An Investigation of the Enhanced Fatigue Performance of Low-porosity Auxetic Metamaterials

Abstract

An experimental and numerical investigation of fatigue life and crack propagation in two-dimensional perforated aluminum structures is presented. Specifically, the performance of positive Poisson’s ratio (PPR) geometries using circular holes is compared to that of auxetic stop-hole and straight-groove hole geometries. Mechanical fatigue testing shows that the considered auxetic structures have more than 20% longer life than the porous PPR structure at the same porosity and peak effective maximum stress despite having holes with larger stress concentrations. Digital image correlation is used to detect crack initiation and damage propagation much earlier than can be detected by the unaided eye. Accompanying finite element analyses reveal that auxetic structures have the advantage over their PPR counterparts by delaying crack initiation, spreading damage over a larger area, and having a stress intensity factor that decreases over a significant range of crack lengths. In addition, numerical simulations suggest that auxetic structures maintain their negative Poisson’s ratios in the presence of cracks.

Appendix A: Approximation of Poisson’s ratio from FE Simulations

We present a brief summary of the approaches used to calculate the effective Poisson’s ratio in both the finite-sized and periodic RVE model based on the authors’ previous work [3].

A.1 Finite-size Model

In order to calculate the effective PR on the finite-size test sample geometries, we extracted the horizontal and the vertical displacement components from the boundary nodes of the porous 40 by 40 mm gauge area of the samples. These displacement are used to compute average displacements at the boundaries 〈u_x〉^L, 〈u_x〉^R, 〈u_y〉^T, and 〈u_y〉^B, where the superscripts L, R, T, and B denote the left, top, right, and bottom boundaries, respectively. From these average displacements, we compute local strain averages

$$ \langle \epsilon_{xx} \rangle = \frac{\langle u_{x} \rangle^{R}-\langle u_{x} \rangle^{L}}{l}, \qquad \langle \epsilon_{yy} \rangle = \frac{\langle u_{y} \rangle^{T}-\langle u_{y} \rangle^{B}}{l} $$

(8)

where l denotes the distance between the top/bottom and left/right boundaries (i.e., l = 40 mm here). The effective Poisson’s ratio ν_eff is given by

$$ \nu_{eff} = -\frac{\langle \epsilon_{xx} \rangle }{\langle \epsilon_{yy} \rangle } $$

(9)

A.2 Infinite Periodic Model

Consider a 2D periodic structure composed of an array of RVE’s. The displacement vector at any point (x, y) in the structure can be expressed as [50, 51]

$$ \mathbf{u}(x,y) = \bar{\mathbf{\varepsilon}} \mathbf{x} + \mathbf{u}^{*}(x,y) $$

(10)

where, in general, $\bar {\mathbf {\varepsilon }}$ is the constant average displacement gradient, x is the position vector and u^∗(x, y) is a periodic vector representing local deviation away from the average due to inhomogeneities (e.g., voids). We take $\bar {\mathbf {\varepsilon }}$ to be symmetric so that it is equivalent to the average infinitesimal strain. Continuity of the periodic structure requires that both the top/bottom boundary pairs and left/right boundary pairs deform identically. Thus, for the top and bottom boundaries, we can write

$$ \mathbf{u}^{T} = \bar{\mathbf{\varepsilon}} \mathbf{x}^{T} + \mathbf{u}^{*} $$

(11)

$$ \mathbf{u}^{B} = \bar{\mathbf{\varepsilon}} \mathbf{x}^{B} + \mathbf{u}^{*}, $$

(12)

where u^∗ in equation (11) and (12) are the same. Subtracting these two equations gives us the continuity constraint

$$ \mathbf{u}^{T}-\mathbf{u}^{B} = \bar{\mathbf{\varepsilon}} \left( \mathbf{x}^{T}-\mathbf{x}^{B}\right). $$

(13)

A similar process for the left and right boundaries yields

$$ \mathbf{u}^{R}-\mathbf{u}^{L} = \bar{\mathbf{\varepsilon}} \left( \mathbf{x}^{R}-\mathbf{x}^{L}\right). $$

(14)

To apply these conditions in Abaqus, we utilize two virtual nodesVp_x and Vp_y. These are points unconnected to the RVE and whose spatial placement is arbitrary. The numerical values of the displacements of the two virtual nodes are set to be equivalent to components of the average strain we want to impose. Thus we write (13) and (14) as constraint equations (i.e., *EQUATION) for each periodic node pair as

$$ {u_{x}^{R}} - {u_{x}^{L}} = u_{x}^{Vp_{x}}\left( x^{R}-x^{L}\right) $$

(15)

$$ {u_{y}^{R}} - {u_{y}^{L}} = u_{y}^{Vp_{x}}\left( x^{R}-x^{L}\right) $$

(16)

$$ {u_{x}^{T}} - {u_{x}^{B}} = u_{x}^{Vp_{y}}\left( y^{T}-y^{B}\right) $$

(17)

$$ {u_{y}^{T}} - {u_{y}^{B}} = u_{y}^{Vp_{y}}\left( y^{T}-y^{B}\right), $$

(18)

where we set $u_{x}^{Vp_{x}} = \bar {\varepsilon }_{xx}$, $u_{y}^{Vp_{x}} = \bar {\varepsilon }_{yx}$, $u_{x}^{Vp_{y}} = \bar {\varepsilon }_{xy}$, and $u_{y}^{Vp_{y}} = \bar {\varepsilon }_{yy}$. Since the displacements of our virtual points correspond to the average strain in the structure, we can use them to compute an average (or effective) Poisson’s ratio for the entire structure

$$ \nu_{eff} = -\frac{\bar{\varepsilon}_{xx}}{\bar{\varepsilon}_{yy}} = -\frac{u_{x}^{Vp_{x}}}{u_{y}^{Vp_{y}}}, $$

(19)

where $u_{y}^{Vp_{y}}$ is a specified displacement (we use − 0.005 in this analysis) and $u_{x}^{Vp_{x}}$ is computed by Abaqus as part of the FE analysis.