eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
225
238
531817
Dynamic Instability of Visco-SWCNTs Conveying Pulsating Fluid Based on Sinusoidal Surface Couple Stress Theory
A Ghorbanpour Arani
aghorban@kashanu.ac.ir
1
R Kolahchi
2
M Jamali
3
M Mosayyebi
4
I Alinaghian
5
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran--- Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, University of Kashan, Kashan, Iran
In this study, a realistic model for dynamic instability of embedded single-walled nanotubes (SWCNTs) conveying pulsating fluid is presented considering the viscoelastic property of the nanotubes using Kelvin–Voigt model. SWCNTs are placed in longitudinal magnetic fields and modeled by sinusoidal shear deformation beam theory (SSDBT) as well as modified couple stress theory. The effect of slip boundary condition and small size effect of nano flow are considered using Knudsen number. The Gurtin–Murdoch elasticity theory is applied for incorporation the surface stress effects. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The motion equations are derived based on the Hamilton's principle. The differential quadrature method (DQM) in conjunction with Bolotin method is used in order to calculate the dynamic instability region (DIR) of visco-SWCNTs induced by pulsating fluid. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface stress and fluid velocity on the dynamic instability of SWCNTs. The results depict that increasing magnetic field and considering surface effect shift DIR to right. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems.
http://jsm.iau-arak.ac.ir/article_531817_2b4c0bbf3ab33ab0c6fe603cbcc9b367.pdf
Dynamic instability
Pulsating fluid
Visco-SWCNTs
Surface effect
Modified couple stress theory
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
239
248
531818
Evaluation of Buckling and Post Buckling of Variable Thickness Shell Subjected to External Hydrostatic Pressure
A.R Ghasemi
ghasemi@kashanu.ac.ir
1
M.H Hajmohammad
2
Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
Faculty of Mechanical Engineering, Department of Solid Mechanics, University of Kashan, Kashan, Iran
Buckling and post buckling of cylindrical shells under hydrostatic pressure is regarded as important issue in structure of submarines. These cylindrical shells have variable thickness due to construction process which effected by pressure of buckling and its destruction. In this paper, effects of changing thickness on buckling and destruction pressure under external hydrostatic pressure of a shell are studied. Results of buckling pressure of cylindrical shell have been obtained with theoretical relations and finite element method. Then, using machining process a sample of cylindrical shell with variable thickness has been produced. Buckling pressure and post buckling of the constructed sample have been obtained with the reservoir under closed-ended hydrostatic pressure. Changes of the test sample size have been considered with closed-ended testing apparatuses which are used for new evaluation of buckling. In this research, results of the pressure have been obtained in terms of the volume change. At the end, results of the finite element method have been compared with results of the analytical solutions and experimental data. Results show that the shell with variable thickness has buckling pressure close to shell bucking pressure with mean thickness.
http://jsm.iau-arak.ac.ir/article_531818_b9c01dae275ee843b5182dc7418228e9.pdf
External pressure
Buckling, Cylindrical shell
Variable thickness
Post buckling
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
249
262
531819
Damping Ratio in Micro-Beam Resonators Based on Magneto-Thermo-Elasticity
A Khanchehgardan
a.khanchehgardan@gmail.com
1
G Rezazadeh
2
A Amiri
amiri.ahd@gmail.com
3
Mechanical Engineering Department, Urmia University, Urmia, Iran
Mechanical Engineering Department, Urmia University, Urmia, Iran
Mechanical Engineering Department, Urmia University, Urmia, Iran
This paper investigates damping ratio in micro-beam resonators based on magneto-thermo-elasticity. A unique aspect of the present study is the effect of permanent magnetic field on the stiffness and thermo-elastic damping of the micro resonators. In our modeling the theory of thermo-elasticity with interacting of an externally applied permanent magnetic field is taken into account. Combined theoretical and numerical studies investigate the permanent magnetic field effect on the damping ratio in clamped-clamped and cantilever micro-beams. Furthermore, the influence of the magnetic field intensity on the frequency of the micro-beams with thermo-elastic damping effect is evaluated. Such evaluations are used to determine the influence of magnetic field on the vibration amplitude of the resonators. The meaningful conclusion is that the magnetic field increases the equivalent stiffness and thermo-elastic damping and consequently the energy consumption of the resonators.
http://jsm.iau-arak.ac.ir/article_531819_9d9e1d0e9b535cc2896780f18998b5ba.pdf
Thermo-elasticity
MEMS
Maxwell stress tensor
Lorentz force
Magnetic Field
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
263
275
531820
Stress Waves in a Generalized Thermo Elastic Polygonal Plate of Inner and Outer Cross Sections
R Selvamani
selvam1729@gmail.com
1
Department of Mathematics, Karunya University, Coimbatore-641 114, Tamil Nadu, India
The stress wave propagation in a generalized thermoelastic polygonal plate of inner and outer cross sections is studied using the Fourier expansion collocation method. The wave equation of motion based on two-dimensional theory of elasticity is applied under the plane strain assumption of generalized thermoelastic plate of polygonal shape, composed of homogeneous isotropic material. The frequency equations are obtained by satisfying the irregular boundary conditions along the inner and outer surface of the polygonal plate. The computed non-dimensional wave number and wave velocity of triangular, square, pentagonal and hexagonal plates are given by dispersion curves for longitudinal and flexural antisymmetric modes of vibrations. The roots of the frequency equation are obtained by using the secant method, applicable for complex roots.
http://jsm.iau-arak.ac.ir/article_531820_d6654b66378824f5de9b30901b279f81.pdf
Waves in thermal plate
Piezoelectric plate
Layered plate
collocation method
Thermal relaxation times
Temperature sensors
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
276
290
531821
A New Finite Element Formulation for Buckling and Free Vibration Analysis of Timoshenko Beams on Variable Elastic Foundation
A Mirzabeigy
1
M Haghpanahi
2
R Madoliat
madoliat@iust.ac.ir
3
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran--- Young Researchers and Elite Club, Kermanshah Branch, Islamic Azad University, Kermanshah, Iran
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran
In this study, the buckling and free vibration of Timoshenko beams resting on variable elastic foundation analyzed by means of a new finite element formulation. The Winkler model has been applied for elastic foundation. A two-node element with four degrees of freedom is suggested for finite element formulation. Displacement and rotational fields are approximated by cubic and quadratic polynomial interpolation functions, respectively. The length of the element is assumed to be so small, so that linear variation could be considered for elastic foundation through the length of the element. By these assumptions and using energy method, stiffness matrix, mass matrix and geometric stiffness matrix of the proposed beam element are obtained and applied to buckling and free vibration analysis. Accuracy of obtained formulation is approved by comparison with the special cases of present problem in other studies. Present formulation shows faster convergence in comparison with conventional finite element formulation. The effects of different parameters on the stability and free vibration of Timoshenko beams investigated and results are completely new.
http://jsm.iau-arak.ac.ir/article_531821_521227bfafc59bc1b71f69959c575b3e.pdf
Buckling
Vibration
Timoshenko beam
Variable elastic foundation
Finite element formulation
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
291
301
531822
Experimental and Numerical Simulation Investigation on Crushing Response of Foam-Filled Conical Tubes Stiffened with Annular Rings
M.J Rezvani
m.rezvani@semnaniau.ac.ir
1
Department of Mechanical Engineering, Semnan Branch, Islamic Azad University, Semnan, Iran
In this paper, crashworthiness characteristics of conical steel tubes stiffened by annular rings and rigid polyurethane foam are investigated. For this purpose, wide circumferential rings are created from the outer surface of the conical tube at some determined areas along tube length. In fact, this method divides a long conical tube into several tubes of shorter length. When this structure is subjected to axial compression, folds are shaped within the space of these annular rings. In this study, several numerical simulations using ABAQUS 5.6 finite element explicit code are carried out to study of crashworthiness characteristics of the empty and the foam-filled thin-walled conical tubes. In order to verify these numerical results, a series of quasi-static axial compression tests are performed. Moreover, load-displacement curves, deformation mechanism of the structure, energy absorption, crush force efficiency (CFE), initial peak load with different number of rings are described under axial compression. The results show that a conical tube with stiff rings as a shock absorber could be improved or adjusted the crushing mode of deformation and energy absorption ability.
http://jsm.iau-arak.ac.ir/article_531822_6170d78b5d97b5ce8bd0dccc6e9e835e.pdf
Conical
Annular rings
Polyurethane Foam
energy absorption
Maximum crushing load
CFE
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
302
318
531823
Thermal Creep Analysis of Functionally Graded Thick-Walled Cylinder Subjected to Torsion and Internal and External Pressure
S Sharma
sanjiit12@rediffmail.com
1
S Yadav
2
R Sharma
3
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India
Department of Mathematics, Jaypee Institute of Information Technology, Noida, India
Department of Mathematics, School of Basic Sciences and Research Sharde University , Greater Noida, India
Safety analysis has been done for the torsion of a functionally graded thick-walled circular cylinder under internal and external pressure subjected to thermal loading. In order to determine stresses the concept of Seth’s transition theory based on generalized principal strain measure has been used. This theory simpliﬁes the set of mechanical equations by mentioning the order of the measure of deformation. This theory helps to achieve better agreement between the theoretical and experimental results. Results have been analyzed with or without thermal effects for functionally graded and homogeneous cylinder with linear and nonlinear strain measure.From the analysis, it has been concluded that in creep torsion cylinder made up of less functionally graded material (FGM) under pressure is better choice for designing point of view as compared to homogeneous cylinder. This is due to shear stresses which are maximum for cylinder made up of functionally graded material as compared to homogeneous material.
http://jsm.iau-arak.ac.ir/article_531823_85fcd51d7247de27dfeaca6e0a6f6b67.pdf
Thermal
Creep
Torsion
Strain measure
Functionally graded material
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
319
337
531824
Vibration Analysis of a Rotating Nanoplate Using Nonlocal Elasticity Theory
M Ghadiri
ghadiri@eng.ikiu.ac.ir
1
N Shafiei
2
S Hossein Alavi
3
Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran
Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran
School of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
The nanostructures under rotation have high promising future to be used in nano-machines, nano-motors and nano-turbines. They are also one of the topics of interests and it is new in designing of rotating nano-systems. In this paper, the scale-dependent vibration analysis of a nanoplate with consideration of the axial force due to the rotation has been investigated. The governing equation and boundary conditions are derived using the Hamilton’s principle based on nonlocal elasticity theory. The boundary conditions of the nanoplate are considered as free-free in y direction and two clamped-free (cantilever plate) and clamped-simply (propped cantilever) in <em>x</em> direction. The equations have been solved using differential quadrature method to determine natural frequencies of the rotating nanoplate. For validation, in special cases, it has been shown that the obtained results coincide with literatures. The effects of the nonlocal parameter, aspect ratio, hub radius, angular velocity and different boundary conditions on the first three frequencies have been investigated. Results show that vibration behavior of the rotating nanoplate with cantilever boundary condition is different from other boundary conditions.
http://jsm.iau-arak.ac.ir/article_531824_c4e4e72f55b3a3a2cde7fda2f9b20ed3.pdf
Rotating nanoplate
Cantilever nanoplate
Propped cantilever nanoplate
Nonlocal elasticity theory
DQM
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
338
350
531825
Axisymmetric Buckling Analysis of Porous Truncated Conical Shell Subjected to Axial Load
M Zarghami Dehaghani
1
M Jabbari
m_jabbari@azad.ac.ir
2
Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University, Tehran, Iran
This paper studied Buckling analysis of porous truncated conical shell subjected to axial load. It is considered that a fluid undrained between porous material and the Porous material properties vary across the thickness of shell with a specific function also assumed that the edge of the shell is simply supported. The governing equations are based on the Sanders kinematics equations and the first-order shell theory and by using of variational formulations. The general mechanical non-linear equilibrium and linear stability equations are derived. At the end, the result of dimensionless buckling critical load ratio dimensionless thickness in different condition such as variation in thickness, porosity and angle of conical shell is investigated. The mechanical load results are verified by the known results in the literature.
http://jsm.iau-arak.ac.ir/article_531825_6ecc21691f185295dadad4cdaa20c266.pdf
Axisymmetric
Porous material
Buckling analysis
Conical shell
Axial load
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
351
369
531826
Finite Element Instability Analysis of the Steel Joist of Continuous Composite Beams with Flexible Shear Connectors
H Bakhshi
h.bakhshi@hsu.ac.ir
1
H.R Ronagh
2
Engineering Faculty, Hakim Sabzevari University, Sabzevar, Iran
Instiutute for Infrastructure Engineering, Western Sydney University, Penrith, NSW 2751, Australia
Composite steel/concrete beams may buckle in hogging bending regions. As the top flange of I-beam in that arrangement is restricted from any translational deformation and twist, the web will distort during buckling presenting a phenomenon often described as restricted distortional buckling. There are limited studies available in the literature of restricted distortional buckling of composite steel/concrete I-beams subjected to negative or hogging bending. There is none however that includes the effect of partial shear interaction. In this paper, finite element models for in-plane analysis and out-of-plane buckling of continuous composite I-beams including the effects of partial shear interaction are presented.
http://jsm.iau-arak.ac.ir/article_531826_d4868ada6bc9b08fbee745ca9fe91a1d.pdf
Finite element model
In-plane analysis
Out-of-plane buckling
Restricted distortional buckling
Partial shear interaction
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
370
383
531827
Design and Analysis of Graded Honeycomb Shock Absorber for Increasing the Safety of Passengers in Armored Vehicles Exposed to Mine Explosion
S. A Galehdari
ali.galehdari@gmail.com
1
H Khodarahmi
2
A Atrian
3
Modern Manufacturing Technologies Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Department of Mechanical Engineering, I H University, Tehran, Iran
Modern Manufacturing Technologies Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran
Protecting armored vehicles from mine explosion can lead to the survival of thousands of people exposed to this risk. Very purpose, shock absorbers such as honeycomb structures can be applied for crashworthiness improvement. In this study, graded honeycomb structure is primarily introduced as a shock absorber, followed by the introduction of its absorbed energy and the force and acceleration applied to the occupant which is numerically simulated and measured in Abaqus software. In order to validate the numerical simulation results, a low-velocity experimental test has been conducted on a prototype, and the obtained results indicate well agreement with empirical results. In the meanwhile of mine explosion under a armored vehicle, it has been considered that the vehicle will be thrown upwards with a velocity of 10 m/s and will hit to the ground thereafter. For this case, a shock absorber has been designed, optimized and analyzed. According to the obtained results, the designed shock absorber meets all of the standard requirements. The applied simulation and design method can be further extended for miscellaneous shock absorbers.
http://jsm.iau-arak.ac.ir/article_531827_136104d4c4ab2879a6f35d218ab8b188.pdf
Graded honeycomb structure (GHS)
shock absorber
Armored vehicle
Mine explosion
crashworthiness
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
384
395
531828
Transient Nonlinear Vibration of Randomly Excited Cylindrical Shallow Panels in Non Aging Viscous Medium
A Asnafi
asnafi@shirazu.ac.ir
1
Hydro-Aeronautical Research Center, Shiraz University, Shiraz, Iran
<span>In this paper, the nonlinear transient vibration of a cylindrical shallow panel under lateral white noise excitation is studied. The panel is in contact with a non aging viscoelastic medium. Since the external load is a time varying random wide band process, deterministic and conventional approaches cannot be used. Instead, the evolution of the probability density function of the response is investigated. To compute the density function, the famous Monte Carlo simulation is employed while its correctness for this specific application is validated with another work in literature. The governing equation is rewritten to a non dimensional format; so that the results can be applied to a wide range of panels. Specifically, the transient behavior is investigated with respect to the quasi slenderness ratio and the non dimensional mean value of lateral load. As expected, both the simple damped oscillation and unstable jumping phenomenon are seen relative to the values of prescribed parameters. Finally, the joint probability density function of the response is drawn that give someone an idea about the quality of the response in the phase plane. </span>
http://jsm.iau-arak.ac.ir/article_531828_0647a4c7b6ee5e056f9226f5e7e76fe0.pdf
Circular panels
Shallow shells, Monte carlo simulation
Random vibration
Non aging materials
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
396
410
531829
Static and Free Vibration Analyses of Orthotropic FGM Plates Resting on Two-Parameter Elastic Foundation by a Mesh-Free Method
H Momeni-Khabisi
h.momeni@ujiroft.ac.ir
1
Department of Mechanical Engineering, University of Jiroft , Jiroft , Iran
In this paper, static and free vibrations behaviors of the orthotropic functionally graded material (FGM) plates resting on the two-parameter elastic foundation are analyzed by the a mesh-free method based on the first order shear deformation plate theory (FSDT). The mesh-free method is based on moving least squares (MLS) shape functions and essential boundary conditions are imposed by transfer function method. The orthotropic FGM plates are made of two orthotropic materials and their volume fractions are varied smoothly along the plate thickness. The convergence of the method is demonstrated and to validate the results, comparisons are made with finite element method (FEM) and the others available solutions for both homogeneous and FGM plates then numerical examples are provided to investigate the effects of material distributions, elastic foundation coefficients, geometrical dimensions, applied force and boundary conditions on the static and vibrational characteristics of the orthotropic FGM plates.
http://jsm.iau-arak.ac.ir/article_531829_eb9114628e2e853c9b9b60ae752c649f.pdf
Static
Free Vibration
FGM plate
Orthotropic
Mesh-Free
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
411
422
531830
Reflection of Plane Wave at Traction-Free Surface of a Pre-Stressed Functionally Graded Piezoelectric Material (FGPM) Half-Space
P.K Saroj
pksaroj.ism@gmail.com
1
S.A Sahu
2
Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad-826004,India
Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad-826004,India
This paper is devoted to study a problem of plane waves reflection at a traction-free surface of a pre-stressed functionally graded piezoelectric material (FGPM). The effects of initial stress and material gradient on the reflection of plane waves are studied in this paper. Secular equation has been derived analytically for the pre-stressed FGPM half-space and used to show the existence of two coupled waves namely and Continuity condition of stress, electrical potential and electrical displacement at traction free surface is used to obtain the reflection coefficient of and waves. Results of the problem are shown graphically and effects of initial stress and material gradient are discussed for a particular case of Lithium niobate material.
http://jsm.iau-arak.ac.ir/article_531830_6454ea2284da610fed383a953c28ef91.pdf
Piezoelectricity
Reflection
Traction-free surface
Reflection coefficient
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
423
433
531831
Variational Principle and Plane Wave Propagation in Thermoelastic Medium with Double Porosity Under Lord-Shulman Theory
R Kumar
rajneesh_kuk@rediffmail.com
1
R Vohra
richavhr88@gmail.com
2
M.G Gorla
3
Department of Mathematics, Kurukshetra University, Kurukshetra, Haryana, India
Department of Mathematics& Statistics, H.P.University, Shimla, HP, India
Department of Mathematics& Statistics, H.P.University, Shimla, HP, India
The present study is concerned with the variational principle and plane wave propagation in double porous thermoelastic infinite medium. Lord-Shulman theory [2] of thermoelasticity with one relaxation time has been used to investigate the problem. It is found that for two dimensional model, there exists four coupled longitudinal waves namely longitudinal wave (P), longitudinal thermal wave (T), longitudinal volume fractional wave corresponding to pores (PV1), and longitudinal volume fractional wave corresponding to fissures (PV2), in addition to, a transverse wave (S) which is not affected by the volume fraction fields and thermal properties. The different characteristics of the wave such as phase velocity and attenuation quality factor are computed numerically and depicted graphically. Some special cases are also deduced from the present investigation.
http://jsm.iau-arak.ac.ir/article_531831_4204e81a530f1f107875455364b773ae.pdf
Double porosity
Lord-shulman theory
Variational principle
Plane waves
Phase velocity
Attenuation quality factor
eng
Islamic Azad University Arak Branch
Journal of Solid Mechanics
2008-3505
2008-7683
2017-06-30
9
2
434
444
531832
Theoretical, Numerical and Experimental Investigation on Formability of Al3105-St14 Two-Layer Sheet
H Deilami Azodi
hdazodi@arakut.ac.ir
1
R Darabi
2
Department of Mechanical Engineering, Arak University of Technology, P.O.Box 38135-1177, Arak, Iran
Department of Mechanical Engineering, Arak University of Technology, P.O.Box 38135-1177, Arak, Iran
Two-layer metallic sheets have wide applications in various industries due to their superlative characteristics. This paper presents analytical model to investigate the formability of two-layer sheets based on Marciniak and Kuckzinsky (M-K) method using Barlat and Lian non-quadratic yield criterion. FEM simulation is also performed to calculate the forming limits based on bifurcation theory. Forming limit diagrams (FLDs) and forming limit stress diagrams (FLSDs) determined by analytical and numerical approaches are compared with experimental results of Al3105-St14 two-layer sheet to verify the validity of theoretical models. The formability of two-layer sheet is also compared with the formability of its components. The results show that the forming limit diagram of two-layer sheet is located between the FLDs of separate layers. The effects of the anisotropy and the orientation of layers on formability of two-layer sheet are studied. The higher formability will be achieved in the case of coincidence of rolling directions of layers.
http://jsm.iau-arak.ac.ir/article_531832_3b6b90d87e73b0c83dd4b5a143229a3e.pdf
Two-layer metallic sheet
Forming limit diagram (FLD)
Forming limit stress diagram (FLSD)
M-K theory
Bifurcation theory