טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentBen-Shmuel Yaron
SubjectModeling Plasticity by Non-Continuous Deformation
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Emeritus Eli Altus
Full Thesis textFull thesis text - English Version


Abstract

Plasticity and failure theories are still subjects of intense research. Although much theoretical work has been done in the past half century to explain metal plasticity phenomena in the micro scale, no comprehensive theory has been achieved.

This study is based on heterogeneity and non-continuous deformations.  It includes three main parts, each serve as a stepping stone towards the main research objective: a macro continuum equation for solids which incorporates higher order correlation terms analogous to the Reynolds stress term in turbulent fluid mechanics.

The first part of the research examines whether simple models based on micro stochastic rules can exhibit the complex behavior expected on the macroscale. The classical linear 1D diffusion equation is derived based on micro probabilistic rules and analytical solution is compared to simulation results. Generalizing this concept, a non-linear diffusion equation is derived in which a shock wave with discontinuous density is obtained.

A 2D cellular automata model of a perfectly plastic material under compression is simulated. At the micro level, plastic deformation is formed by switching neighbors in a stochastic manner. Averaging leads to a macro continuous smooth-like deformation.

In the second part, simulations on 2D specimens subjected to uniaxial loading are modelled by a set of point elements (particles) with controlled stochastic heterogeneity by a disorder parameter l. A comparison between real material morphologies and “l morphologies” is made and strong similarities are observed.

The interacting (by force) particles can detach and/or connect from their neighbors during plastic deformation. By ensemble averaging of 200 specimens, it was found that plastic deformation takes place only when both detachments and attachments are allowed. Well-known plastic behaviors are obtained: dependency of the loading/unloading elastic modulus with l, shakedown effects, Bauschinger effect and energy of cold work.  Macro-micro relations explored in the elastic region revealed a strong dependency of the macro elastic moduli with average number of interactions. The convexity and normality of the yield surface stems naturally from the model. Yield envelopes are qualitatively comparable with concrete.

The statistical distribution of yield stress fits very well with the Weibull distribution. The dispersion increases asymptotically with l and the most probable yielding stress decreases linearly with l>0.

Plastic energy and residual plastic strain are proportional to the number of detachments and attachments, respectively. The attachment-detachment mechanisms lead to plastic deformation which depends on the loading direction. Anisotropy is inherently developed.

The third part of the study presents analytical relations connecting morphology and macro properties. The affine approximation limits the moduli and predicts upper bounds for the stiffness. The non-affine analysis showed how the micro correlations enter the calculations for the macro results, similar to the well-known Reynolds stress in fluid mechanics.

The thesis offers a new perspective on modelling plasticity in solid mechanics. The advantage of building a model from simple micro mechanism and adopting tools from statistical mechanics to establish macro properties is clearly demonstrated. It has been proven that the essential “building blocks” of heterogeneity and switching neighbours are the driving forces of modelling plastic deformation.