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

Abstract

Limitations of the finite element method (FEM) in some cases involving large deformations as in forging or high compression tests are overcome nowadays by meshless methods such as the smoothed particle hydrodynamic (SPH) method. This paper presents a corrected total Lagrangian SPH formulation for problems encountering large deformations in solid mechanics. The continuum is modeled as a Hamiltonian system of particles (energy-based framework). The total Lagrangian formulation proposed overcomes some problems faced by standard SPH in simulating solid mechanic problems such as tensile instability. Numerical applications compared with experimental results are presented to show the capabilities of this novel formulation.

1. Introduction

The use of the smoothed particle hydrodynamic (SPH) method [1, 2, 3, 4, 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, 9, 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, 13, 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.

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].

Total Lagrangian formulation reduces also the computational time by avoiding the search for neighboring particles for the construction of the kernel function at 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, 18, 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

$$V_i=m_i/{\rho }_i$$E1

where m_i is the mass of the particle and ρ_i its density.

To proceed with a variational formulation of the equations of motion of the continuum, the kinetic K, internal ${\prod }_{\mathit{int}}$, and external energy ${\prod }_{\mathit{ext}}$ 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]:

$$\frac{d}{\mathit{dt}}\frac{\partial L}{\partial {\mathbf{v}}_i}-\frac{\partial L}{\partial {\mathbf{x}}_i}=-\frac{\partial {\mathrm{\Pi }}_{\mathit{dis}}}{\partial {\mathbf{v}}_i}$$E2

where ${\mathbf{x}}_i$ and ${\mathbf{v}}_i$ are the spatial position and velocity of the particle i, ${\mathrm{\Pi }}_{\mathit{dis}}$ is the dissipative energy, and L is the Lagrangian given by

$$L\left({\mathbf{x}}_i,{\mathbf{v}}_i\right)=K\left({\mathbf{v}}_i\right)-{\mathrm{\Pi }}_{\mathit{ext}}\left({\mathbf{x}}_i\right)-{\mathrm{\Pi }}_{\mathit{int}}\left({\mathbf{x}}_i\right)$$E3

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

$$\frac{d}{\mathit{dt}}\frac{\partial K}{\partial {\mathbf{v}}_i}=-\frac{\partial {\mathrm{\Pi }}_{\mathit{ext}}}{\partial {\mathbf{x}}_i}-\frac{\partial {\mathrm{\Pi }}_{\mathit{int}}}{\partial {\mathbf{x}}_i}-\frac{\partial {\mathrm{\Pi }}_{\mathit{dis}}}{\partial {\mathbf{v}}_i}$$E4

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

$$K=\frac{1}{2}\sum _i{m}_i\left({\mathbf{v}}_i.{\mathbf{v}}_i\right)$$E5

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

$${\Pi }_{\mathit{ext}}=-\sum _i{m}_i\left({\mathbf{x}}_i.\mathbf{g}\right)$$E6

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 $\pi$ that depends on the deformation, density, or other constitutive parameters:

$${\Pi }_{\mathit{int}}=\sum _i{m}_i\pi \left({\rho }_i,\dots \right)$$E7

The dissipative energy can also be expressed as

$${\mathrm{\Pi }}_{\mathit{\text{disp}}}=\sum _{i=1}{m}_i{\pi }_{\mathit{\text{disp}}}\left(\mathbf{d}\right)$$E8

where ${\pi }_{\mathit{\text{disp}}}$ is the dissipative energy per unit mass and $\mathbf{d}$ is the rate of deformation tensor.

$$\mathbf{d}=\frac{1}{2}\left(\nabla \mathbf{v}+\nabla {\mathbf{v}}^T\right)$$E9

with the velocity gradient given by

$$\nabla {\mathbf{v}}_i=\sum _j\frac{m_j}{{\rho }_j}\left({\mathbf{v}}_j-{\mathbf{v}}_i\right)\nabla W\left({\mathbf{x}}_i-{\mathbf{x}}_j,h\right)$$E10

where $W\left({\mathbf{x}}_i-{\mathbf{x}}_j,h\right)$ is the SPH kernel function and h 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 $\mathbf{X}$ 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 $\nabla W$ [19].

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

$$\mathbf{F}=\frac{\partial \mathbf{x}}{\partial \mathbf{X}}=\frac{\partial \left(\mathbf{X}+\mathbf{u}\right)}{\partial \mathbf{X}}$$E11

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

$$⟨{\mathbf{F}}_i⟩=\left(-\sum _j\left({\mathbf{u}}_j-{\mathbf{u}}_i\right)\otimes {\nabla }_{{\mathbf{X}}_j}W\left({\mathbf{X}}_i-{\mathbf{X}}_j,h_0\right)V_j^0\right)\mathbf{B}+I$$E12

where ${\nabla }_{{\mathbf{X}}_j}$ is the gradient with respect to a material point X, $V_j^0$ is the initial volume of particle j, $h_0$ is the initial smoothing length, and I is the identity matrix.

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

$$\mathbf{B}={\left(\sum _j\frac{m_j}{{\rho }_j}\left({\mathbf{X}}_i-{\mathbf{X}}_j,h_0\right)\otimes {\nabla }_{{\mathbf{X}}_i}W\left({\mathbf{X}}_i-{\mathbf{X}}_j,h_0\right)\right)}^{-1}$$E13

The corrected mass conservation equation for particle i is

$${\rho }_{0i}={\rho }_{i}J={\rho }_{i}det{\mathbf{F}}_i$$E14

where $J$ and ${\rho }_0$ are the Jacobian and the initial density.

The corrected momentum equation for a particle i is

$$⟨{\mathbf{a}}_i⟩=\left(-\sum _j\left({\mathbf{P}}_j-{\mathbf{P}}_i\right)\otimes {\nabla }_{{\mathbf{X}}_j}\tilde{W}\left({\mathbf{X}}_i-{\mathbf{X}}_j,h_0\right)V_j^0+{\mathbf{f}}_i\right):\mathbf{B}$$E15

where $\mathbf{a}$, $\tilde{W}$, and ${\mathbf{f}}_i$ are the acceleration, the normalized smoothing function, and the body force.

P is the first Piola-Kirchhoff stress.

$$\mathbf{P}=J{\mathbf{\sigma F}}^{-T}$$E16

where $\mathbf{\sigma }$ is the Cauchy stress tensor and ${\mathbf{F}}^{-T}$ is the inverse of the transpose of the gradient of deformation tensor.

The corrected energy conservation equation for particle i is

$$⟨{\dot{\mathbf{e}}}_i⟩={\mathbf{P}}_j:\left[\left(\begin{array}{l}-\sum _j\frac{m_j}{{\rho }_i{\rho }_j}\left({\mathbf{v}}_j-{\mathbf{v}}_i,h_0\right)\otimes {\nabla }_{{\mathbf{X}}_i}W\left({\mathbf{X}}_i-{\mathbf{X}}_j,h_0\right)V_j^0\\ +k\nabla T_i+r_{\mathit{pl}}\end{array}\right)\mathbf{B}\right]$$E17

where $\dot{\mathbf{e}}$ is the energy rate, $r_{\mathit{pl}}$ 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:

$${\mathbf{a}}_i={\ddot{\mathbf{u}}}_i=\frac{1}{{\mathbf{m}}_i}\left({\mathbf{f}}_{\mathit{ex}\mathrm{t}\left(i\right)}-{\mathbf{f}}_{int\left(i\right)}\right)$$E18

$${\dot{\mathbf{T}}}_i=\frac{1}{{\mathbf{C}}_i}\left({\mathbf{h}}_{\mathit{ext}\left(i\right)}^k-{\mathbf{h}}_{\mathit{int}\left(i\right)}^k\right)$$E19

where ${\mathbf{f}}_{int\left(i\right)}$ and ${\mathbf{f}}_{\mathit{ex}\mathrm{t}\left(i\right)}$ are the internal and external forces, ${\mathbf{h}}_{\mathit{int}\left(i\right)}^k$ and ${\mathbf{h}}_{\mathit{ext}\left(i\right)}^k$ 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

$${\mathbf{f}}_{int\left(i\right)}=\sum _{j=1}^n{V}_j^0{\mathbf{P}}_j{\mathbf{G}}_i\left({\mathbf{X}}_j\right)$$E20

where G is the gradient function and contains the corrected kernel gradients $\stackrel{\sim }{\nabla }W$ at the initial reference configuration.

$${\mathbf{G}}_i\left({\mathbf{X}}_j\right)=V_i{\stackrel{\sim }{\nabla }}_0{\mathbf{W}}_i\left({\mathbf{X}}_j\right)$$E21

Internal heat flux can be expressed as

$${\mathbf{h}}_{\mathit{int}\left(i\right)}={\mathbf{k}}_i{\mathbf{T}}_i$$E22

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.

Figure 2.

SPH code structure [23].

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

$${\mathbf{T}}_{\left(t+\mathit{\Delta t}\right)}={\mathbf{T}}_{\left(t\right)}+\Delta t_{\left(t+1\right)}{\dot{\mathbf{T}}}_{\left(t\right)}$$E23

${\dot{\mathbf{T}}}_t$ is computed at the beginning of the increment by

$${\dot{\mathbf{T}}}_{\left(t\right)}={\mathbf{C}}^{-1}\left({\mathbf{h}}_{\mathit{ext}}^t-{\mathbf{h}}_{\mathrm{int}}^t\right)$$E24

The stability time is given by

$$\Delta t_T\approx \frac{\Delta r_{\min }^2}{2\alpha }$$E25

where $\Delta r_{\min }$ is the smallest interparticle distance and $\alpha$ is the diffusivity of the material.

$$\alpha =\frac{k}{\mathit{\rho s}}$$E26

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 $\ddot{\mathit{u}}$at time t is given by

$${\ddot{\mathbf{u}}}_{\left(t\right)}={\mathbf{M}}^{-1}\left({\mathbf{P}}_{\left(t\right)}-{\mathbf{I}}_{\left(t\right)}\right)$$E27

where M, ${\mathbf{P}}_{\left(t\right)}$, and ${\mathbf{I}}_{\left(t\right)}$ represent the mass matrix and the external and internal forces.

The integration leads to the nodal velocity $\dot{\mathbf{u}}$ (Eq. 28) and the nodal displacement $\mathbf{u}$ (Eq. 29).

$${\dot{\mathbf{u}}}_{\left(t+\frac{\mathit{\Delta t}}{2}\right)}={\dot{\mathbf{u}}}_{\left(t-\frac{\mathit{\Delta t}}{2}\right)}+\frac{\left(\Delta t_{\left(t+\mathit{\Delta t}\right)}+\Delta t_{\left(t\right)}\right)}{2}{\ddot{\mathbf{u}}}_{\left(t\right)}$$E28

$${\mathbf{u}}_{\left(t+\mathit{\Delta t}\right)}={\mathbf{u}}_{\left(t\right)}+\Delta t_{\left(t+\mathit{\Delta t}\right)}{\dot{\mathbf{u}}}_{\left(t+\frac{\mathit{\Delta t}}{2}\right)}$$E29

The stable time is calculated as follows:

$$\mathit{\Delta t}=min\left(\frac{L_e}{c_d}\right)$$E30

where $L_e$ and $c_d$ are, respectively, the characteristic length of the element and the dilatational wave speed of the material.

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).

5. Material behavior

Johnson-Cook model [24, 25, 26] is used in this work, and the flow stress is expressed as follows:

$${\sigma }_f=\left[A+B{\left({\epsilon }_p\right)}^n\right]\left[1+Cln\left(\frac{{\dot{\epsilon }}_p}{{\dot{\epsilon }}_{p0}}\right)\right]\left(1-{\left(\frac{T-T_r}{T_m-T_r}\right)}^m\right)$$E31

where $\epsilon$ is the plastic strain, $\dot{\epsilon }\left(s^{-1}\right)$ is the plastic strain rate, ${\dot{\epsilon }}_0\left(s^{-1}\right)$ is the reference plastic strain rate, $T_m$is the melting temperature, $T_r$ 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.

A (MPa)	B (MPa)	c	n	m	${\dot{\epsilon }}_0$ (s^−1)	Tm (°C)	Tr (°C)
420	465	0.862	0.5088	0.081	0.1	641	25

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⁻¹) 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 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.

Figure 3.

Axial compression test setup in SPH.

Figure 4.

Stress-strain curves: experimental vs. SPH.

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⁻¹, 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 initial diameter of the sample was reduced over 50% during the test. See comparison of results at Table 2.

Figure 5.

Lateral compression test: Eulerian vs. Lagrangian.

	Simulation time	Number of particles
Eulerian SPH	4 h 04 min	5313
Lagrangian SPH	1 h 36 min	5313

Table 2.

Simulation time comparison.

6.3 Discussion

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.

Acknowledgments

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