure-interaction of a shear wall on a rigid circular foundation embedded in a linear elastic half-space, excited by an incident plane SV wave, is solved by relaxing the zero-stress boundary conditions on the ground surface. An approximate solution of this model was previously presented for the cylindrical approximation of the ground surface. This paper formally neglects such boundary conditions, and as a result, the complexity of the formulation is significantly reduced. Solutions for foundation and building motions in frequency space are compared to the previous results and it is shown that the present approximate solution provides excellent and almost identical results. The objective of this paper is to provide further insight on the role and the effects of surface boundary conditions for this type of problem.

Shale/claystone response to air and liquid exposure, and implications for handling, sampling and testing

This paper documents the results of scientific studies to quantify the impact on shales and claystones of exposure to air, and to different liquids. These studies show that the water content of such rocks is controlled by the relative humidity of the surrounding air. Most shales/claystones will lose water, shrink and desaturate if exposed to air, unless the relative humidity is sufficiently high (usually 80–96%), causing desiccation damage. While exposure to brine will prevent desiccation, it results in water uptake and swelling, and a different type of alteration and damage. Confining stress will limit this swelling. Well-preserved shales are generally fully saturated under zero-stress conditions, but they have a negative pore water pressure. Hydrocarbon will not normally enter a fully saturated shale. Based on these findings, it is recommended that shale water content should not be altered unless it is done so under controlled conditions; to accomplish this one must minimize exposure to air, and avoid contact with water or brine unless the shale is under stress. Specific guidelines are provided for core handling and preservation, sampling, and testing. Gas shales are both similar to, and different than, claystone shales, and this alters slightly the findings and guidelines.

Effect of intrinsic and extrinsic factors on the simulated D-band length of type I collagen.

A signature feature of collagen is its axial periodicity visible in TEM as alternating dark and light bands. In mature, type I collagen, this repeating unit, D, is 67 nm long. This periodicity reflects an underlying packing of constituent triple-helix polypeptide monomers wherein the dark bands represent gaps between axially adjacent monomers. This organization is visible distinctly in the microfibrillar model of collagen obtained from fiber diffraction. However, to date, no atomistic simulations of this diffraction model under zero-stress conditions have reported a preservation of this structural feature. Such a demonstration is important as it provides the baseline to infer response functions of physiological stimuli. In contrast, simulations predict a considerable shrinkage of the D-band (11-19%). Here we evaluate systemically the effect of several factors on D-band shrinkage. Using force fields employed in previous studies we find that irrespective of the temperature/pressure coupling algorithms, assumed salt concentration or hydration level, and whether or not the monomers are cross-linked, the D-band shrinks considerably. This shrinkage is associated with the bending and widening of individual monomers, but employing a force field whose backbone dihedral energy landscape matches more closely with our computed CCSD(T) values produces a small D-band shrinkage of < 3%. Since this force field also performs better against other experimental data, it appears that the large shrinkage observed in earlier simulations is a force-field artifact. The residual shrinkage could be due to the absence of certain atomic-level details, such as glycosylation sites, for which we do not yet have suitable data.

Two-dimensional scattering and diffraction of P- and SV-waves around a semi-circular canyon in an elastic half-space: An analytic solution via a stress-free wave function

A well-defined boundary-valued problem of wave scattering and diffraction in elastic half-space should have closed-form analytic solutions. This two-dimensional (2-D) scattering around a semi-circular canyon in elastic half-space subjected to seismic plane and cylindrical waves has long been a challenging boundary-value problem. In all cases, the diffracted waves will consist of both longitudinal (P-) and shear (S-) rotational waves. Together at the half-space surface, these in-plane longitudinal P- and shear SV-waves are not orthogonal over the infinite half-space flat-plane boundary. Thus, to simultaneously satisfy both the zero normal and shear stresses at the flat-plane boundary, some approximation of the geometry and/or wave functions often has to be made, or in some cases, relaxed (disregarded). This paper re-examines this two-dimensional (2-D) boundary-value problem from an applied mathematics points of view and redefines the proper form of the orthogonal cylindrical-wave functions for both the longitudinal P- and shear SV-waves so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite-canyon surface are then comparatively less complicated to solve. This is now a closed-form analytic solution of the 2-D boundary-valued problem satisfying the half-space zero-stress boundary conditions exactly.

A note on three-dimensional scattering and diffraction by a hemispherical canyon–I: Vertically incident plane P-wave

The three-dimensional scattering by a hemi-spherical canyon in an elastic half-space subjected to seismic plane and spherical waves has long been a challenging boundary-value problem. It has been studied by earthquake engineers and strong-motion seismologists to understand the amplification effects caused by surface topography. The scattered and diffracted waves will, in all cases, consist of both longitudinal (P-) and shear (S-) shear waves. Together, at the half-space surface, these waves are not orthogonal over the infinite plane boundary of the half-space. Thus, to simultaneously satisfy both zero normal and shear stresses on the plane boundary numerical approximation of the geometry and/or wave functions were required, or in some cases, relaxed (disregarded). This paper re-examines this boundary-value problem from the applied mathematics point of view, and aims to redefine the proper form of the orthogonal spherical-wave functions for both the longitudinal and shear waves, so that they can together simultaneously satisfy the zero-stress boundary conditions at the half-space surface. With the zero-stress boundary conditions satisfied at the half-space surface, the most difficult part of the problem will be solved, and the remaining boundary conditions at the finite canyon surface will be easy to satisfy.

Theory and simulation of the diffusion of kinks on dislocations in bcc metals

Isolated kinks on thermally fluctuating (1/2)<111> screw, <100> edge and (1/2)<111> edge dislocations in bcc iron are simulated under zero stress conditions using molecular dynamics (MD). Kinks are seen to perform stochastic motion in a potential landscape that depends on the dislocation character and geometry, and their motion provides fresh insight into the coupling of dislocations to a heat bath. The kink formation energy, migration barrier and friction parameter are deduced from the simulations. A discrete Frenkel-Kontorova-Langevin (FKL) model is able to reproduce the coarse grained data from MD at a fraction of the computational cost, without assuming an a priori temperature dependence beyond the fluctuation-dissipation theorem. Analytic results reveal that discreteness effects play an essential r\^ole in thermally activated dislocation glide, revealing the existence of a crucial intermediate length scale between molecular and dislocation dynamics. The model is used to investigate dislocation motion under the vanishingly small stress levels found in the evolution of dislocation microstructures in irradiated materials.

The Ground Motion of Isosceles Triangular Hill in Right-Angle Plane Impacted by SH-Wave

The analytical solution to the problem of the scattering of SH-wave by isosceles triangular hill in right-angle plane is given by using the methods of complex function and multiple coordinate. Firstly, the solution region is divided into two domains, where domain I involves isosceles triangular hill and a semi-circular bottom, domain II involves a semi-circular hollow in right-angle plane. And a standing wave function is constructed which satisfies the zero-stress conditions at the triangular wedges. In domain II, the scattering wave functions which satisfy the stress free boundary conditions at the free surfaces for the right-angle plane are constructed. Secondly, based on the conditions of the displacement continuity and stress continuity at the “common border” in the domains, a series of infinite algebraic equations are given and solved by truncation. Finally, some examples for amplitude of displacement on the surface are given. Numerical results show that amplitude of displacement on the surface is influenced by isosceles triangular hill.