Continuum Approximations

Abstract

The continuum approximation is a mathematical idealization for modeling the collective response of discrete systems. While seemingly inapplicable to nanoscale structures, the use of continuum mechanics at the nanoscale is still a useful approximation with careful consideration of the assumptions inherent in the theory and with the inclusion of scale-dependent physical phenomena such as surface effects, micromorphic and strain-gradient effects, as well as nonlocal phenomena. Continuum mechanics may be applied to both discrete and heterogeneous media through the use of homogenization theory, which provides a mathematically elegant and rigorous framework for replacing a discrete collection of interacting entities by an equivalent homogenous continuum. The interaction energy and forces existing within a system are upscaled to an effective constitutive model and a set of partial-differential equations. The resulting boundary-value problem can then be efficiently solved using a number of numerical techniques, the most popular of which is arguably the finite-element method. Given the continuum approximation of a system, homogenization theory further provides a method for recovering the solution of the original discrete or heterogeneous system. In this chapter, we briefly review continuum mechanics, homogenization theory and computational homogenization, and constitutive modeling including crystal-plasticity.

Appendix

Example: Error in the Continuum Approximation

Discrete systems have an inherent length scale governed fundamentally by the interaction distance between entities. The accuracy of the continuum approximation depends critically on the size of the structure compared to the size of this intrinsic length scale. To illustrate the continuum approximation of a discrete system and its accuracy, consider a simple one-dimensional chain of N atoms of total initial length L with initial atomic spacing l subjected to a body force F per atom as shown in Fig. 3.12a. The initial position X _I of each atom is given by X _I  = l ⋅ I, I = 0, 1, 2, 3, …, N with X _N  = L. For simplicity, we take the interatomic potential to be harmonic with spring constant K, pair-wise additive, with only nearest-neighbor interactions.

Fig. 3.12

(a ) One-dimensional chain of atoms of total initial length L with initial atomic spacing l subjected to a body force F per atom. The initial position X _I of each atom is given by X _I  = l ⋅ I, I = 0, 1, 2, 3, …, N with X _N  = L. The interatomic forces are governed by local interactions and a spring constant K. (b ) One-dimensional chain of atoms with nonlocal interatomic forces. The nearest-neighbor spring constant is K ₁, and next-nearest spring constant is K ₂

The forces between the atoms are linear with respect to the relative displacements so that

$$\displaystyle{ K(u_{N} - u_{N-1}) = F }$$

(3.71a)

$$\displaystyle{ K(u_{N-1} - u_{N-2}) = 2F }$$

(3.71b)

$$\displaystyle{\vdots}$$

$$\displaystyle{ K(u_{I} - u_{I-1}) = F(N - I + 1) }$$

(3.71c)

$$\displaystyle{\vdots}$$

$$\displaystyle{ K(u_{1} - u_{0}) = FN }$$

(3.71d)

where u _I is the displacement of atom I with u ₀ = 0. Since this set of equations telescopes starting with $u_{1} = FN/K$, we have

$$\displaystyle{ u_{I} = \frac{F} {K}\left [I \cdot N -\sum _{J=1}^{I}(J - 1)\right ] = \frac{F} {K}\left [I \cdot N -\frac{1} {2}I(I - 1)\right ]\,. }$$

(3.72)

Since $N = L/l$ and $I = X_{I}/l$, we can write Eq. (3.72) as

$$\displaystyle{ u_{I} = \frac{(F/l)L^{2}} {(Kl)} \left [\frac{X_{I}} {L} \left (1 + \frac{l} {2L}\right ) -\frac{1} {2}\left (\frac{X_{I}} {L} \right )^{2}\right ]\,. }$$

(3.73)

Note that the natural length-scale ratio for this problem is l∕L.

To obtain the continuum version of this problem, we first use Eq. (3.13) reduced to the one-dimensional form, along with a linear-elastic constitutive model, $\sigma _{xx} = Edu/dX$. This results in the following equation for the continuum displacement field in a one-dimensional bar,

$$\displaystyle{ \frac{d} {dX}\left (k \frac{du} {dX}\right ) + f(X) = 0\,, }$$

(3.74)

where f is the body force per unit length along the bar, and k is the cross-sectional stiffness. The solution to this equilibrium equation with the boundary condition u(0) = 0 is given by

$$\displaystyle{ u(X) = \frac{fL^{2}} {k} \left [\frac{X} {L} -\frac{1} {2}\left (\frac{X} {L} \right )^{2}\right ]\,. }$$

(3.75)

Note that there is no intrinsic length scale in this continuum solution. If we identify k = K ⋅ l and F = f ⋅ l, then Eq. (3.73) converges to Eq. (3.75) in the limit as l∕L → 0. The absolute error in the continuum approximation is given by,

$$\displaystyle{ e_{I}\doteq \vert u(X_{I}) - u_{I}\vert = \frac{fL^{2}} {k} \left (\frac{X_{i}} {L} \right )\left ( \frac{l} {2L}\right ) }$$

(3.76)

This error is proportional to the length-scale ratio l∕L. Also, the error varies linearly along the chain, from e ₀ = 0 at X ₀ = 0 to a maximum value at X _N  = 0. Thus, in the limit of infinitesimally small intrinsic length scale, the discrete solution converges to the continuum solution.

The normalized displacement solution for the discrete atom chain, given by Eq. (3.73) with $\bar{u}\doteq u/(FL^{2}/Kl^{2})$, is shown in Fig. 3.13a as a function of normalized initial position, $\bar{X}\doteq X/L$, for several values of the length-scale ratio l∕L. The continuum approximation for the displacement field, given by Eq. (3.75), is also shown. The error in the continuum approximation, given by Eq. (3.76) with $\bar{e}\doteq \vert \bar{u} -\bar{ u}_{I}\vert$, is shown in Fig. 3.13b, and is seen to approach zero as l∕L → 0. Only for l∕L < 0. 02 (N = 50) is the maximum error less than 2 % of the peak displacement.

Fig. 3.13

(a ) Normalized displacement $\bar{u}$ as a function of normalized initial position $\bar{X}$ of a one-dimensional chain of atoms subjected to a uniform body force and fixed at the end $\bar{X} = 0$ for various values of the ratio of atom spacing to total initial chain length, l∕L. The continuum approximation is also shown. (b ) Normalized error in the continuum approximation. Only for l∕L < 0. 02 (N = 50) is the maximum error less than 2 % of the peak displacement

Example: Absence of a Surface Effect in Classical Continuum Mechanics

Discrete systems can also display surface effects that are not present in classical continuum theories. To illustrate the continuum approximation of a discrete system and its accuracy, consider a simple one-dimensional chain of N atoms of total initial length L with initial atomic spacing l subjected to a body force F per atom as shown in Fig. 3.12b. The initial position X _I of each atom is given by X _I  = l ⋅ I, I = 0, 1, 2, 3, …, N with X _N  = L. We take the interatomic potential to be harmonic, pair-wise additive, with both nearest-neighbor interactions with spring constant K ₁, and nonlocal interactions with spring constant K ₂. Note that the atoms at the end of the chain experience a distinctly different force environment than those atoms in the interior of the chain due to the number of interacting neighbors.

The forces between the atoms are linear with respect to the relative displacements so that

$$\displaystyle{ K_{1}(u_{N} - u_{N-1}) + K_{2}(u_{N} - u_{N-2}) = F }$$

(3.77a)

$$\displaystyle{ -K_{1}(u_{N} - u_{N-1}) + K_{1}(u_{N-1} - u_{N-2}) + K_{2}(u_{N-1} - u_{N-3}) = F }$$

(3.77b)

$$\displaystyle\begin{array}{rcl} -K_{2}(u_{N} - u_{N-2}) - K_{1}(u_{N-1} - u_{N-2}) + K_{1}(u_{N-2} - u_{N-3})& & \\ + K_{2}(u_{N-2} - u_{N-4})& =& F{}\end{array}$$

(3.77c)

$$\displaystyle{\vdots}$$

$$\displaystyle{ -K_{2}(u_{I+2} - u_{I}) - K_{1}(u_{I+1} - u_{I}) + K_{1}(u_{I} - u_{I-1}) + K_{2}(u_{I} - u_{I-2}) = F }$$

(3.77d)

$$\displaystyle{\vdots}$$

$$\displaystyle{ -K_{2}(u_{3} - u_{1}) - K_{1}(u_{2} - u_{1}) + K_{1}(u_{1} - u_{0}) = F }$$

(3.77e)

where u _I is the displacement of atom I with u ₀ = 0. This system of equations results in a matrix equation Ku = F where K is N × N banded matrix of the Toeplitz type and can be solved using standard methods.

The normalized displacement solution for the discrete atom chain with $\bar{u}\doteq u/(FL^{2}/K_{1}l^{2})$ is shown in Fig. 3.14a as a function of normalized initial position, $\bar{X}\doteq X/L$, for several values of the length-scale ratio l∕L. For this example, we have chosen K ₂ = 0. 5K ₁. In order to use the continuum approximation given by Eq. (3.75), we must first define an effective spring stiffness, K _eff. To this end, we isolate a unit cell of length 2l surrounding one interior atom. Within each cell, there are two K ₁ springs acting in series thus contributing a value of $\frac{1} {2}K_{1}$ to K _eff. There is a full K ₂ spring acting in parallel thus contributing a value of K ₂ to K _eff. There are also two K ₂ springs that effectively act in parallel to the unit cell, thus contributing a value of $\frac{1} {2}K_{2} + \frac{1} {2}K_{2}$ to K _eff. Thus, $K_{\mathrm{eff}} = \frac{1} {2}K_{1} + 2K_{2}$, and $k\doteq K_{\mathrm{eff}} \cdot (2l)$. Also, the effective force per unit length is $f = 2F/2l = F/l$. The continuum approximation for the displacement field is also shown in Fig. 3.14a. There is some noticeable surface effect on the atoms near the ends of the chain, particularly near $\bar{X} = 0$. This effect is more noticeable if we plot the error in the continuum approximation, $\bar{e}\doteq \vert \bar{u} -\bar{ u}_{I}\vert$, as shown in Fig. 3.14b. Notice that the surface effect near $\bar{X} = 0$ affects several atoms. The surface effect is absent in the chosen continuum approximation.

Fig. 3.14

Illustration of the surface effect for a one-dimensional discrete chain of atoms subjected to a body force F applied to each atom. The atoms are connected to their nearest neighbors by linear springs with spring constant K ₁, and to their next-nearest neighbors by linear springs with spring constant K ₂. The chain is fixed at X = 0. The total initial length of the chain is L, and the atomic spacing is l so that the number of atoms $N = L/l$. (a ) Normalized displacement, $\bar{u}\doteq u/(FL^{2}/K_{1}l^{2})$, plotted as a function of the normalized initial position, $\bar{X}\doteq X/L$. Results are shown for $l/L = 0.1$, 0. 05, 0. 02 (N = 10, 20, 50, respectively) with K ₂ = 0. 5K ₁. The continuum approximation, l∕L → 0, is also shown with $\bar{u}\doteq u/(fL^{2}/k)$ where $f\doteq F/l$, and $k\doteq K_{1}\,l$. (b ) Error in the continuum approximation with $\bar{e}\doteq \vert \bar{u} -\bar{ u}_{i}\vert$