**5. Material behavior**

**T**\_

where *k* is the conductivity and *s* is the specific heat.

**<sup>u</sup>**\_ *<sup>t</sup>*þ*Δ<sup>t</sup>* ð Þ<sup>2</sup>

The stable time is calculated as follows:

dilatational wave speed of the material.

<sup>¼</sup> **<sup>u</sup>**\_ *<sup>t</sup>*�*Δ<sup>t</sup>* ð Þ<sup>2</sup>

The stability time is given by

material.

*Progress in Relativity*

forces.

**Figure 2.**

**110**

*SPH code structure [23].*

**u** (Eq. 29).

ð Þ*<sup>t</sup>* <sup>¼</sup> **<sup>C</sup>**�<sup>1</sup> **<sup>h</sup>***<sup>t</sup>*

*<sup>Δ</sup>tT* <sup>≈</sup> *<sup>Δ</sup>r*<sup>2</sup>

where *Δr*min is the smallest interparticle distance and *α* is the diffusivity of the

*<sup>α</sup>* <sup>¼</sup> *<sup>k</sup> ρs*

For the mechanical part, an explicit central-difference integration rule is used to integrate the equation of motion. The nodal accelerations *u*€ at time t is given by

**<sup>u</sup>**€ð Þ*<sup>t</sup>* <sup>¼</sup> **<sup>M</sup>**�<sup>1</sup> **<sup>P</sup>**ð Þ*<sup>t</sup>* � **<sup>I</sup>**ð Þ*<sup>t</sup>*

where **M**, **P**ð Þ*<sup>t</sup>* , and **I**ð Þ*<sup>t</sup>* represent the mass matrix and the external and internal

The integration leads to the nodal velocity **u**\_ (Eq. 28) and the nodal displacement

*Δt*ð Þ *<sup>t</sup>*þ*Δ<sup>t</sup>* þ *Δt*ð Þ*<sup>t</sup>* 

þ

**<sup>u</sup>**ð Þ *<sup>t</sup>*þ*Δ<sup>t</sup>* <sup>¼</sup> **<sup>u</sup>**ð Þ*<sup>t</sup>* <sup>þ</sup> *<sup>Δ</sup>t*ð Þ *<sup>t</sup>*þ*Δ<sup>t</sup>* **<sup>u</sup>**\_ *<sup>t</sup>*þ*Δ<sup>t</sup>* ð Þ<sup>2</sup>

*<sup>Δ</sup><sup>t</sup>* <sup>¼</sup> min *Le*

*cd* 

where *Le* and *cd* are, respectively, the characteristic length of the element and the

*ext* � **<sup>h</sup>***<sup>t</sup>* int

min

(24)

<sup>2</sup>*<sup>α</sup>* (25)

(27)

<sup>2</sup> **<sup>u</sup>**€ð Þ*<sup>t</sup>* (28)

(26)

(29)

(30)

Johnson-Cook model [24–26] is used in this work, and the flow stress is expressed as follows:

$$\sigma\_f = \left[A + B\left(\varepsilon\_p\right)^n\right] \left[\mathbf{1} + \text{Cln}\left(\frac{\varepsilon\_p^\cdot}{\varepsilon\_{p0}}\right)\right] \left(\mathbf{1} - \left(\frac{T - T\_r}{T\_m - T\_r}\right)^m\right) \tag{31}$$

where *ε* is the plastic strain, *ε*\_ *s* �<sup>1</sup> ð Þ is the plastic strain rate, *<sup>ε</sup>*\_ <sup>0</sup> *<sup>s</sup>* �<sup>1</sup> ð Þ is the reference plastic strain rate, *Tm* is the melting temperature, *Tr* is the reference temperature, *T* is the current temperature, *A* is the yield stress, *B* is the coefficient of strain hardening, *C* is the coefficient of strain rate hardening, n is the strain hardening exponent, and m is the thermal softening exponent.

The material used for the simulations (see Section 6) is an Al-Zn-Mg-Cu aluminum alloy. The Johnson-Cook material parameters are shown in **Table 1**.


**Table 1.**

*Johnson-Cook material parameters [12].*

## **6. Applications**

#### **6.1 Axial compression test**

A cylindrical sample (diameter, 25.4 mm; length, 25.4 mm) was subjected to the uniaxial compression test at constant velocity (2.54 mm s�<sup>1</sup> ) and high temperature (400°C). Both experimental and numerical tests were performed (**Figure 3**). The aim of this test is to demonstrate the efficiency of the proposed total Lagrangian SPH formulation. We compared the numerical stress-strain curve with the

**Figure 3.** *Axial compression test setup in SPH.*

experimental ones to verify the accuracy and the stability of the code (see **Figure 4**). To confirm the validity of the experimental result, the tests were repeated three times.

initial diameter of the sample was reduced over 50% during the test. See compari-

*Hot Compression Tests Using Total Lagrangian SPH Formulation in Energy-Based Framework*

**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

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.

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

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

© 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,

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

provided the original work is properly cited.

son of results at **Table 2**.

*DOI: http://dx.doi.org/10.5772/intechopen.85930*

**6.3 Discussion**

is reached.

**7. Conclusion**

**Acknowledgements**

**Author details**

Québec, Canada

Kadiata Ba

**113**
