**6.2 Lateral compression test**

Lateral compression test is performed in this section (**Figure 5**). Eulerian and Lagrangian simulation time are compared, and tensile instability is verified. The test is carried out at 30 mm s<sup>1</sup> , and cylindrical sample (diameter 25.4 mm, length 25.4 mm) with 5313 particles was used. This is a case of a large deformation test; the

**Figure 4.** *Stress-strain curves: experimental vs. SPH.*

**Figure 5.**

*Lateral compression test: Eulerian vs. Lagrangian.*


**Table 2.** *Simulation time comparison.*

*Hot Compression Tests Using Total Lagrangian SPH Formulation in Energy-Based Framework DOI: http://dx.doi.org/10.5772/intechopen.85930*

initial diameter of the sample was reduced over 50% during the test. See comparison of results at **Table 2**.

## **6.3 Discussion**

experimental ones to verify the accuracy and the stability of the code

repeated three times.

*Progress in Relativity*

**6.2 Lateral compression test**

is carried out at 30 mm s<sup>1</sup>

*Stress-strain curves: experimental vs. SPH.*

*Lateral compression test: Eulerian vs. Lagrangian.*

**Figure 4.**

**Figure 5.**

**Table 2.**

**112**

*Simulation time comparison.*

(see **Figure 4**). To confirm the validity of the experimental result, the tests were

Lateral compression test is performed in this section (**Figure 5**). Eulerian and Lagrangian simulation time are compared, and tensile instability is verified. The test

25.4 mm) with 5313 particles was used. This is a case of a large deformation test; the

, and cylindrical sample (diameter 25.4 mm, length

**Simulation time Number of particles**

Eulerian SPH 4 h 04 min 5313 Lagrangian SPH 1 h 36 min 5313

**Figures 4** and **5** and **Table 2** gather the tests results. From **Figure 4** (axial compression test), we can see that the SPH result is very accurate compared to the experimental ones. Less than 5% of error is noted between the curves. The simulated sample shows no clustered particles, meaning there is no tensile instability.

**Figure 5** and **Table 2** show the results of the lateral compression test and confirm the previous result. Even in very large deformation test, particles keep their initial neighbors and do not suffer from tensile instability. In addition, the simulation time is very interesting compared to classical SPH formulation; simulation time is reduced drastically (from 4 h 04 min to 1 h 36 min); a good numerical efficiency is reached.

## **7. Conclusion**

A corrected SPH particle approximation in energy-based framework is presented. Stability (no tensile instability), accuracy, and fast result production are shown leading to the conclusion that the total Lagrangian SPH formulation is very well suited to simulate solid mechanic problems. This is particularly interesting in simulating large deformation problems with physical fragmentation where the numerical fragmentation (tensile instability) will not corrupt the results.

#### **Acknowledgements**

The author wishes to acknowledge Augustin Gakwaya for his appreciated help.

#### **Author details**

Kadiata Ba Department of Applied Sciences, Université du Québec à Chicoutimi (UQAC), Québec, Canada

\*Address all correspondence to: kadiata\_ba@uqac.ca

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
