A note on incremental equilibrium equations and approximate constitutive relations in large inelastic deformations

1974 ◽  
Vol 19 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
P. K. Larsen ◽  
E. P. Popov
Keyword(s):  
Constitutive Relations ◽  
Equilibrium Equations

In-Plane Nonlinear Buckling of Funicular Arches

2015 ◽  
Vol 15 (05) ◽  
pp. 1450073 ◽  
Author(s):  
Jianbei Zhu ◽  
Mario M. Attard ◽  
David C. Kellermann
Keyword(s):  
Numerical Technique ◽  
Constitutive Relations ◽  
Finite Strains ◽  
Equilibrium Path ◽  
Shear Rigidity ◽  
Equilibrium Equations ◽  
Buckling Mode ◽  
Nonlinear Buckling ◽  
Shear Component ◽  
Longitudinal Stretch

This paper presents a numerical technique to determine the full pre-buckling and post-buckling equilibrium path for elastic funicular arches. The formulation includes the effects of shear deformations and geometric nonlinearity due to large deformations. The Timoshenko beam hypothesis is adopted for incorporating shear. Finite strains are considered without approximation. The finite strains are defined in terms of the normal and shear component of the longitudinal stretch. The constitutive relations for the internal actions are based on a hyperelastic constitutive model. Using the differential equilibrium equations and the constitutive laws, the nonlinear buckling behavior of some typical funicular arches are investigated using the trapezoid method with Richardson extrapolation enhancement. The results are validated by using the finite element package ANSYS and solutions available in the literature. Examples include parabolic arches under a uniformly distributed gravity load, a catenary under a distributed load along the arch and a catenary arch under an overburden load. Parametric studies are performed to identify the factors that influence the nonlinear buckling of funicular arches. The axial to shear rigidity ratio is shown to have a significant effect on the buckling load and the buckling mode shape.

Structural Behavior of Umbilicals: Part I—Mathematical Modeling

2010 ◽  
Author(s):  
Celso Pupo Pesce ◽  
Roberto Ramos ◽  
Lauro Massao Yamada da Silveira ◽  
Rafael Loureiro Tanaka ◽  
Clo´vis de Arruda Martins ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Cross Sections ◽  
Constitutive Relations ◽  
Development Program ◽  
Structural Behavior ◽  
Local Analysis ◽  
Equilibrium Equations ◽  
Steel Tubes ◽  
Single Structure ◽  
Geometric Compatibility

Umbilicals for offshore application are very complex, since they combine elements of different mechanical behavior, such as steel tubes, thermoplastic hoses and power cores in a single structure, not to mention helically laid-up armouring layers and polymeric sheathes. This motivates continuous research on their mechanical modeling. This paper presents research undertaken in the structural behavior of umbilicals and focus on the mathematical modeling of the elements, which are gathered into concentric layers. Interaction between layers is included as well as helical lay-up of elements. The model here presented will be compared to a set of experimental results in a separate paper (Part II). An analytical model was developed combining equilibrium equations, geometric compatibility and constitutive relations to obtain a set of equations that describe the umbilical behavior under external loads. This set of equations is numerically solved to obtain contact pressures (or gaps) among layers, radial variations and strains in the elements (used to calculate the stresses). The model was built to be general in order to be able to cope with complex cross-sections designs often encountered. It was then implemented in a tailor made local analysis software, called UTILFLEX®. Both modeling and software resulted from a development program partnership between Prysmian Cables & Systems and University of Sa˜o Paulo. The paper will present modeling hypotheses and structural models that were used for steel tubes, hoses (thermoplastic and reinforced) and power cables and how the interaction among them has been treated.

Combined polarization field gradient and strain field gradient effects in elastic flexoelectric materials

2015 ◽  
Vol 22 (5) ◽  
pp. 938-951 ◽  
Author(s):  
K Enakoutsa ◽  
D Del Vescovo ◽  
D Scerrato
Keyword(s):  
Field Gradient ◽  
Strain Gradient ◽  
Strain Field ◽  
Constitutive Relations ◽  
Cylindrical Tube ◽  
Polarization Field ◽  
Equilibrium Equations ◽  
Elastic Energy Density ◽  
Gradient Fields ◽  
Gradient Effects

We consider a constitutive model for the behavior of elastic flexoelectric materials including strain gradient fields and polarization gradient fields. This model is based on a stored elastic energy density function which depends on four independent variables: the polarization field and the polarization field gradient as well as the strain field and the strain field gradient. A generalized Toupin variational approach is utilized to find the governing equations (constitutive relations, equilibrium equations and boundary conditions) of the material. The present model is then applied to the problem of a thick walled cylindrical tube of elastic isotropic flexoelectric material, subjected to axisymmetric loading. The resulting radial displacement field noticeably differs from the elastic and strain gradient elastic cases.

On Saint-Venant's problem for an elastic strip

1988 ◽  
Vol 110 (1-2) ◽  
pp. 161-181 ◽  
Author(s):  
Alexander Mielke
Keyword(s):  
Constitutive Relations ◽  
Solution Space ◽  
Elliptic Systems ◽  
Constitutive Law ◽  
Whole Body ◽  
Infinite Strip ◽  
Dimensional Manifold ◽  
Field Equations ◽  
Equilibrium Equations ◽  
Dimensional Solution

SynopsisThe equilibrium equations for elastic deformations of an infinite strip are considered. Under the assumption of sufficiently small strains along the whole body, it is shown that all solutions lie on a six-dimensional manifold. This is achieved by rewriting the field equations as a differential equation in a function spaceover the cross-section, the axial variable taken as time. Then the theory of centre manifolds for elliptic systems applies. Thus the local Saint-Venant's problem is solved. Moreover, the structure of the finite-dimensional solution space is analysed to reveal exactly the two-dimensional rod equations of Kirchhoff. The constitutive relations for this rod model are calculated in a mathematically rigorous way out of the constitutive law of the material forming the strip.

Modeling of a Shape Memory Alloy Circular Bar Under Combined Axial-Torsional Loading

2012 ◽  
Author(s):  
Masood Taheri Andani ◽  
Amin Alipour ◽  
Ahmadreza Eshghinejad ◽  
Mohammad Elahinia
Keyword(s):  
Shape Memory Alloy ◽  
Shape Memory ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Torsional Loading ◽  
Equilibrium Equations ◽  
Dimensional Finite Element ◽  
Loading And Unloading ◽  
Circular Bar ◽  
Three Dimensional Finite Element

In this paper, a semi-analytical analysis of the pseudoelastic response of shape memory alloy rods and tubes subjected to combined axial and torsional loading is proposed. A three-dimensional phenomenological SMA constitutive model is simplified to obtain the corresponding two-dimensional constitutive relations. The rod is partitioned into a finite number of narrow annular regions and the equilibrium equations are found in each annular region for both loading and unloading paths. Several numerical examples are presented to demonstrate the efficiency of the proposed method, and the results are compared with three-dimensional finite element simulations.

scholarly journals Elastoplastic finite element analysis for wet multidisc brake during lasting braking

2015 ◽  
Vol 19 (6) ◽  
pp. 2205-2217 ◽  
Author(s):  
Zhanling Ji ◽  
Yunhua Li ◽  
Sujun Dong ◽  
Peng Zhang ◽  
Yunze Li
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Thermal Structure ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Contact Ratio ◽  
Equilibrium Equations ◽  
Start Time ◽  
Element Analysis ◽  
Dimensionless Stress

Addressed to serious heat degradation problem of the braking continuously performed in the drag brake application for a long time, finite element analysis for bidirectional thermal-structure coupling is adopted to investigate temperature and stress when material properties are temperature-dependent. Based on the constitutive relations of heat transfer and strain-stress, three-dimensional transient finite element equilibrium equations with many kinds of boundary conditions for bidirectional thermal-structure coupling were derived. And it was originally presented that start time, location, severity and evolution laws of plastic deformation were depicted using dimensionless stress distribution contour with the yield limit related to temperature. The change laws of plastic element number and contact area versus braking time were expressed by plasticity ratio and contact ratio curves, respectively. The laws revealed by the numerical calculation results are in accordance with the objective perception and reasoning.

Using phase field and third-order shear deformation theory to study the effect of cracks on free vibration of rectangular plates with varying thickness

2020 ◽  
Vol 71 (7) ◽  
pp. 853-867
Author(s):  
Phuc Pham Minh
Keyword(s):  
Free Vibration ◽  
Shear Deformation ◽  
Phase Field ◽  
Deformation Theory ◽  
Plate Thickness ◽  
Shear Deformation Theory ◽  
Rectangular Plates ◽  
Equilibrium Equations ◽  
Third Order ◽  
Free Vibration Frequency

The paper researches the free vibration of a rectangular plate with one or more cracks. The plate thickness varies along the x-axis with linear rules. Using Shi's third-order shear deformation theory and phase field theory to set up the equilibrium equations, which are solved by finite element methods. The frequency of free vibration plates is calculated and compared with the published articles, the agreement between the results is good. Then, the paper will examine the free vibration frequency of plate depending on the change of the plate thickness ratio, the length of cracks, the number of cracks, the location of cracks and different boundary conditions

Determination and prediction of the stress-strain state of pipeline, taking into account soil changes during operation

2020 ◽  
Vol 10 (4) ◽  
pp. 372-378
Author(s):  
Aydar К. Gumerov ◽  
◽  
Rinat M. Karimov ◽  
Robert М. Askarov ◽  
Khiramagomed Sh. Shamilov ◽  
...  
Keyword(s):  
Strain State ◽  
Mathematical Framework ◽  
Equilibrium Equations ◽  
Stress Strain ◽  
Optimal Method ◽  
Initial Stress State ◽  
Soil Response ◽  
Stress Strain State ◽  
Support Reaction ◽  
Key Factor

The key factor determining the strength, reliability, service life and fail-safe operation of the main pipeline is its stress-strain state. The purpose of this article is to develop a mathematical framework and methodology for calculating the stress-strain state of a pipeline section laid in complex geotechnical conditions, taking into account all planned and altitude changes and impacts at various points of operation, as well as during repair and after its completion. The mathematical framework is based on differential equations reflecting the equilibrium state of the pipeline, taking into account the features of the sections (configuration, size, initial stress state, acting forces, temperature conditions, interaction with soil, supports, and pipe layers). The equilibrium equations are drawn up in a curvilinear coordinate system – the same one that is used for in-pipe diagnostics. According to the results of the solution, all stress components are determined at each point both along the length of the pipeline and along the circumference of any section. At the same time, transverse and longitudinal forces, bending moments, shearing forces, pipeline displacements relative to the ground and soil response to displacements are determined. As an example, a solution is given using the developed mathematical framework. During the course of calculation, the places where the lower form of the pipe does not touch the ground and the places where the support reaction becomes higher than a predetermined limit are determined. A comparative analysis was accomplished, and the optimal method for section repair has been selected.

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