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Stress When a metal is subjected to a load (force), it is distorted or deformed, no matter how strong the metal or light the load. If the load is small, the distortion will probably disappear when the load is removed. The intensity, or degree, of distortion is known as strain. If the distortion disappears and the metal returns to its original dimensions upon removal of the load, the strain is called elastic strain. If the distortion disappears and the metal remains distorted, the strain type is called plastic strain. Strain will be discussed in more detail in the next chapter. When a load is applied to metal, the atomic structure itself is strained, being compressed, warped or extended in the process. The atoms comprising a metal are arranged in a certain geometric pattern, specific for that particular metal or alloy, and are maintained in that pattern by interatomic forces. When so arranged, the atoms are in their state of minimum energy and tend to remain in that arrangement. Work must be done on the metal (that is, energy must be added) to distort the atomic pattern. (Work is equal to force times the distance the force moves.) Stress is the internal resistance, or counterforce, of a material to the distorting effects of an external force or load. These counterforces tend to return the atoms to their normal positions. The total resistance developed is equal to the external load. This resistance is known as stress. Although it is impossible to measure the intensity of this stress, the external load and the area to which it is applied can be measured. Stress (s) can be equated to the load per unit area or the force (F) applied per cross-sectional area (A) perpendicular to the force as shown in the Equation below Where: s = stress (psi or lbs of force per in.2) F = applied force (lbs of force) A = cross-sectional area (in.2) Types of Stress Stresses occur in any material that is subject to a load or any applied force. There are many types of stresses, but they can all be generally classified in one of six categories: residual stresses, structural stresses, pressure stresses, flow stresses, thermal stresses, and fatigue stresses.

Residual Stress Residual stresses are due to the manufacturing processes that leave stresses in a material. Welding leaves residual stresses in the metals welded.

Structural Stress Structural stresses are stresses produced in structural members because of the weights they support. The weights provide the loadings. These stresses are found in building foundations and frameworks, as well as in machinery parts. Pressure Stress Pressure stresses are stresses induced in vessels containing pressurized materials. The loading is provided by the same force producing the pressure. Flow Stress Flow stresses occur when a mass of flowing fluid induces a dynamic pressure on a conduit wall. The force of the fluid striking the wall acts as the load. This type of stress may be applied in an unsteady fashion when flow rates fluctuate. Water hammer is an example of a transient flow stress. Thermal Stress Thermal stresses exist whenever temperature gradients are present in a material. Different temperatures produce different expansions and subject materials to internal stress. This type of stress is particularly noticeable in mechanisms operating at high temperatures that are cooled by a cold fluid. Fatigue Stress Fatigue stresses are due to cyclic application of a stress. The stresses could be due to vibration or thermal cycling. The importance of all stresses is increased when the materials supporting them are flawed. Flaws tend to add additional stress to a material. Also, when loadings are cyclic or unsteady, stresses can effect a material more severely. The additional stresses associated with flaws and cyclic loading may exceed the stress necessary for a material to fail. Stress intensity within the body of a component is expressed as one of three basic types of internal load. They are known as tensile, compressive, and shear. Figure 1 illustrates the different types of stress. Mathematically, there are only two types of internal load because tensile and compressive stress may be regarded as the positive and negative versions of the same type of normal loading. However, in mechanical design, the response of components to the two conditions can be so different that it is better, and safer, to regard them as separate types. As illustrated in Figure 1, the plane of a tensile or compressive stress lies perpendicular to the axis of operation of the force from which it originates. The plane of a shear stress lies in the plane of the force system from which it originates. It is essential to keep these differences quite clear both in mind and mode of expression. Tensile Stress Tensile stress is that type of stress in which the two sections of material on either side of a stress plane tend to pull apart or elongate as illustrated in Figure 1(a). Compressive Stress Compressive stress is the reverse of tensile stress. Adjacent parts of the material tend to press against each other through a typical stress plane as illustrated in Figure 1(b). Shear Stress Shear stress exists when two parts of a material tend to slide across each other in any typical plane of shear upon application of force parallel to that plane as illustrated in Figure 1(c). Assessment of mechanical properties is made by addressing the three basic stress types. Because tensile and compressive loads produce stresses that act across a plane, in a direction perpendicular (normal) to the plane, tensile and compressive stresses are called normal stresses. The shorthand designations are as follows. For tensile stresses: "+SN" (or "SN") or "s" (sigma) For compressive stresses: "-SN" or "-s" (minus sigma) The ability of a material to react to compressive stress or pressure is called compressibility. For example, metals and liquids are incompressible, but gases and vapors are compressible. The shear stress is equal to the force divided by the area of the face parallel to the direction in which the force acts, as shown in Figure 1(c) above. Two types of stress can be present simultaneously in one plane, provided that one of the stresses is shear stress. Under certain conditions, different basic stress type combinations may be simultaneously present in the material. An example would be a reactor vessel during operation. The wall has tensile stress at various locations due to the temperature and pressure of the fluid acting on the wall. Compressive stress is applied from the outside at other locations on the wall due to outside pressure, temperature, and constriction of the supports associated with the vessel. In this situation, the tensile and compressive stresses are considered principal stresses. If present, shear stress will act at a 45° angle to the principal stress. Strain / Deformation In continuum mechanics, deformation or strain is the change in the metric properties of a continuous body B in the displacement from an initial placement κ0(B) to a final placement κ(B). A change in the metric properties means that a curve drawn in the initial body placement changes its length when displaced to a curve in the final placement. If all the curves do not change length, it is said that a rigid body displacement occurred. A strain field associated with a displacement is defined, at any point, by the change in length of the tangent vectors representing the speeds of arbitrarily parametrized curves passing through that point. A basic geometric result, due to Fréchet, von Neumann and Jordan, states that, if the lengths of the tangent vectors fulfill the axioms of a norm and the parallelogram law, then the length of a vector is the square root of the value of the quadratic form associated, by the polarization formula, with a positive definite bilinear map called the metric tensor. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined in the initial or in the final placement and on whether the metric tensor or its dual is considered. In a continuous body, a deformation field results from a stress field induced by applied forces or is due to changes in the temperature field inside the body. The relation between stresses and induced strains is expressed by elastic constitutive equations, e.g., Hooke's law for linear elastic materials. Deformations which are recovered after the stress field has been removed, are called elastic deformations. In this case, the continuum completely recovers its original configuration. On the other hand, irreversible deformations, which remain even after stresses have been removed, are called plastic deformations. Such deformations occur in material bodies after stresses have attained a certain threshold value known as the elastic limit or yield stress, and are the result of slip, or dislocation mechanisms at the atomic level. Deformation is measured in units of length. Strain Strain is the geometrical measure of deformation representing the relative displacement between particles in the material body. It measures how much a given displacement differs locally from a rigid-body displacement. Strain defines the amount of stretch or compression along a material line elements or fibers, the normal strain, and the amount of distortion associated with the sliding of plane layers over each other, the shear strain, within a deforming body. Strain is a dimensionless quantity, which can be expressed as a decimal fraction, a percentage or in parts-per notation. This could be applied by elongation, shortening, or volume changes, or angular distortion. The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the normal strain, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the shear strain, radiating from this point. However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions. If there is an increase in length of the material line, the normal strain is called tensile strain, otherwise, if there is reduction or compression in the length of the material line, it is called compressive strain. Strain measures Depending on the amount of strain, or local deformation, the analysis of deformation is subdivided into three deformation theories: 1.	Finite strain theory, also called large strain theory, large deformation theory, deals with deformations in which both rotations and strains are arbitrarily large. In this case, the undeformed and deformed configurations of the continuum are significantly different and a clear distinction has to be made between them. This is commonly the case with elastomers, plastically-deforming materials and other fluids and biological soft tissue. 2.	Infinitesimal strain theory, also called small strain theory, small deformation theory, small displacement theory, or small displacement-gradient theory where strains and rotations are both small. In this case, the undeformed and deformed configurations of the body can be assumed identical. The infinitesimal strain theory is used in the analysis of deformations of materials exhibiting elastic behavior, such as materials found in mechanical and civil engineering applications, e.g. concrete and steel. 3.	Large-displacement or large-rotation theory, which assumes small strains but large rotations and displacements. In each of these theories the strain is then defined differently. The engineering strain is the most common definition applied to materials used in mechanical and structural engineering, which are subjected to very small deformations. On the other hand, for some materials, e.g. elastomers and polymers, subjected to large deformations, the engineering definition of strain is not applicable, e.g. typical engineering strains greater than 1%, thus other more complex definitions of strain are required, such as stretch, logarithmic strain, Green strain, and Almansi strain. The Cauchy strain or engineering strain is expressed as the ratio of total deformation to the initial dimension of the material body in which the forces are being applied. The engineering normal strain or engineering extensional strain e of a material line element or fiber axially loaded is expressed as the change in length ΔL per unit of the original length L of the line element or fibers. The normal strain is positive if the material fibers are stretched or negative if they are compressed. Thus, we have where ℓ is the final length of the fiber. The engineering shear strain is defined as the change in the angle between two material line elements initially perpendicular to each other in the undeformed or initial configuration. The stretch ratio or extension ratio is a measure of the extensional or normal strain of a differential line element, which can be defined at either the undeformed configuration or the deformed configuration. It is defined as the ratio between the final length ℓ and the initial length L of the material line. The extension ratio is related to the engineering strain by This equation implies that the normal strain is zero, so that there is no deformation when the stretch is equal to unity. The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastometers, which can sustain stretch ratios of 3 or 4 before they fail. On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios. The logarithmic strain ε, also called natural strain, true strain or Hencky strain. Considering an incremental strain (Ludwik) the logarithmic strain is obtained by integrating this incremental strain: where e is the engineering strain. The logarithmic strain provides the correct measure of the final strain when deformation takes place in a series of increments, taking into account the influence of the strain path. The Green strain is defined as The Green strain is addressed in more detail in the article on finite strain theory. The Euler-Almansi strain is defined as The Euler-Almansi strain is addressed in more detail in the finite strain theory. Description of deformation It is convenient to identify a reference configuration or initial geometric state of the continuum body which all subsequent configurations are referenced from. The reference configuration need not to be one the body actually will ever occupy. Often, the configuration at t = 0 is considered the reference configuration, κ0(B). The configuration at the current time t is the current configuration. For deformation analysis, the reference configuration is identified as undeformed configuration, and the current configuration as deformed configuration. Additionally, time is not considered when analyzing deformation, thus the sequence of configurations between the undeformed and deformed configurations are of no interest. The components Xi of the position vector X of a particle in the reference configuration, taken with respect to the reference coordinate system, are called the material or reference coordinates. On the other hand, the components xi of the position vector x of a particle in the deformed configuration, taken with respect to the spatial coordinate system of reference, are called the spatial coordinates There are two methods for analysing the deformation of a continuum. One description is made in terms of the material or referential coordinates, called material description or Lagrangian description. A second description is of deformation is made in terms of the spatial coordinates it is called the spatial description or Eulerian description. There is continuity during deformation of a continuum body in the sense that: •	The material points forming a closed curve at any instant will always form a closed curve at any subsequent time. •	The material points forming a closed surface at any instant will always form a closed surface at any subsequent time and the matter within the closed surface will always remain within.

Strength / Mechanics of Materials The following are basic definitions and equations used to calculate the strength of materials. Stress (normal) Stress is the ratio of applied load to the cross-sectional area of an element in tension and is expressed in pounds per square inch (psi) or kg/mm2. Load		 L Stress,   	=	   	= Area		 A Strain (normal) A measure of the deformation of the material that is dimensionless. change in length		  L Strain,   	=	   	= original length		 L

Modulus of elasticity Metal deformation is proportional to the imposed loads over a range of loads. Since stress is proportional to load and strain is proportional to deformation, this implies that stress is proportional to strain. Hooke's Law is the statement of that proportionality. Stress =	  	=	  E   Strain The constant, E, is the modulus of elasticity, Young's modulus or the tensile modulus and is the material's stiffness. Young's modulus is in terms of 106 psi or 103 kg/mm2. If a material obeys Hooke's Law it is elastic. The modulus is insensitive to a material's temper. Normal force is directly dependent upon the elastic modulus. Proportional limit The greatest stress at which a material is capable of sustaining the applied load without deviating from the proportionality of stress to strain. Expressed in psi (kg/mm2). Ultimate strength (tensile) The maximum stress a material withstands when subjected to an applied load. Dividing the load at failure by the original cross sectional area determines the value. Elastic limit The point on the stress-strain curve beyond which the material permanently deforms after removing the load. Yield strength Point at which material exceeds the elastic limit and will not return to its origin shape or length if the stress is removed. This value is determined by evaluating a stress-strain diagram produced during a tensile test. Poisson's ratio The ratio of the lateral to longitudinal strain is Poisson's ratio. lateral strain =	  		 longitudinal strain Poisson's ratio is a dimensionless constant used for stress and deflection analysis of structures such as beams, plates, shells and rotating discs. Bending stress When bending a piece of metal, one surface of the material stretches in tension while the opposite surface compresses. It follows that there is a line or region of zero stress between the two surfaces, called the neutral axis. Make the following assumptions in simple bending theory: 1.	The beam is initially straight, unstressed and symmetric 2.	The material of the beam is linearly elastic, homogeneous and isotropic. 3.	The proportional limit is not exceeded. 4.	Young's modulus for the material is the same in tension and compression 5.	All deflections are small, so that planar cross-sections remain planar before and after bending. Using classical beam formulas and section properties, the following relationship can be derived: 3 PL Bending stress,   b	= 2 w t 2 P L3 Bending or flexural modulus, E  b	= 4 w t 3 y	Where:	P	=	normal force l	=	beam length w	=	beam width t	=	beam thickness y	=	deflection at load point The reported flexural modulus is usually the initial modulus from the stress-strain curve in tension. The maximum stress occurs at the surface of the beam farthest from the neutral surface (axis) and is: M c		 M Max surface stress,   max	=	   	= I		 Z Where:	M	=	bending moment c	=	distance from neutral axis to outer surface where max stress occurs I	=	moment of inertia Z	=	I/c = section modulus

For a rectangular cantilever beam with a concentrated load at one end, the maximum surface stress is given by: 3 d E t max 	= 2 l 2 the methods to reduce maximum stress is to keep the strain energy in the beam constant while changing the beam profile. Additional beam profiles are trapezoidal, tapered and torsion. Where:	d	=	deflection of the beam at the load E	=	Modulus of Elasticity t	=	beam thickness l	=	beam length

Yielding Yielding occurs when the design stress exceeds the material yield strength. Design stress is typically maximum surface stress (simple loading) or Von Mises stress (complex loading conditions). The Von Mises yield criterion states that yielding occurs when the Von Mises stress,    exceeds the yield strength in tension. Often, Finite Element Analysis stress results use Von Mises stresses. Von Mises stress is:

( 1-  2 )2 +  (  2-  3 )2 +  (  1-  3 )2   	=		   	 2 where  1,  2,  3 are principal stresses. Safety factor is a function of design stress and yield strength. The following equation denotes safety factor, fs. Y S fs	= D S Where Y S is the Yield Strength and D S  is the Design Stress Stress & its effects on Materials We have already seen in Mechanical properties of metals that stress on materials results in strain – first elastic strain and then, sometimes (depending on the material), plastic strain. This is shown in Figure 1. Figure 1: Typical stress-strain curves The initial straight-line sections in each graph represent the Modulus of Elasticity (or Young’s Modulus, or stiffness) of the material. If the stress is removed during this elastic strain period, the material will revert to its original shape. However, if the load, or stress, goes beyond this linear section, we enter the area of plastic strain and the material will now never revert to its original shape. One would think that as long as we keep the stress within this linear elastic region, we would be quite safe – the material would not fail. However, this is not always the case, and we will shortly look at three ways in which materials, such as solder joints, can fail under low applied stresses. But first, let us look at the two common types of fracture, ductile and brittle.

Types of failure Fracture is the result of an applied stress and this stress can be tensile, compressive, shear and/or torsional. There are two fracture modes possible – ductile and brittle – depending on the ability of the material to experience plastic deformation. Obviously we do not want failures to occur. However, if they are inevitable, it is far better that we have some warning – for example a gradual change of dimension (strain) occurring in response to the stress – instead of a sudden catastrophic failure with no warning. When a material does strain significantly before failure, it will exhibit a ductile fracture. Ductile fracture Ductile fracture involves plastic deformation in the vicinity of an advancing crack, and is a slow process. It is stable, and will not continue unless there is an increase in the level of applied stress. It normally occurs in a trans-granular manner (across the grains) in metals that have good ductility and toughness. Often, a considerable amount of plastic deformation – including necking – is observed in the failed component. This deformation occurs before the final fracture. Ductile fractures are normally caused by simple overloads or by applying too high a stress to the material, and exhibit characteristic surface features with a significant portion of the fracture surface having an irregular, fibrous face. They also have a small shear lip, where the fracture surface is at a 45° angle to the applied stress. The shear lip, indicating that slip occurred, gives the fracture the cup-and-cone appearance shown in Figure 2. Simple macroscopic observation of this fracture may be sufficient to identify the ductile fracture mode. Figure 2: Macroscopic image of a ductile fracture (x29.7) Examination of the fracture surface at a high magnification – using a scanning electron microscope (SEM) – reveals a dimpled surface. Under a normal tensile stress, these dimples (Figure 3 left) are usually round or equiaxed (having the same dimensions in all directions) – while if shear stress has been dominant, the dimples are oval-shaped or elongated, with the ovals pointing towards the origin of fracture (Figure 3 right). Figure 3: Microscopic images of ductile fractures (x1000) Brittle fracture In brittle fractures, cracks spread very rapidly, with little or no plastic flow, and are so unstable that crack propagation occurs without further increase in applied stress. They occur in high strength metals, in metals with poor ductility and toughness, and in ceramics. Even metals that are normally ductile may fail in a brittle manner at low temperatures, in thick sections, at high strain rates (such as impact), or when flaws play an important role. Brittle fractures are frequently observed when impact rather than overload causes failure. Brittle fracture can be identified by observing the features on the failed surface. Normally, the fracture surface is flat and perpendicular to the applied stress in a tensile test. If a failure occurs by cleavage, each fractured grain is flat and differently oriented, giving a shiny, crystalline appearance to the fracture surface (Figure 4). Figure 4: Macroscopic image of a brittle fracture (x6.5) Initiation of a crack normally occurs at small flaws which cause a concentration of stress. Normally, the crack propagates most easily along specific crystallographic planes by cleavage. However, in some cases, the crack may take an inter-granular (along the grain boundaries) path, particularly when segregation or inclusions weaken the grain boundaries (Figure 5). Note that a crack may propagate at a speed approaching the speed of sound in the material! Figure 5: Microscopic image of intergranular brittle fracture (x1000) Question Obviously we don’t want failures to occur but if they do have to happen, explain why ductile fracture is a preferable failure mode to brittle fracture. Failure by ductile fracture is preferable to brittle fracture because it will involve some plastic deformation and can give a warning that failure is about to occur. Fatigue stresses Fatigue is a form of failure that occurs in materials subjected to fluctuating stresses – for example, solder joints under temperature cycling. Under these circumstances, it is possible for failure to occur at a stress level considerably lower than the tensile or yield strength for a static load. The term ‘fatigue’ is used because this type of failure normally occurs after a lengthy period of repeated stress cycling. It is the single largest cause of failure (approximately 90%) of metallic materials, and polymers and ceramics (other than glasses) are also susceptible to this type of failure. Although failure is slow in coming, catastrophic fatigue failures occur very suddenly, and without warning. Fatigue failure is brittle-like in nature – even in normally ductile metals – in that there is very little, if any, gross plastic deformation associated with failure. The process occurs by the initiation and propagation of cracks, and the fracture surface is usually perpendicular to the direction of an applied stress. A major problem with fatigue is that it is dominated by design. Whilst it is possible to assess the inherent fatigue resistance of a material, the effects of stress-raisers such as surface irregularities and changes in cross-section, as well as the crucial area of jointing (solder joints!) can be a major problem. Failure by fatigue is the result of processes of crack nucleation and growth, or, in the case of components which may contain a crack introduced during manufacture, the result of crack growth only brought about by the application of cyclical stresses. The appearance of a fatigue fracture surface is distinctive and consists of two portions, a smooth portion, often possessing conchoidal, or ‘mussel shell’, markings showing the progress of the fatigue crack up to the moment of final rupture, and the final fast fracture zone. This is shown in Figure 6. Figure 6: Macroscopic image showing fatigue beachmarks (x6.5) The bands visible in the smooth portion, are often referred to as beachmarks. These beachmarks (so called because they resemble ripple marks on a beach) are of macroscopic dimensions – they can be observed with the unaided eye. Each beachmark band represents a period of time over which crack growth occurred. At higher magnifications, using a scanning electron microscope, fatigue striations can be observed (Figure 7). Each striation is thought to represent the advance distance of the crack front during a single load cycle. Figure 7: Scanning electron microscope image of fatigue striations (x1000) An important point regarding fatigue failure is that beachmarks do not occur on the region over which the final rapid failure occurs. This region will exhibit either ductile or brittle failure – evidence of plastic deformation being present for ductile, and absent for brittle failure. The number of cycles that a component can survive without failure depends on the stress amplitude applied. Obviously, the greater the stress amplitude the lower the number of cycles to failure and this is reflected in Figure 8 which exhibits the “Stress amplitude – Number of cycles to failure (or S-N) curve” typical of non-ferrous materials such as solders. Figure 8: S-N curve for non-ferrous materials such as solders The fatigue strength is the maximum permitted stress amplitude for a given number of cycles. The fatigue strength of a solder joint is always less than the fatigue strength of the basic material. This is because it is determined by: •	the size and distribution of defects within the solder alloy and •	the magnitude of the stress concentration factor at the junction of the solder alloy and the parent metal. This important topic will be examined more closely later. Corrosion fatigue is the development and propagation of cracks in a material that is subjected to alternating or fluctuating cycles of load. The presence of a corrosive environment will accelerate the formation and growth of fatigue cracks, thus reducing the fatigue life of the material. Self Assessment Question Explain the difference between the fatigue behaviour of materials and that of components. The fatigue performance of materials is assessed on a polished, defect-free specimen while that of components will include parameters such as service surface finish, discontinuities (such as solder joint interfaces) and environmental issues, The former is a fundamental property of the material while the latter is more useful in real situations. Creep Materials are often placed in service at ‘relatively high temperatures’ and exposed to static mechanical stresses. These stresses are less than the yield strength of the material but nevertheless can cause plastic deformation to take place – particularly over a long period of service time. This phenomenon is known as creep. Note that the term ‘relatively high temperatures’ means high homologous temperatures (Tservice/Tmelting) and is a measure of how near the temperature is to the melting point of the material concerned, as shown in Figure 9 of Mechanical properties of metals. So room temperature (20°C) is a low homologous temperature for steel (melting point around 1600°C), but is a high homologous temperature for tin-lead solder (melting point around 180°C). Creep is observed in all material types – in metals it only becomes important at temperatures greater than about 0.4Tm (where Tm is the melting point in Kelvin). Soft metals such as tin and lead creep at room temperature while aluminium and its alloys creep around 250°C. Steel creeps at about 450°C while nickel-based alloys (nimonics) creep at around 650°C. A typical creep curve is shown in Figure 9. Figure 9: Creep strain versus time graph

The temperature and length of service determines whether or not creep must be considered as a possible mode of failure for a given material component. For example one particular aluminium-copper alloy was used for the forged impellers in the jet engine for the Gloster Meteor aircraft. In this application, the temperature of operation was up to 200°C and the stress was high enough to limit the life to a few hundred hours. Clearly, in this case, 200°C is a creep-producing temperature. The same alloy is used as the skin for Concorde, and over most of this structure the temperature does not exceed 120°C. However, the Concorde airframe is designed for a life in service of 20,000–30,000 hours, and this is a long enough period for 120°C to constitute a possible hazard. Creep is thus important in both applications, even though the temperatures are different.

Combined creep and fatigue Since neither fatigue nor creep while acting on their own is fully understood, the mechanisms involved when they act together are even less well understood. However: •	There is evidence of a synergistic relationship i.e. the sum of their joint effects is greater than their individual contributions •	Factors of little importance to fatigue at room temperature – such as frequency, wave shape and recovery – can become important at high temperatures. Similarly, the mode of failure alters from transcrystalline to intercrystalline •	Uneven heating can lead to thermal fatigue, while oxidation and corrosion result in degradation of creep and fatigue resistance.

Stress raisers Fatigue and creep are two modes of low stress failure. A third mode of failure is caused by the presence of ‘stress raisers’. First investigated by Griffiths in the 1920s, these are microscopic flaws or cracks which always exist, both on the surface and internally, and result in an amplification or concentration of the applied stress at the crack tip. In service, the stress concentrators of importance are crack-like defects and examples include: •	discontinuities in soldered joints, and •	cracks which have grown by fatigue or stress-corrosion mechanisms. It is usually possible to detect such defects, using ultrasonic inspection or radiography, to determine the maximum size of defect in the region of interest. At positions far removed from cracks, the stress is just the nominal stress, that is, the load divided by the cross-sectional area, and this does not pose a problem if the applied stress is below the elastic limit. However, in the vicinity of small cracks or flaws, the situation can be serious. Because of their ability to amplify an applied stress in their locale, such flaws are called ‘stress raisers’. Figure 10: Schematic representation of stress-raising defect With reference to Figure 10, the maximum stress at the crack tip, sm may be approximated by: where so is the applied stress, rt is the radius of curvature of the crack tip and a represents the length of a surface crack or half the length of an internal one. The ratio sm/so is termed the stress concentration factor Kt and is a measure of the degree of stress amplification at the tip of a small crack. While hairline cracks (with a large length to crack tip ratio) are most undesirable, stress amplification also occurs on a macroscopic scale, for example, sharp corners in solder joints. The effects of stress raisers are more significant in brittle than in ductile materials. In ductile materials, plastic deformation allows a more uniform distribution of the stress in the vicinity of the stress raiser and the resultant stress concentration factor is appreciably less than those in brittle materials. In the 1920s, Griffith proposed that fracture occurs when the theoretical cohesive strength is exceeded at the tip of one of the numerous flaws existing in most materials. If no flaws were present the fracture strength would be equal to the cohesive strength of the material. Very small, virtually defect-free metallic and ceramic whiskers have been grown with fracture strengths approaching theoretical values. In many engineering situations where there is a measure of cyclical stressing (including electronic solder joints), we must be aware that tiny, sub-critical cracks can grow and become critical. Fast fracture will then occur. Self Assessment Question Real metallic components often contain internal flaws of 8 mm in length and 2 mm tip radius. Calculate the ‘stress concentration factor’ due to such a defect. The stress concentration factor, Kt is the ratio sm/so and the equation we are looking for is: For the flaw dimensions given, we will get a value for sm of 2so.v[8/2] i.e. 4so So the stress concentration factor is 4so / so = 4. References 	www.engineersedge.com/mechanics_material_menu.shtml 	www.ami.ac.uk/courses/topics/0124_seom/index.html