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We formulate a macroscopic particle modeling analysis of metallic materials (aluminum and copper, etc.) based on theoretical energy and atomic geome tries derivable from their interatomic potential. In fact, particles in this framework are presenting a large mass composed of huge collection of atoms and are interacting with each other. We can start from cohesive energy of metallic atoms and basic crystalline unit (e.g. face-centered cubic). Then, we can reach to interparticle (macroscopic) potential function which is presented by the analytical equation with terms of exponent of inter-particle distance, like a Lennard-Jones potential usually used in molecular dynamics simulation. Equation of motion for these macroscopic particles has dissipative term and fluctuation term, as well as the conservative term above, in order to express finite temperature condition. First, we determine the parameters needed in macroscopic potential function and check the reproduction of mechanical behavior in elastic regime. By using the present framework, we are able to carry out uniaxial loading simulation of aluminum rod. The method can also reproduce Young’s modulus and Poisson’s ratio as elastic behavior, though the result shows the dependency on division number of particles. Then, we proceed to try to include plasticity in this multi-scale framework. As a result, a realistic curve of stress-strain relation can be obtained for tensile and compressive loading and this new and simple framework of materials modeling has been confirmed to have certain effectiveness to be used in materials simulations. We also assess the effect of the order of loadings in opposite directions including yield and plastic states and find that an irreversible behavior depends on different response of the particle system between tensile and compressive loadings.

Recently, multi-scale modeling of materials behavior with hierarchical approach has attracted much interest in research and development of materials science. In surveying various computational methodologies, it is recognized that molecular dynamics (MD) has been well established based on microscopic view [^{0} - 10^{−10} meter (in space) or 10^{0} - 10^{−15} second (in time) respectively. These differences are too wide to make concurrently combined computation model easily. Only in the case when the mechanical behavior of a solid material can be limited to just small strain or small deformation regime, it may become quite possible by constructing the model with elastic and linear constitutive law and parameters. Such approaches include quasi-continuum method [

In conventional approach so far to construct a multi-scale view including both the atomistic and the continuum, there is a try that potential energy function between atoms in solid (metal and crystal) is directly connected to macroscopic elastic constants. In that case, a harmonic approximation that is valid only in small displacement of elastic regime should be assumed. However, for plastic deformation appearing in much larger loading level, the simple approximation rather becomes quite hard task to link directly between microscopic behavior (i.e. slip motion between atoms or atomic diffusion) and macroscopic constitutive law. Plasticity, super-elasticity as well, which is usually observed in many solid materials, is one of difficult issues, and so adequate multi-scale modeling in theory and computational set-up should be required.

Nowadays, it is recognized that not only macroscopic modeling but also microscopic (atomistic) one is inevitable for further development of engineering materials [

In recent years, particle methods (or sometimes called “mesh-less” methods), as an auxiliary method for on-mesh methods such like FE method, have been developed and used extensively, by virtue of the enhancement of computation power. In those new methods, “particles” are likely to be treated in various senses, which correspond to their size scales. On the smallest scale, particles should be real substances as atoms or molecular groups (which is as in MD method). On the other hand, in larger scale, particles are only representative points on continuous body of the material, which are needed for the approximation of discrete-type computation. In every case of particle methods, an essential feature, on the whole, is that we are solving equations of motion and will chase the motion of particle step by step in the time development. Therefore, we can expect that the combination between different scales in particle methods would be nicely accomplished by unified theory along with adequate modeling using particles.

The authors recognize that there are two approaches having opposite direction in modeling with particles as follows. One approach is that macroscopic variables are immediately connected to the evaluation of energies and deformations obtained in atomistic simulation. For example, stress and strain are absolutely macroscopic evaluations that are only defined under continuum hypothesis and relation, but they can be transferred into atomic simulations with the name of “atomic stresses” [

In this paper, we propose a new framework of MPM method stated above. This MPM will be constructed on microscopic (atomic) potential energetics and dynamics which are obtained from atomistic information. This method is capable of reproducing macroscopic materials behavior by adopting Langevin-type equation of motion, where dissipative force, random force indicating thermal fluctuation, and conservative potential force derived from atomic system are well integrated. In the context of our study, the conservative potential force is being expressed by a simple power-law relevant in atomistic simulation of condensed matter, such as the Lennard-Jones potential [

Generally, in particle modeling, the material which occupies a certain space is replaced by assembly of discrete particles. This concept is universal and common to that in the former literature [

M i = ρ V N , (1)

so that the original density of the material ρ is retained. Motion of each particle is described by Newtonian equation of motion,

M i d 2 r i d t 2 = F i , (2)

where F i includes interparticle force as well as additional forces induced by viscous friction and thermal fluctuation. Indeed, this equation is the same as that of molecular dynamics (MD) simulation. Interparticle interaction can be configured from pairwise potential function φ E ( r ) , and then the conservative force is derived by

F i = ∑ j = 1 N e i g h b o r F i j = ∑ j = 1 N e i g h b o r ∂ φ E , i j ∂ r i j r i j r i j , (3)

where φ E , i j = φ E ( r = r i j ) means potential energy acting between two particles i and j. Separation between particles r i j is the absolute value of interparticle difference vector r i j = r j − r i .

From the former study of the original “particle modeling” method [

F i j = G r i j p + 1 − H r i j q + 1 (4)

has been used for interparticle force. In atomic simulation, the famous Lennard-Jones potential has the same function form as Equation (4) (corresponding to p= 12, q= 6). These macroscopic potential parameters, G,H,p,q, have to be determined. The units of parameters, G and H, will be determined after two non-dimensional parameters, p and q, are determined so as that F i j in equation (4) should have the unit of force, i.e. Newton (N). From energy-conservative mechanics, a potential function integrated from force function Equation (4) is given by

φ E , i j = − G p ( r i j ) p + H q ( r i j ) q . (5)

In atomic system, Equation (2) is supposed to have only conservative force so that the total energy of the system should be conserved in principle. However, macroscopic particles’ system should have additional treatment due to invisible effect from many disappeared degrees of freedom. Therefore, Langevin-type equation of motion can be applied to this particles’ system, where energy dissipation and production of thermal energy by fluctuation are integrated. This type of equations have already used to dissipative particle dynamics (DPD) [

M i d 2 r i d t 2 = F i , C + F i , D + F i , R = − ∂ Φ ∂ r i + F i , D + F i , R , (6)

where F i , D and F i , R are force vectors called dissipative and random forces, respectively. The conservative force on one particle is derived from

Φ = ∑ i = 1 N ∑ i > i + 1 N φ E , i j , (7)

which is total conservative energy of the system. The dissipative force term can be formulated by considering inherent resistance of a moving particle. In the meso- or micro-scale (relatively in smaller scale), that term is supposed to be caused by interparticle friction. When a particular viscosity μ is provided, the dissipative force should be estimated, from particles’ velocity v i , v j , by

F i , D = ∑ j = 1 N e i g h b o r μ ( v j − v i ) . (8)

On the other hand, in meso- and macro-scale (relatively in larger scale), the motion of particle should be through some other media, it is supposed that the dissipative force depends only on absolute velocity. In such a case, the friction force may be estimated by

F i , D = γ v i . (9)

Random force induces fluctuation of particles and is formulated based on its randomness.

Thus, in computation, numerically produced random numbers (pseudo-random numbers) with Gaussian distribution are utilized. The dissipative and random forces are working for a particle motion, so to speak, as brake and accelerator, respectively. All these three types of force in Langevin equation (Equation (6)) balance each other in equilibrium, or they evolve to make structural change (deformation and catastrophic fracture) in finite temperature condition.

In the present MPM method, conservative force almost determines elastic properties. Therefore, based on existing elastic constants available from experiment or abinitio (first principle quantum chemistry) calculation, macroscopic potential parameters are determined. At first, total absolute amount of potential energy contained in one macroscopic particle is estimated. When atoms are condensed as solid material with density ρ (its unit is kg/m^{3}), a lot of atoms are assigned to one macroscopic particle. For example, in solid state of fcc metal, each atom has cohesive energy e_{0} which is calculated as the summation of twelve pairwise potential energies (its unit is J). If the number of atoms in one macroscopic particle is N_{atom}, collective energy in this macroscopic particle should be,

P atom = N atom e 0 . (10)

when all the macroscopic particle have a spherical shape and they gather to make fcc structure of lattice constant a_{0} (its unit is meter), N_{atom} is given by,

N atom = ρ π 6 r 0 3 m 1 0.74 , (11)

where r_{0} is equilibrium distance between macroscopic particles (its unit is m), m is atomic mass (its unit is kg), the factor 0.74 means occupancy rate in fcc lattice, which becomes non-dimensional. In most cases, e_{0} and a_{0} as well as structural type of crystal are easily available from experimental fact or abinitio evaluation. Therefore, these e_{0}, a_{0} and ρ are recognized as microscopic parameters, but N_{atom} and P_{atom} come to macroscopic variables to be configured next. When macroscopic particles are separated by equilibrium distance r = r 0 , their interparticle force should all vanish, and Equation (4) gives the condition,

F i j ( r = r 0 ) = G r 0 p + 1 − H r 0 q + 1 = 0 . (12)

Equivalence between microscopic and macroscopic energies (Equation (10)) and equilibrium condition (Equation (12)) are both used to solve unknown potential parameters G, H by following equations.

G = f ( p , q ) r 0 p , H = f ( p , q ) r 0 q , f ( p , q ) = P atom 6 p q p − q . (13)

Two undetermined exponents (powers) p, q which determine net shape of potential curve are needed and will be determined next.

Thus, energetic link between microscopic and macroscopic systems has been carried out. However, for mechanical response, the curve shape of potential function is crucial. Although there may be many choices for function type, we think that Equation (5) is reasonable because it furnishes three basic characteristics necessary for macroscopic particles. They are: 1) divergent feature of energy in closer separation than r_{0}; 2) equilibrium distance (energy minimum) at r_{0}; and 3) convergence to zero-energy in far larger separation. The undetermined parameters p, q can be adjusted from elastic constant (i.e. Young’s modulus, in the unit of Pa) in the vicinity of equilibrium distance r_{0} as follows.

Young’s modulus E can be estimated as the curvature of potential curve at equilibrium distance r_{0}, then,

E = 1 2 r 0 ∂ 2 φ E , i j ∂ r 2 ( r = r 0 ) . (14)

Substituting Equation (5) into this formula, it gives,

E = π ρ 1 0.74 p q 72 e 0 m . (15)

This means that Young’s modulus depends on the product between p and q. For the fcc metals (aluminum, copper, etc.), using each cohesive energy e_{0}, Young’s modulus is estimated. They are tabulated on dependency on p, q as shown in

Actual Young’s modulus obtained by uniaxial tensile testing are approximately 70 - 73 GPa for aluminum and 110 - 130 GPa for copper. Referring these values,

p [−] | q [−] | E_{Al} [GPa] | E_{Cu} [GPa] |
---|---|---|---|

3 | 9 | 51.6 | 76.2 |

4 | 8 | 61.1 | 90.4 |

4 | 9 | 68.8 | 101.6 |

4 | 10 | 76.4 | 112.9 |

5 | 9 | 95.5 | 117.2 |

experiment | 70 - 73 | 110 - 130 |

p = 4 and q = 9 may be the most suitable. Of course, we have another choice, but by this procedure we are able to set p, q to configure elastic response of macroscopic particles system.

It is generally difficult to express both microscopic and macroscopic plasticity mechanisms at the same time. In macroscopic view, constitutive relation (sometimes it exhibits very complicated formula) can be responsible for plasticity. But, it contains lots of macroscopic parameters which are not immediately connected to microscopic dynamics or parameters.

Energy dissipation or heat transport in plasticity, for example, has been one of difficult issues. We are challenging to introduce plasticity mechanism into macroscopic particles’ system based on microscopic parameters, extending elastic method described above.

φ U ( r i j ) = − G p ( r i j − Δ r 0 ) p + H q ( r i j − Δ r 0 ) q + C φ , (16)

where Δ r 0 and C φ , which have units of length (m) and energy (J) respectively, are undetermined values until unloading starts. So, many different φ U functions exist per unloading events. The functions φ U connect each other as well as elastic potential φ E by another function φ P , which prescribes a plastic route for between pairs of particles. The function φ P used here is just a polynomial of interparticle separation r i j . The function φ P should be determined microscopically (from MD simulation) or empirical values obtained by experimental elastic-plastic testing.

Actually, only the elemental framework of simulating in plastic regime has been shown here. It is true that it requires further verification and discussion, but let us to omit those detailed description in the present paper at this time, to avoid complication of the context.

In the field of MD study, the Langevin equation is sometimes used and is including Debye model for thermal conduction [

F D = − α r i , (17)

and random force is given by

F R = F R ( σ ) , (18)

where α is a viscosity for atom and σ is the square root of variance. In MD scale, α and σ have the relation,

σ m i c r o = ( 2 α m i c r o k B T Δ t ) 1 2 , (19)

and

α m i c r o = m π 6 ω D , (20)

where ω D = k B θ h ¯ is Debye frequency ( θ , h ¯ are Debye temperature and Planck’s constant, respectively), k_{B} is Boltzmann’s constant, Δt is time increment of MD and T is an equilibrium temperature. This MD system has to be coarsened and transformed to macroscopic particles’ system, by virtue of scale factors. There is a trend of study to link microscopic parameters to macroscopic relation [

The scale factors are η C G for α and ζ C G for σ, and resulted expressions are

σ m a c r o = η C G ( 2 α m a c r o k B T Δ t ) 1 2 , (21)

and

α m a c r o = ζ C G m π 6 ω D . (22)

Verifying these coarsening parameters, η C G and ζ C G , should be required, but it will be done in further studies.

In this paper, as a first stage of this study, we think that the potential force (conservative force) is the most important factor for evaluation of material properties in solid. Therefore, we can omit those dissipative and random forces temporarily. It means that MPM simulation results discussed in the following section are carried out in zero Kelvin condition.

However, in principle, the MPM simulation in finite temperature T is also feasible by the basic framework shown in this section.

In this study, it is assumed that a metallic material is subjected to the external loading. The specimen is composed of pure aluminum and cylindrical shape. It is applied uniaxial tensile or compressive loading along the longitudinal direction. The macroscopic particles are arranged in face-center-cubic (fcc) lattice as shown in _{0} and lattice constant a_{0}. They have a certain relation, r 0 = a 0 / 2 . The longitudinal length of the specimen l x is configured by l x = r 0 + n x ⋅ a 0 , where n x is the number of division in x-direction and it also determine the length of the fcc unit cell. First, as shown

the dimensions l x and l y is arranged by particles, and then a cylinder with diameter of l x = D is sculpted from it. Once the diameter of cylinder l y is determined, division number by particles in diameter (width) direction is also determined by the same interparticle distance as the longitudinal direction. In loading simulations, the strain increase is prescribed by the constant velocity of moving particles which are located at two ends along longitudinal (z) direction. Only two centered particles at each end part are fixed in their initial position also in x and y directions as well, so as to realize uniaxial loading to the specimen. The longitudinal strain ε is estimated from updated distances between particles at both ends. Stress tensor components are estimated from virial formula, which is conceptually an average of the product of interparticle forces and difference of position vectors of all interacting particles inside the system. This evaluating method of stress is usually used in molecular dynamics calculation, which deals with atomistic particles’ system.

The cylindrical specimen with the same diameter of 14 mm made of aluminum is being divided neatly into macroscopic particles. In dividing, we designate two integers, n φ and n h , which are numbers of division in the direction of diameter (width) and height, respectively. These numbers of division are chosen as the power of 2 as expressed in Equations (23) and (24) using other integers i and j, and they are excluding cases of the division number less than 2.

n ϕ = 2 i ( i = 1 ~ 4 ) (23)

n h = 2 j ⋅ n ϕ ( j = − 3 ~ 3 ) (24)

The numbers of division and calculation conditions are summarized in

Division number n ϕ | - | 2^{1} | 2^{2} | 2^{3} | 2^{4} | |
---|---|---|---|---|---|---|

Particle equilibrium distance | mm | 3.65 | 2.10 | 1.13 | 5.92 × 10^{−1} | |

Time increment | s | 2.82 × 10^{−7} | 1.62 × 10^{−7} | 8.78 × 10^{−}^{8} | 4.58 × 10^{−}^{8} | |

Number of steps | - | 1.0 × 10^{5} | ||||

Total strain | % | 0.1 | ||||

Strain rate | 1/s | 3.53 × 10^{−2} | 6.15 × 10^{−2} | 1.13 × 10^{−}^{1} | 2.18 × 10^{−}^{1} | |

Constant G | J∙m^{4} | 2.40 × 10^{+1} | 5.00 × 10^{−}^{1} | 6.74 × 10^{−}^{3} | 7.04 × 10^{−}^{5} | |

Constant H | J∙m^{9} | 1.57 × 10^{−}^{1} | 2.05 × 10^{−}^{4} | 1.28 × 10^{−}^{7} | 5.04 × 10^{−}^{11} |

n h / n ϕ | n ϕ = 2 1 | n ϕ = 2 2 | n ϕ = 2 3 | n ϕ = 2 4 |
---|---|---|---|---|

2^{−3} | - | - | - | 1999 |

2^{−}^{2} | - | - | 495 | 3593 |

2^{−}^{1} | - | 123 | 889 | 6781 |

2^{0} | 35 | 221 | 1677 | 13,157 |

2^{1} | 61 | 417 | 3253 | 25,909 |

2^{2} | 113 | 809 | 6405 | 51,413 |

2^{3} | 217 | 1593 | 12,709 | 102,421 |

n h / n ϕ | n ϕ = 2 1 | n ϕ = 2 2 | n ϕ = 2 3 | n ϕ = 2 4 |
---|---|---|---|---|

2^{−3} | - | - | - | 2.26 |

2^{−}^{2} | - | - | 4.35 | 3.94 |

2^{−}^{1} | - | 8.05 | 7.56 | 7.29 |

2^{0} | 14.0 | 14.0 | 14.0 | 14.0 |

2^{1} | 24.3 | 25.8 | 26.8 | 27.4 |

2^{2} | 45.0 | 49.6 | 52.5 | 54.2 |

2^{3} | 86.4 | 97.2 | 104 | 107 |

Stress-strain curves obtained for ( n h / n ϕ ) = 2 0 = 1 are shown in

n ϕ is 2^{1} (=2), the largest modulus is obtained. As shown in

The dependence of Poisson’s ratio ν on the division ratio n h / n ϕ is shown in

Next, stress distributions comparing between strain value 0% and 0.1% are shown in

The set-up of compression simulations is the same as that of tensile testing, except for the moving direction of end regions. The dependence of Young’s modulus on the numbers of division n ϕ , n h and their ratio n h / n ϕ in these compression testing is shown in

The strain values when compressive strain is 0.1% are compared as in

The stress distribution inside the specimen for the ratio between the numbers of division, n h / n ϕ = 4 , is shown in

The tensile simulation including not only elastic regime but also plastic one is conducted. The specimen is the same as that used in elastic simulation, which is built with particle division parameters n ϕ = 2 4 = 16 and n h / n ϕ = 4 . The time increment d T = 4.58 × 10 − 8 s is used for all these cases. The calculation conditions are summarized in

The obtained stress-strain curves including plastic regime is shown in

The number of steps | - | 5.0 × 10^{5} |
---|---|---|

Maximum strain | % | 5.0, −5.0 |

Strain rate | 1/s | +2.18, −2.18 |

strain is increasing, which means strain hardening occurs. The stress distribution when the specimen is committing strain hardening is shown in

inside the specimen increases almost uniformly, but it is always lower than those in surface particles.

The compression testing including not only elastic regime but also plastic one is also conducted. The obtained stress-strain curves for the selected division case are shown in

The stress distribution obtained at each strain is shown

Conceptual explanation of force acting between particles inside of the specimen is displayed in

segment does compressive forces. In tensile simulation, component of bonding force between neighbor particles which is parallel to the loading direction is responsible for plastic strain, but that perpendicular to tensile direction becomes compressive. To the contrary, in compressive simulation, the direction of bonding force is reversed. Due to the formulation of plasticity we have established, only tensile strain accompanies stress relaxation in yielding event. In compressive deformation as shown in

It is experimentally known that metals will usually show a kind of Bauchinger’s effect, where reduction of yield stress occurs when the direction of applied plastic stress or strain is reversed. But, simple constitutive modeling in numerical simulation is formulated with a fixed yield condition, and it is agreed that very complicated theoretical formulation must be required for reproducing Bauchinger’s effect. At this point, we will see whether Bauchinger’s effect is reproduced by using the present particle modeling method.

First, uniaxial tensile strain up to 3% is applied to the cylindrical specimen, and then subsequently uniaxial compressive strain up to 3% is applied to the same specimen. The sequence of stress-strain diagrams is shown in

To the contrary, the reversed case, in that compressive load is first applied then tensile one is applied, is resulted as shown in

These simulations with reversed deformation display that any symmetrical yielding behavior is not included in the present formulation which uses shift of potential energy between neighbor particles. It is concluded that an accurate repetitive yielding behaviors for cyclic loading and Bauchinger’s effect must take non-symmetrical interaction between particles into account and the formulation of potential function should be reorganized. These considerations will be studied further as our future work.

In this study, we propose a computational framework by using macroscopic particles method (MPM). Although the proposed MPM method is simply constructed from microscopic parameters such as cohesive energy and density of metallic materials, it provides high feasibility on implementing a simple framework for multi-scale simulation and in discussing the deformation of metallic materials. As seen in the examples of rod-shaped models we presented here, the evaluation by the present MPM method depends on the division number of particles. In the elastic regime, Young’s modulus and the Poisson’s ratio can be sufficiently reproduced. On the other hand, it is understood that the plastic modeling in MPM method still remains the difficult issue. The plasticity mechanism is absolutely important for material modeling, and as the first challenge, we have discussed an irreversible change of the configuration of particles. We found that there is different stability in between tensile and compressive loadings for the total elastic-plastic regime. We can conclude that further sophistication and improvement of the MPM method especially for plastic mechanism are quite expectable and they are worthwhile studying continuously in the future.

The authors declare no conflicts of interest regarding the publication of this paper.

Saitoh, K.-I. and Hanashiro, N. (2021) Particle Modeling Based on Interatomic Potential and Crystal Structure: A Multi-Scale Simulation of Elastic-Plastic Deformation of Metallic Material. World Journal of Nano Science and Engineering, 11, 45-68. https://doi.org/10.4236/wjnse.2021.113003