**1. Introduction**

The use of the smoothed particle hydrodynamic (SPH) method [1–5] (**Figure 1**) in solid mechanics is quite recent (in the 1990s) compared to the SPH fluid formulation. Libersky and Petschek [6] and Libersky et al. [4] are cited as the first to use SPH in solid mechanics, for impacts modeling at high speeds and phenomena of rupture, perforation, and fragmentation. As SPH is a meshless method, there is no mesh to distort; it can efficiently handle large deformations. SPH is an efficient numerical method for applications in forging processes [7], machining [8–10], and welding [11]. Classical approach is widely used to describe SPH equations but faces drawbacks such as lack of completeness and tensile instability (numerical fragmentation). Total Lagrangian corrected SPH formulation is then used to fix the cited problems. In this paper, a Hamiltonian formulation is proposed for dynamic and steady-state problem simulation focusing on numerical efficiency such as accuracy and simulation time. The governing equations are derived following a Lagrangian variational principle leading to a Hamiltonian system of particles [12–14]. With the Hamiltonian SPH formulation, local conservation of momentum between particles is respected, and they remain locally ordered during the process as wanted in solid mechanic problems.

Total Lagrangian formulation reduces also the computational time by avoiding the search for neighboring particles for the construction of the kernel function at

where *mi* is the mass of the particle and *ρ<sup>i</sup>* its density.

*d dt ∂L ∂***v***i*

dissipative energy, and L is the Lagrangian given by

By substituting Eq. (3) into Eq. (2), it leads to

*d dt ∂K ∂***v***i*

deformation, density, or other constitutive parameters:

The dissipative energy can also be expressed as

∇**v***<sup>i</sup>* ¼ ∑ *j mj ρj*

with the velocity gradient given by

continuum, the kinetic K, internal Q

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

kinetic energy of each particle:

the total external energy is

tion tensor.

**107**

to be defined.

To proceed with a variational formulation of the equations of motion of the

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

� *∂L ∂***x***i*

¼ � *<sup>∂</sup>*Π*ext ∂***x***i*

*<sup>K</sup>* <sup>¼</sup> <sup>1</sup> 2 ∑ *i*

Π*ext* ¼ � ∑

Π*int* ¼ ∑ *i*

Π*disp* ¼ ∑ *i*¼1

> **<sup>d</sup>** <sup>¼</sup> <sup>1</sup> 2

where **x***<sup>i</sup>* and **v***<sup>i</sup>* are the spatial position and velocity of the particle i, Π*dis* is the

With dissipative effects such as plasticity, the equations of motion of the system of particles representing the continuum can be evaluated following the classical Lagrangian formalism. For more details, readers can refer to Ba and Gakwaya [12]:

*int*, and external energy Q

¼ � *<sup>∂</sup>*Π*dis ∂***v***i*

� *<sup>∂</sup>*Π*int ∂***x***i*

The total kinetic energy of the system can be approximated as the sum of the

For a common case where the external forces result from a gravitational field **g**,

*i*

The total internal energy can be expressed as the sum of the products of particle masses by the amount of energy accumulated per unit mass *π* that depends on the

where *πdisp* is the dissipative energy per unit mass and **d** is the rate of deforma-

**v***<sup>j</sup>* � **v***<sup>i</sup>*

*L* **x***<sup>i</sup>* ð Þ¼ *;* **v***<sup>i</sup> K*ð Þ� **v***<sup>i</sup>* Π*ext*ð Þ� **x***<sup>i</sup>* Π*int*ð Þ **x***<sup>i</sup>* (3)

� *<sup>∂</sup>*Π*dis ∂***v***i*

*mi* **v***<sup>i</sup>* ð Þ *:***v***<sup>i</sup>* (5)

*mi* **x***i:***g** � � (6)

*miπ ρ<sup>i</sup>* ð Þ *;* … (7)

*miπdisp*ð Þ **d** (8)

<sup>∇</sup>**<sup>v</sup>** <sup>þ</sup> <sup>∇</sup>**v***<sup>T</sup>* � � (9)

� �∇*<sup>W</sup>* **<sup>x</sup>***<sup>i</sup>* � **<sup>x</sup>***j; <sup>h</sup>* � � (10)

*ext* of the system need

(2)

(4)

**Figure 1.**

*(a) Schematic representation of the discretization of the domain Ω by a set of points i [15] and (b) seen in the space of a B-spline [16].*

each time step. Through axial and lateral compression tests, the accuracy of the new formulation is shown. Results are compared to a classical formulation based on differential equations for solid mechanic applications.

#### **2. Discrete equations of motion from energy-based formulation**

The governing equations are derived following a Lagrangian variational principle leading to a Hamiltonian system of particles (energy-based) [12, 17–19] where the motion of each particle is given by the classical Lagrange equations. Therefore, as explained by Bonet et al. [18], constants of the motion such as linear and angular momentum are conserved.

For each particle, the physical quantities are calculated through interpolation over neighbor particles. Every particle is considered as a moving thermodynamic subsystem [12]. The volume of each particle is given by

$$V\_i = m\_i / \rho\_i \tag{1}$$

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

where *mi* is the mass of the particle and *ρ<sup>i</sup>* its density.

To proceed with a variational formulation of the equations of motion of the continuum, the kinetic K, internal Q *int*, and external energy Q *ext* of the system need to be defined.

With dissipative effects such as plasticity, the equations of motion of the system of particles representing the continuum can be evaluated following the classical Lagrangian formalism. For more details, readers can refer to Ba and Gakwaya [12]:

$$\frac{d}{dt}\frac{\partial L}{\partial \mathbf{v}\_i} - \frac{\partial L}{\partial \mathbf{x}\_i} = -\frac{\partial \Pi\_{dis}}{\partial \mathbf{v}\_i} \tag{2}$$

where **x***<sup>i</sup>* and **v***<sup>i</sup>* are the spatial position and velocity of the particle i, Π*dis* is the dissipative energy, and L is the Lagrangian given by

$$L(\mathbf{x}\_i, \mathbf{v}\_i) = K(\mathbf{v}\_i) - \Pi\_{\text{ext}}(\mathbf{x}\_i) - \Pi\_{\text{int}}(\mathbf{x}\_i) \tag{3}$$

By substituting Eq. (3) into Eq. (2), it leads to

$$\frac{d}{dt}\frac{\partial \mathbf{K}}{\partial \mathbf{v}\_i} = -\frac{\partial \Pi\_{\text{ext}}}{\partial \mathbf{x}\_i} - \frac{\partial \Pi\_{\text{int}}}{\partial \mathbf{x}\_i} - \frac{\partial \Pi\_{\text{dis}}}{\partial \mathbf{v}\_i} \tag{4}$$

The total kinetic energy of the system can be approximated as the sum of the kinetic energy of each particle:

$$K = \frac{1}{2} \sum\_{i} m\_i(\mathbf{v}\_i, \mathbf{v}\_i) \tag{5}$$

For a common case where the external forces result from a gravitational field **g**, the total external energy is

$$\Pi\_{\text{ext}} = -\sum\_{i} m\_{i}(\mathbf{x}\_{i}.\mathbf{g})\tag{6}$$

The total internal energy can be expressed as the sum of the products of particle masses by the amount of energy accumulated per unit mass *π* that depends on the deformation, density, or other constitutive parameters:

$$\Pi\_{int} = \sum\_{i} m\_{i} \pi(\rho\_{i}, \dots) \tag{7}$$

The dissipative energy can also be expressed as

$$\Pi\_{disp} = \sum\_{i=1} m\_i \pi\_{disp}(\mathbf{d}) \tag{8}$$

where *πdisp* is the dissipative energy per unit mass and **d** is the rate of deformation tensor.

$$\mathbf{d} = \frac{1}{2} \left( \nabla \mathbf{v} + \nabla \mathbf{v}^T \right) \tag{9}$$

with the velocity gradient given by

$$\nabla \mathbf{v}\_i = \sum\_j \frac{m\_j}{\rho\_j} \left( \mathbf{v}\_j - \mathbf{v}\_i \right) \nabla W(\mathbf{x}\_i - \mathbf{x}\_j, h) \tag{10}$$

each time step. Through axial and lateral compression tests, the accuracy of the new formulation is shown. Results are compared to a classical formulation based on

*(a) Schematic representation of the discretization of the domain Ω by a set of points i [15] and (b) seen in*

**2. Discrete equations of motion from energy-based formulation**

The governing equations are derived following a Lagrangian variational principle leading to a Hamiltonian system of particles (energy-based) [12, 17–19] where the motion of each particle is given by the classical Lagrange equations. Therefore, as explained by Bonet et al. [18], constants of the motion such as linear

For each particle, the physical quantities are calculated through interpolation over neighbor particles. Every particle is considered as a moving thermodynamic

*Vi* ¼ *mi=ρ<sup>i</sup>* (1)

differential equations for solid mechanic applications.

subsystem [12]. The volume of each particle is given by

and angular momentum are conserved.

**Figure 1.**

**106**

*the space of a B-spline [16].*

*Progress in Relativity*

where *<sup>W</sup>* **<sup>x</sup>***<sup>i</sup>* � **<sup>x</sup>***j; <sup>h</sup>* � � is the SPH kernel function and *<sup>h</sup>* is the smoothing length.

## **3. Corrected total Lagrangian SPH formulation for solid mechanics**

Total Lagrangian formulation [20, 21] is well suited for solid mechanic problems as the SPH particles change less often their neighbors than in fluid mechanics [12]. The SPH kernels and their gradients are then expressed in the initial configuration (material coordinates **X** are used). The proposed corrected kernel is to address the lack of completeness and interpolation consistency; the smoothing length *h* is considered as a functional variable in the calculation of the gradient of the kernel function ∇*W* [19].

Lagrangian and spatial coordinates are connected through the gradient of deformation tensor **F**:

$$\mathbf{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \frac{\partial (\mathbf{X} + \mathbf{u})}{\partial \mathbf{X}} \tag{11}$$

**<sup>e</sup>**\_ h i*<sup>i</sup>* <sup>¼</sup> **<sup>P</sup>***<sup>j</sup>* : � <sup>∑</sup>

particle.

forms:

as

0 @

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

2 4

*j*

*mj ρiρ<sup>j</sup>*

þ*k*∇*Ti* þ *rpl*

**v***<sup>j</sup>* � **v***i; h*<sup>0</sup> � �⊗∇**<sup>X</sup>***<sup>i</sup>*

**<sup>a</sup>***<sup>i</sup>* <sup>¼</sup> **<sup>u</sup>**€*<sup>i</sup>* <sup>¼</sup> <sup>1</sup>

where **<sup>f</sup>**intð Þ*<sup>i</sup>* and **<sup>f</sup>***ex*tð Þ*<sup>i</sup>* are the internal and external forces, **<sup>h</sup>***<sup>k</sup>*

the internal and the external heat flux, and **C** is the capacitance matrix.

**f**intð Þ*<sup>i</sup>* ¼ ∑

**G***<sup>i</sup>* **X***<sup>j</sup>*

velocity, position, and temperature of each SPH particle.

*<sup>t</sup>* is computed at the beginning of the increment by

∇�*<sup>W</sup>* at the initial reference configuration.

Internal heat flux can be expressed as

**4. Temporal integration scheme**

ory during computation.

time integration rule [22].

**T**\_

**109**

**T**\_ *<sup>i</sup>* <sup>¼</sup> <sup>1</sup> **C***i* **h***k*

where **e**\_ is the energy rate, *rpl* is the mechanical contribution (heat generated by

The equation of motion and the equation of the thermal energy of each particle can be put after discretization and evaluation of all the interactions in the following

**f***ex*tð Þ*<sup>i</sup>* � **f**intð Þ*<sup>i</sup>*

*<sup>j</sup>* **P***<sup>j</sup>* **G***<sup>i</sup>* **X***<sup>j</sup>*

**W***<sup>i</sup>* **X***<sup>j</sup>*

intð Þ*i*

extð Þ*<sup>i</sup>* � **<sup>h</sup>***<sup>k</sup>*

The expression for the internal force for a given particle can be expressed by differentiating the internal energy per unit mass with respect to the nodal positions

> *n j*¼1 *V*0

� � <sup>¼</sup> *Vi*<sup>∇</sup>

where **G** is the gradient function and contains the corrected kernel gradients

�0

where **k** is the heat conductivity matrix and **T** the vector of nodal temperatures. Explicit finite difference method is used to solve numerically the differential equation presented in this section through explicit dynamic algorithm to update the

A typical integration scheme used for integrating SPH equations is the leapfrog algorithm (**Figure 2**), an extension of the Verlet algorithm with low storage mem-

The heat transfer equations are integrated using the explicit forward-difference

**<sup>T</sup>**ð Þ *<sup>t</sup>*þ*Δ<sup>t</sup>* <sup>¼</sup> **<sup>T</sup>**ð Þ*<sup>t</sup>* <sup>þ</sup> *<sup>Δ</sup>t*ð Þ *<sup>t</sup>*þ<sup>1</sup> **<sup>T</sup>**\_

the plastic dissipation), k is the conductivity, and T is the temperature of the

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

**m***<sup>i</sup>*

*W* **X***<sup>i</sup>* � **X***j; h*<sup>0</sup> � �*V*<sup>0</sup>

*j*

� � (18)

� � (19)

1 A**B**

3

intð Þ*<sup>i</sup>* and **<sup>h</sup>***<sup>k</sup>*

� � (20)

� � (21)

ð Þ*<sup>t</sup>* (23)

**h**intð Þ*<sup>i</sup>* ¼ **k***i***T***<sup>i</sup>* (22)

extð Þ*<sup>i</sup>* are

5 (17)

where **u** is the displacement of a material point.

The expression of the corrected gradient of deformation tensor **F**, in total Lagrangian formulation, is given by

$$
\langle \mathbf{F}\_i \rangle = \left( -\sum\_j \left( \mathbf{u}\_j - \mathbf{u}\_i \right) \otimes \nabla\_{\mathbf{X}\_j} \mathcal{W} \left( \mathbf{X}\_i - \mathbf{X}\_j, h\_0 \right) V\_j^0 \right) \mathbf{B} + I \tag{12}
$$

where ∇**X***<sup>j</sup>* is the gradient with respect to a material point **X**, *V*<sup>0</sup> *<sup>j</sup>* is the initial volume of particle j, *h*<sup>0</sup> is the initial smoothing length, and **I** is the identity matrix.

**B** is the expression of the correction of the gradient expressed as [20]

$$\mathbf{B} = \left(\sum\_{j} \frac{m\_j}{\rho\_j} (\mathbf{X}\_i - \mathbf{X}\_j, h\_0) \otimes \nabla\_{\mathbf{X}\_i} \mathcal{W} (\mathbf{X}\_i - \mathbf{X}\_j, h\_0) \right)^{-1} \tag{13}$$

The corrected mass conservation equation for particle i is

$$
\rho\_{0i} = \rho\_i l = \rho\_i \text{det} \mathbf{F}\_i \tag{14}
$$

where *J* and *ρ*<sup>0</sup> are the Jacobian and the initial density. The corrected momentum equation for a particle i is

$$\langle \mathbf{a}\_i \rangle = \left( -\sum\_j \left( \mathbf{P}\_j - \mathbf{P}\_i \right) \otimes \nabla\_{\mathbf{X}\_j} \tilde{W} \left( \mathbf{X}\_i - \mathbf{X}\_j, h\_0 \right) V\_j^0 + \mathbf{f}\_i \right) : \mathbf{B} \tag{15}$$

where **a**, *W*~ , and **f***<sup>i</sup>* are the acceleration, the normalized smoothing function, and the body force.

**P** is the first Piola-Kirchhoff stress.

$$\mathbf{P} = J \boldsymbol{\sigma} \mathbf{F}^{-T} \tag{16}$$

where **σ** is the Cauchy stress tensor and **F**�*<sup>T</sup>* is the inverse of the transpose of the gradient of deformation tensor.

The corrected energy conservation equation for particle i is

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

$$\langle \dot{\mathbf{e}}\_{i} \rangle = \mathbf{P}\_{j} : \begin{bmatrix} \left( -\sum\_{j} \frac{m\_{j}}{\rho\_{i} \rho\_{j}} \left( \mathbf{v}\_{j} - \mathbf{v}\_{i}, h\_{0} \right) \otimes \nabla\_{\mathbf{X}\_{i}} \mathcal{W} \left( \mathbf{X}\_{i} - \mathbf{X}\_{j}, h\_{0} \right) \mathbf{V}\_{j}^{0} \right) \\\ + k \nabla T\_{i} + r\_{pl} \end{bmatrix} \mathbf{B} \tag{17}$$

where **e**\_ is the energy rate, *rpl* is the mechanical contribution (heat generated by the plastic dissipation), k is the conductivity, and T is the temperature of the particle.

The equation of motion and the equation of the thermal energy of each particle can be put after discretization and evaluation of all the interactions in the following forms:

$$\mathbf{a}\_{i} = \ddot{\mathbf{u}}\_{i} = \frac{1}{\mathbf{m}\_{i}} (\mathbf{f}\_{\text{ext}(i)} - \mathbf{f}\_{\text{int}(i)}) \tag{18}$$

$$\dot{\mathbf{T}}\_i = \frac{\mathbf{1}}{\mathbf{C}\_i} \left( \mathbf{h}\_{\text{ext}(i)}^k - \mathbf{h}\_{\text{int}(i)}^k \right) \tag{19}$$

where **<sup>f</sup>**intð Þ*<sup>i</sup>* and **<sup>f</sup>***ex*tð Þ*<sup>i</sup>* are the internal and external forces, **<sup>h</sup>***<sup>k</sup>* intð Þ*<sup>i</sup>* and **<sup>h</sup>***<sup>k</sup>* extð Þ*<sup>i</sup>* are the internal and the external heat flux, and **C** is the capacitance matrix.

The expression for the internal force for a given particle can be expressed by differentiating the internal energy per unit mass with respect to the nodal positions as

$$\mathbf{f}\_{\text{int}(i)} = \sum\_{j=1}^{n} \mathbf{V}\_{j}^{0} \mathbf{P}\_{j} \mathbf{G}\_{i} \left(\mathbf{X}\_{j}\right) \tag{20}$$

where **G** is the gradient function and contains the corrected kernel gradients ∇�*<sup>W</sup>* at the initial reference configuration.

$$\mathbf{G}\_i(\mathbf{X}\_j) = \mathbf{V}\_i \overset{\sim}{\nabla} \mathbf{W}\_i(\mathbf{X}\_j) \tag{21}$$

Internal heat flux can be expressed as

$$\mathbf{h}\_{\text{int}(i)} = \mathbf{k}\_i \mathbf{T}\_i \tag{22}$$

where **k** is the heat conductivity matrix and **T** the vector of nodal temperatures.

Explicit finite difference method is used to solve numerically the differential equation presented in this section through explicit dynamic algorithm to update the velocity, position, and temperature of each SPH particle.

#### **4. Temporal integration scheme**

A typical integration scheme used for integrating SPH equations is the leapfrog algorithm (**Figure 2**), an extension of the Verlet algorithm with low storage memory during computation.

The heat transfer equations are integrated using the explicit forward-difference time integration rule [22].

$$\mathbf{T}\_{(t+\Delta t)} = \mathbf{T}\_{(t)} + \Delta t\_{(t+1)} \dot{\mathbf{T}}\_{(t)} \tag{23}$$

**T**\_ *<sup>t</sup>* is computed at the beginning of the increment by

where *<sup>W</sup>* **<sup>x</sup>***<sup>i</sup>* � **<sup>x</sup>***j; <sup>h</sup>* � � is the SPH kernel function and *<sup>h</sup>* is the smoothing length.

Total Lagrangian formulation [20, 21] is well suited for solid mechanic problems as the SPH particles change less often their neighbors than in fluid mechanics [12]. The SPH kernels and their gradients are then expressed in the initial configuration (material coordinates **X** are used). The proposed corrected kernel is to address the lack of completeness and interpolation consistency; the smoothing length *h* is considered as a functional variable in the calculation of the gradient of the kernel

Lagrangian and spatial coordinates are connected through the gradient of defor-

*<sup>∂</sup>***<sup>X</sup>** <sup>¼</sup> *<sup>∂</sup>*ð Þ **<sup>X</sup>** <sup>þ</sup> **<sup>u</sup>**

*W* **X***<sup>i</sup>* � **X***j; h*<sup>0</sup> � �*V*<sup>0</sup>

*W* **X***<sup>i</sup>* � **X***j; h*<sup>0</sup>

*ρ*0*<sup>i</sup>* ¼ *ρiJ* ¼ *ρi*det**F***<sup>i</sup>* (14)

*<sup>j</sup>* þ **f***<sup>i</sup>*

**<sup>P</sup>** <sup>¼</sup> *<sup>J</sup>***σF**�*<sup>T</sup>* (16)

The expression of the corrected gradient of deformation tensor **F**, in total

!

volume of particle j, *h*<sup>0</sup> is the initial smoothing length, and **I** is the identity matrix. **B** is the expression of the correction of the gradient expressed as [20]

� � !�<sup>1</sup>

*<sup>W</sup>*<sup>~</sup> **<sup>X</sup>***<sup>i</sup>* � **<sup>X</sup>***j; <sup>h</sup>*<sup>0</sup> � �*V*<sup>0</sup>

!

where **a**, *W*~ , and **f***<sup>i</sup>* are the acceleration, the normalized smoothing function, and

where **σ** is the Cauchy stress tensor and **F**�*<sup>T</sup>* is the inverse of the transpose of the

*<sup>∂</sup>***<sup>X</sup>** (11)

*j*

**B** þ *I* (12)

*<sup>j</sup>* is the initial

: **B** (15)

(13)

**<sup>F</sup>** <sup>¼</sup> *<sup>∂</sup>***<sup>x</sup>**

where **u** is the displacement of a material point.

*j*

**u***<sup>j</sup>* � **u***<sup>i</sup>* � �⊗∇**X***<sup>j</sup>*

where ∇**X***<sup>j</sup>* is the gradient with respect to a material point **X**, *V*<sup>0</sup>

**X***<sup>i</sup>* � **X***j; h*<sup>0</sup> � �⊗∇**X***<sup>i</sup>*

The corrected mass conservation equation for particle i is

where *J* and *ρ*<sup>0</sup> are the Jacobian and the initial density. The corrected momentum equation for a particle i is

> **P***<sup>j</sup>* � **P***<sup>i</sup>* � �⊗∇**<sup>X</sup>***<sup>j</sup>*

The corrected energy conservation equation for particle i is

Lagrangian formulation, is given by

h i **F***<sup>i</sup>* ¼ � ∑

**B** ¼ ∑ *j mj ρj*

h i **a***<sup>i</sup>* ¼ � ∑

**P** is the first Piola-Kirchhoff stress.

gradient of deformation tensor.

the body force.

**108**

*j*

**3. Corrected total Lagrangian SPH formulation for solid mechanics**

function ∇*W* [19].

*Progress in Relativity*

mation tensor **F**:

$$\dot{\mathbf{T}}\_{(t)} = \mathbf{C}^{-1} \left( \mathbf{h}\_{\text{ext}}^t - \mathbf{h}\_{\text{int}}^t \right) \tag{24}$$

The whole thermomechanical problem is solved by explicit coupling; both the forward-difference (for the thermal problem) and central-difference (for the

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

The structure of the SPH code is described below (**Figure 2**). The time integration routine is the main subroutine. It calculates the new variables (density,

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

" # !

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

num alloy. The Johnson-Cook material parameters are shown in **Table 1**.

The material used for the simulations (see Section 6) is an Al-Zn-Mg-Cu alumi-

420 465 0.862 0.5088 0.081 0.1 641 25

A cylindrical sample (diameter, 25.4 mm; length, 25.4 mm) was subjected to the

(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

*p ε*˙ *p*0

�<sup>1</sup> ð Þ is the plastic strain rate, *<sup>ε</sup>*\_ <sup>0</sup> *<sup>s</sup>*

<sup>1</sup> � *<sup>T</sup>* � *Tr Tm* � *Tr* � � � �*<sup>m</sup>*

**<sup>0</sup> (s**�**<sup>1</sup>**

(31)

�<sup>1</sup> ð Þ is the

**) Tm (°C) Tr (°C)**

) and high temperature

� �*<sup>n</sup>* � � <sup>1</sup> <sup>þ</sup> *<sup>C</sup>*ln *<sup>ε</sup>*˙

hardening exponent, and m is the thermal softening exponent.

**A (MPa) B (MPa) c n m** *ε*\_

uniaxial compression test at constant velocity (2.54 mm s�<sup>1</sup>

mechanical problem) integrations are explicit.

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

acceleration, external force, internal force).

*σ<sup>f</sup>* ¼ *A* þ *B ε<sup>p</sup>*

where *ε* is the plastic strain, *ε*\_ *s*

**5. Material behavior**

expressed as follows:

**6. Applications**

**Table 1.**

**Figure 3.**

**111**

**6.1 Axial compression test**

*Axial compression test setup in SPH.*

*Johnson-Cook material parameters [12].*

The stability time is given by

$$
\Delta t\_T \approx \frac{\Delta r\_{\text{min}}^2}{2a} \tag{25}
$$

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

$$a = \frac{k}{\rho \mathfrak{s}}\tag{26}$$

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

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

$$\ddot{\mathbf{u}}\_{(t)} = \mathbf{M}^{-1} (\mathbf{P}\_{(t)} - \mathbf{I}\_{(t)}) \tag{27}$$

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

The integration leads to the nodal velocity **u**\_ (Eq. 28) and the nodal displacement **u** (Eq. 29).

$$
\dot{\mathbf{u}}\_{\left(t+\frac{\omega}{2}\right)} = \dot{\mathbf{u}}\_{\left(t-\frac{\omega}{2}\right)} + \frac{\left(\Delta t\_{\left(t+\Delta t\right)} + \Delta t\_{\left(t\right)}\right)}{2}\ddot{\mathbf{u}}\_{\left(t\right)}\tag{28}
$$

$$\mathbf{u}\_{(t+\Delta t)} = \mathbf{u}\_{(t)} + \Delta t\_{(t+\Delta t)} \dot{\mathbf{u}}\_{\left(t+\frac{\Delta t}{2}\right)} \tag{29}$$

The stable time is calculated as follows:

$$
\Delta t = \min \left( \frac{L\_\epsilon}{c\_d} \right) \tag{30}
$$

where *Le* and *cd* are, respectively, the characteristic length of the element and the dilatational wave speed of the material.

**Figure 2.** *SPH code structure [23].*

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

The whole thermomechanical problem is solved by explicit coupling; both the forward-difference (for the thermal problem) and central-difference (for the mechanical problem) integrations are explicit.

The structure of the SPH code is described below (**Figure 2**). The time integration routine is the main subroutine. It calculates the new variables (density, acceleration, external force, internal force).
