Architecture-Driven Digital Volume Correlation: Application to the Analysis of In-Situ Crushing of a Polyurethane Foam

Abstract

Background

Digital Volume correlation (DVC) consists in identifying the displacement fields that allow for the best possible registration of volume images of a sample captured at various loading stages. With cellular materials, the use of DVC faces an intrinsic limit: in the absence of an exploitable texture on (or in) the struts or cell walls, the available speckle pattern will unavoidably be formed by the material architecture itself. This leads to the inability of classical DVC techniques to measure kinematics below the cellular scale, i.e. at the sub-cellular or micro scales.

Objectives

Here, we extend a newly developed architecture-driven DIC technique [1] for the measurement of 3D displacement fields in real cellular materials at the scale of the architecture.

Methods

The proposed solution consists in assisting DVC by a weak elastic regularization using, as support, an automatic finite-element image-based mechanical model.

Results

Complex (locally buckling) kinematics of a polyurethane foam under compression are accurately measured during an in-situ test. The method is essential to evidence the class of dominance (stretching versus bending) of the foam.

Conclusion

The proposed method allows to confirm that the foam used is bending-dominated, which is not possible with a classical mesoscopic DVC approach. This method is a good candidate for the analysis of complex local deformation mechanisms at the architecture scale.

Appendix: FE interrogation for arbitrary points

We present in this appendix a procedure that allows to perform automatically the displacement exchange between two arbitrary finite-element meshes. We consider the following steps for solving this geometric problem:

Step 1: Location of the Nearest Face of the FE Mesh to the Point

Efficient data structures (very common in collision detection algorithms) can be efficiently used to speed up point queries with respect to complex geometric objects represented by faces. In this work, we mainly use a rootine from the CGAL library [69] and PyMesh [76]. Efficient point queries such as intersections, distance computation, ray shooting can be performed using Axis Aligned Bounding Boxes (AABB) trees [77]. This allows to detect the nearest face to an arbitrary point.

Step 2: Location of the Tetrahedral Element Containing the Point

After determining the nearest face, the location test is performed on the tetrahedrons that share this same face (they are at most two). To do so, one can consider two methods:

• Method 1: Computation of the barycentric coordinates by resolution of the linear system:

  $$\begin{aligned} \left\{ \begin{array}{l} x = \displaystyle \sum _{i=1}^m \lambda _i t_i \\ \displaystyle \sum _{i=1}^m\lambda _i = 1 \end{array} \right. \end{aligned}$$

  (12)

  If $\lambda _i>0, \quad \forall i\in \left\{ 1,...,m\right\}$ then the point x belongs to the tetrahedron bounded by the nodes $(t_i)_{i\in \left\{ 1,...,m\right\} }$. m is the number of nodes per convex set (4 in our case).

• Method 2: A faster method which does not consist in solving a linear system can be considered. We only evaluate the signed distance of the point to each of the tetrahedron faces. First, the orientation of the faces must be determined so that all the face normal vectors point towards the same direction. This is given by an orientation matrix denoted $\textbf{O}\in \mathbb {R}^{3\times 4}$ that depends of the used mesh. Each column of $\textbf{O}$ represents the indices of the nodes of the tetrahedron faces. The point x belongs to the tetrahedron if the distances to all the faces have the same sign. This is written as follows:

  $$\begin{aligned} (x- t_{\textbf{O}_{2,i}})^T n_i(t_{\textbf{O}_{2,i}}) < 0, \quad \forall i \in \left\{ 1,..,4 \right\} \end{aligned}$$

  (13)

  where $n_i(t_{\textbf{O}_{2,i}})$ is the normal vector at node $t_{\textbf{O}_{2,i}}$ (which is the second node of the face i). It is defined by $n_i(t_2)=(t_{\textbf{O}_{2,i}}-t_{\textbf{O}_{1,i}}) \times (t_{\textbf{O}_{3,i}}-t_{\textbf{O}_{2,i}})$.

Step 3: Evaluation of the Displacement Field at the Point

Once the tetrahedron containing the point is determined, the isoparametric transformation $(x=\sum _{i}N_i(\xi )t_i)$ is inverted in order to find the isoparametric coordinate $\xi$ of the point x. The finite element interpolation formula can afterwards be applied to evaluate the desired displacement field $(u^{fine}(x)=\sum _{i}N_i(\xi )u^{coarse}(t_i))$, where $t_i$ are again the nodes of the tetrahedron.