Symbolic Algebra and Theorem Proving for Failure Criteria Reduction

2011 ◽  
Author(s):  
John G. Michopoulos ◽  
Athanasios Iliopoulos
Keyword(s):  
Theorem Proving ◽  
Strain Energy ◽  
Failure Criteria ◽  
Orthotropic Materials ◽  
Structural Failure ◽  
Symbolic Algebra ◽  
Symbolic Computing ◽  
Strain Energy Density Function ◽  
Bner Basis ◽  
Strain Space

The present paper reports on recent efforts of utilizing symbolic computing for identifying failure criteria cross reducibility from the perspective of theorem proving. Utilizing equational theorem proving algorithms and Gro¨bner Basis polynomial theorem provers implemented in Mathematica we have proven a number of interesting theorems related to the area of structural failure criteria for anisotropic and particularly orthotropic materials. The main contribution of this work is the demonstration of the tremendous utility of symbolic algebra for engineering applications as well as the demonstration of the idea that all failure criteria presented in the literature up to know can be proven under certain conditions to be special forms of general criteria relating to the strain energy density function associated with material continua. Two specific examples are presented and discussed along with a theorem proving the existence of a dual form of all stress space based criteria to equivalent one expressed in strain space.

The strain energy density function

1986 ◽  
pp. 237-253
Author(s):  
G. C. Sih ◽  
J. G. Michopoulos ◽  
S. C. Chou
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Strain Energy Density Function

scholarly journals Analisis Numerik Part Bulkhead Pada Sub System Wing To Fuselage Joinner Assembly Pesawat Aerobatik Menggunakan Metode Elemen Hingga

2021 ◽  
Vol 6 (1) ◽  
Author(s):  
Enggar Kristian ◽  
Agus Suprianto ◽  
Nurhadi Pramana ◽  
Sahril Afandi ◽  
Endah Yuniarti
Keyword(s):  
Incidence Angle ◽  
Failure Criteria ◽  
Topological Optimization ◽  
Structural Failure ◽  
Topological Approach ◽  
Approach Method ◽  
Margin Of Safety ◽  
Problem Solving Process ◽  
Aluminum Alloy 7075 ◽  
Geometric Variation

Analisis rancangan bulkhead dilakukan untuk memperoleh geometri terbaik untuk mencari berat yang efisien dengan mengubah geometri bentuk pada bulkhead yang merupakan sub system wing to fuselage untuk pesawat berkategori aerobatik dan berat yang optimal yang memenuhi persyaratan regulasi FAR 23 dan mengetahui respon distribusi tegangan, bending yang dihasilkan  dan kriteria kegagalan struktur berdasarkan variasi geometri bentuk bulkhead. Pada penelitian ini untuk analisis statik bulkhead untuk pesawat berkategori aerobatik menggunakan material Aluminium Alloy 7075-T6 dan menggunakan metode pendekatan Schrenk untuk menghitung beban eksternal distrbusi gaya angkat pada sayap. Selain itu dilakukan proses optimisasi berat bulkhead berdasarkan metode pendekatan topologi yaitu perubahan geometri bentuk pada bulkhead untuk mereduksi berat, sudut insiden spar yang berbeda dan menghitung magin of safety. Proses penyelesaian masalah menggunakan perangkat lunak metode elemen hingga (Abaqus CAE). Optimisasi topologi pada part bulkhead sudut insidet 0° dan 4° menghasilkan volume yang berkurang pada benda sehingga mereduksi berat, tetapi nilai dari margin of safety MS = 0. The bulkhead design analysis was carried out to obtain the best geometry to find an efficient weight by changing the shape geometry of the bulkhead which is a sub-system of the wing to the fuselage for an aircraft categorized as aerobatics and an optimal weight that meets the requirements of FAR 23 regulations and sees the stress distribution response, the resulting bending and structural failure criteria based on the geometric variation of bulkhead shapes. In this study, to analyze the bulkhead static for an aerobatic category aircraft using Aluminum Alloy 7075-T6 material and using the Schrenk Approximation method to calculate the external distribution load of lift force on the wing. In addition, the optimization of bulkhead weight based on the topological approach method is to change the shape geometry of the bulkhead to reduce weight, in different spar incidents and calculate margin of safety. The problem solving process uses finite element method software (Abaqus CAE). Topological optimization of the bulkhead part with an incidence angle of 0 ° and 4 ° results in a reduced volume of the object so that it reduces weight, but the value of the margin of safety MS = 0.  

Research Status of Concrete Strength Theory Based on Stress and Strain

2014 ◽  
Vol 670-671 ◽  
pp. 445-448
Author(s):  
Guo Jun Liu
Keyword(s):  
Failure Criterion ◽  
Concrete Strength ◽  
Failure Criteria ◽  
Practical Significance ◽  
Engineering Structures ◽  
Research Status ◽  
Advantages And Disadvantages ◽  
Strength Failure ◽  
Micro Cracks ◽  
Strain Space

In the civil engineering structures, when concrete structure withstand external loads, internal defects such as micro-cracks gradually developed, eventually will lead to the destruction of the concrete. Currently, there are two types of strength failure criterion of concrete, strength failure criterion based on stress-space and strength failure criterion based on strain-space, both of which have advantages and disadvantages, this paper introduces the research status of the two strength failure criteria of concrete, and the problems need further study in the future. It plays a strong practical significance in scientific research.

A Micromechanically Based Constitutive Model for the Inelastic and Swelling Behaviors in Double Network Hydrogels

2015 ◽  
Vol 83 (2) ◽  
Author(s):  
Yin Liu ◽  
Hongwu Zhang ◽  
Yonggang Zheng
Keyword(s):  
Constitutive Model ◽  
Strain Energy ◽  
Polymer Chains ◽  
Loading Path ◽  
Induced Anisotropy ◽  
Double Network ◽  
Strain Energy Density Function ◽  
Damage Process ◽  
Single Polymer Chain ◽  
Swelling Behaviors

This paper presents a micromechanically based constitutive model within the framework of the continuum mechanics to characterize the inelastic elastomeric and swelling behaviors of double network (DN) hydrogels, such as the stress-softening, necking instability, hardening, and stretch-induced anisotropy. The strain-energy density function of the material is decomposed into two independent contributions from the tight and brittle first network and the soft and loose second network, each of which is obtained by integrating the strain energy of one-dimensional (1D) polymer chains in each direction of a unit sphere. The damage process is derived from the irreversible breakages of sacrificial chains in the first network and characterized by the directional stretch-dependent evolution laws for the equivalent modulus and the locking stretch in the non-Gauss statistical model of a single polymer chain. The constitutive model with the optimized-material evolution law predicts stress–stretch curves in a good agreement with the experimental results during loading, unloading, and reloading paths for both ionic and covalent DN hydrogels. The deformation-induced anisotropy is investigated and demonstrated by the constitutive model for the free swelling of damaged specimen. The constitutive model is embedded into the finite-element (FE) procedure and proved to be efficient to model the necking and neck propagation in the plane-strain uniaxial elongation. Based on the procedure, the effects of imperfection and boundary conditions on the loading path and the material evolution during different stages of deformation are investigated.

On Modeling and Simulation of Innovative Ship Rescue System

2018 ◽  
Vol 140 (6) ◽  
Author(s):  
Ilias Zilakos ◽  
Michael Toulios
Keyword(s):  
Strain Energy ◽  
Structural Response ◽  
Dry Land ◽  
Strain Energy Density Function ◽  
Material Evaluation ◽  
Element Simulation ◽  
Rescue System ◽  
Funded Project ◽  
Elasticity Tensors ◽  
Good Agreement

Inflatable devices that provide reserve buoyancy to damaged ships, preventing capsizing and/or sinking, along with lifting wreckages from the seabed, were studied within the framework of the European funded project “SuSy” (Surfacing System for Ship Recovery). Part of the work involved material evaluation and testing as well as simulations of the structural response. This paper first describes an orthotropic hyperelastic constitutive model for a candidate material also used in the fabrication of prototype inflatable devices. A strain energy density function is proposed that is further used to derive the stress and elasticity tensors required for the numerical implementation of the model in the user-defined subroutine (UMAT) of abaqus/standard. The second part of the paper presents the finite element simulation of the latter stages of inflation of two salvage devices inside an actual double bottom structure. The numerical results are in good agreement with tests conducted in dry land and under water, with the structure raised following the inflation of the devices. The evolving stress state in both the devices and the double bottom structure under increased contact interaction leads to useful conclusions for future use in the development of this salvage system.

An adaptive meshing automatic scheme based on the strain energy density function

1997 ◽  
Vol 14 (6) ◽  
pp. 604-629 ◽  
Author(s):  
A. Hernández ◽  
J. Albizuri ◽  
M.B.G. Ajuria ◽  
M.V. Hormaza
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Adaptive Meshing ◽  
Strain Energy Density Function

Large Deformation Analysis of the Arterial Cross Section

1971 ◽  
Vol 93 (2) ◽  
pp. 138-145 ◽  
Author(s):  
B. R. Simon ◽  
A. S. Kobayashi ◽  
D. E. Strandness ◽  
C. A. Wiederhielm
Keyword(s):  
Finite Element ◽  
Energy Density ◽  
Numerical Integration ◽  
Density Function ◽  
Cross Section ◽  
Finite Deformation ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Exponential Form ◽  
Strain Energy Density Function

Possible relations between arterial wall stresses and deformations and mechanisms contributing to atherosclerosis are discussed. Necessary material properties are determined experimentally and from available data in the literature by assuming the arterial response to be a static finite deformation of a thick-walled cylinder constrained in a state of plane strain and composed of an incompressible, nonlinear elastic, transversely isotropic material. Experimental justification from the literature and supporting theoretical considerations are presented for each assumption. The partial derivative of the strain energy density function δW1/δI , necessary for in-plane stress calculation, is determined to be of exponential form using in situ biaxial test results from the canine abdominal aorta. An axisymmetric numerical integration solution is developed and used as a check for finite element results. The large deformation finite element theory of Oden is modified to include aortic material nonlinearity and directional properties and is used for a structural analysis of the aortic cross section. Results of this investigation are: (a) Fung’s exponential form for the strain energy density function of soft tissues is found to be valid for the aorta in the biaxial states considered; (b) finite deformation analyses by the finite element method and numerical integration solution reveal that significant tangential stress gradients are present in arteries commonly assumed to be “thin-walled” tubes using linear theory.

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