The deformation efficiency concept is attractive and has demonstrably led to a better understanding of the important parameters to be developed to give a meaningful data-base for any given material of interest .sx Lyapunov function stability criteria seem to provide powerful tools for delineating stable and unstable regions on these mappings of deformation behaviour .sx Apparently , these regions so discovered do actually correspond to material instabilities well-known to materials scientists dealing with both metals and non-metals .sx One remaining problem , however , is that of adequately accounting for the history of the deformation .sx Modelling of dynamic material behaviour in hot deformation can most easily be included by using this mapping approach .sx The data-base so developed could then be used to provide data as a function of temperature , strain , strain rate and time ( to include the history of deformation ) which can then be fed into existing comprehensive computer solutions for rolling , for example .sx However , many computer solutions are of the slab type ( see later discussion of plastic flow ) and do not include either temperature or strain rate effects .sx Much work has been done in industry on this problem , but it has not been published widely .sx 5.4 PHYSICAL PROPERTIES OF IMPORTANCE IN MANUFACTURE AND USE .sx Analysis of processes such as hot forming , casting or others used at high temperatures requires a knowledge of thermal conductivity kappa , specific heat c , and thermal expansion alpha .sx The last is especially important in resistance to thermal shock .sx A material such as silicon-aluminium-oxy-nitride has a much lower thermal expansion coefficient than , say , aluminium oxide and it is much more resistant to cracking when suddenly heated or cooled .sx A high thermal conductivity is also valuable in reducing any large temperature gradients which might otherwise occur in a material or composite .sx When composites are used it is necessary to consider the compatibility of expansion coefficients , otherwise high stresses and even plastic deformation may occur .sx 5.5 CHEMICAL PROPERTIES .sx Oxidation is clearly important in hot working operations although in an enclosed process like extrusion , for example , there is very little access of air , so oxidation is substantially avoided .sx The book edited by Braithwaite explains how chemical reaction with lubricants plays a major role in the effectiveness of lubricants .sx Titanium , for example , is very difficult to lubricate because of its thin inert oxide skin .sx Schey points out that stainless steels present the same problem .sx Corrosion is also a critical feature .sx A high proportion of all metallic scrap arises from rusting and related electrochemical degradation processes .sx .sx Much attention has been given to plating and other protective coatings to avoid corrosion and also to improve other surface properties such as hardness and wear resistance ( see the handbooks by Swann , Ford and Westwood and also by Peterson and Winer .sx 5.6 WORKABILITY .sx Although industrial producers will readily recognize some alloys as being easier to work than others , there is no simple test or clearly defined set of properties .sx Ductility is a related property , and it is generally considered that a greater reduction of area to failure in a tensile test indicates better workability .sx While this is true in general terms , the actual fracture in a tensile test depends upon the complex stress state in the neck region for all but the most brittle alloys .sx In the comprehensive book by Atkins and Mai , various criteria are mentioned which have often been proposed on the supposition that fracture in tension depends upon the total plastic work or the tensile plastic work expended in the deformation .sx It is very difficult to obtain accurate experimental data because the final stress state before fracture is frequently ill-defined .sx Recent analytical work using finite-element plasticity has given a better insight into these problems .sx Hot workability is often described in terms of the number of revolutions to failure in a torsion test , but again the analytical significance is obscure .sx Torsion testing is also a valuable indicator of cold workability , but the results depend strongly upon the axial constraint , possibly because fracture and rewelding occur during the test , so the conditions must be carefully controlled .sx 5.7 ELEMENTS OF THE THEORY OF PLASTICITY .sx 5.7.1 General Theory .sx To understand in more detail the mechanics of the forming processes which are being discussed , it is necessary to understand the fundamentals underlying the theory of the plastic flow of materials .sx This is the one unifying theme that underlies all forming processes .sx A complete treatment of the mathematical theory of plasticity is beyond the scope or intent of this book ; comprehensive early texts have been written by Nadai , Hill , and Prager and Hodge .sx Later treatments of the theory of plasticity , discussed specifically with the needs of engineers in mind , are given by Ford with Alexander , Johnson and Mellor , and Alexander .sx A simpler , more practical account is given by Rowe .sx What it is intended to do here is to consider the basic fundamentals on which the theory rests and thereby to show how a better understanding of the behaviour of materials subjected to these processes can be achieved .sx In the first place the question may be asked :sx 'What is the main difference between the behaviour of metal when subjected to large deformation in metal-working processes and its behaviour when stressed within the elastic range ?sx ' The simple answer is that in metal-working processes the amount of straining or deformation is many times larger .sx It is this essential difference between metal-working and elastic straining which is all-important in formulating a workable theory .sx The solution of a problem in elasticity is often achieved by making the assumption that the external shape is unaltered during elastic straining .sx That this is justified for most metals can be realized when it is remembered that their elastic range never exceeds about 0.4 per cent strain , corresponding with a change in external dimensions of 0.4 per cent , and is often only 0.1 per cent .sx By their very nature the processes of metal-working demand that strains of more than one hundred times this value be imposed .sx Indeed , in the case of forging or extrusion , reductions in cross-sectional area of the order of 95 per cent are found .sx For such large deformations a more precise definition of strain is necessary than simply that of '(change in length)/(original length)' .sx For example , if a tensile specimen is stretched to double its length the engineer's strain would be unity , or 100 per cent .sx To achieve 100 per cent compressive strain ( engineer's strain ) it would obviously be necessary to compress the specimen until it had zero thickness .sx Clearly , then , this measure of strain is unsuitable for such large deformations , since 100 per cent compression certainly represents much more straining than does 100 per cent extension .sx Thus can be realized the first result of having such large deformations , namely the necessity for specifying some better measure of strain .sx Another consequence of having large strains of this order is that , as far as determining the changes of shape and the forces required is concerned , it will be permissible to neglect elastic strains .sx This considerably simplifies the theoretical approach , ( but it should be remembered that elastic deformation is very important when considering residual stresses) .sx Bearing in mind the large deformations which are imposed in metal-working , another question can be asked :sx 'Why is it that , in a tensile test , the metal will rarely withstand more than about 30 per cent elongation without fracture , whereas in metal - working processes much larger strains can be imposed ?sx ' Consideration of the major processes reveals the answer to this question - they are all predominantly compressive in nature .sx Any stress system can be regarded as the sum of an all-round hydrostatic stress ( usually taken as being equal to the mean stress ) and stresses equal to the differences between the actual stresses and this mean stress .sx It was shown experimentally by Bridgman , and later again by Crossland , that there is no plastic flow occasioned by this hydrostatic stress .sx The plastic flow is caused by the deviatoric stress system , so-called because the stresses take the values by which the actual stresses 'deviate' from this mean stress .sx Although the mean stress is not responsible for any plastic flow it has a profound influence on fracture , and the more compressive its value the more deformation can be imposed before failure occurs owing to fracture .sx The exact dependence is not properly understood , but the tests of Bridgman and Crossland have given us much information .sx A corollary of the observation that the mean stress occasions no plastic flow is that there can be no permanent change of volume .sx Thus , if elastic strains are neglected , it may reasonably be assumed that there is no change of volume at all during plastic flow .sx Now , almost all metal-working processes take place under conditions of complex stressing .sx In other words , if uniaxial tension or compression is regarded as a simple system of stressing , in which there is only one principal stress acting along the axial direction of the prismatic specimen , then most metal - working processes involve more complicated systems of stress .sx Thus , in formulating a theory , a general three-dimensional system of stress must be introduced and yet another question must be asked :sx 'How can the behaviour of metal subjected to a complex system of stress be correlated with its behaviour in simple tension or compression ?sx ' To answer this question it is convenient to invoke the concepts of equivalent stress ( or strain ) and effective stress ( or strain) .sx These are parameters which are functions of the imposed complex stresses or strains which quantify their effectiveness in causing the plastic flow of materials .sx Having discussed the principles on which the theory rests , it is now possible to develop the equations of plastic flow .sx Before so doing , it is well to recall the equations of three-dimensional elasticity , to compare them with those for plastic flow .sx Initially , what may be called the laws of elasticity and the laws of plasticity can be set up , as follows :sx Laws of Elasticity .sx Hooke's Law .sx Equations of Equilibrium ( of stresses and forces ) .sx Equations of Compatibility ( of strains and displacements ) .sx Laws of Plasticity .sx Stress-Strain Relations .sx Equations of Equilibrium .sx Equations of Compatibility .sx Yield Criterion ( function of stresses necessary to initiate and maintain plastic flow ) .sx To these must be added the boundary conditions for both stresses and displacements .sx Because of the magnitude of the strains and also the non - linearity of the relationships involved in metal-working processes it is necessary to formulate the equations in terms of incremental strains or strain rates .sx This difficulty does not arise for elastic straining , since the strains are always sufficiently small for the total strains to be used .sx Thus , Hooke's Law for a three - dimensional system of stress , referred to x,y,z cartesian coordinates , is as follows :sx formulae .sx In these equations , tau xy is the stress in the y direction acting on a plane normal to the x direction , gamma xy is the corresponding engineering shear strain , E is Young's Modulus , G is the shear modulus , nu is Poisson's ratio , and formula .sx Equations ( 5.26 ) are the well-known equations of elasticity representing Hooke's law .sx Considering the first one of these equations , a positive ( tensile ) stress sigma xx in the x direction produces a strain of formula in that direction , whilst the positive tensile stress sigma yy will produce a negative strain ( contraction ) of formula in the x direction , and sigma zz similarly produces a contraction .sx By adding together the first three equations of equations ( 5.26 ) the elastic volume change is determined as :sx formula .sx ( The factor formula is three times the bulk modulus K of the material , since formula is the mean stress , or hydrostatic component of the stress system .sx ) .sx Thus , if the volume change is to be zero in plastic flow , Poisson's ratio nu must be replaced by the factor 1/2 in the stress-strain relations , since the right-hand side of equation ( 5.27 ) then becomes zero .sx Young's Modulus E has no meaning for plastic flow in which the elastic strains are neglected , and must be replaced by an analogous parameter which will be considered later .sx In any element of material subjected to a complex system of stress there are three mutually perpendicular directions in which the local direct stresses attain either maximum or minimum values .sx