Experimental and theoretical analysis of the elasto-plastic oblique impact of a rod with a flat

International Journal of Impact Engineering 86 (2015) 307–317

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International Journal of Impact Engineering j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / i j i m p e n g

Experimental and theoretical analysis of the elasto-plastic oblique impact of a rod with a flat Hamid Gheadnia *, Ozdes Cermik, Dan B. Marghitu Department of Mechanical Engineering, Auburn University, 1418 Wiggins Hall, Auburn, AL 36849-5341, USA

A R T I C L E

I N F O

Article history: Received 2 December 2014 Received in revised form 15 May 2015 Accepted 11 August 2015 Available online 5 September 2015 Keywords: Elasto-plastic oblique impact Contact force Coefficient of restitution Permanent deformation

A B S T R A C T

In this study, the elasto-plastic oblique impact of a rod with a flat has been analyzed experimentally and theoretically. Nine different flattening and indentation contact models have been used to simulate the impact. The models have been compared theoretically in terms of the linear and the angular motion, the contact force during the impact, and the permanent deformation. A 3D infrared camera has been used in order to capture the motion of the rod before and after the impact. Experimental results for the coefficient of restitution and the rebound angular velocity have been compared with the presented models. Selecting the appropriate contact model is important on predicting the motion of the system for the simulations. For the impact angle θ = 45° our previous model matches the experimental results. For the impact angle θ = 17.2° it has been shown that all of the presented contact models show smaller values for coefficient of restitution and rebound angular velocity compare to the experiments. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Indentation contact models

Various analytical, experimental, and numerical studies have been done in order to simulate and predict the motion of the objects after the impact. The most challenging part of an impact problem is to find the contact force. The field of contact mechanics provides various models for the contact properties of elasto-plastic objects. Finding a suitable model for an impact problem is important. However, the Hertzian theory [1] can be applied for the impact of the fully elastic objects. Most of the materials subjected to an impact reach to a point where they deform plastically. Therefore, more advanced contact models are needed. There have been many studies for the contact of a hemisphere with a flat surface. One of the impact objects is considered to be rigid for almost all of the studies. Contact models can be categorized into two main groups: flattening and indentation models. For the flattening models, the flat surface is considered to be rigid, and the impacting hemisphere deforms. On the other hand for the indentation models, the hemisphere is rigid, and the flat deforms. There has been a few models in which both objects are considered to be deformed. A summary of different contact models is presented in the following section.

An indented flat with a hemisphere has been modeled since 1900. It has been used for hardness test. In 1900, Brinell [2] introduced a new test in order to determine the hardness of materials. Meyer [3] developed a new hardness test and introduced the Meyer hardness. A simple theory of the static and dynamics hardness was studied by Tabor [4]. Later, Tabor [5] performed many experiments on the hardness of metals, and his work is recognized as one of the best reference for many mechanical applications. Johnson developed his contact model that is now referred to as the simple plastic contact model [6]. In this work the contact is divided into two phases: the fully elastic and the fully plastic phase. The Hertzian theory was used for the fully elastic phase. Hardness is assumed to be constant, H = 2.8 Sy, for the fully plastic phase. Johnson’s model was compared and verified with Tabor’s work [5]. Finite element analysis (FEA) has been used to analyze the elastoplastic deformations during the contact by many researchers. Hardy et al. [7] analyzed the elasto-plastic indentation using FEA modeling. Follansbee et al. [8] and Sinclair et al. [9] improved Hardy’s work with a better meshing for the FEA. They used empirical equations in order to formulate the contact force and area of the contact. Both studies were compared with Tabor’s [5] experiments, Ishlinskii slipline theory [10], and Richmond adhesive slip-line theory [11]. The indentation of a rigid sphere on a layered flat was studied by Komvoloulos using FEA modeling [12,13]. He also developed a formulation for a homogeneous flat. Many assumptions such as: neglecting the effect of the strain hardening, pile up, sink-in, and friction are considered in the FEA models. Mesarovic and Fleck [14] studied the effects of strain hardening, friction, and pre-existing stresses.

* Corresponding author. Department of Mechanical Engineering, Auburn University, 1418 Wiggins Hall, Auburn, AL 36849-5341, USA. Tel.: +13345243916; Fax:334 534 3916. E-mail address: [email protected] (H. Gheadnia). http://dx.doi.org/10.1016/j.ijimpeng.2015.08.007 0734-743X/© 2015 Elsevier Ltd. All rights reserved.

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It was shown that the friction can have a significant effect on the stress distribution, the contact force, and the contact area. Ye and Komvopoulos developed a new formulation for the indentation of homogeneous and layered material [15]. Later, Kogut and Komvopolous [16] improved the model and developed a new equations for the indentation of a homogeneous flat. Alcalá and Esqué-de los Ojos [17] studied the correlations between the hardness, yield strength, and strain hardening exponent for frictionless and frictional indentations of spherical contact. Jackson et al. [18] developed an analytical expression based on the slip-line theory to calculate the hardness as a function of the radius of contact. In 2012, Brake [19] developed a new analytical model. The contact model is divided into four phases: the fully elastic, the elastoplastic, the fully plastic, and the restitution phase. The fully elastic phase follows the Hertzian theory. The fully plastic phase follows Johnson’s theory. Brake used third order polynomial interpolation in order to find the contact force and the area of the contact between the fully elastic and the fully plastic phases for the elasto-plastic phase. The restitution phase has been considered to follow the Hertzian theory. Later in 2015, Brake [20] presented an new contact model based on the Hertzian theory for the elastic phase and Mayer’s hardness for the plastic phase. The contact is divided into two phases in this model, the elastic phase and the elasto-plastic phase.

1.2. Flattening contact models Flattening models have been studied since 1960s. Flattening models refer to the contact of a deformable hemisphere with a rigid flat. In 1968, Johnson [21] studied experimentally the fully plastic contact between spheres and cylinders. Li et al. [22] investigated a theoretical model for the normal contact force between an elastoplastic sphere and a rigid flat for the loading phase. This model is a modified version of Thornton’s contact model [23,24] with more detailed pressure distribution based on a FEA modeling. Later in 2005, Wu et al. [25] studied the impact of an elasto-plastic sphere with a hard flat and completed a new model for the unloading phase. Kogut and Etsion [26] studied the elasto-plastic contact of a hemisphere with a rigid flat and developed a new empirical formulation. The contact is divided into two sub-phases during the loading: the fully elastic and the elasto-plastic. The hardness is considered to be constant for this study. The Hertzian theory is applied for the elastic phase. The empirical formulation for the elasto-plastic phase is obtained from curve fitting techniques of the FEM results. The unloading phase is improved in another study by Etsion et al. [27]. This work provides a new empirical formulation obtained from FEA results. In this model the unloading phase does not follow the Hertzian theory and is considered to be elasto-plastic. Later Shankar and Mayuram [28] improved the Kogut–Etsion model by considering the effect of the strain hardening. However, the results in this work are only compared with the FEA results. Many impact studies have used Kogut–Etsion model for the contact force. However, there is one important condition that needs to be considered. This model works for a small range of deformations, and it is not suitable for large deformations. Jackson and Green [29] developed a new contact model between an elasto-plastic half sphere and a rigid flat. The empirical formulation works for larger deformations. The hardness is not considered to be constant for this model, and it changes during the contact. This model has been compared and verified with Johnson’s experiments [21]. Later Jackson et al. [30] studied the residual stresses after the unloading phase. Residual deformation after the unloading phase is defined as the permanent deformation. The results from this study [30] were used for new developments in Ref. 31. A new empirical formulation for the permanent deformation after the contact has been provided [31].

1.3. Impact models Contact models have been used in many impact applications. Stronge used Johnson’s [6] elastic–fully plastic model and defined a new energetic coefficient of restitution [32]. Stronge divided the impact into three phases: the fully elastic, the fully plastic, and the restitution [33–36]. The elastic phase follows the Hertzian theory [1]. The fully plastic phase follows Johnson’s expression [6]. The restitution is considered to be fully elastic and follows the Hertzian theory. Thornton and Ning [23] developed a numerical model to study the impact of a sphere with a wall. Thornton also studied elastic– perfectly plastic impact and provided an analytical solution on a simplified model [24]. The restitution phase follows the Hertzian theory. However, Thornton assumes that the radius of curvature for this phase is larger than the radius of curvature for the loading phase. In 1999 and 2000, Kharaz et al. performed a new accurate experimental study on the impact of 5 mm aluminum balls with a massive steel surface [37,38]. They used computer vision in order to measure the motion. Jackson et al. [31] studied the impact of an elasto-plastic sphere with a rigid flat and used Jackson–Green contact model in order to predict the coefficient of restitution after the impact. They compared and verified their results with Kharaz’s experiments. Marghitu et al. [39] applied Jackson–Green contact model for the impact of a rotating link with a massive surface. Brake used his contact models [19,20] for the normal and oblique impacts in order to compare and verify his results with Kharaz’s experiments. It has to be considered that Kharaz experiments were performed with 5 mm diameter aluminum balls and did not result in large deformations. Therefore, comparing the results with Kharaz’s experiments might result in some inaccuracies. In our previous work, we studied the normal impact of an elastoplastic rod with an elasto-plastic flat [40]. Using this model the impact is divided into three phases: the elastic, the elasto-plastic, and the restitution phase. The Jackson–Green model has been modified in order to accomodate the effect of the deformation on both of the objects. A new empirical formula for the permanent deformation was proposed. The model was verified with the experiments for the normal impact. The goal of this study is to apply our previous results [40] for the oblique impact with friction. The results are compared with eight well known contact models for the linear and angular motion of the rod during the impact, and permanent deformation. Experimental results are compared with the proposed model and the established models in terms of the coefficient of restitution and the rebound angular velocity. 2. Dynamics of the oblique impact The equations of motion of a rod during the oblique impact were developed using Newton–Euler equations of motion. MATLAB has been used to find and solve the equations of motion. Because of the complexity and non-linearity of the contact force expressions, the equations of motion have been solved numerically using ODE45. Fig. 1 shows the schematic of the rod during the impact. The impact angle is θ. The center of the mass is point C, and the contact  point is T. The gravitational force is G , the normal force during the  impact is Fn , and the friction force during the impact is Ff . The friction has been considered to be continuous. The position vector of the contact point T:

 rT = xiˆ + δ ˆj,

(1)

where x is the tangential displacement of the contact point and δ is the normal deformation or indentation of the rod for flattening and indentation models respectively.

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divide it into three sub-phases: the fully elastic, the elasto-plastic, and the fully plastic phase.

3. Contact force formulations

Fig. 1. Schematic of the rod during the impact.

The position vector of the center of the mass of the rod:

 L rC = ⎡⎣ xiˆ + (δ + R ) ˆj ⎤⎦ + ⎡⎣cos (θ ) iˆ + sin (θ ) ˆj ⎤⎦ , 2

(2)

where the radius of the tip of the rod is R, and the length of the rod is L as shown in Fig. 1. The angular velocity and acceleration can be written as:

  ω = θkˆ , α = θkˆ ,

(3)

where ω is the angular velocity of the rod, and α is the angular acceleration of the rod. The velocity of the center of the mass of the rod and contact point can be calculated as:

       dr vC = c , vT = vC + ω × (rT − rC ) , dt

In this section, different contact models are reviewed in detail. Different contact force expressions are provided for each model and each phase. Two different flattening models are chosen and presented. The first formulation uses the Jackson–Green [29] modeling for the loading phase and Jackson’s work in Ref. 30 for the unloading phase. This formulation will be called JG model in the rest of the paper for simplification. The second formulation uses the Kogut–Etsion [26] model for the loading phase and Etsion et al. work in Ref. 27 for the unloading phase. This formulation is called KE model for the rest of the paper. Three models have been chosen and used among indentation models. The first formulation uses the Kogut–Komvopoulos [16] (KK model) for the loading and unloading phase. The second formulation uses Ye–Komvopoulos [15] (YK model) formulation for the loading phase. Jackson’s work in Ref. 30 is used for the unloading phase since this work does not provide a study for unloading phase. The third formulation uses Brake [20] contact model for the loading and unloading phases. In Brakes model it is not mentioned that the model is for indentation regime; however, the fully plastic formulation that has been used in the model is for indentation cases. Our last work [40] considers both the rod and the flat to be elastoplastic. The model consists of the loading and the unloading phases. It is a modified version of the JG model and will be called MJG for simplification. Moreover, Brakes model presented on 2012 [19], Stronge [36] and Thornton [24] models have been used for comparison.

(4) 3.1. Hertzian contact theory

 where vC is the velocity of the center of the mass of the rod, and  vT is the velocity of the tip of the rod. The acceleration of the center of the mass of the rod and contact point can be written as:

Most of the models are using the Hertzian theory [1] for the elastic phase and the restitution phase. The Hertzian theory uses the reduced modulus of elasticity and radius:

           d 2r aC = 2c , aT = aC + α × (rT − rC ) + ω × [ω × (rT − rC )], dt

E −1 =

(5)

 where aC is the acceleration of the center of the mass of the rod,  and aT is the acceleration of the tip of the rod. The equations of motion are:

  v iˆ mα C = Fniˆ + ( Ff − mg ) ˆj, Ff = − μk Fn  T , vT iˆ

(6)

   ICα = (rT − rC ) × Fniˆ + Ff ˆj ,

(7)

(

)

where IC is the mass moment of inertia about the center of the mass of the rod, Fn is the normal contact force, Ff is the friction force, and μk is the kinematic coefficient of friction. The impact can be divided into two main phases: the loading and the unloading phase. The loading phase starts when the end of the rod touches the flat and continues until the contact point of the tip stops. At this point, the maximum deformation occurs, and the normal velocity of the tip of the rod is zero. The unloading phase starts at this instance and continues until the rod reaches the permanent deformation. At this instance the contact force is zero. The loading phase is divided into sub-phases depending on the contact model. Some models divide the loading phase into two subphases: the fully elastic and the elasto-plastic phase. Other models

1 − ν12 1 − ν 22 + , Er Ef

1 1 1 = + , R Rr Rf where Rr, Er and Rf, Ef are the radius and the modulus of elasticity of the rod and the flat respectively. R = Rr for our case since Rf = ∞. Using calculated R and E the contact force for the elastic phase of the contact, Fe, is calculated as follows:

Fe =

4 0.5 1.5 ER δ . 3

(8)

For the elasto-plastic phase of the contact each model has different expressions for the contact force. The detailed formulation of the selected models will be given in the following sections of the paper.

3.2. JG flattening model In the JG model the contact of an elasto-plastic deformable hemisphere with a rigid flat has been simulated in Fig. 2. The JG model [29] divides the loading phase into two sub-phases, the elastic and the elasto-plastic phase.

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Rr =

1 ⎛ 3Fm ⎞ ⎜ ⎟ , (δ m − δ r )3 ⎝ 4E ⎠

(13)

where Fm = Fep (δ = δ m ) . Using the new radius of curvature, Rr, and the permanent deformation, δr, the contact force for restitution phase is calculated as below:

Fr =

4 0.5 1.5 ERr (δ − δ r ) . 3

(14)

3.3. KE flattening model Fig. 2. Schematic of the deformed objects for flattening contact models.

The KE model [26] has the same methodology as the JG model; however, the formulation is simpler.

3.2.1. JG elastic phase The elastic phase has been considered to follow the Hertzian [1] theory. It has been considered that the effective elasto-plastic phase starts at δ* ≥ 1.9 where

3.3.1. KE elastic phase The contact is elastic while δ/δc ≤ 1, where δc can be calculated as: 2

2

δ ⎛ π C jSy ⎞ δ* = , δy = ⎜ R, C j = 1.295e0.736ν , ⎝ 2E ⎟⎠ δy

(9)

π KH ⎞ δ c = ⎛⎜ R, ⎝ 2E ⎟⎠

(15)

K = 0.454 + 0.41ν , H = 2.8S y .

and δy is the interference at which the yields start and Sy is the yield strength of the weaker material. The elasto-plastic phase starts theoretically at δ* = 1, however effectively starts at δ* = 1.9 according to Ref. 29. For the contact force in this phase, Eq. (8) is used.

The KE model considered the hardness (average normal pressure) to be constant.

3.2.2. JG elasto-plastic phase For the elasto-plastic phase the JG model considers the following formulation:

3.3.2. KE elasto-plastic phase The elasto-plastic phase has been divided into two sections. For small deformations 1 < δ/δc ≤ 6 the contact force expression is:

5 12 59 ⎡ ⎤ 4 HG Fep = Fc ⎢e −0.25δ *) δ *1.5 + 1 − e −0.04(δ *) δ * ⎥ , C S j y ⎣ ⎦

Fep ⎛δ⎞ = 1.03⎜ ⎟ ⎝ δc ⎠ Fc

(

)

(10)

1.425

, Fc = Fe (δ = δ c ) .

(16)

For large deformations 6 < δ δ c ≤ 110 the contact force expression

where is: B

⎛ δ ⎞ B = 0.14e23 Sy E , a = Rδ y ⎜ , ⎝ 1.9δ c ⎟⎠

Fep ⎛δ⎞ = 1.40 ⎜ ⎟ ⎝ δc ⎠ Fc

HG ⎡ ⎛ a⎞⎤ = 2.84 − 0.92 ⎢1 − cos ⎜ π ⎟ ⎥ , ⎝ R⎠ ⎦ Sy ⎣ 2

.

(17)

It has to be considered that for most of the impact problems δ/δc is larger than 110.

3

4 ⎛ R ⎞ ⎛ π C jSy ⎞ Fc = ⎜ ⎟ ⎜ ⎟ . 3⎝ E⎠ ⎝ 2 ⎠

1.263

(11)

The average normal pressure is HG, the critical force at the instant the yield occurs is Fc, and the contact force during the elastoplastic phase is Fep .

3.3.3. KE restitution Etsion et al. [27] model considers the restitution phase to be elasto-plastic and provides expressions for the permanent deformation after the contact and the contact force during the restitution:

3.2.3. JG restitution phase For the restitution phase of the JG model, Jackson’s work in Refs. 30 and 31 has been used. The restitution has been considered to be elastic and follows the Hertzian theory. The permanent deformation after the impact has been calculated from curve-fitting of the FEM analysis. The empirical formulation provided in Refs. 30 and 31 is as below:

⎡ ⎛ δ ⎞ −0.28 ⎤ ⎡ ⎛ δ m ⎞ −0.69 ⎤ δ r = δ m ⎢1 − ⎜ m ⎟ ⎥ ⎢1 − ⎜⎝ ⎟⎠ ⎥, δc ⎣ ⎝ δc ⎠ ⎦⎣ ⎦

⎛ ⎡ ⎛ δ m δ y + 5.9 ⎞ −0.54 ⎤⎞ δ r = δ m ⎜ 1.02 ⎢1 − ⎜ ⎥⎟ , ⎟⎠ 6.9 ⎝ ⎣ ⎝ ⎦⎠

(12)

where δm is the maximum deformation at the end of the loading phase, and δr is the permanent deformation after the contact. It has been considered that the radius of curvature for the unloading phase is changing as below:

⎛ δ − δr ⎞ Fm = Fep (δ = δ m ) , Fr = Fm ⎜ ⎝ δ m − δ r ⎟⎠

(18)

1.5(δ m δ c )−0.0331

,

(19)

where the maximum interference is δm, and the maximum contact force during the impact is Fm.

3.3.3.1. KK indentation model. For the indentation models δ is defined as the indentation of the rigid sphere in the flat as shown in Fig. 3. The formulations provided by KK model [16] are presented below.

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3.4.1. YK elastic-phase The elastic phase starts at Q = 0 and continues until Q < 1.78. For elastic phase the contact force is formulated as:

Fe 4 2 a′ = QS y , a = , a ′ = π r ′ 2. a 3π 2

(24)

3.4.2. YK elasto-plastic phase For the small deformation when Q ≤ 21, the contact force is defined as:

where r′ is shown in Fig. 3, E is the reduced modulus of elasticity defined in the Hertzian theory, and Sy is the yield strength of the flat. Using these criteria for the elastic phase the contact force is: (20)

(27)

3.4.3. YK restitution phase The YK model does not provide any governing equations for the restitution phase. Therefore, for the sake of the comparison we used the formulation provided by the JG model for the YK model. 3.5. Brake indentation model

a′ 2 , a ′ = π r ′ 2, r ′ = R 2 − ( R − δ ) . 2

3.3.5. KK elasto-plastic phase The elasto-plastic phase starts when δ r ′ > 1.78S y E . The KK model suggests the following expression for the contact force: 0.656 0.839 + ln ⎡⎣( E S y ) (δ r ′ )0.651 ⎤⎦ Fep = . a ′S y 2.192 − ln ⎡( E S y )0.394 (δ r ′ )0.419 ⎤ ⎣ ⎦

3.5.1. Brake elastic phase The elastic phase starts from δ = 0 and continues until δ ≤ δy:

−0.156

.

Brake model [20] divides the contact into two phases, elastic and elasto-plastic phases. The elastic phase follows the Hertzian theory and for the plastic phase Meyer’s hardness has been used. For the elasto-plastic phase transitionary functions have been proposed and showed that are matching the experimental results. The elastoplastic phase in Brake’s model [20] follows Mayer’s hardness model, therefore this model has been categorized in the indentation models.

(21)

3.3.6. KK restitution phase The restitution phase has been considered to follow the Hertzian theory with the following expression for the permanent deformation:

δ r = δ m (1 − E Rδ ) , E Rδ

(26)

Eq. (26) should be used to calculate a for the medium deformations. For the large deformation when Q > 21, Eq. (27) is used with a = a ′ 0.71 .

δ 1.78S y ≤ , r′ E

⎛ E⎞ = 0.591⎜ ⎟ ⎝ Sy ⎠

a′ = (0.05logQ 2 − 0.57logQ + 2.41) . a

Fep = 2.9S y . a

3.3.4. KK elastic phase The elastic phase starts from δ = 0 and continues until:

a=

(25)

For the medium deformation when 21 < Q ≤ 400, the following formulation is provided:

Fig. 3. Schematic of the deformed flat for indentation contact models.

Fe 4 2 ⎛ Eδ ⎞ = , aS y 3π ⎜⎝ S y r ′ ⎟⎠

Fep = (0.7log Q + 0.66) S y , a

(22)

2

δy =

r ⎛ π Sy ⎞ ⎜ ⎟ , F (ν ) ⎝ 2E ⎠

⎛ ⎞ 1 ⎡ z ⎛ r⎞⎤ 3 F (ν ) = max ⎜ − [1 − ν ] ⎢1 − tan−1 ⎜ ⎟ ⎥ + ⎝ z ⎠ ⎦ 2 1 + ( z r )2 ⎟⎠ ⎝ ⎣ r

(28) 2

for

z ≥0

Using these criteria for the elastic phase the contact force is calculated from Eq. (8).

3.4. YK indentation model The YK model [15] defines four different sub-phases for the contact: the elastic phase, small deformations, medium deformations, and large deformations. YK model defines Q as:

⎛ E ⎞⎛ δ ⎞ 2 Q = ⎜ ⎟ ⎜ ⎟ , r ′ = R2 − (R − δ ) , ⎝ Sy ⎠ ⎝ r ′ ⎠ where r′ is shown in Fig. 3.

(23)

3.5.2. Brake elasto-plastic phase The elasto-plastic phase starts when δ > δy. Brake propose the following equation for the contact force:

δ − δy ⎞ 4 ⎛ F = sech ⎜ (1 + nε ) E rδ 1.5 ⎝ δ p − δ y ⎠⎟ 3 δ − δ y ⎞ ⎤⎞ ⎡ ⎛ an + ⎜ 1 − sech ⎢(1 − nε ) ⎥⎟ P0π n−2 ⎟ δ p − δ y ⎠ ⎦⎠ ap ⎝ ⎣

(29)

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⎛ 2 2⎞ P0 = Hg106, H = ⎜ + ⎝ H s H f ⎟⎠

−1

4. Simulation results (30)

where nε = n − 2, and n is Meyer’s strain hardening components, H is Brinell’s hardness and ap can be calculated by solving the following nonlinear equation for the smallest positive root of δ:

⎛ 3P ⎞ a p = 2δ R = ⎜ 0 2n 2 π r (n−1) 2δ (n−3) 2 ⎟ ⎝ 4E ⎠

1 (n−2)

,

and δ p = a 2p 2R .

Nine different contact models have been used in the simulations and compared for the linear and the angular motion during the impact and the permanent deformation. The material properties and the geometry in Table 1 are used for the simulations. Table 2 provides the legend used for comparison plots. The rod has round end and is impacting the flat with initial angle θ = 45° and θ = 17.2° , zero initial tangential and angular velocities. The coefficient of friction has been calculated and averaged for all of the experiments at each angle using the following equation:

μ= 3.5.3. Brake restitution phase The restitution phase has been considered to follow the Hertzian theory with the following expression for the permanent deformation and radius of curvature for the unloading phase:

⎛ ⎞ Fm Fm2 δ r = δ m ⎜1− , Rr = 1.5 ⎟ 2 ⎝ ( 4 3E ) (δ m − δ ) ⎠ (4 3E ) (δ m − δ )3

(31)

3.6. Modified JG model Our previous work modifies the JG model in order to satisfy the effects of deformation on both of the objects for a specific material properties and provides a new empirical formula for the permanent deformation [40].

3.6.1. Elastic phase The elastic phase has been considered to follow the Hertzian [1] theory. This phase is same as the JG model and uses the same formulations. For the contact force in this phase, Eq. (8) is used.

3.6.2. Elasto-plastic phase For the elasto-plastic phase, JG expression for the contact force has been modified [40]. This expression satisfies the effects of deformation on both of the objects for our specific material properties. The modified JG model expression for the contact force is shown below: 1 − (δ *)5 9 ⎞ ⎡ 5 12 4 HG ⎛ 1.1 ⎤ Fep = Fc ⎢e −0.17δ * δ *1.5 + 1 − e 78 ⎜ ⎟⎠ δ * ⎥ . ⎝ C S j y ⎣ ⎦

(32)

Simular to JG model Eqs. (11) are used for the calculations.

3.6.3. Restitution phase The contact force for the restitution phase of this model has been considered to be elastic and follows the Hertzian theory. According to Ref. 31 the radius of curvature Rr changes as in Eq. (13). The permanent deformation, δr, for this model is calculated from experimental results [40]. The following expression is provided: −2

⎡ ⎛ δ m δ y + 5.5 ⎞ ⎤⎞ δr ⎛ = 0.8 ⎢1 − ⎜ ⎟⎠ ⎥⎟ . 6.5 δ m ⎜⎝ ⎣ ⎝ ⎦⎠

(33)

Using the new radius of curvature, Rr, and the permanent deformation δr, the contact force for restitution phase is calculated as below:

Fr =

4 0.5 1.5 ERr (δ − δ r ) . 3

(34)

vCf iˆ − vCiiˆ v ˆj − v ˆj Cf

(35)

Ci

where vCi and vCf are the velocity of the center of the rod before and after the impact respectively. For the initial impact angle θ = 45° the coefficient of friction is μ = 0.13 and for θ = 17.2° the coefficient of friction is μ = 0.2. Fig. 4 shows the normal displacement of the contact point during the impact. The loading phase starts at t = 0 s. For JG, MJG, KE, KK, YK and Stronge models the loading phase continues to t = 1.1 × 10−4 s . Brake (2012), Brake (2015) and Thornton models are showing longer loading phase. At this point the normal displacement is minimum. The restitution phase starts at this instant. Stronge and Thornton models show minimum and maximum permanent deformations respectively. Thornton shows the longest impact duration among the other models. Besides the Thornton model Brake models are showing longer impact duration compare to the other models. Fig. 5 depicts the tangential displacement of the contact point during the impact. All of the models show same trend, while Thornton and Brake models are slightly different from the other models. Brake (2012) models show the largest tangential displacements. The normal velocity of the contact point during the impact is shown in Fig. 6. The initial normal velocity used in the simulation  is vi = −1 j . There is a large difference between Thornton and the other models. When the normal velocity becomes zero, unloading phase starts, and all of the models give different results for this phase. The final rebound normal velocity calculated with Stronge model

Table 1 The material properties of the rod and the flat. Rod (stainless steel AISI 201)

Flat (low-carbon iron AISI 1010)

R L m ρ Er S yr Hr νr

t ρ Ef S yf Hf νf

0.0088 (m) 0.3048 (m) 0.4607 (kg) 7800 (kg m3 ) 200 (GPa) 760 (GPa) 327 kgf m2 0.3

0.03 (m) 7870 (kg m3 ) 200 (GPa) 305 (MPa) 105 (kgf m2 ) 0.3

Table 2 The legends, equations and references used for the graphs. Legend

Name, equations and references Jackson–Green (JG), Eqs. (8–14) [29–31] Ghaednia et al. (MJG), Eqs. (8,32–34) [40] Kogut–Etsion (KE), Eqs. (8,15–19) [26,27] Kogut–Komvopoulos (KK), Eqs. (20–22) [16] Ye–Komvopoulos (YK), Eqs. (14,23–27) [15] Brake (2015) Eqs. (8,28–31) [20] Brake (2012) [19] Stronge [36] Thornton [24] Experimental results

H. Gheadnia et al./International Journal of Impact Engineering 86 (2015) 307–317

Fig. 4. Normal displacement during the impact.

313

Fig. 6. Normal velocity during the impact.

is the largest compare to other models. Brake (2012), Brake (2015) and KE models are showing nearly similar results. Thornton model shows the smallest value for the rebound velocity. Figs. 7 and 8 present the tangential velocity and the angular velocity of the rod during the impact respectively. Similar to the normal velocity results Thornton model is different from the other models. For all of the models the tangential velocity starts at zero and decreases with an increasing rate until the maximum compression happens and then decreases with a decreasing rate until the end of the impact. Stronge shows the maximum rebound angular and tangential velocities. Other than the Thornton model KK and YK are showing the minimum angular and tangential velocities. Thornton, Brake and Stronge models are showing longer impact duration. Fig. 9 represents the permanent deformation after the impact as a function of the initial impact velocity. The permanent deformation increases with the increase of the initial impact velocity with a same trend for all of the models. Thornton model is showing significantly larger permanent deformations than the other models. Besides the Thornton model, the Brake (2015) and Stronge models show the largest and smallest values for the permanent deformation. 5. Experiments

Fig. 5. Tangential displacement during the impact.

The experimental setup, shown in Fig. 10, has been built for the oblique impact of a rod with a flat surface. The rod with the material properties shown in Table 1 have been tested for the experiments. A dropping device has been designed in order to provide accurate and consistent initial conditions for the experiments. The device, shown in Fig. 10, has two electromagnet solenoids attached to it. Electromagnet solenoids are used to keep the rod with desired angle and height. A release button is controlled by the operator. This device has been used to drop the rod from different

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Fig. 9. Permanent deformation function of initial impact velocity.

Fig. 7. Tangential velocity during the impact.

heights and different angles. The setup provides different initial velocities in the range of 0.5–4 m/s. Two sets of experiments for initial angles, θ i = 45° and θ i = 17.2° , and different initial velocities have been done. At least three drops have been done for each height. The flat has been fixed on the massive table by clamps. Impacts have been placed far enough from each other in order to ensure the residual stresses from the previous impacts on the flat do not affect the impact. Three markers have been attached to the rod. One of the markers is located at the center of the mass of the rod. The second marker is attached near the impact point and the third marker is attached to the top of the rod. These markers are used to track the displacement and the angle of the rod. A 3D infrared camera (Optotrak 3020) with 500 frames per seconds has been used in order to capture the motion of the markers on the rod.

5.1. Experimental results

Fig. 8. Angular velocity during the impact.

The motion of the rod measured with Optotrak is analyzed with MATLAB. The position of the center of the mass, the velocity of the center of the mass, and the angular velocity of the rod are determined and shown in Figs. 11–13 respectively. Numerical differentiation has been done in order to calculate linear and angular velocities. Position of the center of the mass of the rod has been calculated before and after the impact for each heights. The tangential and the normal components of the position of the rod are shown in Fig. 11. The impact is occuring nearly at t = 0.256 s. The velocity of the center of the mass of the rod is calculated before and after the impact for all experiments. Fig. 12 shows the experimental results for one of the experiments with θ i = 45° . For this example, the impact occurs at t  0.26 s . The normal b = −2.487 m s and velocities before and after the impact are vCn a vCn = −0.855 m s respectively. The tangential velocities before and b a = 0.067 m s and vCt = −0.14 m s after the impact are calculated as vCt respectively. In order to verify the accuracy of the experimental data, the slope of the velocity graph is calculated and compared with gravitational acceleration, g. The experimental results show g exp = 9.809 m s which has less than 0.1% difference from the standard g for our location.

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Fig. 10. (a) Sketch of the experimental setup, right view. (b) Front view.

The angle of the rod before and after the impact has been also measured and the angular velocity of the rod has been calculated. For this case the rod is initially released with θ = 45° , and the angle of the rod is not changing until the rod impacts the flat. Fig. 13 shows

the angular velocity of the rod before and after the impact for θ i = 45° . The angular velocities before and after the impact are ωb = 0.02 rad/s and ω a = −19.67 rad s respectively. 6. Results and comparison

Fig. 11. Position of the rod before and after the impact.

Fig. 12. Velocity of the center of the mass before and after the impact.

The simulations and the experiments have been compared for the coefficient of restitution and the rebound angular velocity for initial impact angles, θ i = 17.2° and θ i = 45° . It has to be considered that the material properties of the rod and the flat used for the experiments lead to an indentation regime. The kinematic coefficient of restitution is defined as the ratio between the rebound and the initial velocity at the contact point. Fig. 14 shows the coefficient of restitution as a function of the initial velocity for the contact models and the experiments. The experiments in this plot are performed for initial velocities from 0.5 to 4 m/s and initial impact angle, θ = 45° . Experimental results are shown with black crosses. All of the models other than the MJG model show similar trend. For all of these models the coefficient of restitution is decreasing as the initial normal velocity increases. The experiments show nearly constant results for the coefficient of the restitution, e  0.43. The MJG model is in good agreement with the experimental results. The KE, Brake (2012), Brake (2015) and Stronge models are predicting larger values for smaller initial velocities and matching the experimental results for larger initial velocities. The JG, YK, and KK models predict smaller coefficient of

Fig. 13. Angular velocity of the rod before and after the impact.

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Fig. 14. Comparison between the simulations and the experimental results for the coefficient of restitution for θ = 45° .

Fig. 16. Rebound angular velocity as a function of initial impact velocity for θ = 17.2° .

restitution. The Stronge and Thornton models have the largest and smallest values respectively. Fig. 15 shows the rebound angular velocity as a function of initial impact velocity for θ = 45° . The rebound angular velocity increases with the increase of the initial impact velocity. The MJG, KE, Stronge, Brake (2012) and Brake (2015) models match the experiments for small initial velocities and show slightly smaller values for larger velocities. The JG, KK, YK and Thornton models show smaller values for the rebound angular velocities. Fig. 16 shows the coefficient of restitution as a function of the initial velocity for the contact models and the experiments. The experiments are performed for initial velocities from 0.5 to 4 m/s and initial impact angle, θ = 17.2° . Experimental results are shown with black crosses. Compare to the θ i = 45° experiments (Fig. 14), coefficient of restitution for θ i = 17.2° shows larger values. All of the models other than the MJG model show similar trend. For all of these models the coefficient of restitution is decreasing as the initial normal velocity increases. The experiments show almost constant results

for the coefficient of the restitution, e  0.55. The MJG model shows the same trend as the experimental data; however it shows smaller results. Overall none of the models matches the experimental results in this case. Thornton and Stronge models similar to Fig. 14 show the smallest and largest results respectively. Fig. 17 shows the rebound angular velocity as a function of initial impact velocity. The rebound angular velocity increases with the increase of the initial impact velocity. All of the models show smaller values than the experimental results. Stronge and Thornton models are showing the maximum and the minimum results respectively.

Fig. 15. Comparison between the simulations and the experimental results for the coefficient of restitution for θ = 45° .

7. Conclusions In this study, the dynamics of the oblique impact of a rod with a flat has been analyzed experimentally and theoretically. The motion of the rod during the impact has been analyzed. The linear and angular motion of the rod has been calculated for five different contact models. The permanent deformation plays an important role

Fig. 17. Rebound angular velocity as a function of initial impact velocity for θ = 17.2° .

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for all the models. Selecting the appropriate contact model is very important and gives different results especially for the unloading phase. The coefficient of restitution and the rebound angular velocity calculated from the experiments with impact angles, θ = 45° and θ = 17.2° , are compared with the simulations using different contact models. The experimental results lead to larger deformations than the previous experimental studies such as Kharaz’s works [37,38], which help us to be able to compare the contact models for more realistic situations when plastic deformation are dominating. It has been shown that for the impact angle, θ = 45° , our previous work, MJG, matches the experimental results for both the coefficient of restitution and rebound angular velocity. Stronge and Brake models also show good agreements with the experimental results. For the impact angle, θ = 17.2° , it has been shown that all of the presented contact models are predicting smaller values for both the coefficient of restitution and the rebound velocity compare to the experimental results.

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