Moderate Rotations Research Articles

The Sierra Madre Oriental Orocline: Paleomagnetism of the Nazas Province in NE Mexico

AbstractCurved mountain belts are spectacular natural features that contain crucial 3D information about the tectonic evolution of orogenic systems in the absence of other kinematic markers. The Mesozoic units exposed in the Mexican Fold and Thrust Belt in northeastern Mexico show a striking curvature, whose kinematic history has not been studied. The existing tectonic models of the region simply assumed the shape of the tectonic units as an inherent feature to the orogen. We investigated the kinematic history of this curvature through paleomagnetism and rock magnetism analyses, coupled with an exhaustive review of available published literature. The studied data sets indicate a protracted history of (re)magnetizations that occurred during the Late Jurassic‐Paleocene times at least during the Late Jurassic, Cretaceous and early Eocene. More significantly, they show significant counterclockwise rotations in the northern flank of the curvature and moderate clockwise vertical axis rotations along its southern flank. This data set suggests that the Sierra Madre Oriental was a linear belt that experienced oroclinal bending or buckling during the Cretaceous to early Eocene period (120–50 Ma).

The effects of nonlinear soil spatial variability on the nonlinear beam response


 
 
 
 The nonlinear analysis of a beam resting on nonlinear random tensionless soil was studied with the aim of quantifying the influence of the spatial variability of the tension soil characteristics on the behavior of the beam and illustrating the importance of the geometric nonlinear analysis of a beam. The soil–structure interaction mechanism is taken into account where the soil is modeled as nonlinear. Due to large deflections and moderate rotations of the beam, the Von-Kàrman type nonlinearity based on the finite element formulation of nonlinear beam response is adopted and the interfacial shearing resistance along the beam is taken into account. To assess the probabilistic nonlinear behavior of the beam resting on nonlinear random soil, a Monte Carlo approach was used and the response of the beam was statistically analyzed for different beam and soil parameters. The equation of motion is solved using the Newton-Raphson iterative method which is implemented by using a Matlab software. The results indicate that the probabilistic approach of the soil and the geometric nonlinearity of the beam serve a major role in the evaluations of the beam response (of the real response of the buried beam).
 
 
 

Nonlinear bending analysis of variable thickness multi-directional functionally graded plates based on isogeometric analysis

In this study, the nonlinear bending behavior of multi-directional functionally graded plates with variable thickness is investigated. To this regard, the governing equation of the plate problem is developed based on the principle of virtual work, while the third-order shear deformation plate theory proposed by Reddy is employed to describe the kinematic relations. The geometrical nonlinearity is taken into account by adopting the von Kármán’s assumptions for nonlinear problems with small strains and moderate rotations. The plates investigated in this study are assumed to be made from two different material constituents, whose volume fractions vary spatially within the plate domain. The effective material properties are calculated based on the mixture approach. The Isogeometric Analysis approach is used as a numerical method to solve the governing equations. NURBS basis functions are used as interpolation functions for geometrical modeling and discretization procedure, where the C 1-continuity requirement is meet efficiently. Various numerical examples are conducted to validate the accuracy of the proposed approach and investigate the influences of geometrical and material factors on the nonlinear bending responses of the plates.

Thermally induced deformations in multi-layered polymeric struts

Stimulus activated structures that deform from a reference to target configurations are used in various fields such as soft robotics, smart materials, and actuators. Examples of stimuli include electric and magnetic fields, swelling, and temperature. The design of stimulus activated structures requires a comprehensive understanding of the relations between geometry, material properties, structure composition, and the influence of the chosen stimulus. With the aim of better understanding these relations, this work employs moderate rotations theory to model the thermo-mechanical response of multi-layered polymeric struts in a wide range of temperatures. In particular, the transition from a glassy to a rubbery state is explicitly discussed. Implicit equations that relate the material properties and the geometry to the thermally-induced deformation are developed. To illustrate the merit of the model, we investigate the thermo-mechanical behavior of homogeneous and heterogeneous multi-layered struts under two boundary conditions - free-standing and pinned. We show that multi-layered beams with a mismatch in the thermal and elastic properties bend as a result of temperature change. Interestingly, the glass-transition temperature can result in the reversal of bending direction. We follow with simulations of thermally-activated polymer-based bi-layer struts with typical properties to demonstrate the applicability of the proposed design. This framework can be used as a guideline and an inverse design tool for the selection of materials and geometries for thermally-activated shape-morphing lattice-based structures that are capable of achieving multiple shape-shifts with a single stimulus. Additionally, this work can be readily extended to other stimuli and thus serves as a foundation for the effective design of stimulus activated structures capable of polymorphism.

Nonlinear static analysis of plates with arbitrary aspect ratios using Extended Higher Order Sandwich Panel Theory

A nonlinear static analysis of the sandwich plate using variational techniques and the Ritz method is presented. The kinematic description developed for a sandwich plate undergoing small strains and moderate rotations within the framework of Extended Higher Order Sandwich Panel Theory is considered. Employing the Ritz method, the total potential energy of the system is developed. Four different cases and combinations of boundary conditions are studied and assumed solutions satisfying the geometric boundary conditions are developed. The total potential energy results in nonlinear algebraic equations for the unknown coefficients in the assumed solution. The nonlinear equations are then solved using the Newton–Raphson method. A convergence study is then carried out to study the effect of variation of the number of terms in the assumed solution. In order to analyze the effect of inclusion of nonlinear effects in the core, results are computed considering only the facesheets with nonlinear strains and the core is considered to remain in the linear range. A stress analysis is carried out to compare the stresses predicted by the linear theory against the nonlinear results. After calculating and comparing the results for the simply supported case, three more cases for different sets of boundary conditions are also considered and presented. Since EHSAPT allows for analysis of plates with variable aspect ratios a further case where it is assumed that the depth of the plate is half the width is also considered and the results are presented for various geometric configurations.

Development and implementation of a geometrically nonlinear beam theory model for SMA composite beams with asymmetric behavior

This work aims at developing a finite element solution with consideration of the geometric nonlinearity for analyzing the bending behavior of a Shape Memory Alloy multi-layered beam. In this solution, a numerical method based on the Newton–Raphson method is implemented in order to satisfy equilibrium equations. The validation of the model is carried out by means of comparing the simulation results of beam with the semi-analytical solution. The validation outcomes are described for both symmetric and asymmetric models, and it is observed that both methods are in good agreement. The results of simulation of composite SMA beam are reported by both considering and neglecting the geometric nonlinearity and asymmetric behavior. Load-deflection, stress–strain as well as variation of the maximum slope are some of the diagrams that are investigated in this study. In one example, the error between the models which include and neglect the geometric nonlinearity is 107% and also in another case study, the error between asymmetric and symmetric models is 14.9%. The results reveal that considering the geometric nonlinearity and asymmetric behavior have important effects on simulation outcomes. This model is suitable for the design of structures including multilayer SMA composite which undergoes large displacements and moderate rotations.

Dual mesh control domain analysis of functionally graded circular plates accounting for moderate rotations

Nonlinear analysis of the axisymmetric bending of circular plates, accounting for through-thickness power-law variation of a two-constituent material and the von Kármán nonlinearity is presented using the dual mesh control domain methods (DMCDM). The classical and first-order shear deformation theory kinematics are used and displacement and mixed models are developed using the DMCDM. The DMCDM predicts displacements as accurate as the finite element method (FEM), but has the advantage of predicting the stress resultants more accurately than the FEM. The developed computational models are used to determine the effect of the geometric nonlinearity and power-law index on the bending deflections and stress resultants of functionally graded circular and annular plates with different boundary conditions. It is found that the power-law index, which dictates the material distribution through the thickness, has two different regions of response, one with steep increase in deflections followed by relatively slow increase with respect to the power-law index.

Numerical Study of Rotating Thermal Convection on a Hemisphere

Numerical simulations of rotating two-dimensional turbulent thermal convection on a hemisphere are presented in this paper. Previous experiments on a half soap bubble located on a heated plate have been used for studying thermal convection as well as the effects of rotation on a curved surface. Here, two different methods have been used to produce the rotation of the hemisphere: the classical rotation term added to the velocity equation, and a non-zero azimuthal velocity boundary condition. This latter method is more adapted to the soap bubble experiments. These two methods of forcing the rotation of the hemisphere induce different fluid dynamics. While the first method is classically used for describing rotating Rayleigh–Bénard convection experiments, the second method seems to be more adapted for describing rotating flows where a shear layer may be dominant. This is particularly the case where the fluid is not contained in a closed container and the rotation is imposed on only one side of it. Four different diagnostics have been used to compare the two methods: the Nusselt number, the effective computation of the convective heat flux, the velocity and temperature fluctuations root mean square (RMS) generation of vertically aligned vortex tubes (to evaluate the boundary layers) and the energy/enstrophy/temperature spectra/fluxes. We observe that the dynamics of the convective heat flux is strongly inhibited by high rotations for the two different forcing methods. Also, and contrary to classical three-dimensional rotating Rayleigh–Bénard convection experiments, almost no significant improvement of the convective heat flux has been observed when adding a rotation term in the velocity equation. However, moderate rotations induced by non-zero velocity boundary conditions induce a significant enhancement of the convective heat flux. This enhancement is closely related to the presence of a shear layer and to the thermal boundary layer just above the equator.

Geometrically nonlinear Euler–Bernoulli and Timoshenko micropolar beam theories

Two ways of incorporating moderate rotations of planes normal to the axis of a straight beam into the Euler–Bernoulli and the Timoshenko micropolar beam theories are presented. In the first case, the von Karman nonlinear strains are used to incorporate the moderate rotations of normal planes into the beam theories. In the second case, appropriate approximations are made on the nonlinear Cosserat deformation gradient to reflect the condition of moderate rotations of the normal planes. The governing nonlinear differential equations and corresponding natural boundary conditions in both cases are derived using the principle of virtual displacements. A weak-form Galerkin displacement finite element formulation is presented for the developed nonlinear beam theories. The phenomenon of locking usually encountered in beam displacement finite elements is eliminated using higher-order finite elements with nodes located at spectral points. Finally, numerical examples are presented to illustrate the effect of coupling number and bending characteristic length scale on deflections and microrotations when a micropolar beam is modeled with the developed nonlinear beam theories.

On the nonlinear vibration and static deflection problems of actuated hybrid nanotubes based on the stress-driven nonlocal integral elasticity

In the present paper, the effects of material properties, nonlocal parameter, Lorentz and electric forces on maximum static deflections and natural frequencies of actuated hybrid carbon/boron-nitride nanotubes (CBNNT) subjected to thermal loads are studied for the first time. The displacement field of the nanotube satisfies assumptions of the Bernoulli–Euler beam theory. The Green-Lagrange small strains and moderate rotations for geometric nonlinearity of the nanotube are taken into consideration. Two nonlinear governing equations of motion for carbon and boron-nitride parts of the clamped nanotube are derived using the d'Alembert principle and the stress-driven nonlocal integral elasticity theory. Solution to the formulated nonlinear boundary value problem was obtained using the Galerkin modal expansion method. Validation of obtained results and parametric study are comprehensively presented. Obtained results show effect of variation of temperature, nonlocal parameter, ratio of length of carbon and boron-nitride components, direct current (DC) and alternating current (AC) voltages on shifting of the fundamental frequency and maximum deflection of hybrid nanotube.

Developing a beam formulation for semi-crystalline two-way shape memory polymers

In this research, the bending of a two-way shape memory polymer beam is examined implementing a one-dimensional phenomenological macroscopic constitutive model into Euler–Bernoulli and von-Karman beam theories. Since bending loading is a fundamental problem in engineering applications, a combination of bending problem and two-way shape memory effect capable of switching between two temporary shapes can be used in different applications, for example, thermally activated sensors and actuators. Shape memory polymers as a branch of soft materials can undergo large deformation. Hence, Euler–Bernoulli beam theory does not apply to the bending of a shape memory polymer beam where moderate rotations may occur. To overcome this limitation, von-Karman beam theory accounting for the mid-plane stretching as well as moderate rotations can be employed. To investigate the difference between the two beam theories, the deflection and rotating angles of a shape memory polymer cantilever beam are analyzed under small and moderate deflections and rotations. A semi-analytical approach is used to inspect Euler–Bernoulli beam theory, while finite-element method is employed to study von-Karman beam theory. In the following, a smart structure is analyzed using a prepared user-defined subroutine, VUMAT, in finite-element package, ABAQUS/EXPLICIT. Utilizing generated user-defined subroutine, smart structures composed of shape memory polymer material can be analyzed under complex loading circumstances through the two-way shape memory effect.

Corotational elements for thin-walled laminated composite beams with large 3D rotations

This work presents two finite elements for geometric nonlinear analysis of thin-walled laminated composite beams. The element cross-section properties are evaluated through a suitable thin-walled beam theory, yielding a 4×4 constitutive matrix where different couplings between generalized stresses and strains can be considered. In the local coordinate system, one element named CRL is linearly formulated, while the other element named CRTL incorporates moderate rotations. The element independent corotational approach is used in order to deal with large rotations in 3D space. Numerical experiments, considering different cross-sections and layups, demonstrate the accuracy and effectiveness of the proposed finite elements.

Nonlocal vibration of carbon/boron-nitride nano-hetero-structure in thermal and magnetic fields by means of nonlinear finite element method

Abstract Hybrid nanotubes composed of carbon and boron-nitride nanotubes have manifested as innovative building blocks to exploit the exceptional features of both structures simultaneously. On the other hand, by mixing with other types of materials, the fabrication of relatively large nanotubes would be feasible in the case of macroscale applications. In the current article, a nonlinear finite element formulation is employed to deal with the nonlocal vibrational behavior of carbon/boron-nitride nano-hetero-tubes in the presence of magneto-thermal environment. Euler–Bernoulli beam model in conjunction with the Eringen’s nonlocal theory of elasticity is adopted to derive the governing equation of motion. In order to conduct a nonlinear frequency analysis, the von-Kármán nonlinearity associated with moderate rotations is also considered. It is well known that temperature gradients can significantly change the dynamic behavior of nanotubes. On the other hand, the coefficients of thermal expansions of carbon and boron-nitride nanotubes are quite different that may affect the structural stability of hybrid nanotubes. Hence, to explore the vibration characteristic of such composite structures, the influence of magneto-thermal environment is also taken into account. Finally, the eigenvalue analysis is performed to exhibit the nonlinear mode shapes and natural frequencies of the system due to initial displacement. It is expected that the recognition of dynamic behavior of such hybrid nanotubes may open the doors to the creative design of next-generation nano-devices.

Paleomagnetic and magnetic fabric data from Lower Triassic redbeds of the Central Western Carpathians: new constraints on the paleogeographic and tectonic evolution of the Carpathian region

In the Central Western Carpathians (CWC), most published paleomagnetic results from Permo-Mesozoic rocks document extensive remagnetizations and come from thin-skinned thrust units that have undergone multistage deformation. We present results from lower Triassic redbeds from the autochthonous cover overlying the basement that carry a primary magnetization. Petromagnetic results indicate that the dominant ferromagnetic carrier is hematite, while magnetic susceptibility and its anisotropy are controlled by both ferromagnetic and paramagnetic minerals. Magnetic fabrics document weak deformation related to Late Cretaceous shortening. The directions of the high unblocking temperature remanence components pass both reversal and fold tests, attesting to their primary nature. Paleomagnetic inclinations are flatter than expected from reference datasets, suggesting small latitudinal separation between the CWC and stable Europe. Paleomagnetic declinations are mostly clustered within individual mountain massifs, implying their tectonic coherence. They show only minor differences between the massifs, indicating a lack of significant vertical-axis tectonic rotations within the studied central parts of the CWC. The paleomagnetic declinations are therefore representative of the whole of the CWC in terms of regional paleogeographic interpretations, and imply moderate counterclockwise rotations ( c . 26°) of the region with respect to stable Europe since the Early Triassic.

Response sensitivity for geometrically nonlinear displacement-based beam-column elements

Accurate and efficient response gradients are required in structural reliability, optimization, and system identification applications where the structural system is expected to yield under significant deformations. The direct differentiation method (DDM) is applied to the response sensitivity of displacement-based beam-column finite elements with geometric nonlinearity due to moderate rotations in the basic system. The derived sensitivity equations are implemented in the OpenSees software framework. Stand-alone sensitivity analyses using the DDM and finite difference method (FDM) validate the sensitivity equations for example structures with nonlinear static and dynamic response. Finite element reliability analysis of a steel frame shows that, relative to a geometrically linear formulation, using geometrically nonlinear displacement-based elements increases the probability of failure and affects the importance ranking of the random variables. In addition, for a performance function based on residual capacity at a target lateral displacement, the frame strength at the design point is higher than that obtained when using geometrically linear elements. The developed sensitivity equations increase the number of nonlinear finite element formulations available for gradient-based applications in structural engineering.

Nonlinear finite element analysis of lattice core sandwich beams

A geometrically nonlinear finite element model is developed for the bending analysis of micropolar Timoshenko beams using the principle of virtual displacements and linear Lagrange interpolation functions. The nonlinearity enters the model via a nonlinear von Kármán strain term that allows the micropolar beam to undergo moderate rotations. The nonlinear micropolar Timoshenko beam is used as an equivalent single layer model to study four different lattice core sandwich beams. A two-scale energy method is used to derive the micropolar constitutive equations for web, hexagonal, Y-frame and corrugated core topologies. Various bending cases are studied numerically using the developed 1-D finite element model. Reduced integration techniques are used to overcome the shear and membrane locking. The present 1-D results are in good agreement with the corresponding 2-D finite element beam frame results for global bending.

Rotational Diversity Effects in a Triticale-based Cropping System

Research indicates that not all crops respond similarly to cropping diversity and the response of triticale (× Triticosecale ssp.) has not been documented. We investigated the effects of rotational diversity on cereals in cropping sequences with canola (Brassica napus L.), field pea (Pisum sativum L.), or an intercrop (triticale:field pea). Six crop rotations were established consisting of two, 2-yr low diversity rotations (LDR) (continuous triticale (T-T_LDR) and triticale-wheat (Triticum aestivum L.) (T-W_LDR)); three, 2-yr moderate diversity rotations (MDR) (triticale-field pea (T-P_MDR), triticale-canola (T-C_MDR), and a triticale: field pea intercrop (T- in P_MDR)); and one, 3-yr high diversity rotation (HDR) (canola-triticale-field pea (C-T-P_HDR)). The study was established in Lethbridge, Alberta (irrigated and rainfed); Swift Current (rainfed) and Canora (rainfed), Saskatchewan, Canada; and carried out from 2008 to 2014. Triticale grain yield for the 3-yr HDR was superior over the LDR rotations and the MDR triticale-field pea system; however, results were similar for triticale-canola, and removal of canola from the system caused a yield drag in triticale. Triticale biomass was superior for the 3-yr HDR. Moreover, along with improved triticale grain yield, the 3-yr HDR provided greater yield stability across environments. High rotational diversity (C-T-P_HDR) resulted in the highest soil microbial community and soil carbon concentration, whereas continuous triticale provided the lowest. Net economic returns were also superior for C-T-P_HDR ($670 ha–1) and the lowest for T-W_LDR ($458 ha–1). Overall, triticale responded positively to increased rotational diversity and displayed greater stability with the inclusion of field pea, leading to improved profitability and sustainability of the system.

Modeling the non-linear elastic response of periodic lattice materials

Periodic elastic lattice materials are a class of cellular materials with unique properties that cannot be achieved with fully uniform solids. In this work we employ moderate-rotation theory, which is an expansion of linear elasticity that takes into account moderate displacements and rotations, to derive a fundamental multi-scale model that captures the non-linear response of infinitely periodic elastic lattice materials and allows the imposition of local constraints on lattice junctions. We begin by modeling the response of a single strut. Next, an infinitely periodic lattice structure is considered. The periodicity of the structure is accounted for through a unit cell (UC) and periodic lattice directions. A representative volume element (RVE) comprising one or more UCs is then defined. It is assumed that the RVE experiences the macroscopic displacement gradient. To determine the stress, we employ the principle of virtual work and homogenization methods. To illustrate the predictions of the model, we analyze two infinitely periodic lattice materials - a diamond lattice and a triangular lattice with rigid junctions. To validate our predictions, we compute the response for RVEs of several sizes until convergence is reached.

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

SummaryThis paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.

Consideration of Stress Stiffening and Material Reorientation in Modal Space Based Finite Element Solutions

Structural deformations are an important aspect of many engineering tasks. They are typically resolved as “off-line” finite element computations with accuracy set as the primary objective. Though high computational efficiency is always an important aspect, in certain applications its priority is of equal or similar importance as the accuracy itself. This paper tackles the problem of proper extension of linear models with the objective of keeping high numerical efficiency and covering moderate geometric nonlinearities. Modal-space based approach is addressed as one of the standard techniques for robust model reduction. Two extensions are proposed to account for moderate geometric nonlinearities in modal-space based solutions, one accounting for stress stiffening effect and the other for moderate material rigid-body rotations during deformation. Examples are provided to demonstrate the applicability and discuss the aspects of proposed techniques.