A.G. Neeman; R.M. Brannon; B. Jeremic; A. Van Gelderand; A. Pang
Top view (Z from above) of eigentensors for Drucker-Prager material, time step 124, colored by minimum stretch eigenvalue.
Visualization of fourth-order tensors from solid mechanics has not been explored in depth previously. Challenges include the large number of components (3x3x3x3 for 3D), loss of major symmetry and loss of positive definiteness(with possibly zero or negative eigenvalues). This paper presents a decomposition of fourth-order tensors that facilitates their visualization and understanding. Fourth-order tensors are used to represent a solid’s stiffness.The stiffness tensor represents the relationship between increments of stress and increments of strain. Visualizing stiffness is important to understand the changing state of solids during plastification and failure. In this work,we present a method to reduce the number of stiffness components to second-order 3×3 tensors for visualization.The reduction is based on polar decomposition, followed by eigen-decomposition on the polar “stretch”. If any resulting eigenvalue is significantly lower than the others, the material has softened in that eigen-direction. The associated second-order eigentensor represents the mode of stress (such as compression, tension, shear, or some combination of these) to which the material becomes vulnerable. Thus we can visualize the physical meaning of plastification with techniques for visualizing second-order symmetric tensors.
Dynamic indentation of SiC-N ceramic by a tungsten carbide sphere. Left: experimentally observed impact crater and radial cracking (both highlighted for clarity). Middle: BFS model prediction of externally visible damage. Right: prediction of internal damage (suitable for validation against XCT data).
With the relatively recent introduction of quantitative and volumetric X-ray computedtomography (XCT) applied to ballistic impact damage diagnostics, signiﬁcant inroads have beenmade in expanding our knowledge base of the morphological variants of physical impactdamage. Yet, the current state of the art in computational and simulation modeling of terminalballistic performance remains predominantly focused on the penetration phenomenon, withoutdetailed consideration of the physical characteristics of actual impact damage. Similarly, armorceramic material improvements appear more focused on penetration resistance than on improved intrinsic damage tolerance and damage resistance. Basically, these approaches minimizeour understanding of the potential inﬂuence that impact damage may play in the mitigation orprevention of ballistic penetration. Examples of current capabilities of XCT characterization,quantiﬁcation, and visualization of complex impact damage variants are demonstrated anddiscussed for impacted ceramic and metallic terminal ballistic target materials. Potential beneﬁtsof incorporating such impact damage diagnostics in future ballistic computational modeling arealso brieﬂy discussed.