Mechanical Properties of Matter PDF

Chapter One Mechanical Properties of Matter Prepared by Dr. Mohamed Hasabelnaby Lecturer of physics, School of Allied Health Sciences, Badr University.  The properties which determine the applications and behavior of materials are called...

Chapter One Mechanical Properties of Matter Prepared by Dr. Mohamed Hasabelnaby Lecturer of physics, School of Allied Health Sciences, Badr University.  The properties which determine the applications and behavior of materials are called mechanical properties of the materials. Mechanical properties are helpful in identification of materials because every material has its own identical properties and also tell the usefulness of materials. Mechanical properties are a group of physical properties that describe the behavior of material under a force or a load. Often materials are subject to forces (loads) when they are used. These properties are important because most restorative material used in dentistry must withstand force in service i. e. during fabrication or mastication. In general, the properties of a solid under applied load are determined by the nature and/or strength of its atomic binding forces. 2.1 The Concept of Force Everyone has a basic understanding of the concept of force from everyday experience, when you push your empty dinner plate away; you exert a force on it. Similarly, you exert a force on a ball when you throw or kick it. In these examples, the word force is associated with muscular activity and some change in the velocity of an object. Force is basically push and pull. When you push and pull you are applying a force to an object. If you are Appling force to an object you are changing the objects motion. For an example when a ball is coming your way and then you push it away. The motion of the ball is changed because you applied a force. The term force is common in everyday language. Curiously, in its use in physics, force can only be defined by the effect that it has on one or more objects. More specifically, force represents the action of one body on another. Forces do not always cause motion, however. For example, as you sit reading this book, a gravitational force acts on your body and yet you remain stationary. As a second example, you can push (in other words, exert a force) on a large rock and not be able to move it. 2.2 Stress and strain When forces are applied on a material, it’s deformed to a small or large extent depending upon the nature of the material of the body and the magnitude of the deforming force. The deformation may not be noticeable visually in many materials but it is there. When a material is subjected to a force, a restoring force is developed in the material. This restoring force is equal in magnitude but opposite in direction to the applied force. The restoring force per unit area is known as stress.  If F is the force applied and A is the area of cross section of the material, The stress (σ) = F/A (2.1) Stress is difficult to measure directly, so the force and the area to which the force is applied are measured, and stress is calculated from the ratio of force per area. Stress is commonly expressed in SI units as Pascal (1 Pa = I N/m2). Because the stress in a structure varies directly with the force and inversely with area, the area over which the force acts is an important consideration. This is particularly true in dental restorations in which the areas over which the forces are applied often extremely small, for example, cusp areas of contact may have cross-sectional areas of only 0.16 to 0.016 cm2. As a numerical example, if the occlusal force of 440 N is concentrated on an area of 4 mm2, the stress developed is 110 MPa. 2.2.1 Types of Stress A force can be applied from any angle or direction, and often several forces combine to develop complex stresses in a structure. It is rare for forces and stresses to be isolated to a single axis. Individually applied forces can be defined as tension, compression, shear, bending, and torsional. These directional forces are illustrated in a simplified manner in Figure (2.l). Figure (2.l): The different types of stresses.  Tension stress: is defined as the force acting per unit area such that there is an elongation in the length of the object. Tension results from two sets of forces directed away from each other in the same straight line.   Compression stress is defined as the force acting per unit area such that there is a compression of the object along its length. Compression results from two sets of forces directed toward each other in the same straight line.  Shear stress is defined as the two sets of forces directed parallel to each other, but not along the same straight line.  Torsion stress results from twisting of a body, and bending stress or flexure results from an applied bending moment. When tension is applied, the molecules make up the material resist being pulled apart. When compression is applied, they resist being forced more closely together. As a result of a shear stress application, one portion of the material must resist sliding past another. These resistances of a material to deformation represent the basic qualities of elasticity of solid materials. Each type of stress is capable of producing a corresponding deformation in a material as shown in Figure 2.1. The deformation from a tension force is an elongation in the axis of applied force, whereas a compressive force causes compression or shortening of the body in the axis of loading. Strain is the measure of the deformation of an object under stress and is defined as the fractional change of the object’s length (∆L =L- L0) when the object experiences stress. Strain (ε) = ∆L/L0 (2.2) The units of measurement (length/length) cancel in the calculation of strain, so it is unitless. lf a load is applied to a wire with an original length of 2 mm resulting in a new length of 2.02 mm, it has deformed 0.02 mm and the strain is 0.02/2 = 0.01, or 1%. Strain is often reported as a percentage. The amount of strain will differ with each type of material and with the magnitude of the force applied. Note that regardless of the composition or nature of the material, and regardless of the magnitude and type of force applied to the material, deformation and strain result with each stress application. 2.2.2 Stress-Strain Curves The stress-strain curve provides a graphical measurement of strength and elasticity of the material. Also, the behavior of the materials can be studied with the help of the stress-strain curve which makes it easy with the application of these materials. Under different loads, the stress and corresponding strain values are plotted. An example of a stress-strain curve is given below. The stress-strain graph has different points or regions as follows:  Proportional limit  Elastic limit  Yield point  Ultimate stress point  Fracture or breaking point Proportional limit A Stress-strain curve for a material subjected to stress, as the stress is increased, the strain is increased. In the initial portion of the curve, from 0 to A, the stress is linearly proportional to the strain. As the strain is doubled, the stress is also double. At point A, the stress is linearly proportional to the strain. Hence the value of the stress at point A is proportional limit, defined as the highest stress at which the stress-strain curve is a straight line, which is the stress is linearly proportional to strain. Below the proportional limit, no permanent deformation occurs in the material and when the force is removed, the material will return to its original dimensions. Below the proportional limit, the material is elastic in nature. Elastic limit The elastic limit is defined as the maximum stress material will withstand without permanent deformation. Even though the curve is not linear between the proportionality limit and the elastic limit, the material is still elastic in this region and if the load is removed at or below this point the material will return to its original length. For linearly elastic materials, the proportional limit and elastic limit represent the same stress within the structure, and the terms are often used interchangeably in referring to the stress involved  The region of the stress-strain curve before the elastic limit is called the elastic region. When an object experiences a stress greater than the elastic limit, permanent or irreversible strain occurs. The region of stress-strain curve beyond the elastic limit is called the plastic region. Figure (2.2): Stress-Strain Curve. Yield point It is often difficult to explicitly measure the proportional and elastic limits because the precise point of deviation of the stress- strain curve from linearity is difficult to determine. The yield point of a material is property that can be determined readily and often used to describe the stress at which the material begins to function in a plastic region. At this point, a small, defined amount of permanent strain has accrued in the materials. The yield point is defined as the point at which the material starts to deform plastically. After the yield point is passed, permanent plastic deformation occurs. There are two yield points (i) upper yield point (ii) lower yield point. The elastic limit and yield point of material are among its most in important properties because they define the transition from elastic to plastic behavior. Ultimate stress point As the stress on material is increased further, the stress and the strain increases from yield point to a point called ultimate stress point where stress applied is maximum. Thus ultimate stress point can be defined as the highest stress that martials can withstand before failure occurs. Fracture or breaking point In figure (2.2) the materials fractured at point E. This is the fracture point or the break point, which is the point at which the material fails and separates into two pieces. If a bar of material is subjected to an applied force, F, the magnitude of the force and the resulting deformation (σ) can be measured. In another bar of the same material, but different dimensions, the same applied force produces different force - deformation characteristics Figure (3.3, A). However, if the applied force is normalized by the cross-sectional area A of the bar (stress), and the deformation is normalized to the original length of the bar (strain), the resulting stress-strain curve is independent of the geometry of the bar Figure (3.3, B). Figure (2.3): Force-deformation characteristics. A, Force - deformation characteristics for The same material but having different dimensions. B, stress-strain characteristics of the same group of bars. The stress - strain curve is independent of the geometry of the bar. 2.3 Elastic modulus We shall discuss the deformation of solids in terms of the concepts of stress and strain. Stress is a quantity that is proportional to the force causing a deformation; more specifically, stress is the external force acting on an object per unit cross sectional area. The result of a stress is strain, which is a measure of the degree of deformation. It is found that, for sufficiently small stresses, stress is proportional to strain; the constant of proportionality depends on the material being deformed and on the nature of the deformation, we call this proportionality constant the elastic modulus. The elastic modulus is therefore defined as the ratio of the stress to the resulting strain: 𝑠𝑡𝑟𝑒𝑠𝑠 The elastic modulus = 𝑠𝑡𝑟𝑎𝑖𝑛 (2.3) The measurement of elasticity of a material is described by the term elastic modulus, also referred to as modulus of elasticity or Young s modulus, and denoted by the variable E. The word modulus means ratio. The elastic modulus represents the stiffness of a material within the elastic range. The elastic modulus can be determined from a stress strain curve (see Figure 2.2) by calculating the ratio of stress to strain, or the slope of the linear region of the curve. Table (2.1): Elastic modulus for some dental materials. The elastic quantities of a material represent a fundamental property of the material. The interatomic or intermolecular forces of the material are responsible for the property of elasticity. The stronger of the forces, the greater the values of the elastic modulus and the more rigid or stiff the material. Because this property is related to the forces within the material, it is usually the same when the material is subjected to either tension or compression. The property is generally independent of any heat treatment or mechanical treatment that a metal or alloy has received, but is quite dependent on the composition of the material.  Most materials can be classified into two categories:  Ductile materials.  Brittle materials. Each of these materials exhibit different behavior when external force applied to them. We can analyze this behavior by their stress strain curve. Ductile materials Ductile materials are those which are capable of having large strains before they are fractured. Ductile materials can withstand high stress and are also capable of absorbing large amount of energy before their failure. A ductile material has a large percentage of elongation before failure. The ultimate stress point and fracture point are not the same in the stress strain curve, so ductile materials exhibits elastic as well as plastic deformation. Some examples of ductile materials are aluminum, mild steel and some of its alloys i.e. copper, magnesium, brass, nickel, bronze and many others. Brittle materials When an external force applied to brittle materials, it breaks with very small elastic deformation and without elastic deformation. For brittle materials the values of elastic limit, yield point, ultimate stress point and fracture point are the same. In the other words brittle materials absorb relatively small energy prior to fracture. Some examples of Brittle materials are glass, ceramic, graphite, wood and some alloys. Figure (2.4): Ductile materials and Brittle materials stress strain curve comparison. 2.4 The viscosity and Poiseuille’s law When a force is applied to a solid, it will yield slightly, and then resist further movement. However, when we apply force to a fluid, it continues to move away from the force, and we create a velocity gradient. Within a pipe, for example, the fluid in the center moves faster than the fluid at the outside. Viscosity is the resistance to flow experienced as the internal layers of liquid move over each other. Any kind of fluid, liquid or gas has resistance to flow. This is caused at the molecular level by the drag between adjacent molecules. On a wider level, the flow of one layer of fluid against another is subject to ‘drag’, in the same way that surfaces in contact display friction when relative movement takes place. Viscosity describes a fluid's internal resistance to flow and may be thought of as a measure of fluid friction, a fluid like honey is very viscous, and a fluid like water has a low viscosity. If the fluid were ideal, with zero viscosity, it would flow through the tube with a speed that is the same throughout the fluid as shown in figure (2.5a). Real fluids with finite viscosity are found to have flow patterns and the speed of the fluid goes to zero on the walls of the tube and reaches its maximum value in the center of the tube as shown in figure (2.5b). Fig. (2.5): fluid flow through a tube. a. ideal fluid. b. real fluid. What causes flow? The answer, not surprisingly, is pressure difference. In fact, there is a very simple relationship between horizontal flow and pressure. Flow rate Q is in the direction from high to low pressure. The greater the pressure differential between two points, the greater the flow rate. This relationship can be stated as: 𝑃1 −𝑃2 Q= (2.4) 𝑅 Where P1and P2 are the pressures at two points, such as at either end of a tube and R is the resistance to flow. The greater the viscosity of a fluid, the greater the value of R. The resistance R to laminar flow of an incompressible fluid having viscosity coefficient η measured in units of dyn (sec/ cm2) or poise through a horizontal tube of uniform radius r rand lengthℓ is given by: 8𝜋η R= (2.5) 𝜋𝑟 4 From equations (2.8) and (2.9) we have: 𝜋𝑟 4 (𝑃1 −𝑃2 ) Q= cm3/sec (2.6) 8𝜋η This equation describes laminar flow through a tube and is called Poiseuille’s law, French scientist J. L. Poiseuille (1799–1869). Poiseuille’s law applies to laminar flow of an incompressible fluid of viscosity η through a tube of length l and radius r. The direction of flow is from greater to lower pressure. Flow rate Q is directly proportional to the pressure difference P2−P1, and inversely proportional to the length l of the tube and viscosity η of the fluid. Flow rate increases with r4, the fourth power of the radius. The viscosities of some fluids are listed in table (2.2) In general, viscosity is a function of temperature and increases as the fluid becomes colder. Table (2.2): Viscosities of Selected Fluids. The circulatory system provides many examples of Poiseuille’s law in action—with blood flow regulated by changes in vessel size and blood pressure. Blood vessels are not rigid but elastic. Adjustments to blood flow are primarily made by varying the size of the vessels, since the resistance is so sensitive to the radius. During vigorous exercise, blood vessels are selectively dilated to important muscles and organs and blood pressure increases. This creates both greater overall blood flow and increased flow to specific areas. Conversely, decreases in vessel radii, perhaps from plaques in the arteries, can greatly reduce blood flow. If a vessel’s radius is reduced by only 5% (to 0.95 of its original value), the flow rate is reduced to about (0.95)4=0.81(0.95)4=0.81of its original value. A 19% decrease in flow is caused by a 5% decrease in radius. The body may compensate by increasing blood pressure by 19%, but this presents hazards to the heart and any vessel that has weakened walls. Another example comes from automobile engine oil. If you have a car with an oil pressure gauge, you may notice that oil pressure is high when the engine is cold. Motor oil has greater viscosity when cold than when warm, and so pressure must be greater to pump the same amount of cold oil 2.5 Surface Tension A small insect resting on the surface of a pond or lake is a common sight in the summertime. If you look carefully, you can see that the insect creates tiny dimples in the water′s surface, almost as if it were supported by a thin sheet of rubber. In fact the surface of water and other fluids behaves in many respects as if it were an elastic membrane. This effect is known as surface tension. In fact, even a needle or a razor blade can be supported on the surface of water if they are put into place gently, even though they have densities significantly greater than the density of water. Surface tension is the property of the free surface of a liquid at rest to behave like a stretched membrane in order to acquire minimum surface area. Imagine a line AB in the free surface of a liquid at rest (Fig. 5.20). The force of surface tension is measured as the force acting per unit length on either side of this imaginary line AB. The force is perpendicular to the line and tangential to the liquid surface. If F is the force acting on the length L of the line AB, then surface tension is given by; 𝐹 T= (2.7) 𝐿 Surface tension is defined as the force per unit length acting perpendicular on an imaginary line drawn on the liquid surface, tending to pull the surface apart along the line. Its unit is N m–1. It depends on temperature, decreases linearly with temperature. 2.5.1 Molecular Theory of Surface Tension The surface tension of liquid arises out of the attraction of its molecules. Molecules of fluid (liquid and gas) attract one another with a force. If any other molecule is within the sphere of influence of first molecule it will experience a force of attraction. Figure (2.6) illustrates the molecular basis for surface tension by considering the attractive forces that molecules in a liquid exert on one another. Part (a) shows a molecule within the bulk liquid, so that it is surrounded on all sides by other molecules. The surrounding molecules attract the central molecule equally in all directions, leading to a zero net force. In contrast, part (b) shows a molecule in the surface. Since there are no molecules of the liquid above the surface, this molecule experiences a net attractive force pointing toward the liquid interior resulting in a surface of minimum area. Fig. (2.6): (a) A molecule within the bulk liquid is surrounded on all sides by other molecules, which attract it equally in all directions, leading to a zero net force. (b) A molecule in the surface experiences a net attractive force pointing toward the liquid interior, because there are no molecules of the liquid above the surface. Intermolecular force The force between two molecules of a substance is called intermolecular force. This intermolecular force is basically electric in nature. The intermolecular forces are of two types. They are (i) cohesive force and (ii) adhesive force. Cohesive force Cohesive force is the force of attraction between the molecules of the same substance. This cohesive force is very strong in solids, weak in liquids and extremely weak in gases. Adhesive force Adhesive force is the force of attraction between the molecules of two different substances. For example due to the adhesive force, ink sticks to paper while writing. Fevicol, gum etc exhibit strong adhesive property. Water wets glass because the cohesive force between water molecules is less than the adhesive force between water and glass molecules. Whereas, mercury does not wet glass because the cohesive force between mercury molecules is greater than the adhesive force between mercury and glass molecules. 2.6 Mechanical Advantage of Levers Levers are normally used to make physical work easier, to move something that is heavy, or to move something quickly. When we exercise most of our movements will involve the use of levers. For example, when we run, lift weights, kick or throw a ball, all of these actions will involve the use of levers. A lever system within the body would use a lever (bone) to move an object, for example, when we run we are the object being moved, but when kicking a ball, the object being moved is the ball. A lever is a bar that turns on an unmoving point called a fulcrum. When you push or pull on one end of the bar that pressure is called the force and the object that is lifted on the other end is called the load.  There are three parts to all levers: Fulcrum - the point at which the lever rotates. Input force (also called the force) - the force applied to the lever. Output force (also called the load) - The load is the object requiring moving. A lever provides mechanical advantage. Mechanical advantage refers to how much a simple machine multiplies an applied force. The location of the force, load, and fulcrum will determine the type of lever and the amount of mechanical advantage the machine has. The farther the effort is away from the fulcrum, the easier it is to move the load. Mechanical advantage can be calculated using this formula: Distance from the force to the fulcrum 𝑋𝐴 𝑙𝑜𝑎𝑑 𝐹𝐿 Mechanical advantage (MA) = = Distance from the load to the fulcrum 𝑋𝐿 𝑓𝑜𝑟𝑐𝑒 𝑎𝑝𝑝𝑙𝑖𝑒𝑑 𝐹𝐴 Figure (2.7): Mechanical advantage. If the distance from the force to the fulcrum is greater than the distance from the load to the fulcrum, then the lever has a mechanical advantage. There are three types of levers: first class, second class and third class. The difference between the three classes depends on where the force is, where the fulcrum is and where the load is. First Class Lever A first-class lever is a lever in which the fulcrum is located between the force and the load in operation, a force is applied (by pulling or pushing) to a section of the bar, which causes the lever to swing about the fulcrum, overcoming the resistance force (load) on the opposite side. Figure (2.8): First Class Lever. Fulcrum is between FA(force) and FL (load) When the force greater than load, the Mechanical advantag >1 When the force less than the load, the Mechanical Advantage < 1 as shown in figure (1.16). Figure (2.9): First Class Lever. Common examples of first-classlevers include  crowbars.  Scissors.  Pliers.  Seesaws.  Human skull.  Triceps muscle. Fig. (2.10): Common examples of first-classlevers. Second Class Lever A second class lever is a lever in which the load is between fulcrum and force, The fulcrum is at one end of the lever. The fulcrum is usually closer to the load, The closer the load is to the fulcrum, the easier the load is to lift, think of a wheelbarrow the long handles of a wheel barrow are really the long arms of a lever. Figure (2.11): Second class lever. Common examples of second-classlevers include:  Wheelbarrow.  Nutcrackers.  Handle on fingernail clippers.  Gas Pump Handle.  Pop Bottle Opener. Fig. (2.12): Common examples of second-classlevers. Third Class Lever A third class lever is a lever in which the force is between fulcrum and load. The mechanical advantage of a 3rd class lever is always less than 1. For this class of levers, the force is higher than the load, which is different from second class levers and some first-class levers. Such levers do not have good mechanical advantage, in fact they have mechanical disadvantage. The force is closer to the fulcrum than the load. The force is always greater than the load. However, one advantage of such levers is that the distance moved by the load is greater than the distance moved by the force. So the load moves farther than force. Multiplies the distance the force travels and the speed at which it moves.  Figure (2.13): Third class lever. Examples of third-class levers include:  Tweezers.  Arm hammers.  Shovels.  Biceps muscle. Fig. (2.14): Common examples of third-classlevers.  We can classify the mechanical advantage as:  The mechanical advantage of first class lever either larger or smaller than 1,  The mechanical advantage of second class lever is always larger than 1.  The mechanical advantage of third class lever is always less than 1. Figure (2.15): Types of levers.  Chapter Two Problems 1. A patient’s leg was put into traction, stretching the femur from a length of 0.46 m to 0.461 m. The femur has a diameter of 3.05 cm. With the knowledge that bone has a Young’s modulus of ~ 1.6 × 1010 in tension, what force was used to stretch the femur? 2. A wire 2 m long and 2 mm in diameter, when stretched by weight of 8 kg has its length increased by 0.24 mm. Find the stress, strain and Young’s modulus of the material of the wire. g = 9.8 m/s² 3. A wire of length 2 m and cross-sectional area 10-4 m² is stretched by a load 102 kg. The wire is stretched by 0.1 cm. Calculate longitudinal stress, longitudinal strain and Young’s modulus of the material of wire. 4. A metal wire 1 m long and of 2 mm diameter is stretched by a load of 40 kg. If Y = 7 × 10 10 N/m² for the metal, find the (1) stress (2) strain and (3) force constant of the material of the wire. 5. What must be the elongation of a wire 5m long so that the strain is 1% of 0.1? If the wire has cross-selection of 1mm² and is stretched by 10 kg-wt, what is the stress? 6. A brass wire of length 2 m has its one end, fixed to a rigid support and from the other end 4 kg is suspended. If the radius of the wire is 0.35 mm, find the extension produced in the wire. g = 9.8 m/s², Y = 11 × 1010 N/m² 7. Calculate the change in length of the upper leg bone (the femur) when a 70.0 kg man supports 62.0 kg of his mass on it, assuming the bone to be equivalent to a uniform rod that is 40.0 cm long and 2.00 cm in radius 8. A wire of length 1.5 m and of radius 0.4 mm is stretched by 1.2 mm on loading. If the Young’s modulus of its material is 12.5 × 1010 N/m². , find the stretching force. 9. Find the maximum load which may be placed on a tungsten wire of diameter 2 mm so that the permitted strain not exceed 1/1000. Young’s modulus for tungsten = Y = 35 × 1010 N/m². 10. A wire is stretched by the application of a force of 50 kg. What is the percentage increase in the length of the wire? Y = 7 × 1010 N/m² 11. Which point on the stress strain curve occurs after the proportionality limit? A. Upper yield point. B. Lower yield point. C. Elastic limit. D. Ultimate point. 12. Stress is………………………………………………………..….. A. External force. B. Internal resistive force. C. Axial force. D. Radial force. 13. The property of a body by virtue of which it tends to regain its original size and shape when the applied force is removed is called………………………………………….. A. Elasticity. B. Plasticity. C. Rigidity. D. Compressibility 14. Substances which can be stretched to cause large strains are called………………………… A. Steel. B. Rubber. C. Copper. D. Glass. 15. The restoring force per unit area is known as……………………………………….. A. Elasticity. B. Plasticity. C. Rigidity. D. Stress 16. The modulus of elasticity of steel is greater than that of rubber because under the same stress because……………………………… A. the strain in steel is less than rubber B. the strain in steel is BIGGER than rubber C. None of the mentioned. 17. The unit of Young’s modulus of elasticity is……………………………………… A. Pascal. B. Meter. C. Dimension-less. D. Newton.