Effective thermo-magneto-electro-elastic properties of laminates with non-uniform imperfect contact: delamination and product properties

Abstract

Non-uniform contact on the interphases of composites can model a wide variety of situations, ranging from delamination patterns with “islands” of contact to variable surface capacitance due to chemical reactions with the environment. In this work, the non-uniformity is modeled via spring-type imperfect contact conditions whose proportionality constants are functions of the position over the surface of contact. The effective thermo-magneto-electro-elastic coefficients of such a composite are studied via asymptotic homogenization. A modification of the ansatz of Bakhvalov is introduced on the global elliptic problem and the contact conditions in order to separate the global and local scales. This modification allows us to obtain families of local problems that are analytically solvable systems of ordinary differential equations whose solutions depend parametrically on the position variables of the contact surfaces. By integrating the local problems, analytical formulae for the functionally graded-like effective coefficients are obtained. The multiple coupled fields and associated moduli are organized through a matrix notation that facilitates the application of the asymptotic formalism and the derivation of the analytical formulae. The formulae are valid for any finite number of layers and anisotropies. Finally, some numerical examples are provided for composites with crystal symmetry 6 mm. One is related to the effective stiffnesses with a rectangular zone of delamination on the interphase. The second example shows the influence of some ideal non-uniform imperfect contact cases over the product properties of a thermopiezoelectric/thermopiezomagnetic two-phase composite.

Appendix: Existence and uniqueness of periodic solutions

Let $\varvec{A}(y)$, $\varvec{F}_0(y)$ and $\varvec{F}_1(y)$ be 1-periodic matrix-valued functions of $y\in \mathbb {R}$ of sizes $s\times s$, $s\times r$ and $s\times r$, respectively. Suppose that the elements of $\varvec{A}(y)$, $\varvec{F}_0(y)$ and $\varvec{F}_1(y)$ are piecewise differentiable in $\left[ 0,1\right] {\setminus } \Gamma$ where

$$\begin{aligned} \Gamma = \left\{ \theta _1,\ldots \theta _p\right\} \subset \left( 0,1\right) , \end{aligned}$$

is the set of points of finite jump discontinuity of $\varvec{A}$, $\varvec{F}_0$ and $\varvec{F}_1$. Furthermore, we assume that $\varvec{A}$ is invertible for every $y\in \left[ 0,1\right] {\setminus } \Gamma$ and that $\left\langle \left( \varvec{A}(y)\right) ^{-1}\right\rangle$ is also invertible. We define the operator:

$$\begin{aligned} L = \frac{d}{dy}\left[ \varvec{A}\left( y\right) \frac{d}{dy}\right] . \end{aligned}$$

Let $\varvec{N}(y)$ be a $s\times r$ matrix-valued function, consider the following equation and jump conditions:

(17)

where $\varvec{K}^{(q)}$ and $\varvec{R}^{(q)}$ are constant matrices for each $q = 1,\ldots , p$ of sizes $s\times s$ and $s\times r$, respectively. We shall assume that $\varvec{K}^{(q)}$ is invertible for each q.

Lemma

A necessary and sufficient condition for the existence of 1-periodic solutions $\varvec{N}(y)$ of (17) is that:

$$\begin{aligned} \left\langle \varvec{F}_0\right\rangle _y = 0. \end{aligned}$$

Furthermore, the periodic solution is unique up to an additive constant.

Proof

Let us assume that a periodic solution N(y) exists and take the average of Eq. (17):

$$\begin{aligned} \left\langle \varvec{F}_0\right\rangle _y = \left\langle L\varvec{N}- \frac{d\varvec{F}_1}{dy}\right\rangle _y = \sum _{q = 0}^{p}\int _{\theta _q}^{\theta _{q+1}} \left\{ \frac{d}{d\xi }\left[ \varvec{A}(\xi )\frac{d\varvec{N}}{d\xi }(\xi )\right] - \frac{d\varvec{F}_1}{d\xi }(\xi )\right\} d\xi , \end{aligned}$$

where we have designated $\theta _0 = 0$ and $\theta _{p+1} = 1$, even though the functions $\varvec{A}$, $\varvec{N}$, $\varvec{F}_0$ and $\varvec{F}_1$ are differentiable in those points. Then, by applying the fundamental theorem of calculus in each one of the integrals above:

From here, the condition (17)$_2$ and the periodicity of the functions $\varvec{A}$, $\varvec{N}$ and $\varvec{F}_1$, we get that $\left\langle \varvec{F}_0\right\rangle _y = 0$. This proves the necessity of the condition. For proving sufficiency, we are going to simply solve Eq. (17)$_1$ and impose the conditions on the solution. By integrating the equation:

$$\begin{aligned} \varvec{A}(y)\frac{d\varvec{N}}{dy}(y) -\varvec{F}_1(y) = \int _0^y \varvec{F}_0(\xi )d\xi + \varvec{C}_1,\quad y\in \left[ 0,1\right] {\setminus } \Gamma , \end{aligned}$$

(18)

where $\varvec{C}_1$ is a constant matrix of integration. We use now that $\varvec{A}$ is invertible for each y to clear the derivative:

$$\begin{aligned} \frac{d\varvec{N}}{dy}(y) = \left( \varvec{A}(y)\right) ^{-1}\left[ \int _0^y \varvec{F}_0(\xi )d\xi + \varvec{C}_1 + \varvec{F}_1(y)\right] . \end{aligned}$$

We now integrate between zero and $y\in \left[ 0,1\right) {\setminus }\Gamma$:

$$\begin{aligned} \int _0^y\frac{d\varvec{N}}{d\xi }(\xi )d\xi = \sum _{q = 1}^{r(y)}\int _{\theta _{q-1}}^{\theta _{q}}\frac{d\varvec{N}}{d\xi }(\xi )d\xi +\int _{\theta _{r(y)}}^{y}\frac{d\varvec{N}}{d\xi }(\xi )d\xi , \end{aligned}$$

where r(y) is the number of elements in the set $\left[ 0,y\right] \cap \Gamma$ and the sum from $q = 1$ to $q = 0$ should be taken as equal to zero. We write the above expression in terms of the jumps:

$$\begin{aligned} \int _0^y\frac{d\varvec{N}}{d\xi }(\xi )d\xi = \varvec{N}(y) - \varvec{N}(0) - \sum _{q = 1}^{r(y)} \llbracket \varvec{N}(y)\rrbracket \bigg |_{y = \theta _q}. \end{aligned}$$

Then, we introduce the conditions (17)$_3$ in the expression above:

$$\begin{aligned} \int _0^y\frac{d\varvec{N}}{d\xi }(\xi )d\xi = \varvec{N}(y) - \varvec{N}(0) - \sum _{q = 1}^{r(y)} \left( \varvec{K}^{(q)}\right) ^{-1}\varvec{R}^{(q)}. \end{aligned}$$

This and (18) imply that for all $y\in \left[ 0,1\right]$

$$\begin{aligned} \varvec{N}(y) = \int _0^y\left( \varvec{A}(\rho )\right) ^{-1}\left[ \int _0^\rho \varvec{F}_0(\xi )d\xi + \varvec{C}_1 + \varvec{F}_1(\rho )\right] d\rho + \sum _{q = 1}^{r(y)} \left( \varvec{K}^{(q)}\right) ^{-1}\varvec{R}^{(q)} + \varvec{N}(0). \end{aligned}$$

(19)

Let us find the constant $\varvec{C}_1$ that makes $\varvec{N}(y)$ a periodic solution:

$$\begin{aligned} \begin{aligned} \varvec{N}(y+1) - \varvec{N}(y)&= \int _y^{y+1}\left( \varvec{A}(\rho )\right) ^{-1}\left[ \int _0^\rho \varvec{F}_0(\xi )d\xi + \varvec{C}_1 + \varvec{F}_1(\rho )\right] d\rho + \sum _{q = 1}^{p} \left( \varvec{K}^{(q)}\right) ^{-1}\varvec{R}^{(q)}. \end{aligned} \end{aligned}$$

(20)

The function:

$$\begin{aligned} G(y) = \left( \varvec{A}(y)\right) ^{-1}\left[ \int _0^y \varvec{F}_0(\xi )d\xi + \varvec{C}_1 + \varvec{F}_1(y)\right] \end{aligned}$$

is periodic due to $\varvec{A}(y)$, $\varvec{F}_0(y)$ and $\varvec{F}_1(y)$ being periodic and due to the condition $\left\langle \varvec{F}_0\right\rangle _y = 0$. This makes it possible to translate the integration limits of (20):

$$\begin{aligned} \begin{aligned} \varvec{N}(y+1) - \varvec{N}(y)&= \int _0^{1}\left( \varvec{A}(\rho )\right) ^{-1}\left[ \int _0^\rho \varvec{F}_0(\xi )d\xi + \varvec{C}_1 + \varvec{F}_1(\rho )\right] d\rho + \sum _{q = 1}^{p} \left( \varvec{K}^{(q)}\right) ^{-1}\varvec{R}^{(q)}. \end{aligned} \end{aligned}$$

From here, we can determine the constant $\varvec{C}_1$ so that $\varvec{N}(y+1) - \varvec{N}(y) = 0$:

$$\begin{aligned} \varvec{C}_1 = -\left\langle \left( \varvec{A}(y)\right) ^{-1}\right\rangle _y^{-1}\left[ \left\langle \left( A(y)\right) ^{-1}\left( \int ^{y}_0\varvec{F}_0\left( \xi \right) d\xi + \varvec{F}_1(y)\right) \right\rangle _y + \sum _{q = 1}^{p} \left( \varvec{K}^{(q)}\right) ^{-1}\varvec{R}^{(q)}\right] . \end{aligned}$$

(21)

The function N(y) of (19) with the constant $\varvec{C}_1$ defined by (21) satisfies the equation and contact conditions (17). For uniqueness, it is enough to impose, for instance, the null average condition to the solution. $\square$