This document summarizes research characterizing damage in thin acrylonitrile butadiene styrene (ABS) plates under uniaxial loading. Experiments were conducted on smooth and notched ABS plate specimens. Results showed mechanical properties like ultimate stress and stiffness decreased with increasing notch diameter due to stress concentration. A damage model based on the unified damage theory was applied to quantify damage. Damage increased linearly with the fraction of life and reached a maximum of 0.9 at failure. Stress concentration factor was also examined, decreasing with increasing notch to width ratio. In conclusion, the experiments and modeling successfully characterized ABS plate damage and the effects of notches on stress concentration and mechanical properties.
Study of Damage to ABS Specimens Submitted To Uniaxial Loadingtheijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation.
High Strain Rate Testing of Materials
Polymers, composites and some metallic materials are viscoelastic and strain-rate sensitive. Under high strain rates the micro mechanisms by which these materials deform is different than that experienced at low strain rates. Consequently, use of quasi-static stress-strain data may not produce accurate and reliable predictions, when such data is used in simulation and Finite element analysis FEA of engineering components.
Design and analysis of Stress on Thick Walled Cylinder with and with out HolesIJERA Editor
The conventional elastic analysis of thick walled cylinders to final radial & hoop stresses is applicable for the internal pressures up to yield strength of material. The stress is directly proportional to strain up to yield point Beyond elastic point, particularly in thick walled cylinders. The operating pressures are reduced or the material properties are strengthened. There is no such existing theory for the stress distributions around radial holes under impact of varying internal pressure. Present work puts thrust on this area and relation between pressure and stress distribution is plotted graphically based on observations. Here focus is on pure mechanical analysis & hence thermal, effects are not considered. The thick walled cylinders with a radial cross-hole ANSYS Macro program employed to evaluate the fatigue life of vessel. Stresses that remain in material even after removing applied loads are known as residual stresses. These stresses occur only when material begins to yield plastically. Residual stresses can be present in any mechanical structure because of many causes. Residual stresses may be due to the technological process used to make the component. Manufacturing processes lead to plastic deformation. Elasto plastic analysis with bilinear kinematic hardening material is performed to know the effect of hole sizes. It is observed that there are several factors which influence stress intensity factors. The Finite element analysis is conducted using commercial solvers ANSYS & CATIA. Theoretical formulae based results are obtained from MATLAB programs. The results are presented in form of graphs and tables.
LITERATURE REVIEW ON FRACTURE TOUGHNESS AND IMPACT TOUGHNESSijiert bestjournal
The present paper gives a technical revi ew of fracture toughness,and impact toughness for metallic materials in terms of the linear elastic fracture mechanics. This includes the early investigations and recent advances of fracture toughness test methods and practices developed by various agencies and societies. The review describes the most important fracture mechanics parameters:such as the elastic energy release rate G,the stress intensity factor K,the J - integral,the crack - tip opening displacement (CTOD) and the crack - tip opening angle (CTOA) from the basic concept,definition,to,tes t methods. Attention is paid to guidelines on how to choose an appropriate fracture parameter to characterize fracture toughness for the material of interest,and how to measure the fracture toughness value defined either at a critical point or in a resist ance curve format using laboratory specimens. The effects of loading rate,temperature and crack - tip constraint on fracture toughness as well as fracture instability analysis are also reviewed.
There are currently three approaches to characterize and quantify the fatigue behaviour of composite laminates that are, Fatigue Life Modelling and Prediction, Phenomenological and Empirical Modelling, and Progressive Damage Modelling. These approaches constitute the evolution that is driven by ever expanding industrial needs and academic pursuit and assisted by perpetual technological advances in experimentation capabilities. In the first approach of Fatigue Life Modelling and Prediction the individual material degradation mechanisms are not directly concerned with, rather the determination of stress-life relationships based on experimental data is concerned with and the failure criteria or the residual strength determination is established based on these relationships, for the specific composite laminate.
Study of Damage to ABS Specimens Submitted To Uniaxial Loadingtheijes
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
The papers for publication in The International Journal of Engineering& Science are selected through rigorous peer reviews to ensure originality, timeliness, relevance, and readability.
Theoretical work submitted to the Journal should be original in its motivation or modeling structure. Empirical analysis should be based on a theoretical framework and should be capable of replication. It is expected that all materials required for replication (including computer programs and data sets) should be available upon request to the authors.
The International Journal of Engineering & Science would take much care in making your article published without much delay with your kind cooperation.
High Strain Rate Testing of Materials
Polymers, composites and some metallic materials are viscoelastic and strain-rate sensitive. Under high strain rates the micro mechanisms by which these materials deform is different than that experienced at low strain rates. Consequently, use of quasi-static stress-strain data may not produce accurate and reliable predictions, when such data is used in simulation and Finite element analysis FEA of engineering components.
Design and analysis of Stress on Thick Walled Cylinder with and with out HolesIJERA Editor
The conventional elastic analysis of thick walled cylinders to final radial & hoop stresses is applicable for the internal pressures up to yield strength of material. The stress is directly proportional to strain up to yield point Beyond elastic point, particularly in thick walled cylinders. The operating pressures are reduced or the material properties are strengthened. There is no such existing theory for the stress distributions around radial holes under impact of varying internal pressure. Present work puts thrust on this area and relation between pressure and stress distribution is plotted graphically based on observations. Here focus is on pure mechanical analysis & hence thermal, effects are not considered. The thick walled cylinders with a radial cross-hole ANSYS Macro program employed to evaluate the fatigue life of vessel. Stresses that remain in material even after removing applied loads are known as residual stresses. These stresses occur only when material begins to yield plastically. Residual stresses can be present in any mechanical structure because of many causes. Residual stresses may be due to the technological process used to make the component. Manufacturing processes lead to plastic deformation. Elasto plastic analysis with bilinear kinematic hardening material is performed to know the effect of hole sizes. It is observed that there are several factors which influence stress intensity factors. The Finite element analysis is conducted using commercial solvers ANSYS & CATIA. Theoretical formulae based results are obtained from MATLAB programs. The results are presented in form of graphs and tables.
LITERATURE REVIEW ON FRACTURE TOUGHNESS AND IMPACT TOUGHNESSijiert bestjournal
The present paper gives a technical revi ew of fracture toughness,and impact toughness for metallic materials in terms of the linear elastic fracture mechanics. This includes the early investigations and recent advances of fracture toughness test methods and practices developed by various agencies and societies. The review describes the most important fracture mechanics parameters:such as the elastic energy release rate G,the stress intensity factor K,the J - integral,the crack - tip opening displacement (CTOD) and the crack - tip opening angle (CTOA) from the basic concept,definition,to,tes t methods. Attention is paid to guidelines on how to choose an appropriate fracture parameter to characterize fracture toughness for the material of interest,and how to measure the fracture toughness value defined either at a critical point or in a resist ance curve format using laboratory specimens. The effects of loading rate,temperature and crack - tip constraint on fracture toughness as well as fracture instability analysis are also reviewed.
There are currently three approaches to characterize and quantify the fatigue behaviour of composite laminates that are, Fatigue Life Modelling and Prediction, Phenomenological and Empirical Modelling, and Progressive Damage Modelling. These approaches constitute the evolution that is driven by ever expanding industrial needs and academic pursuit and assisted by perpetual technological advances in experimentation capabilities. In the first approach of Fatigue Life Modelling and Prediction the individual material degradation mechanisms are not directly concerned with, rather the determination of stress-life relationships based on experimental data is concerned with and the failure criteria or the residual strength determination is established based on these relationships, for the specific composite laminate.
Modified maximum tangential stress criterion for fracture behavior of zirconi...dentalid
The veneering porcelain sintered on zirconia is widely used in dental prostheses, but
repeated mechanical loadings may cause a fracture such as edge chipping or delamination.
In order to predict the crack initiation angle and fracture toughness of zirconia/veneer bilayered
components subjected to mixed mode loadings, the accuracy of a new and
traditional fracture criteria are investigated. A modified maximum tangential stress
criterion considering the effect of T-stress and critical distance theory is introduced, and
compared to three traditional fracture criteria. Comparisons to the recently published
fracture test data show that the traditional fracture criteria are not able to properly predict
the fracture initiation conditions in zirconia/veneer bi-material joints. The modified
maximum tangential stress criterion provides more accurate predictions of the experimental
results than the traditional fracture criteria
Multi resolution defect transformation of the crack under different anglesIJRES Journal
It is used to analyze the crack of different angles by the method of finite element. In the same material, the same crack is applied different angles, crack with 15 degree Angle differences, and applying Ⅰ type load on the material. The effective elastic modulus under the different angles of crack are obtained by finite element. With comparative judgment method, it provides the relationship between the modulus of elasticity and crack different angles, and a method crack material defects of transformation. On based on the transformation of energy equivalent principle, there are a lot of crack defects of materials for different degrees of defects, so as to simplify the material crack, provide a simple way for material fatigue analysis.
When a ductile material with a crack is loaded in
tension, the deformation energy builds up around the crack tip
and it is understood that at a certain critical condition voids are
formed ahead of the crack tip. The crack extension occurs by
coalescence of voids with the crack tip. The “characteristic
distance” (Lc) defined as the distance b/w the crack tip & the void
responsible for eventual coalescence with the crack tip. Nucleation
of these voids is generally associated with the presence of second
phase particles or grain boundaries in the vicinity of the crack tip.
Although approximate, Lc assumes a special significance since it
links the fracture toughness to the microscopic mechanism
considered responsible for ductile fracture. The knowledge of the
“characteristic distance” is also crucial for designing the size of
mesh in the finite element simulations of material crack growth
using damage mechanics principles. There is not much work
(experimental as well as numerical) available in the literature
related to the dependency of “characteristic distance” on the
fracture specimen geometry. The present research work is an
attempt to understand numerically, the geometry dependency of
“characteristic distance” using three-dimensional FEM analysis.
The variation of “characteristic distance” parameter due to the
change of temperature across the fracture specimen thickness was
also studied. The work also studied the variation of “characteristic
distance”, due to the change in fracture specimen thickness.
Finally, the ASTM requirement of fracture specimen thickness
criteria is evaluated for the “characteristic distance” fracture
parameter. “Characteristic distance” is found to vary across the
fracture specimen thickness. It is dependent on fracture specimen
thickness and it converges after a specified thickness of fracture
specimen. “Characteristic distance” value is also dependent on the
temperature of ductile material. In Armco iron material, it is
found to decrease with the increase in temperature.
Experimental and numerical evaluation of plasticity model with ductile damage...IJERA Editor
This work aims to develop a methodology for numerical evaluation via finite elements applied to projects shields sheet metal. To validate the methodology physical test were conduct and were compare with the numerical simulation. In the simulation, a plasticity material modelit was used at high strain rates, besides the insertion of a damage model through forming limit diagram (FLD) to capture the initiation of damage and energy criteria for propagation of the fracture. The tested shielding design is for the II-A protection level using the type 9mm ammunition.
Examples of Homogenization Techniques in the Material Modeling Under the Effe...Sardasht S. Weli
Explosions become a very attractive research area in the last decades.
This is due to the increase of accidental and intentional explosions.
Historical structure were not designed and built against the extreme loading events.
Homogenization Techniques were developed to assess the masonry response.
Structure failure often occurs in the structure of wall. This failure can adversely affect the comfort level of the structure. Knowing the behavior of structure resulting from the load is important, as it can help to predict the strength of the structure and comfort of the structure being worked on. One way to find out and predict the strength and comfort of the structure as a result of the load received is experimental test and simulation. The simulation VecTor2 used to predict the shear force, crack, and displacement of reinforced concrete wall when applied the load. This simulation considered the effect of bond stress-slip effect of behavior reinforced concrete. Bonds stress-slip gives a great influence on the strength and hysteretic response of the reinforced concrete wall. That is why this study considers the influence of bond stress-slip on reinforced concrete wall. All the result of simulation VecTor2 using bond stress-slip effect would be compared with the result of the experimental test to see the accuracy of the simulation test.
All structures are subjected to degenerative effects that may
cause initiation of structural defects such as cracks which,
as time progresses, lead to the catastrophic failure or
breakdown of the structure. Thus, the importance of
inspection in the quality assurance of manufactured
products is well understood.
Modified maximum tangential stress criterion for fracture behavior of zirconi...dentalid
The veneering porcelain sintered on zirconia is widely used in dental prostheses, but
repeated mechanical loadings may cause a fracture such as edge chipping or delamination.
In order to predict the crack initiation angle and fracture toughness of zirconia/veneer bilayered
components subjected to mixed mode loadings, the accuracy of a new and
traditional fracture criteria are investigated. A modified maximum tangential stress
criterion considering the effect of T-stress and critical distance theory is introduced, and
compared to three traditional fracture criteria. Comparisons to the recently published
fracture test data show that the traditional fracture criteria are not able to properly predict
the fracture initiation conditions in zirconia/veneer bi-material joints. The modified
maximum tangential stress criterion provides more accurate predictions of the experimental
results than the traditional fracture criteria
Multi resolution defect transformation of the crack under different anglesIJRES Journal
It is used to analyze the crack of different angles by the method of finite element. In the same material, the same crack is applied different angles, crack with 15 degree Angle differences, and applying Ⅰ type load on the material. The effective elastic modulus under the different angles of crack are obtained by finite element. With comparative judgment method, it provides the relationship between the modulus of elasticity and crack different angles, and a method crack material defects of transformation. On based on the transformation of energy equivalent principle, there are a lot of crack defects of materials for different degrees of defects, so as to simplify the material crack, provide a simple way for material fatigue analysis.
When a ductile material with a crack is loaded in
tension, the deformation energy builds up around the crack tip
and it is understood that at a certain critical condition voids are
formed ahead of the crack tip. The crack extension occurs by
coalescence of voids with the crack tip. The “characteristic
distance” (Lc) defined as the distance b/w the crack tip & the void
responsible for eventual coalescence with the crack tip. Nucleation
of these voids is generally associated with the presence of second
phase particles or grain boundaries in the vicinity of the crack tip.
Although approximate, Lc assumes a special significance since it
links the fracture toughness to the microscopic mechanism
considered responsible for ductile fracture. The knowledge of the
“characteristic distance” is also crucial for designing the size of
mesh in the finite element simulations of material crack growth
using damage mechanics principles. There is not much work
(experimental as well as numerical) available in the literature
related to the dependency of “characteristic distance” on the
fracture specimen geometry. The present research work is an
attempt to understand numerically, the geometry dependency of
“characteristic distance” using three-dimensional FEM analysis.
The variation of “characteristic distance” parameter due to the
change of temperature across the fracture specimen thickness was
also studied. The work also studied the variation of “characteristic
distance”, due to the change in fracture specimen thickness.
Finally, the ASTM requirement of fracture specimen thickness
criteria is evaluated for the “characteristic distance” fracture
parameter. “Characteristic distance” is found to vary across the
fracture specimen thickness. It is dependent on fracture specimen
thickness and it converges after a specified thickness of fracture
specimen. “Characteristic distance” value is also dependent on the
temperature of ductile material. In Armco iron material, it is
found to decrease with the increase in temperature.
Experimental and numerical evaluation of plasticity model with ductile damage...IJERA Editor
This work aims to develop a methodology for numerical evaluation via finite elements applied to projects shields sheet metal. To validate the methodology physical test were conduct and were compare with the numerical simulation. In the simulation, a plasticity material modelit was used at high strain rates, besides the insertion of a damage model through forming limit diagram (FLD) to capture the initiation of damage and energy criteria for propagation of the fracture. The tested shielding design is for the II-A protection level using the type 9mm ammunition.
Examples of Homogenization Techniques in the Material Modeling Under the Effe...Sardasht S. Weli
Explosions become a very attractive research area in the last decades.
This is due to the increase of accidental and intentional explosions.
Historical structure were not designed and built against the extreme loading events.
Homogenization Techniques were developed to assess the masonry response.
Structure failure often occurs in the structure of wall. This failure can adversely affect the comfort level of the structure. Knowing the behavior of structure resulting from the load is important, as it can help to predict the strength of the structure and comfort of the structure being worked on. One way to find out and predict the strength and comfort of the structure as a result of the load received is experimental test and simulation. The simulation VecTor2 used to predict the shear force, crack, and displacement of reinforced concrete wall when applied the load. This simulation considered the effect of bond stress-slip effect of behavior reinforced concrete. Bonds stress-slip gives a great influence on the strength and hysteretic response of the reinforced concrete wall. That is why this study considers the influence of bond stress-slip on reinforced concrete wall. All the result of simulation VecTor2 using bond stress-slip effect would be compared with the result of the experimental test to see the accuracy of the simulation test.
All structures are subjected to degenerative effects that may
cause initiation of structural defects such as cracks which,
as time progresses, lead to the catastrophic failure or
breakdown of the structure. Thus, the importance of
inspection in the quality assurance of manufactured
products is well understood.
Evaluation of Damage by the Reliability of the Traction Test on Polymer Test ...inventy
In recent decades, polymers have undergone a remarkable historical development and their use has been greatly imposed by gradually dethroning most of the secular materials. These polymer materials have always distinguished themselves by their simple shaping and inexpensive price, their versatility, lightness, and chemical stability but despite their massive use in everyday life as well as in advanced technologies. Generally, these materials still not understood which requires a thorough knowledge of their chemical, physical, rheological and mechanical properties. This paper, we study the mechanical behavior of an amorphous polymer: Acrylonitrile Butadiene Styrene “ABS” by means of uniaxial tensile testing on pierced test pieces with different notch lengths ranging between 1 to 14mm.The proposed approach consists in analyzing the evolution of the global geometry of the obtained strain curves by taking into account the zones and characteristic points of these curves as well as the effect of the damage on the mechanical behavior of the polymer ABS, in order to visualize the evolution of the damage by a static model
This paper addresses the fracture toughness ( ), or also known as critical stress intensity Factor, according to
conditions of Lineal Elastic Fracture Mechanics (LEFM). The characterization of the mechanical properties in
tensile and fracture toughness of structural steel pipes API-5L used in hydrocarbons transportation was
performed. For fracture toughness, the material was tested through fatigue crack propagation on standardized
compact specimen (CT) according to ASTM E-399 norm. A thickness (B) equal to and a crack size (a) equal
to 0.5w were used. With the porpoise of establishing the adequate conditions at the crack tip, the specimens were
subjected to fatigue pre-cracking by application of repeated cycles of load in tensile-tensile and constant load
amplitude with a load ratio of R = 0.1. The experimental Compliance method was used based on data obtained
from load vs. Crack Mouth Opening Displacement (CMOD). The results show a Stress Intensity factor of 35.88
MPa√m for a 25 mm crack size specimen. The device used for testing is a MTS-810 machine with capacity of
100KN and 6 kHz sampling rate, which meets the conditions of the ASTM E-399 standard. The cracking
susceptibility of steel is influenced by the size, morphology and distribution of non-metallic inclusions,
thermochemical interaction with the environment and microstructure.
Thermo mechanical characterization and damage of polymer materials:Applicatio...IJERD Editor
Plastic materials occupy a large part in our daily lives because of their ease of installation and relatively low production costs. The rapid technical development and we live brings more and more mechanical engineers to face the problems of damage to materials. However, these problems are even more serious than fatigue cracking often leads to a sudden break often cause accidents. This unfortunately happens all too frequently, due to insufficient knowledge either room service conditions or even damage parameters. This work presents new developments in the field of fracture mechanics and the objective is the evaluation of defects and thus a better estimate of the reliability of the polymeric material structures
Model Development for Estimation of Failure Loads: A Case Study of Composite ...ijsrd.com
This paper deals with the study of failure loads of glass vinylester composite plates with a circular hole subjected to a traction force by a rigid pin. These are investigated for two variables, the ratio of distance from the free edge of the plate (E) to the diameter of the hole (D) and the ratio of width of the plate (W) to the diameter of the hole (D). The work consists of a numerical study of different specimens using finite element analysis package ANSYS. Also, a mathematical model has been developed to determine the failure loads of different geometry plates. The results obtained from the numerical study and the mathematical models are compared with experimental results from the existing literature and the correlations are observed for both. A comparison of the experimental results with the numerical model shows that the numerical model gives results with correlation co-efficient 0.96. A comparison of the experimental results with the mathematical model shows that the mathematical model gives results with correlation co-efficient 0.99. For estimation of the failure loads within the range of E/D and W/D considered for the study, the mathematical model developed, i.e., Full Cubic Model proves to be more efficient with the observed values of correlation co-efficient, Root Mean Square Error and Maximum Absolute Error.
Under repeated impact composite domes subjected 6 J energy, changes locally with
increasing drop height. The action of the dynamic load generates reactions at the
support and bending moments at points on the surface of the composite. The peak loads
were noted to increase and stabilise about some mean value; and the 150mm diameter
shell was more damage tolerant compared to the 200 mm diameter one.
1Introduction
The purpose of this research work is to study the fatigue related behavior of weld toe
and weld root geometrical parameters in fillet welds based on the effective notch stress
approach.
The fatigue tests of welded structures under fluctuating loads shows that the crack
initiation and propagation until the final failure is carried out mostly on the weld toe and
weld root. Since the geometrical effect on stress distribution over a part plays a
meaningful role in respect to increasing the stress concentration factor value and
consequently the risk of failure, in this research the geometrical variables of welding
which can be recommended in some case of welding procedures such as weld toe
waving and weld root penetration percentage is studied. The fillet weld models under
special case of loading and constraint analyzed by three-dimensional linear static
analyses of finite element method to define the maximum principal stress distribution in
the modeled cases. The fatigue effect of analysis added to model by utilizing the
effective notch stress approach, which models the sharp lines in weld toe, and weld root
by determined rounded radius of 1 mm for steel material to avoid the geometrical
singularity of numerical analysis and take into consideration the fatigue notch factor.
The models of this study focus on the variation of stress concentration factor due to
weld toe waving geometrical effects defining by two variables of waving width and
waving radius in two separate set of models which the weld flank angle has been
changed. This leads an understanding to the benefit of varying stress concentration
factor on the weld toe between waving tips and waving depths so that the significant
decrease of this factor in waving depths can stop the rate of arbitrary initiated crack
propagation.
That is a crack, which initiated in a susceptible location such as wave tips could be
controlled by the waving depths, which have a significantly lower stress.
Meanwhile the study continued to analyze the distribution of stress in fillet weld root in
respect of the percentage of weld penetration into the base material by the same fatigue
9
method and numerical analyzing tools. The result of this part depicts the usability of
analyzing models type applying the effective notch stress approach and can be utilized
to define an optimized penetration percentage in the weld root of fillet-welded joints
2Fatiguebasefracture
Material properties, relate to the quality control of materials and initial material
selection by a designer and employing only a look at the stress-strain analysis will cause
the valuable information is lost. There are factors other than exceeding the yield stress
and causing plastic deformation, which will affect structures. Fracture is concerned with
the initiation and propagation of a crack until the load can no longer be held by the
structure. It is well known that most structures will c.
Experimental and numerical evaluation of plasticity model with ductile damage...IJERA Editor
This work aims to develop a methodology for numerical evaluation via finite elements applied to projects shields sheet metal. To validate the methodology physical test were conduct and were compare with the numerical simulation. In the simulation, a plasticity material modelit was used at high strain rates, besides the insertion of a damage model through forming limit diagram (FLD) to capture the initiation of damage and energy criteria for propagation of the fracture. The tested shielding design is for the II-A protection level using the type 9mm ammunition.
Analysis of Stress in Nozzle/Shell of Cylindrical Pressure Vessel under Inter...IJERA Editor
This work a comparative study of the methods of analysis of stress in vessel/nozzle, due to external loads. The
methods of analysis compared are WRC 107, WRC 297 and Method of Finite Elements. To make the
comparison between the methods, one model of nozzle has been developed without reinforcement plate. In this
nozzle it was applied external loads and after the application of the loads, compared the results of stress for the
three methods of analyses considered in this study.
Computational approach to contact fatigue damage initiation and deformation a...eSAT Journals
Abstract The paper describes a general computational model for the simulation of contact fatigue-damage initiation and deformation in the contact area of meshing gears. The model considers the continuum mechanics approach, where the use of homogenous and elastic material is assumed. The stress field in the contact area and the relationship between the cyclic contact loading conditions and observed contact points on the tooth flank are simulated with moving Hertzian contact pressure in the framework of the finite element method analysis. An equivalent model of Hertzian contact between two cylinders is used for evaluating contact conditions at the major point of contact of meshing gears. For the purpose of fatigue-damage analysis, the model, which is used for prediction of the number of loading cycles required for initial fatigue damage to appear, is based on the Coffin-Manson relationship between deformations and loading cycles. On the basis of computational results, and with consideration of some particular geometrical and material parameters, the initiation life of contacting spur gears in regard to contact fatigue damage can be estimated. Index Terms: Contact fatigue, Deformation, Crack initiation, Numerical modeling and Gear teeth flanks
Computational approach to contact fatigue damage initiation and deformation a...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Abstract. Ensuring of permanent and continuous working process of oil-gas and field equipment alongside with the other factors, depends also on reliability of sealing units. A problem of deterioration modeling of a sealing element of a packer including into an oil field equipment complex is considered in this paper.
“Comparison of Maximum Stress distribution of Long & Short Side Column due to...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Kubernetes & AI - Beauty and the Beast !?! @KCD Istanbul 2024Tobias Schneck
As AI technology is pushing into IT I was wondering myself, as an “infrastructure container kubernetes guy”, how get this fancy AI technology get managed from an infrastructure operational view? Is it possible to apply our lovely cloud native principals as well? What benefit’s both technologies could bring to each other?
Let me take this questions and provide you a short journey through existing deployment models and use cases for AI software. On practical examples, we discuss what cloud/on-premise strategy we may need for applying it to our own infrastructure to get it to work from an enterprise perspective. I want to give an overview about infrastructure requirements and technologies, what could be beneficial or limiting your AI use cases in an enterprise environment. An interactive Demo will give you some insides, what approaches I got already working for real.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
PHP Frameworks: I want to break free (IPC Berlin 2024)
C028011017
1. Research Inventy: International Journal Of Engineering And Science
Vol.2, Issue 8 (March 2013), Pp 11-17
Issn(e): 2278-4721, Issn(p):2319-6483, Www.Researchinventy.Com
11
Damage Characterization of a Thin Plate Made Of ABS under
Uniaxial Solicitation
H. Farid1
, K. Elhad1
, M. Elghorba1,
F. Erchiqui2
, M. Chergui1
1.
Laboratoire De Bioplasturgie Et Nanotechnologie, Université Du Québec Abitibi-Témiscamingue, 445
Boulevard De l’Université, Rouyn-Noranda, Québec J9X 5E4, Canada
2.
Laboratoire De Contrôle Et Caractérisation Mécanique Des Matériaux Et Des Structures, École Nationale
Supérieure d’Électricité Et De Mécanique, Route d'El Jadida. BP 8118 Oasis Casablanca, Maroc
Abstract: There is an important soaring in theuse of polymers in many industrial fields. Although they are
abundantly used both in ordinary and high performance products, their versatility makesthem highly needed for
newer applications; and thisrequires a detailed knowledge of their physical, chemical, rheological and
mechanical properties. The determination of mechanical behavior of those materials becomes very necessary
either during processing or under operation. In this work, we are interested in characterizing a notched
polymeric ABS flat plate under uniaxial solicitation. TheUnified Damage Theory for characterizing and
quantifying the damage is used. Our crucial aim, being thestudy of the stress concentration factor,and a
contribution to the determination of the damage degree of notched structures.
Key words: Damage mechanics, Damage theory, Uniaxial characterization, Softened polymer, ABS.
I. INTRODUCTION
Many engineering studies show that whatever the material (metal, polymers, composites, or wood),in
spite of their various microstructures and physical properties, the qualitative mechanical behavior they exhibit is
substantially identical. These include an elastic behavior, a yield stress, certain plasticity or irreversible
deformation, more or less important deformation anisotropy, a hysteresis ring with damage by monotonic or
cyclic loading, and a crack initiation under static or dynamic loading [1].Additionally, they exhibit many levels
of imperfections, due to processing, post processing handling such as transportation and storage. Such
imperfections include knit lines, flaws, vacuum or inclusions which make polymer-based structures
heterogeneous to a certain degree.
The consequences of the presence of knit lines for example in such materials on the mechanical
properties are discussed by Criens and Moslé [2]. Due to the presence of crazes, scratches, cracksand other
imperfections, the mechanical properties of realpolymeric materials are not as good as they theoretically could
be [3].Damage mechanics aimsat modeling these phenomena at the level of structures calculations. The purpose
of this approach being to build behavior models which once included in a calculation code,could simulate the
fracture and, more generally, to estimate the state of damagesin a structure according to one or many ultimate
states [4].Modeling the structural behavior of polymers has been the subject of a range of researches. In this
regard, several authors have addressed the problem of rheological behavior [5], formingand processing
modeling [6], and the behavior during the failure process [7].This work is a contribution to the study of the
failure mechanism of a flat plate made by ABS (Acrylonitrile-Butadiene-Styrene). Initially, the structural
behavior of a smooth plate has been studied, in aim to study and reveal the mechanical characteristic under
uniaxial solicitation tensile test, followed by a series of tests conducted on notched flat plates, the notch effect
will be discussed and highlighted through experimental results. Finally, through the application of the stress
concentration factor criteria [8], and the Unified Damage Theory [9], the damage evolution in the ABS material
is investigated.
II. THEORY
Generally, damage decreases the stiffness and resistance of materials. For a given stress state, the
deformations are all larger than the damage is high. Hence the importance of coupled calculations, which
calculates simultaneously stress, strain and damage in mechanical properties evolution problems.The main aim
of damage theories is to define a Damage parameter which translates the damage state of the structure.
However, it remainsa delicate problem since nothing (or almost nothing!) distinguishes a damaged volume
element of a virgin (blank)volume element macroscopically. It is then necessary to imagine an internal variable
representing the state of deterioration of the material.
2. Damage Characterization Of A Thin Plate Made...
12
Several modelsare proposed for damage modeling in structural calculation.The most responded modelis the
Minner model of damage [10], [11], however it remains limited in its linear method of description of the
cumulative damage progress, which does not accurately reflect the state of the deterioration of the structure.
Other proposed approaches have tried to adjust this model to a more adequate damage presentation;they include
the models of Miller [12] and Chaboche [13]…
In order to address the various points omitted by the theoretical damage models, Bui-Quoc[9] proposed
a model that involves several loading parameters. The Unified Theory hasthus been developed in this context, to
define damageparameter which is the internal variable that describe the damage state of the structure in terms of
the a life fraction , the fraction of life depends on solicitation conditions (for example the number of
cycle for dynamics loading, notch dimension for static tensile loading, temperature conditioning…).In this
respect, the endurance limit is defined as the stress level below which there is no damage, the part supports an
infinite number of cycles in fatigue tests for example, or its resistance reach the highest level for a notched static
test specimen. The rate of reduction of the endurance limitto the number of applied fatigue cyclesfor a dynamic
solicitation test was suggested as follows:
(1)
Wherethe various terms are successively:
Non-dimensionalendurencelimit
Non-dimensionalcyclic stress
Instantaneous non-dimensional endurance limit
Endurance limit of the virgin material
Applied stress level
and Material constants
Number of applied fatigue cycles
With boundary conditions:
i.
ii.
Where is the number of cycles to failure for a fatigue test; for our study, it represents the critical diameter of
the hole, m is an empirical material constant superior to one.
(2)
With
This equation determines the relationship between stress and applied number of cycles. The normalized damage
is defined by:
(3)
With
By setting posing this theory leads to the following equation:
(4)
For the damage quantification in the material, we used the damage variable for each life fraction, so by
analogy with theory shown above, the life fraction of our study representsthe relationship between the hole
diameter progressionand its critical value, which is the value of the larger hole diameter. Based on the earlier
presented theories, we chose the law of damages given by the unified theory. The mean advantage of this theory
relative to other approaches is to directly link the damage progressof the material to the variation of the ultimate
residual limit, thus directly connecting the material’s damage to its characteristics.
Thedamage variable can thusbe defined as:
3. Damage Characterization Of A Thin Plate Made...
13
(5)
Where
is theultimate residual stress
is the ultimate limit,
is the stress limit
III. Experimental
A series of tests was carried on forthe characterization of material damage, on two series ofrectangular
specimensofABS (Acrylonitrile-Butadiene-Styrene), based on theguidelines prescribed by the ASTM D 882-02
[14] and ASTM D5766M [15]. Thefirst series consisted ofrectangular smooth specimens (without hole) to
characterize the materialand the second series consisted of rectangular specimens with holes of 1 mm to
7mmdiameters. The first series was used as the reference, and the second to monitorandhighlight the influence
of the hole size on the specimenbehavior. All the experimental series was conducted under controlled
displacement. Elongations were determined using the displacement sensor crosshead incorporated in the
machine. Figure 1 below shows the set-up for the tensile test.
Figure 1. Experimental set-up:A test specimen held by the sample holder of the tensile testing machine (left),
anddimensions of the test specimen (right).
IV. RESULTS
In the notched specimen, a stress concentration phenomenon near the hole takes place, and the response
of the structure to the applied effort is affecteddepending on the hole diameter, we talk here on a notch effect.
Figure 2 shows the stress versus strain curves for the specimens with holes.As the tests proceed, these curves
show a strikingdecreasing pace, and the gap between the values.
Figure 2. Evolution of the stress-strain curves foruniaxial tensile tests on rectangular test specimens with 1mm
to 7mm hole diameters, according to ASTM D 882-02 two and ASTM D5766M
4. Damage Characterization Of A Thin Plate Made...
14
In fact, these curves showa remarkable degradation of the mechanical properties of the material with an
increase in hole diameter; these properties include the elastic stress, ultimate stress, the failure stress and the
elongation.We observe an area of stress stabilization, and a high elongation (maximum elongation).
Additionally, rupture is often preceded by local yielding followed byanabrupt failure.The mechanical
characteristics are affected due to the notch effect; the characteristics decrease as long as the notch diameter
increases. Figures 3 showsthe evolution of the ultimate stress thewithhole diameter:
Figure 3. Ultimate stress evolution withhole diameters
Figure 4. Stiffness evolutionwithhole diameters
As already mentioned above, a decrease in the ultimate stress (maximum stress value reached in the
tensile test) with hole diameters was observed.The right branchof the stress-strain curve present the elastic
response of the specimen, this branch allows the calculation of the Young Modulus. However, with the existence
of the hole, we talk more about Stiffnessfor notched specimens’more than Young modulus for smooth specimens.
The stiffness determines then the elastic response limits for notched structures. Figure 4 shows a significant rise
in stiffness until values closer to those given by smooth specimens which represent the Young’s modulus.As
discussed above, the existence of notch affect the mechanical characteristics of the material, the evolution of its
size has also remarkable influence on those properties. Figure 5 shows aschematicalevolution of the ultimate
nondimensional stress versus nondimensional hole diameter.The ultimate nondimensional stress was obtained
by nondimensionalization of the ultimate residual stress (the residual ultimate stress is given as the ultimate
value of stress reached by a sane specimen in a simple tensile test, for our study, its value is ) by
the ultimate stress level obtained for each hole diameter.The same for the nondimensional hole diameter which
was obtained by nondimensionalization of the hole diameter by the critical hole diameter (the larger value of the
hole diameter above which the structure no longer support any effort).
5. Damage Characterization Of A Thin Plate Made...
15
Figure 5. Evolution of the ultimate nondimensional stress versus the nondimensional hole diameter
Figure 5 illustratesthe ultimate nondimensional stress evolution versus thenondimensionalholediameter; the
curve shows a decrease of the material ultimate stress while the nondimensional hole increases. The intersection
with the interval limit allows us to determine a stress limit below which we estimate that the damage occurred to
the material remains acceptable, this stress limit has as value .Figure 6 reports the variation of
the resistance loss with the nondimensional diameter of the holes. It shows that the resistance loss is remarkable
and it becomesincreasingly important when approaching the critical value of the default already prescribed.The
aim of this representation is to show the resistance loss of the material due to the damage accumulation, this
damage itself is due to the notch and its evolution in the structure.Fatigue behavior of the
material,ascharacterized through its endurance limit,allows us to determine the endurance limit that will be used
later for the quantification of the material damage. The endurance limit was determines for the ABS smooth
samples, throughfatigue test, and takes as value [16].
Figure 6. The resistance loss in static tension
Both curves of Figure 6 have the same shapelike the one giving the variation of the ultimate limit. Both
show a resistance decrement while the nondimensional hole diameter keep increasing. Theintersection between
the later curve andthe boundary of the interval of study gives a value which is considered thesafe
endurance limitvaluebelow which nofatigue failure will occur.
6. Damage Characterization Of A Thin Plate Made...
16
V. DAMAGE IN THE ABS NOTCHED PLATES
Figure 7. Evolution of the damage versus the fraction of life
For quantifying the damage occurred into the notched structure, we propose a damage variable based
on presentation demonstrated above in equation (5). The proposition of this variable was taken by analogy with
cumulative damage theory proposed by Gatts [17] and retaked by Bui-Quoc [9]. The proposed approach present
the damage evolution into a notched structure, and link the damage state of the material to an intrinsic property
which is its residual stress.The damage on the notched samples with the non-dimensional hole diameters is
shown in the figure 7. The damagenearlyfollowsa linear variation.At theinitiation, the damage is
negligible for a smooth flat plate. As the hole size increases,the damage acceleratesa maximum value of
0.9is reached where failure occurred.The initiation area is characterized by the highest mechanical
characteristics for the materials, while the point of failure is marked by their remarkable degradation.
VI. STRESS CONCENTRATION IN THE NOTCHED ABS PLATES
The concept of a stress concentration factor is often employed by designers to account for the increase
of stress at a concentration point, the nominal stress being multiplied by a stress concentration factor to obtain
and estimate of the local stress at the point.Generally, Stress concentration is a phenomenon of a local stress
increasing in an area which contains geometric modifications [18].It appears in a discontinuity of a part or a
structure for examplewith the presence of a notch after machining. The area of stress concentration is most often
the site of crack initiation.In many cases, particularly in which the stress is highly localized, a mathematical
analysis proves difficult or impracticable. Then, experimental or mechanical methods of stress analysis are used.
The vicinity of the incident formis characterized by a heterogeneous stress distribution leading to the stress
concentration, thus stress concentration factor is defined as “the quotient of the maximal applied stress to the
corresponding nominal stress” [19]:
(6)
The severity of the stress concentration depends on both the notchgeometry and the notchconfiguration. Stress
concentration is an important factor to be considered during every structural design, as issues such as sudden
break can be avoided either by reducing or by avoiding it [20].
The stress concentration factor variation relative to the specimen ratio hole diameter to width is shown in the
figure 8.
7. Damage Characterization Of A Thin Plate Made...
17
Figure 8.Evolution of the stress concentration factor with the hole diameterbased on the ratio D/W for notched
plate
Thestress concentration factor decreases with the ratio D/W.This means that stress concentration increases with
a reduction of D/W.Additionally, this concentration tends towards stability when the ratio D/W increases, thus
the maximum stress is equal to the nominal stress, and the large size of the defect has no effect on this ratio.
VII. CONCLUSION
The objective of this work was to characterize the damage behavior of a thermoplastic flat plate. It
found that the uniaxial tensile test was the most practical approach and the closest representation of the real state
of deformation during the real solicitation.Thistechnique was used to describe the damage occurring to the
material. The unified damage theory was used in the aim to give a more complete damage model that can more
faithfully describe the state of the material. This approach was applied to several materials; however the results
reported in this article are only based on ABS.The results obtained have highlighted the applicabilityof the
reporteddamage approach. The proposed approach involves the use of theintrinsic parameters ofthematerial,
thusa rigorous description of the damage state of the materials. These preliminary studies are essential steps
towards the full achievement of our mid-term goals of performing and developing tools for modeling and
simulatingof thermoplastic forming processes.
REFERENCES
[1] Sanchez-Santana U., Comportement dynamique des matériaux et structures après sollicitation en fatigue. Thèse de doctorat,
Université des sciences et technologies de Lille, 2007.
[2] Criens R.M. and Moslé H.G., Faillure of plastics, edited by W. Brostow and R.D. Corneliunssen Hauser, 1986.
[3] Brostow W., Kubàt J. and Kubàt M., Physical properties of polymers handbook, Chapter 23: Mechanical properties, American
Institute of physics, James E. Mark editor, 1996.
[4] Lemaitre J. andChaboche J.L., Aspect phénoménologique de la rupture par endommagement, journal de la mécanique appliquée,
Vol 2, 1978.
[5] Shia-Chung C., Rheologicalbehavior of PS polymer melt under ultra-high speed injection molding, Polymer Testing Journal,
Volume 31, Issue 7, October 2012.
[6] Erchiqui1 F., Ozdemir Z., Souli M., Ezzaidi H., Dituba-Ngoma G., Neural networks approach for characterization of viscoelastic
polymers, volume 89, Issue 5, October 2011.
[7] Figueredo R., Cherouat A., Une nouvelle approche dans la prise en compte de la déformation permanente et volumique induite par
l’endommagement dans la mise en forme des structures caoutchoutiques, 19 ème Congrès Français de Mécanique, Marseille 2009.
[8] Davis J.R., Tensile testing second edition, ASM International, The material information society 2004.
[9] Bui-Quoc T., Dubuc J., Bazergui A., Biron A., Cumulative fatigue damage under stress-controlled conditions, Journal of Basic
Engineering, Transactions of the ASME, 1971.
[10] Miner M., Cumulative damage in fatigue, Journal of Applied Mechanics, Volume 12, 1945.
[11] Jardin A., Leblond J.B., Berghezan D., Par-delà la loi de Miner pour la modélisation de la fatigue des élastomères : théorie et
expérimentation, 19 ème Congrès Français de Mécanique, Marseille 2009.
[12] Miller K.J., Zachariah K.P, Cumulative damage fatigue crack initiation and state I propagation, Journal of Strain Analysis, 1977.
[13] Lemaitre J, Chaboche J.L, Mechanics of Solid Materials, Cambridge University Press, Cambridge, UK, 1990.
[14] D 882-02 ASTM Standard test method for tensile properties of thin plastic sheeting.
[15] D5766M ASTM Standard test method for open hole tensile strength of polymer matrix composite laminates.
[16] E606 / E606M - 12 ASTM Standard Test Method for Strain-Controlled Fatigue Testing.
[17] Gatts R., Application of cumulative damage concept to fatigue, ASTM Transactions, Journal of basic engineering, Volume 83,
1961.
[18] LemaignanC., La Rupture des matériaux, EDP Sciences 2003.
[19] Boresi A.P., Sidebottom O.M., Advanced mechanics of materials 4th
edition, John Wiley and sons 1985.
[20] Lemaitre J, Chaboche J.L, Mécanique des matériaux solides 3ème
édition, Edition Dunod 2009.