Page 1 of 8

European Journal of Applied Sciences – Vol. 11, No. 5

Publication Date: October 25, 2023

DOI:10.14738/aivp.115.15448

Uehara, T. (2023). Deformation Simulation of a Soft–sheet Material Using Particle Method. European Journal of Applied Sciences,

Vol - 11(5). 296-303.

Services for Science and Education – United Kingdom

Deformation Simulation of a Soft–sheet Material Using Particle

Method

Takuya Uehara

Department of Mechanical Systems Engineering,

Yamagata University, Yonezawa, Japan

ABSTRACT

In this study, the deformation of a soft–sheet material was simulated using a

particle method. Herein, a square sheet made of soft material was modeled, and a

rigid sphere was indented on the surface of the sheet. Considering the conventional

scheme of the particle method, the sheet was assumed to consist of discrete

particles. A spring-and-dashpot connection was applied as the interaction between

the calculation particles. Consequently, dent deformation on the surface and overall

deformation of the object were successfully realized. Furthermore, various

deformation modes were observed at different interparticle parameters,

confirming the validity of the present model for various materials. Subsequently,

the free fall of a rigid ball on the surface of the sheet was simulated, and thus,

repeated rebounds of the ball were properly reproduced. Therefore, it can be

concluded that the present method can express both static and dynamic properties

of various soft materials despite the simple algorithm and a small number of

parameters.

Keywords: Particle method, Elastic deformation, soft material, large deformation,

Bouncing, Computer simulation.

INTRODUCTION

Soft materials, such as rubber, polymers, and their composites, have a wide range of

applications. Notably, they have been used even in place of hard materials, such as metals and

ceramics in mechanics and structures. In such engineering fields, precise estimation of

distortion and strength prediction are crucial. Thus, the use of computer simulations has

become a common practice for efficient design, and therefore, various techniques have been

developed. For example, the finite-element method is widely employed for engineering

purposes, and many commercial and free software solutions are available. However, the

deformation of soft materials is greater than that of hard materials, and the description of the

deformation mechanics in soft materials is not straightforward. Additionally, the properties of

soft materials may substantially differ depending on the materials. Moreover, their physical

characteristics vary; some materials are porous, and others are gel-like. These microscopic

differences complicate the construction of a general framework for the simulation of

mechanical behavior. Consequently, numerical analyses tend to be case studies. Therefore, in

the present study, we developed a simulation procedure for the deformation of soft materials

based on a simple algorithm that can be applied to a wide range of materials without involving

complicated procedures.

Page 2 of 8

297

Uehara, T. (2023). Deformation Simulation of a Soft–sheet Material Using Particle Method. European Journal of Applied Sciences, Vol - 11(5). 296-

303.

URL: http://dx.doi.org/10.14738/aivp.115.15448

The particle method is considered the most fitting tool for this purpose. There are various

versions in of this method, with the representative ones being smoothed particle

hydrodynamics (SPH) [1] and moving particle semi-implicit (MPS) methods [2]. These were

originally developed for fluid dynamics but were modified to apply to solid mechanics [3,4].

One of their greatest merits is their applicability to large deformation without requiring

complex remeshing, as in the finite-element method. Additionally, the material properties may

be easily changed by varying the interactions between particles. In the present study, the

interaction between the discretized particles was simplified to linear elastic and viscous

damping interactions [5,6]. The target model was set as a sheet-shaped soft material, and a rigid

object was indented on the surface. In addition to the deformation through indentation, the

bouncing of the rigid object was simulated, and the applicability of the proposed method was

validated by exhibiting various static and dynamic deformation modes depending on the

parameters.

SIMULATION METHOD

In the present study, one of the simplest particle methods was used. The target model was a

sheet object made of soft material on which a rigid sphere was indented. The sheet was

discretized into many particles, and corresponding interactions were assumed between the

particles. The form of the interactions is generally very complex depending on the physical,

chemical, and many other properties of the material. However, in the present study, very simple

interactions were assumed because, under such an assumption, the method may be applied to

various materials through facile fitting of the parameters. The following interactions were

imposed:

Fr = −kr(rij − r0) (rij < r0) (1)

Fa = −ka(rij − r0) (r0 < rij < rc) (2)

Fd = −cd(vj − vi) (rij < rc) (3)

Here, Fr

, Fa, and Fd are the repulsive, attractive, and damping forces acting on the i-th particle

as it interacts with the j-th particle, with the corresponding parameters kr

, ka, and cd; rij is the

central distance between the two particles, and vi and vj are the translational velocities of the

i-th and j-th particles, respectively; parameter r0 represents the equilibrium distance or the

sum of the radii of two particles, and rc

is the cut-off distance. It should be noted that the abrupt

break in the interaction is inevitable, but the effect is minimized by adjusting the distance

between the first and second neighbors. The equation of motion with these forces was

numerically solved using the explicit finite difference method. The calculation particles were

initially disposed on the lattice points of the fcc-type crystal structure, and the mass of the

object was equally distributed to every particle.

Indentation Process

Model and Conditions:

Dimensions of the model sheet were set to 30 × 30 × 10 in the nondimensional scale unit L,

where L is the diameter of the discretized particle. For comparison, 40 × 40 × 10 and 20 × 20 ×

Page 3 of 8

Services for Science and Education – United Kingdom 298

European Journal of Applied Sciences (EJAS) Vol. 11, Issue 5, October-2023

5 models were also constructed. The model was initially set in a rectangular box surrounded

by walls, which were removed when the computations started. The sheet deformed due to

gravity during the initial 5000 steps of relaxation. After that, a rigid sphere of the diameter 32

L was indented at the center of the top surface. The indentation process was operated by

applying compulsive displacement at a constant rate until predetermined depth. The repulsive

term, Eq. (1), was assumed as the interaction between the sheet surface and rigid sphere.

Parameters kr

, ka, and cd were varied in the range 100 ≤ kr ≤ 400, 50 ≤ ka ≤ 200, and 0 ≤

cd ≤ 10.

Deformation Result:

Figure 1 represents the sheet model after the initial relaxation for the 30 × 30 × 10 model with

kr = 200, ka = 100, and cd = 10. The calculated particles are drawn as colored spheres, with

the color indicating the height (z-coordinate) of the particle, where red is the top and blue is

the bottom. Overall, the model is slightly sunk due to gravity. In particular, the level of the

central part of the top surface becomes lower than the edge area, and the shape is equilibrated

in the specific form. This shape is often observed in real soft objects.

(a) Side view (b) 3D view after relaxation

Fig. 1: Simulation model at the initial state and after relaxation before indentation.

The result obtained after a rigid sphere was indented is shown in Fig. 2, where Fig. 2(a) shows

the three-dimensional (3D) view, Fig. 2(b) the top view, and Fig. 2(c) the cross-sectional view

along the central line. The arc shown in Fig. 2(c) represents the indented rigid sphere. The

surface of the sheet is dented along the spherical shape and smoothly connected to the surface

line at the rim part. The surface line is inclined toward the center, and the side faces are also

affected by the surface deformation. Consequently, the bottom face is slightly widened. This

result is consistent with natural deformation; thus, the present model is considered valid.

(a) 3D view (b) Top view (c) Cross-sectional view

Fig. 2: Simulation result of the indented sheet for kr = 200, ka = 100, and cd = 10.