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This is the complete lecture note for MME407 Mechanical Behaviour of Materials. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
Course code: MME 407 Course title: Mechanical Behaviour of Materials Course credit load: 2 Units Series 1: Stress, Strain and Tensors: The Language of the Mechanical Behaviour of Solids • Part 1.1: Introduction to the Mechanical Behaviour of Materials o Sub Part 1.1.1: Flow and Fracture of Solids o Sub Part 1.1.2: Material Classes and the Mechanisms of Deformation o Sub Part 1.1.3: Types of Loading • Part 1.2: Stress o Sub Part 1.2.1: Normal and Shear Stress o Sub Part 1.2.2: Stress at a Point and the Stress Tensor o Sub Part 1.2.3: Stresses on Inclined Planes and Principal Stresses • Part 1.3: Strain o Sub Part 1.3.1: Normal and Shear Strain o Sub Part 1.3.2: The Strain Tensor o Sub Part 1.3.3: Principal Strains and Volume Change • Part 1.4: States of Stress and Mohr's Circle o Sub Part 1.4.1: Uniaxial, Biaxial and Triaxial States of Stress o Sub Part 1.4.2: Mohr's Circle for Plane Stress o Sub Part 1.4.3: Hydrostatic and Deviatoric Stress
Introduction When a lorry crosses a bridge or a paper clip is bent open, the material inside does not feel a force in the everyday sense. It feels stress, a measure of the intensity of load at every point, and it responds with strain, a measure of the deformation that results. Before the flow, fracture and fatigue of solids can be understood, these two quantities must be defined carefully, and this Series sets out the language in which the mechanical behaviour of materials is described. The aim of this Series is to equip you to explain the scope of the mechanical behaviour of materials, define and calculate stress and strain, and use tensors and Mohr's circle to describe states of stress. We shall commence with an introduction to the mechanical behaviour of materials, including flow, fracture and the types of loading. Next, we will explore stress, from simple normal and shear stress to the stress tensor. Afterwards, we will turn to strain and the strain tensor. We will conclude with states of stress and Mohr's circle. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1: explain the scope of the mechanical behaviour of materials, including the flow and fracture of solids under different types of loading, • SLO 1.2: compare the general mechanisms of deformation in metals, ceramics and polymers, • SLO 1.3: define normal and shear stress and explain the stress tensor, • SLO 1.4: calculate the stresses on inclined planes and determine principal stresses, • SLO 1.5: define normal, shear, engineering and true strain, • SLO 1.6: explain the strain tensor and volumetric strain, • SLO 1.7: describe uniaxial, biaxial, triaxial, hydrostatic and deviatoric states of stress, • SLO 1.8: construct and interpret Mohr's circle for a state of plane stress. 1.1 Introduction to the Mechanical Behaviour of Materials 1.1.1 Flow and Fracture of Solids
The mechanical behaviour of materials is the study of how solids respond to applied forces, and it explains how much load a material can carry, how it deforms and how it fails. Two broad responses are of interest. In flow, the material changes shape permanently without losing continuity, and in fracture it separates into two or more pieces. Between these lies elastic deformation, which is fully recoverable, and the loading conditions decide which response occurs. This course treats these responses as the consequences of atomic and microstructural mechanisms. A load may be applied once and increased steadily, applied suddenly, held for a long time or repeated many times, and each gives a different response. Understanding the response to each type of loading is the basis of safe design, since a part must carry its loads without excessive deformation, must not fracture, and must last its intended life. Worked example: A steel paper clip that is bent and stays bent has undergone flow, whereas a glass rod that is bent until it breaks into pieces has undergone fracture. Application: Knowing whether a material will flow or fracture under a given load decides how it should be used and how a component is designed. Fill in the blank: The permanent change of shape of a solid without loss of continuity is called __________. 1.1.2 Material Classes and the Mechanisms of Deformation The mechanical response of a material depends on its bonding and structure. In metals, the metallic bond is non-directional and layers of atoms can slide by the motion of dislocations, which are line defects, so metals show extensive flow and are ductile. In ceramics, strong directional ionic and covalent bonds resist sliding, so ceramics show very little flow at room temperature and fracture in a brittle manner. In polymers, strong covalent bonds along the chains are joined by weak secondary bonds, so deformation occurs mainly by the uncoiling and sliding of chains. Despite these differences, the materials share common features. All show elastic deformation at small loads, in which atomic bonds are stretched and then recover, all may fracture at large loads, and all are described by the same quantities of stress and strain. The similarities and differences in response within a class are also important, since the properties of a metal vary with alloying and heat treatment, and those of a polymer vary with temperature and crystallinity. Worked example: A copper wire can be bent and drawn because dislocations move easily, whereas a ceramic plate of alumina under the same bending load fractures with almost no permanent deformation. Application: Comparing the bonding and mechanisms of each class explains why each is chosen for particular loading conditions.
Fill in the blank: In metals, layers of atoms slide by the motion of line defects called __________. 1.1.3 Types of Loading Loads are classified by how they are applied and by the stresses they produce. In tension, the load pulls the material and tends to lengthen it, in compression it pushes and tends to shorten it, in shear it tends to slide one plane of material over another, in torsion it twists a member such as a shaft, and in bending it produces tension on one side of a member and compression on the other. Real components often experience combinations of these. Loads are also classified by their history in time. Monotonic loads increase slowly and steadily to a maximum, impact loads are applied suddenly at a high rate, sustained loads are held for long periods, and repeated loads vary cyclically. The rate and duration of loading, and the temperature, influence the response, since a material that is ductile under slow loading may fracture under impact, and creep and fatigue appear only under sustained and repeated loads. Worked example: A rod hanging from a ceiling under a weight is in tension, a column supporting a floor is in compression, and a drive shaft that transmits power is in torsion. Application: Identifying the type of loading is the first step in choosing the correct stress calculation and the correct test. Fill in the blank: A load that twists a member such as a shaft produces __________. Figure 1.1: Types of loading: tension, compression, shear, torsion and bending.
Pilot questions 1.1 1. Define flow. (Linked to SLO 1.1) 2. What is fracture? (Linked to SLO 1.1) 3. Name two types of loading classified by their history in time. (Linked to SLO 1.1) 4. Why do metals show extensive flow? (Linked to SLO 1.2) 5. By which mechanism do polymers deform mainly? (Linked to SLO 1.2) 1.2 Stress 1.2.1 Normal and Shear Stress Stress is the intensity of internal force per unit area acting within a material, and it is measured in pascals, or newtons per square metre, with megapascals used for most solids. Normal stress, σ, acts perpendicular to a plane, and by the usual convention it is positive if tensile and negative if compressive. For a uniform bar under an axial force F on a cross-sectional area A, σ = F/A. This is the engineering stress, which is based on the original area. Shear stress, τ, acts parallel to a plane, and for a force V acting parallel to an area A it is τ = V/A. Because the area of a loaded bar changes as it deforms, true stress is defined using the instantaneous area, so that true stress is the force divided by the actual area at that moment. The difference is small at low strain but large at large plastic strains, when the area has shrunk considerably. Worked example: A steel bar of 20 millimetres diameter carrying an axial load of 30 kilonewtons has an area of 0.000314 square metre, so σ = 30 000/0.000314, which is about 95.5 megapascals. Application: Dividing force by area lets loads on parts of different sizes be compared through stress. Fill in the blank: Stress that acts parallel to a plane is called __________ stress. 1.2.2 Stress at a Point and the Stress Tensor The state of stress at a point is described by considering a tiny cube of material around the point. Each of its
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