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The Problem
The Problem
Large deformations of solids in close contact with fluids are ubiquitous in engineering applications. Examples include manufacturing processes such as impact welding, and ballistic applications such as impact, perforation and penetration of armour by a bullet. Several challenges arise when one tries to simulate such phenomena.
- Due to large deformations, the material constitutive behaviour (i.e. the relation between stresses and strains) is nonlinear.
- In solids, deformations themselves need to be tracked, in contrast to fluids, where it is sufficient to track the deformation gradients (e.g., the velocity field).
- Given the typical energy levels, shock waves can be formed and can propagate within these solid-fluid systems. These shocks can interact with the material interfaces and can lead to severe distortions.
- Highly distorting interfaces and large property jumps across interfaces pose numerical challenges (spurious oscillations arise in the fields).
Thus, the challenge is to enable robust and accurate simulations of these phenomena.
Approach and Sample Results
Approach and Sample Results
A high-order accurate numerical framework has been developed that tries to overcome the challenges mentioned above. The main ingredients of the method are as follows.
- Constitutive relations based on the thermodynamically consistent theory of hyperelasticity, along with modifications to handle different plastic behaviours.
- Fully Eulerian framework with the inverse deformation gradient tensor as a kinematic (tracking) variable.
- Localized artificial diffusivity (LAD) scheme for capturing shocks and material interfaces.
With this framework, high-order formal accuracy and high-order preservation of curl constraint have been demonstrated. Some sample applications are shown below.
- High-Order Preservation of Curl Constraint
Every measure of strain satisfies certain consistency conditions. In a numerical simulation, these consistency conditions can be violated unless special care is taken. In our case, every row of the inverse deformation gradient tensor must be a gradient. Since the curl of a gradient is identically zero, the curl of the inverse deformation gradient tensor must be a zero tensor. This is often termed as the curl constraint. The ability of our numerical method to satisfy this is tested on a 2D problem (left and center panels). It is seen that the curl constraint is preserved much more accurately (right panel), at eighth-order, as compared to previous methods which did so at first-order.
- Richtmeyer-Meshkov Instability Involving Solids
An initially perturbed interface between Copper and Aluminum (fused together) is subjected to a shock with a pressure ratio of approximately 25. The initial perturbations are amplified due to repeated interactions between shocks that bounce from the rigid wall on the right and the interface. This problem demonstrates the robustness of the method (largely due to the fully Eulerian treatment) in handling severe distortions of the interface.
- Taylor Rod-on-Rod Impact
The Taylor rod-on-rod impact experiment involves two identical cylindrical rods striking each other with high velocities. The final base radius, height, and the shape of the bars are determined and are often used to determine material constitutive relations. Simulations of two impacting Copper rods were carried out and the results are in close agreement with experimental results, and serve to validate our simulation framework.
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