Discrete and continuous models of linear elasticity: history and connections

Abstract

This paper tracks the development of lattice models that aim to describe linear elasticity of solids and the field equations of which converge asymptotically toward those of isotropic continua, thus showing the connection between discrete and continuum. In 1759, Lagrange used lattice strings/rod dynamics to show the link between the mixed differential-difference equation of a one-dimensional (1D) lattice and the partial differential equation of the associated continuum. A consistent three-dimensional (3D) generalization of this model was given much later: Poincaré and Voigt reconciled the molecular and the continuum approaches at the end of the nineteenth century, but only in 1912 Born and von Kármán presented the mixed differential-difference equations of discrete isotropic elasticity. Their model is a 3D generalization of Lagrange’s 1D lattice and considers longitudinal, diagonal and shear elastic springs among particles, so the associated continuum is characterized by three elastic constants. Born and von Kármán proved that the lattice equations converge to Navier’s partial differential ones asymptotically, thus being a formulation of continuous elasticity in terms of spatial finite differences, as for Lagrange’s 1D lattice. Neglecting shear springs in Born–Kármán’s lattice equals to Navier’s assumption of pure central forces among molecules: in the limit, the lattice behaves as a one-parameter isotropic solid (“rari-constant” theory: equal Lamé parameters, or, equivalently, Poisson’s ratio $\upsilon =1/4$). Hrennikoff and McHenry revisited the lattice approach with pure central interactions using a plane truss; the equivalent Born–Kármán’s lattice in plane stress in the limit tends to a continuum with Poisson’s ratio $\upsilon = 1/3$. Contrary to McHenry–Hrennikoff’s truss, Born–Kármán’s lattice leads to a “free” Poisson’s ratio bounded by its “limit’ bound ($\upsilon =1/4$ for plane strain or 3D elasticity; $\upsilon =1/3$ for plane stress elasticity). Unfortunately, Born–Kármán’s lattice model does not comply with rotational invariance principle, for non-central forces. The consistent generalization of Lagrange’s lattice in 3D was achieved only by Gazis et al. considering an elastic energy that depends on changes in both lengths and angles of the lattice. An alternative consistent three-parameter elastic lattice is the Hrennikoff’s, with additional structure in the cell. We also discuss the capability of nonlocal continuous models to bridge the gap between continuum isotropic elasticity at low frequencies and lattice anisotropic elasticity at high frequencies.

Appendix A: Equivalence of the lattice-based gradient elasticity model and Mindlin equation for Gazis et al. model

The continualized lattice-based gradient elasticity equations derived for Gazis et al. lattice [67] are given by Eq. (58) which is here reformulated:

$$\begin{aligned}&\alpha \left( {\frac{\partial ^{2}u}{\partial x^{2}}+\frac{a^{2}}{12}\frac{\partial ^{4}u}{\partial x^{4}}} \right) \nonumber \\&\quad +\frac{\beta \,}{2}\left( {4\frac{\partial ^{2}u}{\partial x^{2}}+2\frac{\partial ^{2}u}{\partial y^{2}}+a^{2}\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{a^{2}}{3}\frac{\partial ^{4}u}{\partial x^{4}}+\frac{a^{2}}{6}\frac{\partial ^{4}u}{\partial y^{4}}+2\frac{\partial ^{2}u}{\partial z^{2}}+a^{2}\frac{\partial ^{4}u}{\partial x^{2}\partial z^{2}}+\frac{a^{2}}{6}\frac{\partial ^{4}u}{\partial z^{4}}} \right) \nonumber \\&\quad +4\,\gamma \,\left( {\frac{\partial ^{2}u}{\partial y^{2}}+\frac{a^{2}}{12}\frac{\partial ^{4}u}{\partial y^{4}}+\frac{\partial ^{2}u}{\partial z^{2}}+\frac{a^{2}}{12}\frac{\partial ^{4}u}{\partial z^{4}}} \right) \nonumber \\&\quad +\left( {\gamma +\frac{\beta }{2}} \right) \left( {4\frac{\partial ^{2}v}{\partial x\partial y}+\frac{2a^{2}}{3}\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{2a^{2}}{3}\frac{\partial ^{4}v}{\partial x^{3}\partial y}+4\frac{\partial ^{2}w}{\partial x\partial z}+\frac{2a^{2}}{3}\frac{\partial ^{4}w}{\partial x\partial z^{3}}+\frac{2a^{2}}{3}\frac{\partial ^{4}w}{\partial x^{3}\partial z}} \right) =\rho a\,\ddot{{u}} \end{aligned}$$

(A.1)

Equation (A.1) can be reformulated in two terms with the zero-th order and the second-order strain gradient expression:

$$\begin{aligned}&\left( {\alpha +2\beta } \right) \frac{\partial ^{2}u}{\partial x^{2}}+\left( {2\beta +4\gamma } \right) \,\left( {\frac{\partial ^{2}v}{\partial x\partial y}+\frac{\partial ^{2}w}{\partial x\partial z}} \right) +\left( {\beta +4\gamma } \right) \,\left( {\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\partial ^{2}u}{\partial z^{2}}} \right) \nonumber \\&\quad +\frac{a^{2}}{12}\left[ {\left( {\alpha +2\beta } \right) \frac{\partial ^{4}u}{\partial x^{4}}+\left( {\beta +4\gamma } \right) \,\left( {\frac{\partial ^{4}u}{\partial y^{4}}+\frac{\partial ^{4}u}{\partial z^{4}}} \right) +6\beta \left( {\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{\partial ^{4}u}{\partial x^{2}\partial z^{2}}} \right) } \right] \nonumber \\&\quad +\frac{a^{2}}{3}\left( {\beta +2\gamma } \right) \left( {\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{\partial ^{4}v}{\partial x^{3}\partial y}+\frac{\partial ^{4}w}{\partial x\partial z^{3}}+\frac{\partial ^{4}w}{\partial x^{3}\partial z}} \right) =\rho a\,\ddot{{u}} \end{aligned}$$

(A.2)

Equation (A.2) can be compared to Navier’s partial differential equation Eq. (13) with cubic symmetry, for the zero-th order medium.

$$\begin{aligned} \left\{ {\begin{array}{l} \;\alpha =a\,\left( {c_{11} -2c_{12} } \right) \\ \;\beta =ac_{12} \\ \;\gamma =\frac{a}{4}\left( {c_{44} -c_{12} } \right) \\ \end{array}} \right. \end{aligned}$$

(A.3)

Injecting the stiffness calibration Eq. (A.3) into the second-order expansion Eq. (A.2) of Gazis et al. lattice equations gives the gradient elasticity cubic model:

$$\begin{aligned}&c_{11} \frac{\partial ^{2}u}{\partial x^{2}}+\left( {c_{12} +c_{44} } \right) \,\left( {\frac{\partial ^{2}v}{\partial x\partial y}+\frac{\partial ^{2}w}{\partial x\partial z}} \right) +c_{44} \,\left( {\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\partial ^{2}u}{\partial z^{2}}} \right) \nonumber \\&\quad +\frac{a^{2}}{12}\left[ {c_{11} \frac{\partial ^{4}u}{\partial x^{4}}+c_{44} \,\left( {\frac{\partial ^{4}u}{\partial y^{4}}+\frac{\partial ^{4}u}{\partial z^{4}}} \right) +6\,c_{12} \left( {\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{\partial ^{4}u}{\partial x^{2}\partial z^{2}}} \right) } \right] \nonumber \\&\quad +\frac{a^{2}}{6}\left( {c_{12} +c_{44} } \right) \left( {\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{\partial ^{4}v}{\partial x^{3}\partial y}+\frac{\partial ^{4}w}{\partial x\partial z^{3}}+\frac{\partial ^{4}w}{\partial x^{3}\partial z}} \right) =\rho \,\ddot{{u}} \end{aligned}$$

(A.4)

This partial differential lattice-based equation exactly coincides with Eq. (10) page 315, of Mindlin [98].

In case of asymptotic linear elastic isotropy, the micro elastic parameters are related to the macro elastic parameters for Gazis et al. lattice by:

$$\begin{aligned} \left\{ {\begin{array}{l} \;\alpha =a\,\left( {c_{11} -2c_{12} } \right) =\left( {2\mu -\lambda } \right) \,a \\ \;\beta =ac_{12} =\lambda a \\ \;\gamma =\frac{a}{4}\left( {c_{44} -c_{12} } \right) =\frac{\mu -\lambda }{4}a \\ \end{array}} \right. \quad \text {with } \quad \left\{ {\begin{array}{l} \;c_{11} =\lambda +2\mu \\ \;c_{44} =\mu \\ \;c_{12} =\lambda \\ \end{array}} \right. \end{aligned}$$

(A.5)

The gradient elasticity lattice-based equations are rewritten using Lamé parameters:

$$\begin{aligned}&\left( {\lambda +2\mu } \right) \frac{\partial ^{2}u}{\partial x^{2}}+\left( {\lambda +\mu } \right) \,\left( {\frac{\partial ^{2}v}{\partial x\partial y}+\frac{\partial ^{2}w}{\partial x\partial z}} \right) +\mu \,\left( {\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\partial ^{2}u}{\partial z^{2}}} \right) \nonumber \\&\quad +\frac{a^{2}}{12}\left[ {\left( {\lambda +2\mu } \right) \frac{\partial ^{4}u}{\partial x^{4}}+\mu \,\left( {\frac{\partial ^{4}u}{\partial y^{4}}+\frac{\partial ^{4}u}{\partial z^{4}}} \right) +6\lambda \left( {\frac{\partial ^{4}u}{\partial x^{2}\partial y^{2}}+\frac{\partial ^{4}u}{\partial x^{2}\partial z^{2}}} \right) } \right] \nonumber \\&\quad +\frac{a^{2}}{6}\left( {\lambda +\mu } \right) \left( {\frac{\partial ^{4}v}{\partial x\partial y^{3}}+\frac{\partial ^{4}v}{\partial x^{3}\partial y}+\frac{\partial ^{4}w}{\partial x\partial z^{3}}+\frac{\partial ^{4}w}{\partial x^{3}\partial z}} \right) =\rho \,\ddot{{u}} \end{aligned}$$

(A.6)

It is worth mentioning that the gradient elasticity lattice-based equation cannot be cast in the linear isotropic strain gradient form which depends on 7 parameters, two classical Lamé parameters and 5 additional non-classical material parameters of the strain gradient form [126,127,128]. In fact, the generalized wave equation of the linear isotropic strain gradient medium can be shown to depend on two macroscopic additional length scales that depend on the 5 additional parameters [126, 127, 129, 130]:

$$\begin{aligned} \left( {1-l_{1}^{2} \Delta } \right) \mu \Delta u+\left( {1-l_{2}^{2} \Delta } \right) \left( {\lambda +\mu } \right) \left( {\partial _{x}^{2} u+\partial _{x} \partial _{y} v+\partial _{x} \partial _{z} w} \right) =\rho \,\ddot{{u}} \quad \text {with} \quad \Delta =\partial _{x}^{2} +\partial _{y}^{2} +\partial _{z}^{2} \end{aligned}$$

(A.7)

where $l_{1}$ and $l_{2}$ are the two additional length scales that depend on the 5 additional length scales of the linear isotropic strain gradient medium. Equation (A.7) can be equivalently rewritten using Mindlin’s notations [126]:

$$\begin{aligned}{} & {} \left( {\lambda +2\mu } \right) \left( {1-\tilde{{l}}_{1}^{2} \Delta } \right) \left( {\partial _{x}^{2} u+\partial _{x} \partial _{y} v+\partial _{x} \partial _{z} w} \right) \nonumber \\{} & {} \quad -\mu \left( {1-\tilde{{l}}_{2}^{2} \Delta } \right) \left( {\partial _{x} \partial _{y} v+\partial _{x} \partial _{z} w-\partial _{y}^{2} u-\partial _{z}^{2} u} \right) =\rho \,\ddot{{u}} \quad \text {with} \nonumber \\{} & {} \quad \left\{ {\begin{array}{l} \;\tilde{{l}}_{1}^{2} =\frac{l_{1}^{2} \mu +l_{2}^{2} \left( {\lambda +\mu } \right) }{\lambda +2\mu } \\ \;\tilde{{l}}_{2}^{2} =l_{1}^{2} \\ \end{array}} \right. \end{aligned}$$

(A.8)

Equation (A.7) can be also rewritten, with the full expression of the differential operators:

$$\begin{aligned}{} & {} \left( {\lambda +2\mu } \right) \partial _{x}^{2} u+\mu \left( {\partial _{y}^{2} u+\partial _{z}^{2} u} \right) +\left( {\lambda +\mu } \right) \left( {\partial _{x} \partial _{y} v+\partial _{x} \partial _{z} w} \right) \nonumber \\{} & {} \quad -\left[ {l_{1}^{2} \mu +l_{2}^{2} \left( {\lambda +\mu } \right) } \right] \partial _{x}^{4} u-\left[ {2l_{1}^{2} \mu +l_{2}^{2} \left( {\lambda +\mu } \right) } \right] \left( {\partial _{x}^{2} \partial _{y}^{2} u+\partial _{x}^{2} \partial _{z}^{2} u} \right) \nonumber \\{} & {} \quad -l_{1}^{2} \mu \left( {\partial _{y}^{4} u+\partial _{z}^{4} u} \right) -2l_{1}^{2} \mu \partial _{y}^{2} \partial _{z}^{2} u \nonumber \\{} & {} \quad -l_{2}^{2} \left( {\lambda +\mu } \right) \left[ {\partial _{x} \partial _{y}^{3} v+\partial _{y} \partial _{x}^{3} v+\partial _{x} \partial _{z}^{3} w+\partial _{z} \partial _{x}^{3} w+\partial _{x} \partial _{y}^{2} \partial _{z} w+\partial _{x} \partial _{y} \partial _{z}^{2} v} \right] =\rho \,\ddot{{u}} \end{aligned}$$

(A.9)

It is not possible to express the lattice-based gradient elasticity model Eq. (A.6) with the isotropic gradient elasticity equation Eq. (A.9). The zero-th order continuous medium asymptotically derived from Gazis et al. lattice can be enforced to be linear isotropic, but the second-order gradient elasticity medium cannot be isotropic (cubic gradient elasticity medium, as already characterized by Mindlin, 1968 for strain gradient elasticity models of the crystal class m3m). This also explains the loss of isotropy in the high frequency regime, even if the cubic lattice may behave isotropically in the low frequency regime.