01 Introduction 2018.Pdf

1 2 Constitutive models or material models comprise our knowledge on the physical behavior of given material systems. It combines forces with resulting displacements, what is the fundament of every structural calculation, elasticity and plasticity theory and hence calculation of deformations and capacities of structures. In the respective material lectures you learned about numerous intrinsic processes in materials considering the micro‐structural composition in a phenomenological way. Their mathematical description is however the precondition for the practical assessment with numerical methods. Starting from a short intro into the mechanics of deformable bodies, we will consider 3D material laws for elasticity, visko‐elasticity, plasticity, materials with intrinsic length scales, and damage continuum mechanics with internal variables. Of course these are of general nature, however we will always exemplify for building materials like metal, concrete, wood or other composites, heterogeneous materials. We will see how own material models can be implemented in Abaqus using its material interface. 3 Teaching goals: This introductory course aims to bridge the gap between phenomenological, qualitative comprehension of processes in building materials, their characterization in mechanical testing and the ability to apply those for practical design purposes via constitutive models. Students gain knowledge on deriving and applying static an dynamic constitutive equations for 3D bodies and their reduction to simplified rheological models. You will see how material modeling comprises multiple disciplines, that you only studied in an isolated fashion up to now. Material modeling is truly interdisciplinary. 4 The list of accidents on constructions, bridges collapsing, collapse of roofs or buildings is very long. Up to the 20th century a lack of understanding of strength problems, theoretical foundations of mechanics of materials and of course also of practical engineers trained to interpret their designs this way was missing. Many of these findings are set out in codes, in the form of safety factors or principles of construction. However we need to work as good engineers beyond the standards. Therefore we must understand the behavior of structures and materials, which are loaded beyond the typical design limit. Correctly assessing the non‐linear regime of material behavior is essential for the assessing the real safety of buildings in the event of an emergency. Empirical work following rules of thumb even up to date result in too many collapses ‐ of older structures, but also new ones. The understanding of the processes inside the building materials, as well as their appropriate numerical description are in my opinion the key skills for aspiring engineers who want to help shape the challenging architecture of tomorrow, and by no means things are linear. 5 On this slide I placed terms we deal with today. As you can see they originate from diverse disciplines ranging from materials science to strength of materials, material theory, thermodynamics, continuum mechanics up to numerics. It is a truly interdisciplinary lecture. Since we only meet once a week, it is clear that we cannot dig into everything to the same detailing. The course is composed in a way, that in the beginning we quickly refer to many known things to create a common basis for the further lectures. This is to remind you about all the things you already know or have known, or should look at. I will hand out exercises to you quite often. Take it serious. But for now, let’s dive into the world of materials. 6 Let’s start with a material volume that is supposed to be representative. We have external loads (mechanical, thermal, hygric) that result in an effect, here a deformation. Let’s consider the material volume as a black‐box and just determine the deformations due to external load. 7 One of the first men, who thought about how to describe what happens to a body under load was Robert Hooke. He was prone to an excessive lifestyle, celebrated, drank, smoked and loved conversation with his friends till late at night. He also jumped from discipline to discipline driven by curiosity. Everything had to be examined and understood. As it was hard to just survive by one science, he was physicist, astronomer, architect, engineer (reconstruction of London after a devastating fire), biologist and so on. He had a friend, that clock maker Thomas Tompion (1639‐ 1713) with whom he discussed from time to time about springs and pendulums. Like all his contemporaries, Hook knew nothing about atomistic bondings, but he had the idea that a spring, like the ones built into clocks, is maybe nothing different than a very special case of an elastic body, and that a thing like rigid material behavior simply does not exist. The spring stiffness would then be a property inherent to every solid. Well Robert Hook was a man who did not suffer from exaggerated modesty. He was fighting with many researchers and inventors of his time, because he loved to adopt their ideas as his own. At ETH he would most likely be expelled due to unethical behavior. To secure priority claims, he published a manuscript titled «A decimate of the Centesme of the Inventions I Intended to Publish» ‐ How arrogant! In there you find a chapter on «The true theory of elasticity or springiness». Ceiinosssttuu is the only word printed – an anagram. For two years the world had to puzzle what this meant, until Hooke resolved it in an manuscript called »De Potentia Restitutiva , or of a Spring». The anagram meant «Ut tensio sic uis» Like the strain so the force. Well «Tensio» is not really consistently defined by romans or literary scholars. Sometimes it is translated as stress, sometimes as strain. Anyway, the first constitutive law was born and said that force was proportional to elongation and vice versa. For small deformations (5‐10%) this is a quite good approximation and it worked for a long time, since materials were so bad, that 10% was almost never reached. However there was a flaw in thinking. It worked 8 excellent for structures, but other structures resulted in different elongations or it was a different force need for the same elongation, in short, it was not a real constitutive law and it took another 100 years for one to emerge. 8 To get to a constitutive law, it needed a real giant, a true universal scholar – Thomas Young. This man was unbelievable. He spoke 10 languages – at the age of 14! He in principle made the foundation to decipher Egyptian hieroglyphics. In his linguistic research he compared the grammar and vocabulary of 400 languages – without a computer. However this is not the reason why Helmholtz, Maxwell or even Einstein adored him. He had also strong interest in natural sciences. He could prove the wave theory of light by peculiar phenomena, that could not be explained by the corpuscular theory of light, that regarded light as a photon stream (e.g. Newton rings). 1807 he postulated first the three color theory of sight, that was picked up and refined by Hermann von Helmholtz to the Young‐Helmholtz‐theory. And then there is the Young’s modulus or modulus of elasticity (MOE). Supposedly it was around 1800, when Thomas Young realized that when stresses and strains are used, instead of deformations and forces, the Hook’s law results in a constant. He further postulated that this constant was a characteristic property of specific matter, called material stiffness. Young published his ideas in a rather cryptic written manuscript. He might have been a giant, but in 1807 the university released him from his chair, since according to their view, his work was «not sufficiently applied». Anyway, strain is without unit, and the MOE has the unit of a stress – it is exactly the stress at which the length of a sample doubles, if it did not break or deform non‐linearly by then. 9 But how small is a material volume allowed to be, so we can still measure the MOE of the material – hence it’s constitutive behavior? At further inspection of any construction material, we realize that they are all but homogeneous. They can be composites of fibers, particles, aggregates, polycrystalls or even amorphous. However all of them have inherent characteristic length scales in common, and disorder. Strictly speaking materials are not homogeneous matter, but microscopic structures that react upon loading in diverse ways by micro‐structural failure. Think of a knife. When pointing its sharp tip on a ceramic plate, stresses emerge, that are way beyond all known yield stresses and strengths. However the tip of the knife remains entirely intact and not deformed. This can be explained by the fact that the tip is not representative for the metal polycrystall. What is also puzzling is that knowing about non‐linear inter‐atomistic forces, e.g. the Lennard‐Johnes potential, it remains unclear how a linear material law can emerge. In fact on whiskers, one measures non‐linear elastic behavior that relates to the inter‐ atomic potentials. However in Hooks time materials were of such quality that they failed before reaching strains where the non‐linearity could show, or so large, that one always dealt with polycrystalls. 10 Consequently we have some conditions for the material volume that have to be met: The sample always has to be larger than the RVE or RUC. This sounds banal, and is true for many materials like metals, polymers etc. For particulate matter like concrete, however it is far from being a matter of course. 11 Let’s look at some principles for material laws that we will meet over and over again. If we make a list of all dependent variables and balance equations for solving a thermo‐ mechanical field problem, we lack 16‐5=11 equations.

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