# An efficient and stabilised SPH method for large strain metal plastic deformations

## Abstract

Due to its simplicity and robustness, smooth particle hydrodynamics (SPH) has been widely used in the modelling of solid and fluid mechanics problems. Through the years, various formulations and stabilisation techniques have been adopted to enhance it. Recently, the authors developed JST–SPH, a mixed formulation based on the SPH method. Originally devised for modelling (nearly) incompressible hyperelasticity, the JST–SPH formulation is mixed in the sense that linear momentum and a number of strain definitions, instead of the displacements, act as main unknowns of the problem. The resulting governing system of conservation laws conveniently enables the application of the Jameson–Schmidt–Turkel (JST) artificial dissipation term, commonly employed in computational fluid dynamics, to solid mechanics. Coupled with meshless SPH discretisation, this novel scheme eliminates the shortcomings encountered when implementing fast dynamics explicit codes using traditional mesh-based methods. This paper focuses on the applicability of the JST–SPH mixed formulation to the simulation of high-rate, large metal elastic–plastic deformations. Three applications—including the simulation of an industry-relevant metal forming process—are examined under different loading conditions, in order to demonstrate the reliability of the method. Results compare favourably with both data from the previous literature, and simulations performed with a commercial finite elements package. Most noticeably, these results demonstrate that the total Lagrangian framework of JST–SPH, fundamental to reduce the computational effort associated with the scheme, retains its accuracy in the presence of large distortions. Moreover, an algorithmic flow chart is included at the end of this document, to facilitate the computer implementation of the scheme.

## Keywords

Metal plasticity Equal channel angular extrusion (ECAE) Solid dynamics Mixed formulations SPH JST## 1 Introduction

Displacement-based explicit dynamic codes, implemented using low-order finite element methods, are commonly used for advanced numerical simulations in aerospace, automotive, biomedical, defence and manufacturing applications. However, difficulties arise in modelling high-speed impacts or large material deformations, often leading to simulation failure. In these scenarios, most computational codes prefer to employ the 8-noded underintegrated hexahedral element to model solid components [5]. Nonetheless, for many practical applications (e. g. crashworthiness, ballistics and metal forming), the extremely large deformations still result in severe mesh distortions. These lead to irregular and entangled elements, unless some form of adaptive remeshing is employed.

In addition, difficulties currently encountered by the widely employed, displacement-based, low-order finite element analysis include lower order of convergence for relevant field variables; excessive element distortion under large deformations, requiring periodic remeshing; locking behaviour in bending scenarios; non-physical pressure instabilities, and high-frequency noise due to Newmark-type time integrators.

Recent developments in computational methods for fast solid dynamics [1, 2, 7, 8, 9, 18, 19, 21, 23, 24, 25] recommend the representation of motion and deformation of a given body via a system of first-order, mixed formulation conservation laws. The partial differential equations (PDEs) that form this system do not present the displacement as the main unknown to be evaluated, instead yielding a set of other relevant quantities (i. e. velocity, deformation gradient, its cofactor matrix and scalar Jacobian), depending on the number of conservation equations being considered. This choice of the field variables will be driven on the one hand, by the need to reproduce certain features of material behaviour (e. g. near or full incompressibility) while ensuring polyconvexity of the strain energy function and, on the other hand, by the computational efficiency of the resulting scheme.

Mixed conservation laws have already been proven [1, 21, 25] to achieve second order of accuracy for stresses and strains. The two-field mixed formulation, composed of linear momentum \(\vec {p}\) and deformation gradient \({{\mathbf {\mathsf{{F}}}}}\), was later augmented by including a governing equation to conserve the Jacobian *J* of the deformation gradient [18]. This enabled the scheme to effectively solve nearly and fully incompressible materials. Further enhancement of this \(\{\vec {p}, {{\mathbf {\mathsf{{F}}}}}, J \}\) framework has also been recently reported in [7, 19] for modelling compressible, nearly incompressible and fully incompressible materials governed by a polyconvex constitutive law. In particular, this requires the presence of the matrix of co-factors of the deformation gradient \({{\mathbf {\mathsf{{F}}}}}\) (denoted here by \({{\mathbf {\mathsf{{H}}}}}\)), leading to an extended system of field variables. The \(\{ \vec {p}, {{\mathbf {\mathsf{{F}}}}}, {{\mathbf {\mathsf{{H}}}}}, J \}\) system can be reformulated to an alternative description in terms of entropy conjugates, as in [7, 19], due to the existence of a generalised convex entropy function. This readily facilitates the proof of existence of associated real wave speeds, which in turn establishes existence and uniqueness of real solutions.

A novel computational framework was devised to improve the numerical simulation behaviour during large material distortions, and to address the other numerical issues (e. g. locking, spurious oscillations) highlighted earlier. In a total Lagrangian perspective, the proposed methodology combines the use of smooth particle hydrodynamics (SPH), a meshless spatial discretisation technique, with an explicit, two-stage, total variation diminishing (TVD) Runge–Kutta temporal scheme.

To date, the application of mixed formulation has been largely focused on nearly incompressible hyperelasticity [1, 2, 21, 25], where many of the aforementioned shortcomings are encountered while using linear finite element methods. The work presented in this paper aims to investigate the feasibility of using the developed mixed formulation technique in the field of metal plasticity, by exploring some applications involving large material deformations. These will include a high-velocity impact scenario, where material is subjected to a non-uniform deformation and thus develops a large geometric discontinuity; a constrained boundary problem under heavy distortion, and the simulation of severe plastic deformation in a metallurgical process of industrial relevance.

This paper is organised as follows: a description of the mixed hyperbolic system of governing partial differential equations is presented in Sect. 2, while the adopted SPH discretisation methodology, that incorporates a stabilisation procedure based on a Jameson–Schmidt–Turkel (JST) artificial dissipation term, widely used in the field of computational fluid dynamics (CFD), will be detailed in Sect. 3. Numerical applications in metal plasticity are presented in Sect. 4, along with an assessment of the accuracy and stability of the methodology. These numerical examples are mainly chosen to demonstrate the potential and the accuracy of the developed computational strategy. Concluding remarks are then presented in Sect. 5. Finally, for the purpose of completeness, a brief outline of the chosen elasto-plastic constitutive model, including relevant computational procedures for evaluating the material deformation, is provided in “Appendix A”.

## 2 Conservation laws

In this paper, a vector will be expressed with the notation \(\vec {a}\), and a tensor with \({{\mathbf {\mathsf{{A}}}}}\). The material and spatial coordinates are denoted by \(\vec {X}\) and \(\vec {x}\), respectively. The gradient operator that refers to initial spatial coordinates will be denoted by \(\vec {\nabla }_0\). The symbol \(\pmb {\varvec{\times }}\) represents the tensorial cross product, a key operation utilised in obtaining a concise mathematical representation of the proposed methodology. This notation was introduced in a series of papers devoted to the development of mixed formulation techniques [7, 8, 9, 19]. More details on the properties of the tensorial cross product can be found in the cited references.

*J*can be written as:

In Eq. (2.1a), \(\rho \) and \(\vec {b}\), respectively, represent the material density and the body force per unit mass. System (2.1) is solved for \(\vec {p}\), \({{\mathbf {\mathsf{{F}}}}}\), \({{\mathbf {\mathsf{{H}}}}}\), *J* with respect to time *t*. It is a set of hyperbolic PDEs similar in structure to those widely used in CFD [42]. This similarity enables one to employ well-established artificial dissipation numerical techniques from CFD to improve the computational stability of Eqs. (2.1a) and (2.1d) [1, 23, 24]. It is worth noting that, in the presence of non-smooth solutions, the above system (2.1) of local conservation laws must be accompanied by suitable jump conditions, as described in [42].

^{1}as, respectively,

In the place of the full set of conservation variables, denoted here by \(\varvec{\mathcal {U}} = \{ \vec {p}, {{\mathbf {\mathsf{{F}}}}}, {{\mathbf {\mathsf{{H}}}}}, J\}\), reduced systems based on \(\{ \vec {p}, {{\mathbf {\mathsf{{F}}}}}, J\}\), or only \(\{ \vec {p}, {{\mathbf {\mathsf{{F}}}}} \}\) formulations have been adopted in the past. The robustness of these reduced systems has been positively ascertained by testing them against a thorough variety of numerical regimes and external conditions [1, 18, 24].

The most stringent convexity condition is the notion of polyconvexity [3, 27]. More precisely, the polyconvexity condition states that the potential strain function \(\varPsi \left( {{\mathbf {\mathsf{{F}}}}}, {{\mathbf {\mathsf{{H}}}}}, J \right) \) has to be convex in the function space formed by the components of \({{\mathbf {\mathsf{{F}}}}}\) and \({{\mathbf {\mathsf{{H}}}}}\), and by *J* [3, 8]. This also ensures a one-to-one mapping between the stress \({{\mathbf {\mathsf{{P}}}}}\) and strain measures \(\{ {{\mathbf {\mathsf{{F}}}}}, {{\mathbf {\mathsf{{H}}}}}, J \}\) [3, 16, 27, 31].

## 3 Discretisation

The governing systems of Eqs. (2.1) and (2.3) can be discretised by the SPH numerical scheme incorporating JST stabilisation components (JST–SPH) [21, 25].

spatial discretisation with the SPH scheme;

a JST-based numerical dissipation tool, adopted from CFD, to improve the accuracy and the stabilisation of the overall discretisation procedure;

an explicit, two-stage total variation diminishing Runge–Kutta time integrator scheme to follow the time evolution of the solutions during dynamic simulations.

### 3.1 The SPH method

Being a meshfree technique, SPH can be effectively employed in the simulation of high-velocity impacts and high strain-rate deformations [10, 15, 20, 26, 28, 29, 34, 39, 40]. In fact, beyond the yield point, SPH discretisation becomes attractive because large distortions make standard Lagrangian finite elements analyses difficult to pursue, without resorting to remeshing techniques that increase the computational cost [5, 38]. This subsection briefly outlines the background for the SPH method and details the total Lagrangian SPH discretisation of the governing system of equations. The adoption of a total Lagrangian SPH formulation allows to avoid numerical tensile instabilities [4, 11, 33, 40]. However, it was proven in [32] that for simulations involving material discontinuities (such as cracks and fragmentation), stabler results are achieved when switching to Eulerian kernels over the discontinuity regions while retaining Lagrangian kernels elsewhere.

*h*is the smoothing length, and \(W(\vec {x}_b - \vec {x}, h)\) is usually a polynomial function with compact support \(\mathcal {D}\) of radius 2

*h*(that is,

*W*vanishes for \(\Vert \vec {x}_b - \vec {x} \Vert \ge 2h\)). Although studies introducing adaptivity via a variable radius of support

*h*exist in the particle methods literature (see [35]), here

*h*and consequently \(\mathcal {D}\) and

*V*for in-field particles are kept constant for simplicity.

*d*is the number of spatial dimensions of the problem, and \(\alpha \) is a normalising constant that depends on

*d*as,

*b*of a particle

*a*, at which the gradient is evaluated. The mixed \(\{ \vec {p}, {{\mathbf {\mathsf{{F}}}}}, {{\mathbf {\mathsf{{H}}}}}, J \}\) system described by Eqs. (2.1) and (2.3) can now be discretised in space using the corrected SPH interpolation as described above.

*V*, the above expression (3.7) is approximated to:

*a*located at or near the domain boundary. The density is assumed to be homogeneous throughout the domain. Equation (3.8) can now be expressed in terms of SPH discretisation as

*a*:

*J*, can also be discretised in a similar manner to what has been done for (3.9), with the linear momentum \(\vec {p}\) evaluated through SPH:

Depending on the chosen materials, the above semi-discrete formulation (3.9)–(3.11) may still suffer from accumulated numerical instabilities over a long span of time response. Therefore, it is necessary to address any such instability via suitable computational procedures.

### 3.2 JST artificial dissipation

In the context of SPH, various artificial dissipation techniques have been used in the past [28, 29, 30] to alleviate the aforementioned numerical instabilities. Alternatively, SPH can be stabilised by introducing stress points in the formulation (see [33]). In this case, however, a background mesh would be needed for the computation and assignment of fractional volumes. In the present work, an adapted nodally conservative JST stabilisation term \(\varvec{\mathcal {D}}^\mathrm{JST}(\varvec{\mathcal {U}}_a)\) will be incorporated into (3.9), mirroring CFD techniques. The hyperbolic nature of the first-order conservation laws in system (2.1), reflecting that of the Euler equations in fluid dynamics, makes it possible to introduce the JST term as a dissipative component [1].

In (3.13), \(c_p\) is the pressure wave speed, \(\Delta x_\mathrm{min}\) is the particle spacing, \(\kappa ^{(2)}\) and \(\kappa ^{(4)}\) are user-defined parameters, while the \(\tilde{\nabla }^2_0\) symbol represents a corrected Laplacian operator, applied to kernel *W* as detailed in [10].

### 3.3 Semi-discrete governing equations

Out of the four unknowns in the vector of state \(\varvec{\mathcal {U}}\), only \(\vec {p}\) and *J* will see JST dissipation terms appear in their respective conservation laws (3.9) and (3.11c). \(\varvec{\mathcal {D}}^\mathrm{JST} ({{\mathbf {\mathsf{{F}}}}}) = \varvec{\mathcal {D}}^\mathrm{JST} ({{\mathbf {\mathsf{{H}}}}}) = {{\mathbf {\mathsf{{0}}}}}\) are imposed in order to respect conditions (2.2), set to ensure these strain measures preserve compatibility with the domain motions. This is because \({{\mathbf {\mathsf{{F}}}}}\) and \({{\mathbf {\mathsf{{H}}}}}\) are now main independent variables of the problem, and are not associated with displacements \(\vec {x}\). In the context of the standard displacement-based method, it is worth recalling that \({{\mathbf {\mathsf{{F}}}}}\) is directly computed from \(\vec {x}\), which acts as main independent variable of the problem.

### 3.4 Two-stage TVD Runge–Kutta temporal integration

To obtain a fully discretised system, that is able to describe the time evolution of the solution, the set of particle equations (3.14) has to be now explicitly integrated between chosen time intervals.

The TVD Runge–Kutta scheme as defined in (3.16) has the advantage of being able to control the amount of spurious energy introduced by the explicit time integration, while allowing it to retain its qualities of speed and simplicity, especially when compared to implicit time-stepping methods.

Current particles positions, \(\vec {x}_a\) for \(a=1,\dots ,N\,\), are obtained monolithically from the two-stage Runge–Kutta process (3.16), given \(\varvec{\mathcal {U}} = \vec {x}\) and \(\varvec{\mathcal {R}} (\varvec{\mathcal {U}}) = \dfrac{\vec {p}}{\rho }\).

## 4 Applications

In this section, the proposed JST–SPH algorithm will be used to perform three numerical examples. The first amongst these will be the classic Taylor bar test for high-speed-impact plasticity effects: a three-dimensional cylindrical metal bar that impacts a horizontal wall at a high velocity. In the second example, the performance of the method is assessed by examining the elasto-plastic compression of a constrained rectangular body. The last example involves the simulation of the equal channel angular extrusion (ECAE) cold metal forming process, where a workpiece is forced to pass through two 90° turns inside an extrusion die. The localised distortions caused by the substantial shear stresses experienced by the workpiece at these 90° turns should constitute an ideal test for assessing the capability of the developed methodology under large strain, high-velocity conditions.

### 4.1 Taylor bar high-speed impact

Elastic behaviour is governed by the material energy function presented in Eq. (A.1). Equation (A.1) is coupled with the plastic yield model described in “Appendix A”, using the linear isotropic hardening law presented in (A.14). The material is assumed to be copper, with the following properties: Young’s modulus \(E = 117\) GPa, Poisson ratio \(\nu = 0.35\), yield stress \(\sigma ^{Y}_{0} = 0.4\) GPa, hardening modulus \(H = 0.1\) GPa and density \(\rho = 8930\)\(\text {kg}/\text {m}^3\). The bar has initial height \(h_0 = 32\) mm and radius \(r_0 = 3.2\) mm, and is discretised by a set of 4131 particles arranged in a regular pattern. The impact is supposed frictionless and takes place at an initial speed of the bar of \(\textit{v}_{0z} = -227\,\text {m}/\text {s}\) in the *z* direction, normal to the rigid wall. Due to the extensive plastic dissipation, the JST stabilising term is set to a very low value.

Figure 3 presents results of the Taylor bar simulation performed by JST–SPH. As observed in experimental results [44], in Fig. 3 the plastic front appears closer to the bottom wall at the early stages of the simulation (red contour regions). It then slowly moves towards the top of the bar, as more kinetic energy dissipates into plastic strain.

To investigate the stress distribution across the bar, the von Mises equivalent stress is plotted in Fig. 4. A gradual increase in the plastic stress flow over time can be clearly observed.

Plots of the total internal energy and of energy dissipation associated with plastic deformation and JST artificial dissipation are reported in Fig. 5.

It is shown in Fig. 5 that artificial dissipation introduced by the JST algorithm constitutes a negligible quantity.

### 4.2 Highly constrained problem with moving discontinuity

An example of the use of JST–SPH in a highly constrained setting is presented in [21, 25] for a fully elastic, nearly incompressible material. The effects of plasticity will be studied in this similar example, where a bidimensional rectangular block with dimensions \(1 \, \text {m} \, \times \, 0.5 \, \text {m}\) is considered under the assumption of plane strain. The material is copper, with properties chosen to be the same as those in the Taylor bar example in Sect. 4.1.

These specific values of \(\kappa ^{(2)}\) and \(\kappa ^{(4)}\) were chosen in order to provide the simulation with a small amount of JST numerical dissipation, useful to suppress any spurious pressure oscillations at the initial stages of the process.

The purpose of this example is to demonstrate the capability of the total Lagrangian framework of the JST–SPH method, in the presence of high, localised distortions. The evolution of the von Mises equivalent plastic strain \(\epsilon _{(p)}\) during the analysis is shown in the top row of Fig. 7. For comparison, the same simulation has been run using a commercial finite element method package, without any remeshing, and the results are presented in the bottom row of Fig. 7.

It is observed from Figs. 7 and 8 that the basic finite element method is not able to handle the excessive distortions, whereas the JST–SPH method is able to simulate severe material deformation without exhibiting any numerical instability.

Plots of the total internal energy and of energy dissipation associated with plastic deformation and JST artificial dissipation are reported in Fig. 9. Energy quantities have been calculated using (4.1). It is shown in Fig. 9 that artificial dissipation introduced by the JST algorithm is a negligible quantity. Roller constraints imposed at the boundaries of the block are at the origin of irregularities appearing in the internal energy plot [calculated as in (4.1a)].

### 4.3 Simulation of the equal channel angular extrusion (ECAE) process

The simulations are performed with a regular mesh of 400 particles. Contact between workpiece and die is assumed to be lubricated, and considered frictionless during the simulation. For the purpose of simplicity, a contact algorithm based on a reflective, bouncing back procedure [22] is adopted to simulate the interaction between the die walls and the particles representing billet material.

Figures 11 and 12, respectively, depict the distributions of pressure and plastic strains during the deformation process of the billet, at various instants in time. For the purpose of comparison, Figs. 11 and 12 also present the corresponding pressure and plastic strain distributions obtained with a commercial finite element software (Abaqus/Explicit) using linear quadrilateral elements.

It can be noted that the results from the JST–SPH method correlate well with those obtained with the finite element method. Results from Figs. 11 and 12 are also consistent with observations reported in [36]. It is evident from the figures that the particle distribution in the vicinity of sharp corners of the channel is not smooth at the later stages of the simulation. This uneven distribution of the particles can be attributed to boundary effects. The plastic strain contour illustrated in Fig. 12 compares well with results reported in [36]. Further, it is clearly evident that the plastic strain distribution increases significantly as the billet material passes through each bend of the channel in the die. This feature is what is mainly exploited in the ECAE process to achieve the desired level of plastic strain in the workpiece.

Investigation of the energy patterns throughout the simulation confirms that plastic deformation is the main mechanism of energy dissipation, being on average two orders of magnitude larger than the energy dissipation introduced by activating the JST stabilisation term. Figure 13 compares the energies associated with plastic deformation, JST artificial dissipation, and the total internal energy of the bar during the simulation of the entire ECAE process. Energy quantities in Fig. 13 were calculated using (4.1).

As shown in Fig. 13, the values of total internal energy and of total plastic dissipation follow specific trends that can be linked to the motion of the billet inside the channel of the die. For example, a peak value for both energies was observed around \(0.00142 \, \text {s}\), corresponding to the instant in which the billet comes into contact with the bottom wall of the middle (horizontal) part of the channel. Following that, the energy values begin to decline steeply during the expansion of the billet material in the mid-channel. A similar, but less pronounced energy pattern can be observed once the bar reaches, and then begins to flow into the second channel, with a peak value attained around \(0.035 \, \text {s}\). Moreover, it transpires from Fig. 13 that the energy values are highly oscillatory when the billet initially passes through the angular sections of the channel. However, the frequency and the intensity of these oscillations gradually decrease, as the material moves away from the angular sections.

In addition, Fig. 13 shows the gradual decrease in energy dissipation associated with the JST term at the onset of plastic deformation. This effect follows from the definition of the JST terms in (4.2), and further demonstrates that higher artificial dissipation values are only required at the very early stages of the simulation, that is, before any plastic deformation takes place.

In order to verify the accuracy of the simulation, a convergence study was performed by varying the number of particles in the discretised domain.

The mesh with 640 particles has also been utilised for closer comparison of the degree of similarity between the JST–SPH ECAE test conducted here with the analysis found in [36]. For this purpose, the strain contour in the direction normal to the billet movement has been investigated towards the end of the test, at simulation time \(t = 0.04 \, \text {s}\). The two locations selected are “Section 1” positioned at the centre of the middle channel, and “Section 2” at \(5 \, \text {mm}\) from the second corner of the die. The above locations are consistent with those of the results reported in [36].

Parametric analyses reported in [36] provided an opportunity for further verification of the validity of the results.

In [36], geometrical properties of the die are modified in order to investigate the possibility of enhancing ECAE metallurgical performances. One of these parameters was the length of the middle channel of the die. In [36], it was noted that interesting deformed shapes develop, when the middle channel is shortened from 24 to 16 mm. The upper part of second bend has a large unfilled area, while the billet experiences more stress due to bending than due to shear. This effect is particularly severe around the inner region of the billet in the vicinity of the bottom part of the second bend. Both of these phenomena could be identified in an analogous simulation performed with JST–SPH, as presented in Fig. 16 at time \(t = 0.027 \, \text {s}\). Results obtained with the finite elements commercial solver Abaqus/Explicit, using linear quadrilateral elements, are also shown in Fig. 16.

It is evident from the numerical examples that the JST–SPH scheme eliminates the excessive element distortions usually associated with mesh-based methods during the simulation of large plastic strains. In addition, the total Lagrangian framework adopted here provides an efficient modelling tool to gain better understanding of the physics underlying large material deformations.

## 5 Conclusions

This paper has focused on the application of JST–SPH mixed formulation in metal plasticity. This methodology has proved to be a valid tool for computing plastic deformations under dynamic regimes. It has been shown that there is good agreement between results reported in the previous section and data for the same tests found in the literature. The Taylor bar impact case provides an ideal application for JST–SPH, since SPH performs particularly well in high-velocity, large deformation settings. On the other hand, the ECAE test confirmed the robustness of JST–SPH: the total Lagrangian framework was able to withstand high local gradients of stresses, strains and displacements, while the presence of a constant loading in time did not lead to any kind of instability. This robustness was further evidenced in the constrained block simulation, where it can be seen that JST–SPH is able to achieve accurate results while preserving the initial connectivity under very large distortions. The same outcome cannot be obtained with standard mesh-based methods, without remeshing. It was noted, in the case of the ECAE simulation, that the results in the vicinity of the boundaries were slightly affected by sharp corners. These boundary effects can be easily eliminated by using more sophisticated boundary treatments. This will be one of the focal points in future publications.

## Footnotes

- 1.
\(\mathcal {E}_{ijk}\) assumes value \(+1\) in case

*ijk*is an even permutation of the sequence [1, 2, 3], and \(-1\) in case is odd.

## Notes

### Acknowledgements

On behalf of all authors, the corresponding author states that there is no conflict of interest. The authors would like to acknowledge the financial support provided by the Sêr Cymru National Research Network (Grant No. NRN038). The authors also wish to thank Dr. C.H. Lee, Prof. A. Gil and Prof. J. Bonet for the useful insights that have inspired many of the concepts developed in this work.

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