# THEORY OF PLASTICITY

**(i)Experimental data from low cycle fatigue**

Here we are explaining the process of the fatigue test;

The equation for defining engineering stress;

S = *P**A0* ; in the following equation, P is the axial force and A0 is representing the cross-sectional area of the uniaxial specimen.

Now, the equation for measuring true stress;

Similar to the above equation, P is the axial force and A defines the cross-sectional area. This equation is given in the following way;

? = *P**A*

For comparing these two types of stress, it can be said that true stress is the cross-axial tension is comparatively higher than the engineering stress.

The equation which has been presented gives out the relationship between the stress in tension and length of the gage.

**Figure 1: Equation for stress and length**

(Source: Sevenois*et al. *2019)

In the above-given equation, l represents the instantaneous length and the change in length is represented by *?l* represents the change in the instantaneous length. An equation is also given for measuring the true stress and the equations are given in the following way;

**Figure 2: Equation for measuring true stress**

(Source:Wang, Zhang &Li, 2017)

## The assumptions which are considered here are as follows;

- Engineering stress can be considered to be more or less equal to the true stress if the small strains are less than or equal to 2%.
- There are no distinctions are such between engineering stress and true stress on the account of these small strains
- Differences between these two stress can be noticed for larger stress

Further, a relationship between volume condition and stress can be established by using the following equations;

**Figure 3: Equations between volume and stress**

(Source: de Krijger *et al. *2017)

The above equations are used when the utilizations of strengths are done to its maximum and necking take place. The process of necking if crucial since it leads to the localization of plastic deformation and there is no uniformity of strain in the gage section.

**Figure 4: Tests for Monotonic Strengths**

(Source: De Barros *et al. *2017)

In the above figure, E is the elasticity modulus, yield length and the ultimate tensile strength are given by *S**y* and *S**u*respectively. For calculating the reduction in the area, the volume has been taken into consideration with *?**f* denoting the true fracture strength. The ductility is represented by *?**f*after measuring the percentage of elongation.

In most of the cases, the true fracture strengths are considered in place of necking under the presence of stress in the biaxial state at the surface of the neck and triaxial state at the interior of the neck.

*cfopen,run1,mac

a=power(ii)

b=scale(jj)

*vwrite,a,b

build_solve,%G,%G

*cfclose

On dealing with cylindrical specimens, the factor for Bridgeman Correction is used for compensating the stress occurring at triaxial state. This equation is given in the following way;

**Figure 5: Correction factor**

(Source: De Barros *et al. *2017)

In the aboveequation, R represents the radius of the curvature and *D**min* denotes the cross-sectionaldiameters at the thinnest part of the neck.

It should be noted that necking is not exhibited by materials which project a behaviour of brittle tensile and hence this correction factor is not applicable for those materials.

**(ii)Perform the optimization task by a non-linear least square**

Strain is controlled by the tests which are always conducting from axial loading. Deflections are used for converted and controlled into the strain. The result is required to measure from applied stress for computing the forces. It is generally makes the work in a more efficient manner. P = axial Force, AO = Original cross area for various sections. The true stress is required to be given in the cross sectional area.

*cfopen,run1,mac

a=power(ii)

b=scale(jj)

*vwrite,a,b

build_solve,%G,%G

*cfclose

The general problem is to make non-linear functions which depends upon various additional parameters. It is called the non-linear programming problem and these are mostly based on the applied mathematics. It has various programming and studies which are depends on the comprehensive survey of various models. It has local properties and optimization in each mapping (Azmi *et al. *2019). The study is based on local approximation for optimizing the mapping parts. Most of the parameters are used in computational algorithm and it has also the high complexity for determining it. Therefore, the ate vector-valued for non-linear mapping is Y = f(u), Y = f (U, p)

The problem is to find the argument on those which are depends upon the matter. This is generally having the maximum value in the regression time. Regression time generally makes the word more commendable and needs .to analysis the matter in right way.

*dim,power,array,4

power(1)=10,20,40,80

*dim,scale,array,6

scale(1)=1,2,3,5,10,20

In this paper, we consider the value such as:

e |
s |

/1/ |
/MPa/ |

-0.02451 |
-464.454 |

-0.02456 |
-464.408 |

-0.02461 |
-463.801 |

-0.02466 |
-462.943 |

-0.02471 |
-462.29 |

-0.02475 |
-462.035 |

-0.02479 |
-462.29 |

-0.02482 |
-463.045 |

-0.02485 |
-463.852 |

-0.02488 |
-464.204 |

-0.0249 |
-463.903 |

-0.02492 |
-463.045 |

-0.02493 |
-461.836 |

-0.02493 |
-460.626 |

-0.02492 |
-459.213 |

The formula is required to be in the given file. It is generally needs to mention on given formulas which are required to be done in the method.

e |
s |

/1/ |
/MPa/ |

-0.02451 |
-464.454 |

-0.02456 |
-464.408 |

-0.02461 |
-463.801 |

-0.02466 |
-462.943 |

-0.02471 |
-462.29 |

-0.02475 |
-462.035 |

-0.02479 |
-462.29 |

-0.02482 |
-463.045 |

-0.02485 |
-463.852 |

-0.02488 |
-464.204 |

-0.0249 |
-463.903 |

-0.02492 |
-463.045 |

-0.02493 |
-461.836 |

-0.02493 |
-460.626 |

-0.02492 |
-459.213 |

-0.02489 |
-457.197 |

-0.02484 |
-454.477 |

-0.02478 |
-450.599 |

-0.02472 |
-445.409 |

-0.02466 |
-439.311 |

-0.02458 |
-432.656 |

-0.02451 |
-426.359 |

The true fracture strength is required to be calculated for mapping the stress in the connection in different environments.The bridgeman factor is also required for making the compensation on triaxial state which is applied in the specimens to cylindrical. The function R is required for measuring the radius of curvature of the neck. It is required for making different factors on the field.

**(iii)In Ansys APDL mechanical simulate the hysteresis loops**

**Commands**

*dim,power,array,4

power(1)=10,20,40,80

*dim,scale,array,6

scale(1)=1,2,3,5,10,20

Most of the control.mac commands will be put inside of nested *do loops. There will be a *do loop for each of parameters being varied.

*do, ii, 1, 4

*do, jj, 1, 6

Next, use * cfopen to set up the arguments to be passed to build_solve. Each time through the *do loops will create a new run1.mac

*cfopen ,run1 , mac

a= power (ii)

b= scale (jj)

*vwrite, a,b

build_solve, %G, %G

*cf close

/INQUIRE, dir_, DIRECTORY

*dim,power,array,4

power(1)=10,20,40,80

*dim,scale,array,6

scale(1)=1,2,3,5,10,20

*dim,power,array,4

*cfopen,temp1,mac

*vwrite,a,b

dirnam='power_%G_scale_%G'

*cfclose

/input,temp1,mac

It uses the string for more prposes which are required to be in the factor.

*dim, power, array,4

*cfopen,temp1,mac

*vwrite,a,b

dirnam='power_%G_scale_%G'

*cfclose

/input,temp1,mac

he code for the windows batch file is:

*cfopen,rfile,bat

*vwrite,dir_(1),dirnam

MKDIR "%C\%S"

*vwrite,dir_(1),dirnam

COPY *.mac "%C\%S"

*vwrite,dir_(1),dirnam

CD "%C\%S"

*vwrite,

The process is:

(i) Create a new directory for doing the project.

(ii) It needs to copy all the macro files from the directory of working and give it to the new directory.

(iii) Change the directory mode in CD and convert the file in different manner.

(iv) Launch the ansys in the batch mode, in case it using the GPU and 12 cpu files. It needs to run on the mac system which is run1.mac and get output from the executed file.

(v) Change the working directory and make the file more accurate.

*cfopen,run1,mac

a=power(ii)

b=scale(jj)

*vwrite,a,b

build_solve,%G,%G

*cfclose

One of the key features of this approach is to run anywhere and build directories below the working directory. Use the /inquire command to store the current directory name.

/INQUIRE,dir_,DIRECTORY

Use *cfopen is valid for creating a string that is used for the main path/ directory name. By the use of different types of variables as part of the string, the directories have unique names. It also has time or date stamp which could also be included in this string.

*cfopen,run1,mac

a=power(ii)

b=scale(jj)

*vwrite,a,b

build_solve,%G,%G

*close

The result is required to be run in the close file which is generally executing the main function.

The result is required to measure from applied stress for computing the forces. It generally makes the work in a more efficient manner. P = axial Force, AO = Original cross area for various sections. The true stress is required to be given in the cross-sectional area. The general problem is to make non-linear functions that depend upon various additional parameters. It is called the non-linear programming problem and these are mostly based on the applied mathematics. It has various programming and studies which depend on the comprehensive survey of various models. It has local properties and optimization in each mapping. The study is based on the local approximation for optimizing the mapping parts. Most of the parameters are used in the computational algorithm and it has also a high complexity for determining it. This macro environment is executed immediately to create the string Birnam for use in the commands subsequently.

*dim, power, array,4

*cfopen, temp1,mac

*vwrite, a,b

dirnam='power _%G_scale_%G'

*cf close

/input,temp1,mac

From the mkdir file, user can get the ansys file which needs for Ansys APDL mechanical simulation.

**References**

Azmi, M. M., Fujii, T., Tohgo, K., Hashim, M. S. M., Ismail, A. H., & Razlan, Z. M. (2019, November). Evaluation of ?J-integral for a shallow crack in steel for pressure vessels under large scale yielding (LSY) condition. In IOP Conference Series: Materials Science and Engineering (Vol. 670, No. 1, p. 012017). IOP Publishing. Retrieved on 8^{th} January 2020 from: https://iopscience.iop.org/article/10.1088/1757-899X/670/1/012017/pdf

De Barros, S., Kenedi, P. P., Ferreira, S. M., Budhe, S., Bernardino, A. J., & Souza, L. F. G. (2017). Influence of mechanical surface treatment on fatigue life of bonded joints. *The Journal of Adhesion*, *93*(8), 599-612. Retrieved on 4^{th} January 2020 from: https://www.researchgate.net/profile/Silvio_De_Barros/publication/285728924_Influence_of_Mechanical_Surface_Treatment_on_Fatigue_Life_of_Bonded_Joints/links/571e343208aeaced7889dcc4/Influence-of-Mechanical-Surface-Treatment-on-Fatigue-Life-of-Bonded-Joints.pdf

de Krijger, J., Rans, C., Van Hooreweder, B., Lietaert, K., Pouran, B., & Zadpoor, A. A. (2017). Effects of applied stress ratio on the fatigue behavior of additively manufactured porous biomaterials under compressive loading. *Journal of the mechanical behavior of biomedical materials*, *70*, 7-16. Retrieved on 5^{th} January 2020 from: http://calvinrans.com/wp-content/uploads/2017/03/JMBBM-D-16-00373R1-submitted-paper.pdf

Sevenois, R. D. B., Garoz, D., Gilabert, F. A., Hochard, C., & Van Paepegem, W. (2019). Influence of tab debonding on measured stiffness evolution in Compression-Compression and Tension-Compression fatigue testing of short gauge length coupons. *Composites Science and Technology*, *180*, 1-13. Retrieved on 4^{th} January 2020 from: https://biblio.ugent.be/publication/8615781/file/8616105.pdf

Wang, S., Zhang, G., & Li, A. (2017). The feasibility evaluation of using the automated ball indentation technique to obtain mechanical properties of the steels used in nuclear vessels. Retrieved on 5^{th} January 2020 from: https://repod.lib.ncsu.edu/bitstream/handle/1840.20/31760/SMiRT-24_02-12-03.pdf?sequence=1

Xu, L. Y., Tao, M. X., Nie, X., Fan, J. S., & Taciroglu, E. (2017). Modeling Techniques for Strain-Range-Dependent Hardening Behavior of Low-Yield-Point Steel Shear Panel Dampers. Journal of Structural Engineering, 143(12), 04017172. Retrieved on 8^{th} January 2020 from: https://www.researchgate.net/profile/Liyan_Xu6/publication/320689772_Modeling_Techniques_for_Strain-Range-Dependent_Hardening_Behavior_of_Low-Yield-Point_Steel_Shear_Panel_Dampers/links/5a3c8a59458515f7ea530267/Modeling-Techniques-for-Strain-Range-Dependent-Hardening-Behavior-of-Low-Yield-Point-Steel-Shear-Panel-Dampers.pdf