scholarly journals The eigenbuckling analysis of hexagonal lattices: closed-form solutions

2021 ◽  
Vol 477 (2251) ◽  
pp. 20210244
Author(s):  
S. Adhikari
Keyword(s):  
Closed Form ◽  
Elastic Properties ◽  
Axial Force ◽  
Hexagonal Lattice ◽  
Force Parameter ◽  
Closed Form Solutions ◽  
Analysis And Design ◽  
Lattice Materials ◽  
Design Calculations ◽  
Analytical Expressions

Elastic instability such as the buckling of cellular materials plays a pivotal role in their analysis and design. Despite extensive research, the quantifi- cation of critical stresses leading to elastic instabi- lities remains challenging due to the inherent nonlinearities. We develop an analytical approach considering the spectral decomposition of the elasticity matrix of two-dimensional hexagonal lattice materials. The necessary and sufficient condition for the buckling is established through the zeros of the eigenvalues of the elasticity matrix. Through the analytical solution of the eigenvalues, the conditions involving equivalent elastic properties of the lattice were directly connected to the mathematical requirement of buckling. The equivalent elastic properties are expressed in closed form using geometric properties of the lattice and trigonometric functions of a non-dimensional axial force parameter. The axial force parameter was identified for four different stress cases, namely, compressive stress in the longitudinal and transverse directions separately and together and torsional stress. By solving the resulting nonlinear equations, we derive exact analytical expressions of critical eigenbuckling stresses for these four cases. Crucial parameter combinations leading to minimum buckling stresses are derived analytically. The exact closed-form analytical expressions derived in the paper can be used for quick engineering design calculations and benchmarking related experimental and numerical studies.

Torsion of Composite Elastic Bars of Arbitrary Cross Section

1968 ◽  
Vol 90 (3) ◽  
pp. 435-440 ◽  
Author(s):  
E. M. Sparrow ◽  
H. S. Yu
Keyword(s):  
Exact Solution ◽  
Closed Form ◽  
Elastic Properties ◽  
Cross Section ◽  
Excellent Agreement ◽  
Solution Method ◽  
Circular Cross Section ◽  
High Accuracy ◽  
Arbitrary Cross Section ◽  
Closed Form Solutions

A method of analysis is presented for determining closed-form solutions for torsion of inhomogeneous prismatic bars of arbitrary cross section, the inhomogeneity stemming from the layering of materials of different elastic properties. It is demonstrated that the solution method is very easy to apply and provides results of high accuracy. As an application, solutions are obtained for the torsion of a bar of circular cross section consisting of two materials separated by a plane interface. The results are compared with those of various limiting cases and excellent agreement is found to exist. Among the limiting cases, an exact solution was derived by Green’s functions for the problem in which the interface between the materials coincides with a diameter of the circular cross section.

Relaxation and creep in twist and flexure

2014 ◽  
Vol 10 (3) ◽  
pp. 304-327 ◽  
Author(s):  
V. Kobelev
Keyword(s):  
Strain Rate ◽  
Closed Form ◽  
Stress Function ◽  
Constitutive Models ◽  
Analytic Description ◽  
Secondary Creep ◽  
Common Procedure ◽  
Content Type ◽  
Closed Form Solutions ◽  
Analytical Expressions

Purpose – The purpose of this paper is to derive the exact analytical expressions for torsion and bending creep of rods with the Norton-Bailey, Garofalo and Naumenko-Altenbach-Gorash constitutive models. These simple constitutive models, for example, the time- and strain-hardening constitutive equations, were based on adaptations for time-varying stress of equally simple models for the secondary creep stage from constant load/stress uniaxial tests where minimum creep rate is constant. The analytical solution is studied for Norton-Bailey and Garofalo laws in uniaxial states of stress. Design/methodology/approach – The creep component of strain rate is defined by material-specific creep law. In this paper the authors adopt, following the common procedure Betten, an isotropic stress function. The paper derives the expressions for strain rate for uniaxial and shear stress states for the definite representations of stress function. First, in this paper the authors investigate the creep for the total deformation that remains constant in time. Findings – The exact analytical expressions giving the torque and bending moment as a function of the time were derived. Research limitations/implications – The material isotropy and homogeneity preimposed. The secondary creep phase is considered. Practical implications – The results of creep simulation are applied to practically important problem of engineering, namely for simulation of creep and relaxation of helical and disk springs. Originality/value – The new, closed form solutions with commonly accepted creep models allow a deeper understanding of such a constitutive model's effect on stress and deformation and the implications for high temperature design. The application of the original solutions allows accurate analytic description of creep and relaxation of practically important problems in mechanical engineering. Following the procedure the paper establishes closed form solutions for creep and relaxation in helical, leaf and disk springs.

Elasticity Approach to Load Transfer in Cord-Composite Materials

2009 ◽  
Vol 76 (6) ◽  
Author(s):  
Anthony J. Paris
Keyword(s):  
Composite Materials ◽  
Closed Form ◽  
Axial Force ◽  
Load Transfer ◽  
Axial Loading ◽  
Shear Stresses ◽  
Matrix Interface ◽  
Closed Form Solutions ◽  
Apparent Modulus ◽  
Reinforced Composite

An elasticity approach to the mechanics of load transfer in cord-reinforced composite materials is developed. Finite cords embedded in an elastic matrix and subjected to axial loading is considered, and the extension-twist coupling of the cords is taken into account. Closed form solutions for the axial force and twisting moment in the cord, the shear stresses at the cord-matrix interface in the axial and circumferential directions, the effective axial modulus of the cord, and the apparent modulus of the cord composite are presented. An example of a cord composite typical of what can be found in steel-belted-radial tires is used to illustrate the results. It was found that large shear stresses occur at the cord-matrix interface in both the axial and circumferential directions at the cord ends and that the effective modulus of the cords may be greatly reduced. As a result, the apparent modulus of the composite may be significantly less than that found by a conventional application of the rule-of-mixtures approach.

Gravity anomalies of arbitrary 3D polyhedral bodies with horizontal and vertical mass contrasts up to cubic order

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. G1-G13 ◽  
Author(s):  
Zhengyong Ren ◽  
Yiyuan Zhong ◽  
Chaojian Chen ◽  
Jingtian Tang ◽  
Kejia Pan
Keyword(s):  
Closed Form ◽  
Gaussian Quadrature ◽  
Analytical Formula ◽  
Gravity Anomalies ◽  
Surface Integral ◽  
Third Order ◽  
Closed Form Solutions ◽  
Order Polynomial ◽  
Relative Errors ◽  
Analytical Expressions

A new singularity-free analytical formula has been developed for the gravity field of arbitrary 3D polyhedral mass bodies with horizontally and vertically varying density contrast using third-order polynomial functions. First, the observation sites are moved to the origin of the coordinate system. Then, the volume and surface integral theorems are invoked successively to transform the volume integrals into surface integrals over polygonal faces and into line integrals over the edges of the polyhedral mass bodies. Furthermore, singularity-free closed-form solutions are derived for these line integrals over the edges. Thus, the observation sites can be located inside, on, or outside the 3D distributions. A synthetic prismatic mass body is adopted to verify the accuracy and singularity-free property of our newly developed analytical expressions. Excellent agreements are obtained between our solutions and other published closed-form solutions with relative errors in the order of [Formula: see text] to [Formula: see text]. In addition, an octahedral model and a near-Earth asteroid model are used to verify the accuracy of the presented method for complicated target structures by comparing the results with those from a high-order Gaussian quadrature approach.

Closed-Form Solutions to the Kinematics of a Parallel Locomotion Mechanism With Actuated Spoke Wheels

2011 ◽  
Author(s):  
Ping Ren ◽  
Dennis Hong
Keyword(s):  
Motion Planning ◽  
Numerical Simulations ◽  
Closed Form ◽  
Real Time ◽  
Convex Surface ◽  
Closed Form Solutions ◽  
Time Motion ◽  
Discrete Contact ◽  
Parallel Configuration ◽  
Analytical Expressions

A parallel locomotion mechanism can be defined as “a mechanism with parallel configuration that has discrete contact with respect to the ground which renders a platform the ability to move”. The actuated spoke wheel robot IMPASS (Intelligent Mobility Platform with Active Spoke System) presented in this paper serves as an example of such locomotion mechanisms. The current prototype of IMPASS has two actuated spoke wheels and one passive tail with its lower portion designed as convex surface. The robot is considered as a mechanism with variable topologies (MVTs) because of its metamorphic configuration. Closed-form solutions to the kinematics of the variable topologies are developed and verified with numerical simulations. The analytical expressions to these solutions allow themselves to be used directly in the real-time motion planning and monitoring of the robot.

scholarly journals Thermoelastic Closed-Form Solutions of FGM Plates Subjected to Temperature Change in Axial and Thickness Directions

2020 ◽  
Author(s):  
Yen-Ling Chung
Keyword(s):  
Closed Form ◽  
Temperature Change ◽  
Axial Force ◽  
Maximum Stress ◽  
Closed Form Solution ◽  
Bending Moment ◽  
Bottom Surface ◽  
Thermal Loads ◽  
Element Analysis ◽  
Closed Form Solutions

Abstract An analytical solution of simply supported FGM plates under thermal loads is developed based on medium-thick plate assumption. Further to assume constant Poisson’s ratio and thermal expansion coefficient, the closed-form solutions of the FGM plates with through-the-thickness Young’s modulus under temperature change in x - and z -directions are evaluated, expressed in terms of the thermal axial force and thermal bending moment. The closed-form solutions confirmed by finite element analysis give a complete insight into the thermal-mechanical behavior of FGM plates. Hence, the deflection, strain, stress, axial force, and bending moment of the FGM plate under thermal loads in axial and thickness directions are discussed. Results show that the use of FGM makes the maximum stress from the top or bottom surface move to the inner portion of the FGM plate, and significantly reduces the maximum stress of the plates. Moreover, although the FGM plate is subjected to thermal load in the thickness direction, the deflection of the FGM plate can be zero by properly choosing the steep material gradation, directly derived from the obtained closed-form solution.

Closed Form Solutions of Joint Water-Filling for Coordinated Transmission

2010 ◽  
Vol E93-B (12) ◽  
pp. 3461-3468 ◽  
Author(s):  
Bing LUO ◽  
Qimei CUI ◽  
Hui WANG ◽  
Xiaofeng TAO ◽  
Ping ZHANG
Keyword(s):  
Closed Form ◽  
Closed Form Solutions ◽  
Water Filling
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