Comparative study of different material models for modelling Elasto-Plastic materials

Comparison of results of the material models Law 1, Law 2 and its failure modes, Law 27 and law 36.

Objective:

The objective of this exercise is to compare the results obtained by modelling the material with different material models and determining the material model that best imitates/simulates the behavior of the material that is used in real life.

Material modelling:

Understanding material behavior:

1. There are different types of materials based on whether they are metals or non-metals, which are then categorized into different sub groups.

2. One such category is ductile materials and brittle materials.

3. The ductile materials undergo ductile failure. It’s the kind of failure in which, when stressed beyond the ultimate tensile stress, the material first deforms elastically (can regain its original shape), and then undergo yielding (beginning of permanent deformation by change in cross section) and then gradually fails

4. The brittle materials undergo brittle fracture. It’s the kind of failure in which, when stressed beyond the ultimate tensile stress, the material undergoes very little elastic and plastic deformation and then ruptures suddenly

Correlation between Internal Energy and phases of failure:

1. Let’s consider the ductile failure for an example. In this, when stressed, first the material deforms elastically which means, it gradually absorbs the supplied energy.

2. This is energy absorbed within the elastic region is called resilience. After the material starts yielding, only part of the energy is stored while some part of the energy is spent for the permanent plastic deformation of the material and some part of the energy is liberated as heat.

3. Hence fluctuations in energy or erratic energy change can be observed in plastic region unlike the gradual energy change in the elastic region

4. Hence for all the cases explained here, a graph of Internal energy in addition to stress or strain contours would be the most appropriate to understand the mechanism involved in failure of the material.

Failure and failure criteria:

1. In engineering, all the components are designed to operate within specified stress limits. It’s commonly depicted in the form of Factor of Safety

2. Factor Of Safety = Yield stress /Maximum Working stress

3. As the components are designed at specified FOS, the failure is assumed to occur when the maximum working stress approaches the yield stress of that material

4. If a simulation is run without specifying a failure mode corresponding to the maximum working stress, the simulation will keep running even when the stress exceeds the yield stress, which is a waste of time, money and computational resources.

5. Hence to easily identify when, where and how a component fails, a criteria called Epsilon P Max is used to describe the strain value corresponding to the maximum working stress.

6. This strain value helps to identify when, where and how the component fails thus saving time, money and computational resources.

Case 1: LAW2 - Johnson-Cook Material with EpsPmax Failure and Fail Johnson Card

1. This is law is commonly used to model elasto-plastic materials.

2. For the calculation of the material’s stress, this takes into account the temperature, strain and strain rate.

3. As mentioned earlier, if the conditions such as strain hardening, temperature effect, and strain rate are to be considered, this law is used to model the material.

4. The von mises stress is 7.232E+01 g/mm2 which corresponds to the EPS_p_max value of 0.151 beyond which the elements fail.

Strain:

5. It can be observed from internal energy graph that there is a sharp increase in the internal energy due to the material absorbing the external energy in the plastic region.

6. Then the absorption of energy becomes a bit gradual after some of the energy starts plastic deformation of the material

 

Case 2: LAW2 - Johnson-Cook Material with EpsPmax Failure criteria and FAIL/JOHNSON Card (Crack propagation)

1. This is the same as the case 1 but in fail Johnson card, the option IXFEM and DADV is set to 1 to simulate the failure of material in the form of crack generation

2. The pictures shown above depict the cracking of elements.

3. Here, the von mises stress is 2.747E+02 g/mm2 which corresponds to the EPS_p_max value of 0.151 beyond which the elements fail and get cracked.

Strain

4. It can be observed from the internal energy graph that energy absorbed is slightly more than the energy absorbed in the case 1.

5. Because of the cracking of elements, the energy absorbed is slightly higher than case 1.

6. It should be noted that, to visually verify the cracking of the material easily, in this case, the model is set to wireframe view.

7. To capture the exact propagation of crack, the time interval of animation must be very low which will consume more computational power.

8. Also, in the model browser, a new component called “CRACKED SHELL LAYER” will be created which depicts crack formation.

 

Case 3: LAW2 - Johnson-Cook Material with EpsPmax Failure criteria and no FAIL/JOHNSON Card

1. This is the same as the first case but there is no FAIL/JOHNSON card defined.

2. The FAIL/JOHNSON is used to provide the user more control over the failure.

3. It is only because of this card, conditions like tension or compression or shear could be distinguished from each other.

4. The von mises stress is 1.084E+02 g/mm2 which corresponds to the EPS_p_max value of 0.151 beyond which the elements fail.

Strain:

5. The stress higher than the stress in case 2 is because of the lack of clear and precise failure parameters.

6. The internal energy is more or less equal to the internal energy observed in case 1.

 

Case 4: LAW2 - Johnson-Cook Material without both EpsPmax Failure criteria and FAIL/JOHNSON Card

1. In this case, there are no failure criteria mentioned. There is also no FAIL/JOHNSON Card.

2. This means that the elements won’t be deleted or element stress won’t be reduced by any factor.

3. This case represents just plain simulation of loading of an elasto-plastic material.

4. The von mises stress is 4.247E+02 g/mm2

Strain:

5. In this, the material absorbs some of the external energy, deforms elastically and then starts deforming plastically.

6. As we don’t know for sure if the force applied is enough to induce stress more than the yield stress of the material, it can be concluded that the internal energy keeps increasing without any drop or gradual increase as seen in the previous cases

7. The internal energy is also more in this case because, in all the previously studied cases, the failure criteria causes the material to fail which means the material no longer absorbs and transfers the external energy but in this case ,lack of failure criteria stops the elements from failing.

8. This can be clearly observed in the contour images and animation.

 

Case 5: LAW1 Linear Elastic Material

1. This law is used to model elastic materials only. Modelling any material other than elastic materials using this material model will produce incorrect results.

2. This is because, elastic materials have a linear stress strain curve governed by Hooke’s Law which signifies that they return to their original shape after the load is withdrawn.

3. External energy supplied is never utilized for permanent deformation unlike elasto-plastic materials.

4. Only if any material is stressed within the yield strength, this model is applicable.

5. The von mises stress is 1.086E+04 g/mm2

6. In this case, the material keeps absorbing the external energy without undergoing any permanent deformation. Hence, the stress in the material is very high.

Strain:

7. The internal energy plot also conveys that the material absorbs massive amount of energy

8. Since there is no plastic deformation, this absorbed energy(resilience) brings the material back to its original shape after the load is withdrawn.

 

Case 6: LAW27 Elasto-Plastic Material with Brittle Failure

1. This law is very similar to the Johnson-Cook law2 but instead of ductile failure model like LAW2, LAW27 has brittle failure model

2. Most commonly used to model brittle materials like glass,etc

3. The von mises stress is 2.153E+02 g/mm2 which is almost equal to the von mises stress seen in case 2 (Johnson Cook model with crack propagation)

Strain:

4. One small difference between both the cases is, in case 2, plate develops cracks all over it and only in the portion where stress is maximum, the elements are deleted and shatters while in this case, the plate breaks away into pieces along the fracture with no cracking of elements.

5. The contour can help better illustrate the difference between crack formation in case 2 and the brittle fracture in this case

Brittle Fracture (Case 6)

Element cracking (Case 2)

 6. Since this is a brittle material model, the the stress strain curve of the brittle materials will give a great idea about the energy absorbed.

7. Unlike ductile materials, brittle materials will undergo very small deformation. The fracture is sudden and rapid and the strain is small compared to ductile materials

8. The internal energy graph shows that the energy absorbed is more when compared to the other material models that are based on LAW2. Also, the energy absorption is more streamlined than other material models because of the brittle nature of the failure.

 

Case 7: LAW36 Elasto-Plastic Piecewise Linear Material

1. This law is also used to model elasto-plastic materials just like in LAW2 and Case 1 but the difference is, many curves can be used as input to define the work hardening or strain hardening portion of stress- strain graph.

2. This is particularly useful in cases where there is also emphasis on work hardening and different strain rates are anticipated during the loading or unloading conditions.

3. While strain hardening is also taken into account in LAW2, curves obtained through lab testing can be used in LAW36 and many number of curves can be defined for different strain rates which improves the accuracy of results.

4. The von mises stress is 8.719E+02 g/mm2 which corresponds to the EPS_p_max value of 0.16 beyond which the elements fail.

Strain:

5. The internal energy is slightly more than the internal energy recorded in case 1 despite both being more or less the same because unlike Case 1, a custom curve obtained from experimental lab testing is used for stress and strain calculation after the elastic region.

 

Result:

1. The material given was modelled using different material models and failure parameters and the observations are listed below.

	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7
Internal Energy(J)	27237.656	30531.468	27165.554	64648.171	852796.12	39759.445	45461.582
Von- Mises Stress (g/mm2)	7.232E+01	2.747E+02	1.084E+02	4.247E+02	1.086E+04	2.153E+02	8.719E+02
Von- Mises Strain	1.355E-01	1.423E-01	1.349E-01	3.109E-01	2.010E-01	1.528E-01	1.202E-01
Simulation Run time (S)	228.81	234.80	217.58	231.71	216.02	225.15	248.5
Number of cycles	49480	49397	49405	48737	47969	49356	52208
Energy error (%)	0.2	4.0	0.8	3.0	0.3	1.0	-1.3

2. Based on the results, cases 1 and 7 are the the cases that resemble the real life scenario.

3. Case 1 would be most appropriate for situations where the strain rate is already known.

4. Case 7 would be most appropriate for situations where the exact strain rate is not known and many curves for stress and strain are provided for factoring strain hardening in calculation and the rest of the values for curves are extrapolated by the solver.

5. Case 7 is also preferred when most accurate results are required.

6. Case 4 can also be used if the failure is not taken into consideration.

 

Learning outcome:

1. The modelled materials are only as good as the supporting material lab tests. Hence the material testing data is very crucial for modelling the material accurately to imitate the material in real life.

2. Before modelling the materials, an engineer should know how the material behaves in real life.

3. The behaviors may be isotropy, anisotropy, ductility, brittleness, etc

4. These behaviors are the ones that help an engineer select the most appropriate material model.

5. If modelling crack is the objective, care should be taken to choose appropriate time step to capture the propagation of cracks, mesh must be very fine and the animation intervals must be small.

6. Before choosing a material model, the engineer must also take factors such as time, money, computational power and other resources into consideration.

7. For example, LAW36 requires many curves for various strain rates for precision.

8. This increases the cost of computation, cost of material testing, time, etc.

9. Hence in this case, if the most economical decision is to be made, LAW2 should be used at the risk of little inaccuracy in results.  Hence, a well balanced approach must be taken while selecting material models.

Conclusion:

The given objectives were completed successfully and the observations were documented. Several material models were studied in detail and a material was modelled using different material models and the differences in results were studied.

 

Google drive link - https://drive.google.com/file/d/1MHtnG8Mp-QD5gw-m3iKGYUezWM18-iuv/view?usp=sharing