Summary for ME2151

Chapter 2      

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Mechanical properties are dependent on (1) ambient temperature and (2) the rate at which the force is applied. Engineering strain = (change in length)/(original length) Tensile test – thick cross section use cylindrical specimen to test, thin cross-section use flat specimen to test Note that the tensile test is used for metals and polymers (ceramics use another test called hardness testing to determine the strength) Output of tensile test is recorded as load vs elongation, then normalized to stress vs stain When the deformation is elastic, most metals exhibit a linear relationship between the stress and strain. This is known as hookes law. E is the slope of the linear segment of the stress-strain curve. The higher the young modulus E, the greater is the stiffness. Stiff material has a high modulus and its deformation under an elastic load is small. (High modulus – steep slope – small elastic deformation) In metals and ceramics, E depends on the nature of the bonding of atoms within the material. Stiffness may ONLY be altered by physically combining to 2 or more materials of different stiffness together as composite. (E of a given material is always the same) Plastic deformation follows upon elastic deformation when applied stress exceeds the critical value (elastic limit) Yield strength is specified as the stress level at which a noticeable amount of plastic deformation has occurred, which is a strain of 0.2%. (0.2% offset yield strength) The 0.2% offset yield strength is the stress at which the stress strain curve intersects a line originating at 0.2% strain and parallel to the linear portion of the stress strain curve. Yield strength in all classes of materials depends on the characteristic of atomic bondinging as well as the microstructure. (i.e same material can have different internal microstructure leading to different strength, but young modulus still the same) UTS – maximum stress on the stress –strain curve. Necking occurs after that. Deformation up to UTS is uniform throughout the length of the specimen. True stress is always greater than the engineering stress, while the true strain is always smaller than the engineering strain. True strain = ln(1 + engineering strain) Strength of material increases with increasing plastic deformation (strain hardening) Ductility is the measure of the amount of plastic deformation sustained at fracture. (%EL or %RA)

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%EL =

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%EL varies with different gauge lengths, hence need to standardize tensile test specimens. Brittle – plastic strains of less than 5% The ductility of a material is sensitive to the ambient temperature. Materials that are ductile at room temp may be brittle at low temperatures. The toughness of a material is the ability of a material to absorb energy up to fracture. Tensile test – toughness is the area under engineering stress strain curve up to fracture.(work done per unit volume in causing the material to fracture.

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x 100% = plastic strain after fracture x 100%

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Tough material will exhibit much plastic deformation before fracture. (i.e ductile). But the strength of the material must be high too. (e.g polymer has high ductility but low strength – low toughness) Hardness is the measure of the resistance of a material to surface indentation. Hardness is related to the size or depth of the depression. Larger depression – softer the material. Positive linear relationship between hardness and strength for a particular metal. Hardness testing preferred because it is inexpensive and non-destructive relative to tensile test.

Chapter 3   

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Composition refers to the types of elements that make up the material and the proportion in which they exist. The most stable (lowest energy) electron configuration is that in which the electron states within the outermost shell are completely filled. Valence electrons are most likely to enter into chemical reactions because of their distance away from the nucleus reduces their binding energy to the nucleus and the closed inner shells shield the outermost electrons from the positive nucleus. Ionic bonding arises from the electrostatic attraction between the positive and negative ion. Ionic bonds are non-directional because the electrostatic attraction is omnidirectional and long ranged. Covalent bonding arises between the negatively charged pair of shared electrons and the two positively charged nucleuses. Covalent bonds have fixed directional relationship due to the strong mutual repulsion resulting in each valence electron cloud to assume equilibrium position as far from one another. Metallic bonding is the electrostatic attraction between the positive ions and the sea of delocalized negative electrons. Metallic bonds are non-directional. Most metals can deform a considerable amount without fracturing (high ductility) because the metal atoms can slide past one another without completely disrupting the metallically bonded structure. Note that plastic deformation of a material requires atoms to slip and slide past one another. Number of valence electrons increase – bonding energy increase – higher MP and more energy needed to break the bond Van der waals bonding – secondary bonding due to the attraction between molecular dipoles. As the size of the noble gas increases, melting point increases because of the stronger van der waals bonds due to the electrons having more freedom to create stronger dipole moments. Hydrogen bonding arises from the electrostatic attraction between the positive and negatives ends of the permanent dipoles. Covalent bonding between hydrogen and an element concentrates the electron away from the hydrogen nucleus, leaving a positive charge at the hydrogen end. Thus forming a permanent dipole. Ceramics – mixed covalent/ionic bonding Transition metals – mixed metallic/covalent bonding due to partially filled inner shells resulting in some covalent bonding

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When two atoms are brought close, the attractive force is due to the electrostatic attraction between the nucleus of one atom and the electron cloud of the other. And there is a repulsive force due to the electron clouds repelling one another. The repulsive energy is the energy absorbed as ions come close together and is positive (convention) and the attractive energy is the energy released when the ions come close and its negative. At the equilibrium bond length, there is no net force, and the potential energy is a minimum. The energy separation curve is also known as the potential well and it represents the potential energy of an atom due to the influence of its neighboring atoms. By convention potential energy is set at zero when separation, r = infinite, since atoms would have zero interactions. If the total energy becomes greater than zero, the material exists in the gaseous state. Thermal Expansion – as temperature rises above 0K, K.E results in the atom shifiting up the potential well. Due to the asymmetric shape of the well, the mean position of the atom changes and the atoms tend to move apart as T increases. Material with deep and narrow, the change in interatomic distance for a given rise in T is smaller (smaller thermal expansion coefficient) than that of a broader shallower well. Ionic/covalent materials tend to have deep and narrower wells and hence have good thermal stability; while metals tend to have broader wells will have greater thermal expansion. Melting Point – The melting point is a measure of bond strength given by the depth of the potential well. The deeper the well, the greater is the thermal energy required to reach levels where bonding energy is low. Primary bonds generally have deep wells. Elastic Modulus – it is related to the slope of the force - separation curve near the equilibrium separation distance. The steeper the slope, the stiffer the material. (Greater force req) Strength of atomic bond also depends on the value of Fmax, peak of the force separation curve. Force greater than Fmax, the atomic bond breaks.

Chapter 4      

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Amorphous – lacking in long range order Crystalline materials exhibit both short range and long range order. Repetitive pattern formed by atoms in a crystalline is called the lattice. Properties of a unit cell are the same as those of the whole crystal. Hard sphere model – atoms are assumed to be solid spheres that touch one another when describing crystal structures. And the centers of solid spheres coincide with the lattice points. Coordination Number – number of nearest neighbors (touching atoms) to any atoms. For non-directional materials (ionic and metallic), the lowest energy is obtained when the atoms packed as closely as possible, separated only by their equilibrium bond length. Hence, the no. of neighbors (C.N) would be as high as possible. Atomic packing factor (APF) – fraction of space occupied by atoms in a unit cell. APF = Vol of atoms in unit cell / Vol of unit cell FCC – CN = 12; Atoms per cell = 4; APF =0.74 BCC – CN = 8; Atoms per cell = 2; APF =0.68 FCC - CN = 12; Atoms per cell = 6; APF = 0.74

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Metals do not crystallize into simple hexagonal crystal structure because the APF is too low. Instead, atoms can attain a lower energy and more stable condition by forming HCP. Allotropy is the ability of an element to assume more than one crystal structure in the solid state, depending on the external conditions. Close packed planes and directions are the planes or directions of a crystal where the atoms are in direct contact. Plastic deformation occurs most readily along closed-packed directions in those planes. The number and relative positions of these planes and directions affects properties such as ductility. Size of the interstitial site is defined by radius ratio r/R Miller Indices of direction – subtract start point from end point. E.g. [110] Family of directions – members in the same family of directions share the same set of miller indices but in different permutations, including negative indices. Family of directions along cube edge - Family of directions along face diagonals - Family of directions along body diagonals - Miller-bravis: *u’v’w’+ to *uvtw+ (u = n/3(2u’ – v’); v = n/3(2v’ – u’); t = -n/3(u’ +v’); w = w’) For plane indices, clear fraction but do not reduce to lowest! E.g. (110) Family of planes E.g. {111} Single crystals have same orientation and alignment of unit cells maintained throughout the entire crystal. Single crystals do not contain grain boundaries and they exhibit directionality in properties. The directionality is called anisotropy. Polycrystalline materials are composed of many small crystals with the same structure but different orientation. Hence, they have grain boundaries. Their measured properties are independent of crystallographic direction – isotropic. Atoms at grain boundaries are displaced from their equilibrium position and hence these atoms at grain boundaries possess higher energy. This interfacial energy makes the grain boundaries preferential sites for chemical reactions.

Chapter 5   

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Linear defects are one dimensional imperfection. Collectively known as dislocations. Plane defects are two dimensional imperfections that serve as boundaries between regions having different crystal structure and/or crystallographic orientations. E.g. grain boundaries. The external surfaces are considered as planar defects because the atoms on the surface are only bonded to one side. Hence, the surface atoms have a lower number of neighbors. As a result, they have greater energy and the higher energy state makes them more susceptible to erosion. The number of atoms that possess enough k.e to overcome the activation energy barrier (E>Q) and make successful jumps is found from the kinetic energy distribution of atoms. The rate at which atoms makes successful jumps are proportional to the number of atoms with energy greater than Q. Rate is described by Arrhenius equation. Arrhenius Equation: Rate = C exp(-Q/RT)

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Number of vacancies depends on the total number of lattice sites. Number of vacancies increases with temperature. Self-diffusion which is the constant random movement of atoms of the same type within pure materials and it occurs in the absence of a concentration and there is no net flow. Inter-diffusion affects material properties and requires a concentration gradient. Vacancy diffusion is the diffusion of an atom from its lattice position to an adjacent vacancy. Vacancy diffusion is the mechanism for self-diffusion(same material) as well as for interdiffusion (unlike material) if the host atoms are of the same size to the diffusing atoms. Rate of diffusion is limited by the number of vacancies. In general, a material with higher melting point will have higher activation energy because they tend to have stronger bonds. Hence more energy is required to break the bond to make a successful jump. Interstitial diffusion – atoms move from one interstitial site to an adjacent empty one. At lower temperature, interstitial diffusion is faster than vacancy diffusion because interstitial sites are more abundant than vacancies. Also, interstitial atoms are more mobile. Rate of mass transfer is measured by the diffusion flux. Diffusion flux is defined as the number of atoms passing through a unit cross sectional area of the solid per unit time. J = -D x (instantaneous concentration gradient), where D is the diffusion coefficient or diffusivity. Magnitude of D is indicative of the rate at which atoms diffuse. D is described by arrhenuis equation as it is directly related to the frequency at which atoms makes successful jumps. The higher the temperature, the greater the number of atoms with kinetic energy greater than the activation energy. Hence, the frequency of jumps increases and the larger is the D, and the faster is the rate of diffusion. In vacancy diffusion, the activation energy Q consists of the energy required for vacancy diffusion plus the energy for the jump whereas in interstitial diffusion, energy is only need for the atom to jump since interstitial spaces are available. Hence, Q(vacancy) > Q(interstitial). The atomic bond strength also affects the rate of diffusion, which is reflected in the activation energy Q. Materials with strong atomic bonds have high activation energy Q because an atom requires high k.e in order to the break the bonds and make a jump. Influence of atomic packing in crystal structure is reflected in Q and Do. As it is more difficult for interstitial atoms to squeeze through densely packed regions, Q is higher/Do is smaller in close-packed structures such as FCC. Resulting in a lower D and slower rate of diffusion. (i.e Q is higher in FCC metals as compared to BCC metals) Crystal defects such as grain boundaries are disordered regions where atoms can move easily. Q for diffusion along such short circuit is lower than through the bulk of the crystal and so D is higher and diffusion faster. Only at low temperatures does short circuit diffusion becomes dominant. Nanocrystalline materials have large proportion of grain boundaries and hence grain boundary diffusion can dominate. As activation energy is lower for diffusion along the disordered regions of the grain boundaries as compared to the bulk of the crystal, the larger is the D and hence the faster the diffusion in these materials.

Chapter 6  

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The combination of slip plane and slip direction forms a slip system. The slip planes and the slip direction in which slip occurs are those in which the atoms are densely packed. This is because slip occurs in steps of one atomic spacing. Hence moving the atom from one equilibrium position to the next would involve least energy when atoms are closest together. Unlike FCC, BCC structure does not contain close packed planes. Hence atoms have to move a greater distance from one equilibrium position to the next during slip and thus greater amount of energy and hence a higher shear stress is needed for slip in BCC than FCC. This means BCC metals have higher strength. Slip will begin on a slip system when the resolved shear stress acting on the slip plane in the slip direction reaches a critical value. In a polycrystalline, the grains of various orientations have to slip on 5 independent systems simultaneously to maintain the coherency along grain boundaries. Metals with FCC and BCC structures are ductile because they possess a relatively large number of slip systems (12 in FCC and up to 48 in BCC) and the slip systems in FCC and BCC are also well distributed in space. Hence, cross-slip can occur when as one system is constrained, slip can continue on a different intersecting slip system. On the other hand, HCP structure has fewer slip systems (3 slip systems) and the slip systems are parallel, hence they do not intersect and cross slip is not possible. Thus HCP metals are ductile. Actual yield strength is much lower because in metal crystals, slip occurs via the movement of dislocations, during which only a small fraction of atomic bonds are broken and reformed at any one time. Hence, less energy is needed for the movement of dislocations. Shear stresses in ceramics are higher than metals because covalent and ionic bonds are stronger and the ions have to move a greater distance between equilibrium positions. Also for ceramics which are ionic, if slip occurs in some directions, ions of the same charge will be brought together to generate strong electrostatic repulsion to resist slip. For ceramics with high covalent bonding, the directionality of the bonds makes the displacement of the atoms from their lattice sites difficult. Hence the required shear stress for slip (yield strength) is much higher than that to cause fracture (fracture strength). Hence ceramics are hard and brittle and do not undergo plastic deformation by slip, except at higher temperature. This is because at higher temperature, thermal activation for dislocation movement increases and hence the yield strength decreases and slip can occur at lower applied shear stress. If the yield strength is lower than the fracture strength, plastic flow ahead of the crack tip will occur. The capacity of a metal for plastic deformation depends on the dislocations to move The distortion of atomic bonds around any dislocation increases the potential energy because of the non-equilibrium interatomic separations. This energy is called strain energy. When a dislocation is in close proximity with another dislocation, the stress fields of the two dislocations will interact. If compressive and tensile stress fields of the two dislocations lie on the same sides of the slip plane, the overall strain energy will increase if they overlap and this results in mutual repulsion.

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As most dislocations are randomly curved mixed dislocations, there is low probability that all the conditions for dislocation annihilation to occur will be fulfilled. Thus dislocation interactions tend to be mutually repulsive. During plastic deformation, dislocations encounter other dislocations and their stress field interacts. Since their interactions are generally repulsive, this results in higher strain energy and hence a greater applied stress to overcome this mutual repulsion for dislocation movement to continue. Also, new dislocations are continuously created and this increases the dislocation density. Crystals with intersecting slip systems strain hardens rapldly because slip tends to occur in more than one slip system. Solute atoms distort the surrounding lattice and increase the strain energy. The solute stress field interacts with that of an approaching dislocation such that repulsion arises. A higher applied stress is needed. On the other hand, they could meet such that the overall strain energy is reduced. However, once such low strain configuration is established, further dislocation movements will raise the strain energy and this requires a higher applied stress. Hence the metal strengthens The more the solute added (without exceeding the solubility) the greater is the strengthening effect. Degree of solid solution depends on the relative sizes of the solute and solvent atoms. The larger the difference, the greater is the distortion of the lattice and hence the stronger the strengthening effect. Substitutional solute atom don’t have shear component, hence cannot interact with screw dislocations. Interstitial solute atoms in BCC crystals can cause tetragonal distortion. Hence can interact with both screw and edge dislocations. Annealing is the process in which the effect of work hardening is removed by heating the metal to a sufficiently high temperature. The annealing process replaces the highly distorted work-hardened grain with new grains with few dislocations. Recrystallization occurs whereby new grains of relatively few dislocations nucleate and replaces the strain hardened grains.