The role of epistemic uncertainty of contact models in the design and optimization of mechanical systems with aleatoric uncertainty

Abstract

Epistemic uncertainty, the uncertainty in the physical model used to represent a phenomenon, has a significant effect on the predictions of simulations of mechanical systems, particularly in systems with impact events. Impact dynamics can have a significant effect on a system’s functionality, stability, wear, and failure. Because high-fidelity models of systems with impacts often are too computationally intensive to be useful as design tools, rigid body dynamics and reduced order model simulations are used often, with the impact events modeled by ad hoc methods such as a constant coefficient of restitution or penalty stiffness. The choice of impact model, though, can have significant ramifications on design predictions. The effects of both epistemic and aleatoric (parametric) uncertainty in the choice of contact model are investigated in this paper for a representative multiple-degree of freedom mechanical system. Six contact models are considered in the analysis: two different constant coefficient of restitution models, a piecewise-linear stiffness and damping (i.e., Kelvin–Voight) model, two similar elastic-plastic constitutive models, and one dissimilar elastic-plastic constitutive model. Results show that the optimal mechanism design for each contact model appears extremely different. Further, the effects due to epistemic uncertainty are differentiated clearly in the response from the effects due to aleatoric uncertainty. Lastly, when the mechanisms are optimized to be robust against aleatoric uncertainty, the resulting designs show some robustness against epistemic uncertainty.

Appendix: Contact model descriptions

For each of the contact models, the geometry shown in Fig. 17 is used: the models are derived based on the contact between two spheres (or a sphere and a flat, which is represented as a sphere with an infinite radius), and the primary degree of freedom is the contact displacement $\delta$, which is the relative displacement of the center of mass of the two spheres. In Fig. 17, the illustrative drawing uses a reference frame in which the center of mass of the second sphere is not changed, and thus $\delta$ is defined as the displacement of the center of mass of the first sphere. The elastic-plastic contact models assumes that the two elastic-perfectly plastic spheres (labeled by $i=1$, 2) with radii $r_i$ shown in Fig. 17 have elastic modulus $E_i$, Poisson’s ratio $\nu _i$, yield strength $\sigma _{Yi}$, density $\rho _i$, and Brinell hardness $H_i$, are smooth such that their asperities are small compared to the displacements (specifically, that the asperity heights are much smaller than the yield displacement $\delta _y$, defined in what follows), and that friction is negligible. The normal displacement between the two spheres is $\delta$, and the contact radius is $a$. The bulk parameters used to represent the spheres are given as the effective elastic modulus $E=((1-\nu _1^2)/E_1+(1-\nu _2^2)/E_2)^{-1}$ and radius $r=(r_1^{-1}+r_2^{-1})^{-1}$ [41], and the yield strength $\sigma _y=\min \left[ \sigma _{y1},\sigma _{y2}\right]$.

Fig. 17

The geometry used for the derivation of the impact models. a Two spheres in contact illustratively showing a set of undeformed (dashed line) and deformed (line) shapes, and b close up of the contact region. Note that while sphere flattening is depicted, indentation is also considered in the model

1.1 Coefficient of restitution contact model

The coefficient of restitution $e$ is a useful measure of the energy dissipation from an impact event at a given velocity. For two bodies with velocities just before an impact of $v_{1i}$ and $v_{2i}$, the velocities just after the impact $v_{2o}$ and $v_{1o}$ are governed by

$$\begin{aligned} \begin{aligned}&v_{1o}=\frac{m_1v_{1i}+m_2v_{2i}+m_2e(v_{2i}-v_{1i})}{m_1+m_2} \\&v_{2o}=\frac{m_1v_{1i}+m_2v_{2i}+m_1e(v_{1i}-v_{2i})}{m_1+m_2}. \end{aligned} \end{aligned}$$

(11)

This equation assumes that the two bodies have mass $m_1$ and $m_2$, that the impact is instantaneous, and that the energy dissipation of impact is independent of impact velocity (thus a constant $e$).

1.2 Piecewise-linear contact model

The penalty method of contact is a Kelvin–Voight model described by a piecewise-linear stiffness and damping

$$\begin{aligned} F_C\left( \delta ,\dot{\delta }\right) =K_P\delta +C_P\dot{\delta }, \end{aligned}$$

(12)

where $\dot{\delta }$ is the derivative of $\delta$ with respect to time. The stiffness $K_P=10\hbox { kN/mm}$ and damping $C_P=280\hbox { Ns/m}$ are chosen to represent a sufficiently stiff impact that is tuned to match the energy dissipation predictions of Brake’s contact model. The value of 10 kN/mm is a commonly used approximation in many finite element simulations and is several orders of magnitude larger than the other stiffnesses in the system investigated. The damping coefficient is calibrated to match the energy dissipation predictions of Brake’s contact model at velocities above 4 m/s.

1.3 Brake’s contact model

The formulation for Brake’s contact model is given in [1] and summarized here. Contact is divided into four regimes: elastic, mixed elastic-plastic, fully plastic, and unloading.

1.3.1 The elastic regime

The elastic regime for each of the elastic-plastic models discussed all use Hertz’s solution [42] for the constitutive relationship between the spheres

$$\begin{aligned} F_C(\delta )=\frac{4}{3}E\sqrt{r}\delta ^{3/2}\qquad \qquad 0\le \delta \le \delta _y,\quad 0\le \dot{\delta }, \end{aligned}$$

(13)

with the contact radius determined by $a=\sqrt{r\delta }$. The elastic regime spans from the initiation of contact ($\delta =0$) until yield ($\delta =\delta _y$). In Brake’s contact model, yield is determined based on the stress field that develops in elastic contact between two bodies [41] and the von Mises’ criterion. Defining the maximum amplitude of the stress field

$$\begin{aligned} f(\nu )=\max _{z\in \mathbb {R}} \left( -(1+\nu )\left( 1-z\,\mathrm {atan}\left( \frac{1}{z}\right) \right) +\frac{3}{2}\frac{1}{1+z^2}\right) ^2,\nonumber \\ \end{aligned}$$

(14)

over the distance $z$ into the surface, where $\nu$ is Poisson’s ratio for the material that will yield first,

$$\begin{aligned} \delta _y=\frac{r}{f(\nu )}\left( \frac{\pi \sigma _y}{2E}\right) ^2. \end{aligned}$$

(15)

1.3.2 The plastic regime

In the plastic regime, it is assumed that the pressure $p_0$ over the contact area is uniform and there is a linear relationship between the contact force and $\delta$

$$\begin{aligned}&F=p_0\pi a^2\end{aligned}$$

(16)

$$\begin{aligned}&a^2=2r\delta +\xi \end{aligned}$$

(17)

$$\begin{aligned}&F_C(\delta )=p_0\pi \left( 2r\delta +\xi \right) \qquad \delta _p\le \delta ,\quad 0\le \dot{\delta }. \end{aligned}$$

(18)

Since both the contact radius $a_p$ and displacement $\delta _p$ at the fully plastic regime’s inception are fixed in Eqs. 16 and 17, $\xi$ is chosen for compatibility. The maximum contact pressure is based on the definition of hardness given in [43] $p_0=Hg10^6$, where $g$ is acceleration due to gravity and is necessary for the unit conversion from the Brinell hardness $H$ (given in $\hbox {kgf/mm}^2$) to Pa, and is defined by the hardness of both materials involved in the contact

$$\begin{aligned} H = \left( \frac{2}{H_1}+\frac{2}{H_2}\right) ^{-1}. \end{aligned}$$

(19)

The contact radius $a_p$ and displacement $\delta _p$ at the onset of fully plastic behavior are then defined via

$$\begin{aligned}&\delta _p=\left( \frac{p_0}{\sigma _y}\right) ^2\delta _y,\end{aligned}$$

(20)

$$\begin{aligned}&a_p=\sqrt{2r\delta _p+\xi },\end{aligned}$$

(21)

$$\begin{aligned}&\xi =\left( r\frac{3\pi p_0}{4E}\right) ^2-2\left( \frac{p_0}{\sigma _y}\right) ^2\delta _p. \end{aligned}$$

(22)

The incipient plastic contact force is thus defined as $F_p=p_0\pi a_p^2$.

1.3.3 The mixed elastic-plastic regime

To model the mixed elastic-plastic regime, the set of cubic Hermite polynomials [44] is used to enforce continuity between the end of the elastic regime and the beginning of the plastic regime

$$\begin{aligned} F_C(\delta )&= \!\left( 2F_y-2F_p+\left( \delta _p-\delta _y\right) \left( F_y^\prime +F_p^\prime \right) \right) \nonumber \\&\quad \times \,\left( \frac{ \delta -\delta _y}{\delta _p-\delta _y}\right) ^3 +\left( -3F_y+3F_p +\left( \delta _p-\delta _y\right) \right. \nonumber \\&\left. \quad \times \, \left( -2F_y^\prime -F_p^\prime \right) \right) \left( \frac{\delta -\delta _y}{\delta _p-\delta _y}\right) ^2 \nonumber \\&\quad +\left( \delta _p-\delta _y\right) F_y^\prime \left( \frac{\delta -\delta _y}{\delta _p-\delta _y}\right) +\,F_y\nonumber \\&\quad \delta _y\le \delta \le \delta _p,\quad 0\le \dot{\delta } \end{aligned}$$

(23)

$$\begin{aligned} a&= \!\left( 2a_y-2a_p\!+\!\left( \delta _p\!-\!\delta _y\right) \left( a_y^\prime \!+\! a_p^\prime \right) \right) \left( \frac{\delta -\delta _y}{\delta _p-\delta _y}\right) ^3\nonumber \\&\quad +\left( -3a_y+3a_p+\left( \delta _p-\delta _y\right) \right. \nonumber \\&\quad \times \left. \left( -2a_y^\prime -a_p^\prime \right) \right) \left( \frac{\delta -\delta _y}{\delta _p-\delta _y}\right) ^2 \nonumber \\&\quad +\left( \delta _p-\delta _y\right) a_y^\prime \left( \frac{\delta -\delta _y}{\delta _p-\delta _y}\right) +a_y. \end{aligned}$$

(24)

These polynomials are chosen for convenience; other sets of splines that are $\mathcal {C}^1$ continuous would also be suitable. The net effect of these equations is to enforce both $\mathcal {C}^0$ and $\mathcal {C}^1$ continuity for the relationship between $\delta$ and both $F_C$ and $a$. The $\cdot ^\prime$ indicates a derivative with respect to $\delta$.

1.3.4 The restitution phase

Unloading is assumed to be an elastic process [15, 17, 45] with no reverse yielding. In each regime, unloading follows

$$\begin{aligned} F_C(\delta )=\frac{4}{3}E\sqrt{\bar{r}}\left( \delta -\bar{\delta }\right) ^{3/2}\qquad \dot{\delta }\le 0, \end{aligned}$$

(25)

and $\bar{\delta }$ is related to $\bar{r}$ by enforcing continuity. Defining the maximum displacement and maximum force during the loading phase as $\delta _m$ and $F_m$, respectively,

$$\begin{aligned} \bar{\delta }=\delta _m-\left( \frac{3F_m}{4E\sqrt{\bar{r}}}\right) ^{2/3}. \end{aligned}$$

(26)

The physical interpretation of $\bar{r}$ and $\bar{\delta }$ is that the contacting surfaces deform both in terms of the radius of curvature over the contact area ($\bar{r}$), and in terms of a residual deformation ($\bar{\delta }$).

Unloading from the elastic regime: Impacts that stay purely within the elastic regime result in no plastic deformation. As a result $\bar{r}=r$ and $\bar{\delta }=0$.

Unloading from the plastic regime: The contact radius is assumed to be approximately the same as the radius of a spherical indentation with depth $\delta _m$ and width $2a_m$. From the trigonometric properties of a chord

$$\begin{aligned} \bar{r}=\frac{1}{2}\left( \delta _m+\frac{a_m^2}{\delta _m}\right) , \end{aligned}$$

(27)

and $\bar{\delta }$ is found via Eq. 26.

Unloading from the mixed elastic-plastic regime: The deformed radius is approximated using the cubic Hermite polynomials again. In this case, the deformed radius at $\delta _m=\delta _y$ is $r$ by definition and has the derivative with respect to $\delta$ of 0. Likewise, the deformed radius at $\delta _m=\delta _p$ is defined as $\bar{r}_p$ and has derivative with respect to $\delta$ of $\bar{r}_p^\prime$, which is found numerically using a forward difference calculation. This yields

$$\begin{aligned} \bar{r}&= \left( 2r-2\bar{r}_p+\left( \delta _p-\delta _y\right) \bar{r}_p^\prime \right) \left( \frac{\delta _m-\delta _y}{\delta _p-\delta _y}\right) ^3\nonumber \\&\quad +\left( -3r+3\bar{r}_p-\left( \delta _p-\delta _y\right) \bar{r}_p^\prime \right) \left( \frac{\delta _m-\delta _y}{\delta _p-\delta _y}\right) ^2\nonumber \\&\quad +\, r.\nonumber \\ \end{aligned}$$

(28)

1.4 Etsion’s contact model

Etsion’s contact model [18, 19] uses the same Hertzian loading process as Brake’s contact model in the elastic zone. Following the onset of yield, correlations derived from high-fidelity finite element simulations are used to model the mixed elastic-plastic and the plastic regimes. Thus, the coefficients in Eqs. 30 through 36 are determined from best-fit curves applied to the results of high-fidelity finite element simulations in [18, 19]. The displacement at the inception of yield

$$\begin{aligned} \delta _y=\left( \frac{KH}{2E}\right) ^2r \end{aligned}$$

(29)

is defined in terms of the hardness $H$ of the more compliant material and the hardness coefficient $K=0.454+0.41\nu$ (where $\nu$ is the Poisson’s ratio of the more compliant material).

1.4.1 Loading after yield

The analysis in [18] observes that there are three distinct regions following the inception of yield: the development of the plastic region below the contact surface in which the entire contact surface is still elastic (for $1\le \delta /\delta _y\le 6$), the expansion of the plastic region to include an annular plastic region on the contact surface ($6\le \delta /\delta _y\le 68$), and a fully plastic contact area ($68\le \delta /\delta _y$). In the regime where the extent of the plastic deformation is entirely below the surface

$$\begin{aligned}&F_C(\delta )\!=\!1.03F_y\left( \frac{\delta }{\delta _y}\right) ^{1.425}\quad 1\!\le \!\delta /\delta _y\!\le \!6,\quad 0\!\le \!\dot{\delta }\nonumber \\\end{aligned}$$

(30)

$$\begin{aligned}&\left( \frac{a}{a_y}\right) \!=\!0.93\left( \frac{\delta }{\delta _y}\right) ^{1.136}, \end{aligned}$$

(31)

where $a_y$ and $F_y$ are the contact radius and contact force at yield ($\delta =\delta _y$), and for the regimes where the plastic deformation extends to the contact surface

$$\begin{aligned}&F_C(\delta )=1.40F_y\left( \frac{\delta }{\delta _y}\right) ^{1.263}\qquad 6\le \delta /\delta _y,\quad 0\le \dot{\delta }\end{aligned}$$

(32)

$$\begin{aligned}&\left( \frac{a}{a_y}\right) =0.94\left( \frac{\delta }{\delta _y}\right) ^{1.146}. \end{aligned}$$

(33)

1.4.2 Unloading

Similar to the loading model is developed, the unloading model is based on best-fit correlations to numerical simulations [19]. Defining the residual interference $\delta _r$ as the permanent deformation due to plasticity, the correlations

$$\begin{aligned}&\frac{\delta _r}{\delta _m}=\left( 1-\frac{1}{\left( \delta _m/\delta _y\right) ^{0.28}}\right) \left( 1-\frac{1}{\left( \delta _m/\delta _y\right) ^{0.69}}\right) \end{aligned}$$

(34)

$$\begin{aligned}&F_C(\delta )=F_m\left( \frac{\delta -\delta _r}{\delta _m-\delta _r}\right) ^{1.5\left( \delta _m/\delta _y\right) ^{-0.0331}}\qquad \dot{\delta }\le 0 \end{aligned}$$

(35)

$$\begin{aligned}&a=a_m\left( \frac{\delta -\delta _r}{\delta _m-\delta _r}\right) ^{\left( \delta _m/\delta _y\right) ^{-0.12}} \end{aligned}$$

(36)

are developed with respect to the maximum contact force $F_m$, contact radius $a_m$, and deformation $\delta _m$ during loading.

1.5 Thornton’s contact model

Thornton’s contact model [20] is divided into two loading regimes: elastic and plastic. The contact model uses the same Hertzian solution for the elastic regime as both Brake’s and Etsion’s contact models. The onset of yield is determined by equating the kinetic energy of a sphere before impact with the work done during elastic loading to the point of yield, which is defined by the contact pressure equalling the yield stress

$$\begin{aligned} \delta _y=\left( \frac{\pi \sigma _y}{2E}\right) ^2r. \end{aligned}$$

(37)

1.5.1 Loading in the plastic regime

Thornton’s model does not consider a transition from elastic to plastic behavior. The elastic regime is defined to abruptly transition to the plastic regime for the yield force

$$\begin{aligned} F_y=\frac{4}{3}E\sqrt{r}\delta _y^{3/2}=\frac{\pi ^3r^2\sigma _y^3}{6E^2}. \end{aligned}$$

(38)

The plastic contact force is then found by considering a Hertzian pressure distribution that is truncated for all pressures above the yield stress

$$\begin{aligned} F_C(\delta )=F_y+\pi \sigma _yr\left( \delta -\delta _y\right) \quad \delta _y\le \delta ,\,0\le \dot{\delta }. \end{aligned}$$

(39)

1.5.2 Unloading

Unloading for Thornton’s contact model is an elastic process. In the elastic regime, there is no plastic deformation, and thus the unloading curve is identical to the loading curve. In the plastic regime, the deformed radius of curvature $\bar{r}$ and residual deformation $\bar{\delta }$ are related to the maximum contact force during loading $F_m$ and maximum displacement $\delta _m$ via

$$\begin{aligned}&\bar{r} = r\frac{4/3E\sqrt{r}\delta _m^{3/2}}{F_m}\end{aligned}$$

(40)

$$\begin{aligned}&\bar{\delta } =\delta _m-\left( \frac{3F_m}{4E\sqrt{\bar{r}}}\right) ^{2/3}\end{aligned}$$

(41)

$$\begin{aligned}&F_C(\delta ) = \frac{4}{3}E\sqrt{\bar{r}}\left( \delta -\bar{\delta }\right) ^{3/2}\qquad \dot{\delta }\le 0. \end{aligned}$$

(42)