Tag Archives: shocks

PUBLICATION: Influence of nonclassical plasticity features on shock wave existence and spectral solutions

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A scanned copy is available here.

ABSTRACT (from OCR so has some typos): The influence of non-classical elastic-plastic constitutive features on dynamically moving discontinuities in stress, strain, and material velocity is investigated. Non-classical behavior here includes non-normality of the plastic strain increment to the yield surface, plastic compressibility, pressure sensitivity of yield, and dependence of the elastic moduli on plastic strain. DRUGAN and SHEN’s (1987) analysis of dynamically moving discontinuities with strain as well as stress jumps in classical materials is shown to be valid for a broad class of non-associative material models until deviation from normality exceeds a critical (non-infinitesimal) level. For these non-classical materials, an inequality that bounds the magnitude of the stress jump is derived, which is information not obtainable from a standard spectral analysis of a shock. For the special case of stress discontinuities with continuous strain, or for quasi-static deformations, this inequality is shown to rule out jumps in specific projections of the stress tensor unless the non-normality is sufficiently large. These results invalidate a … claim in the literature that an infinitesimal amount of non-normality permits moving surfaces of discontinuity in stress (with no strain jump) near the tip of a dynamically advancing crack tip.

Using a very general plastic constitutive law that subsumes most non-classical (and classical) descriptions currently in use, a complete closed-form solution is obtained for the plastic wave speeds and eigenvectors. A novel feature of the analysis is the clarity and completeness of the solutions. If the elastic part of the response is isotropic, one plastic wave speed equals the elastic shear wave speed, while the other two possible wave speeds depend in general on the stress and plastic strain within the shock transition layer. Concise necessary and sufficient conditions for real eigenvalues and for vanishing eigenvalues are derived. The real eigenvalues are classified by numerical sign and ordering relative to the elastic eigenvalues. The geometric multiplicity of plastic eigenvectors associated with elastic eigenvalues is shown to depend on the stress state within the shock transition layer. These solutions, several of which hold for arbitrary elastic anisotropy, are also applicable to acceleration waves and localization problems, and to materials with dependence of the elastic moduli on plastic strain. Such elastic–plastic coupling is shown to imply a non-self-adjoint fourth order tangent stiffness tensor even if the plastic constitutive law is associative.

A scanned copy is available here.

Publication: Elements of Phenomenological Plasticity: Geometrical Insight, Computational Algorithms, and Topics in Shock Physics

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This 2007 Book Chapter on the basics of plasticity theory reviews the terminology and governing equations of plasticity, with emphasis on amending misconceptions, providing physical insights, and outlining computational algorithms. Plasticity theory is part of a larger class of material models in which a pronounced change in material response occurs when the stress (or strain) reaches a critical threshold level. If the stress state is subcritical, then the material is modeled by classical elasticity. The bound- ary of the subcritical (elastic) stress states is called the yield surface. Plasticity equations apply if continuing to apply elasticity theory would predict stress states that extend beyond this the yield surface. The onset of plasticity is typically characterized by a pronounced slope change in a stress–strain dia-gram, but load reversals in experiments are necessary to verify that the slope change is not merely nonlinear elasticity or reversible phase transformation.
The threshold yield surface can appear to be significantly affected by the loading rate, which has a dominant effect in shock physics applications.

In addition to providing a much-needed tutorial survey of the governing equations and their solution (defining Lode angle and other Lode invariants and addressing the surprisingly persistent myth that closest-point return satisfies the governing equations), this book chapter includes some distinctive contributions such as a simple 2d analog of plasticity that exhibits the same basic features of plasticity (such as existence of a “yield” surface with associative flow and vertex theory), an extended discussion of apparent nonassociativity, stability and uniqueness concerns about nonassociativity, and a  summary of apparent plastic wave speeds in relation to elastic wave speeds (especially noting that non-associativity admits plastic waves that travel faster than elastic waves).

For the full manuscript with errata, click 2007 Book Chapter on the basics of plasticity theory.