CHAPTER 15 THEORIES OF ELASTIC FAILURE Summary TABLE 15.1 Theory Value in tension Value in complex Criterion for test at failure stress system failure Maximum principal stress (Rankine) 01 61=0, Maximum shear stress (Guest-Tresca) 扣， (a1-) 01m03=0, Maximum principal strain (Saint-Venant) E 01-VG2-03=0, Total strain energy per unit volume 2i++时 c1+c3+a3-2va102+0203 (Haigh) 2E +0301)=0 -2v(G12+0203+a301)] Shear strain energy [(c1-022+(a2-032 per unit volume +(o3-01]=时 Distortion energy theory (Maxwell-Huber- 器 12G[a,-P+a2-, von Mises) +(o-0] Modified shear stress 4+2=1 Internal friction theory 0y0. (Mohr) Introduction When dealing with the design of structures or components the physical properties of the constituent materials are usually found from the results of laboratory experiments which have only subjected the materials to the simplest stress conditions.The most usual test is the simple tensile test in which the value of the stress at yield or at fracture(whichever occurs first) is easily determined.The strengths of materials under complex stress systems are not generally known except in a few particular cases.In practice it is these complicated systems of stress which are more often encountered,and therefore it is necessary to have some basis for 401
CHAPTER 15 THEORIES OF ELASTIC FAILURE Maximum principal stress (Rankine) (Guest-Tresca) strain (Saint- Venant) Maximum sbear stress Maximum principal Total strain energy per unit volume (Haigh) Summary TABLE 15.1 UY +% UY E - - U: 2E Value in tension 1 1 test at failure Theory I I Sbear strain energy per unit volume Distortion energy tbeory (Maxwell-Huhuon Mises) 4 6G Modified sbear stress Internal friction theory Value in complex stress system 1 2E - [Uf + u: +a: - 2V(UlU2 + u2u3 + u3u1)] 1 12G -[(a1 - Criterion for failure u1 =uy 61 - 03 = 6, u, - vu2 - vu3 = cry u: + u: + u: - 2v(u,u2 + u2u3 + u3u1) = u; Introduction When dealing with the design of structures or components the physical properties of the constituent materials are usually found from the results of laboratory experiments which have only subjected the materials to the simplest stress conditions. The most usual test is the simple tensile test in which the value of the stress at yield or at fracture (whichever occurs first) is easily determined. The strengths of materials under complex stress systems are not generally known except in a few particular cases. In practice it is these complicated systems of stress which are more often encountered, and therefore it is necessary to have some basis for 40 1
402 Mechanics of Materials §15.1 determining allowable working stresses so that failure will not occur.Thus the function of the theories of elastic failure is to predict from the behaviour of materials in a simple tensile test when elastic failure will occur under any condition of applied stress. A number of theoretical criteria have been proposed each seeking to obtain adequate correlation between estimated component life and that actually achieved under service load conditions for both brittle and ductile material applications.The five main theories are: (a)Maximum principal stress theory (Rankine). (b)Maximum shear stress theory (Guest-Tresca). (c)Maximum principal strain (Saint-Venant). (d)Total strain energy per unit volume(Haigh). (e)Shear strain energy per unit volume (Maxwell-Huber-von Mises). In each case the value of the selected critical property implied in the title of the theory is determined for both the simple tension test and a three-dimensional complex stress system. These values are then equated to produce the so-called criterion for failure listed in the last column of Table 15.1. In Table 15.1 o,is the stress at the yield point in the simple tension test,and 2 and 3 are the three principal stresses in the three-dimensional complex stress system in order of magnitude.Thus in the case of the maximum shear stress theory o-3 is the greatest numerical difference between two principal stresses taking into account signs and the fact that one principal stress may be zero. Each of the first five theories listed in Table 15.1 will be introduced in detail in the following text,as will a sixth theory,(f)Mohr's modified shear stress theory.Whereas the previous theories (a)to (e)assume equal material strength in tension and compression,the Mohr's modified theory attempts to take into account the additional strength of brittle materials in compression. 15.1.Maximum principal stress theory This theory assumes that when the maximum principal stress in the complex stress system reaches the elastic limit stress in simple tension,failure occurs.The criterion of failure is thus 01=0y It should be noted,however,that failure could also occur in compression if the least principal stress o3 were compressive and its value reached the value of the yield stress in compression for the material concerned before the value of ay was reached in tension.An additional criterion is therefore 3=o,(compressive) Whilst the theory can be shown to hold fairly well for brittle materials,there is considerable experimental evidence that the theory should not be applied for ductile materials.For example,even in the case of the pure tension test itself,failure for ductile materials takes place not because of the direct stresses applied but in shear on planes at 45 to the specimen axis. Also,truly homogeneous materials can withstand very high hydrostatic pressures without failing,thus indicating that maximum direct stresses alone do not constitute a valid failure criteria for all loading conditions
402 Mechanics of Materials $15.1 determining allowable working stresses so that failure will not occur. Thus the function of the theories of elastic failure is to predict from the behaviour of materials in a simple tensile test when elastic failure will occur under any condition of applied stress. A number of theoretical criteria have been proposed each seeking to obtain adequate correlation between estimated component life and that actually achieved under service load conditions for both brittle and ductile material applications. The five main theories are: (a) Maximum principal stress theory (Rankine). (b) Maximum shear stress theory (Guest-Tresca). (c) Maximum principal strain (Saint-Venant). (d) Total strain energy per unit volume (Haigh). (e) Shear strain energy per unit volume (Maxwell-Huber-von Mises). In each case the value of the selected critical property implied in the title of the theory is determined for both the simple tension test and a three-dimensional complex stress system. These values are then equated to produce the so-called criterion for failure listed in the last column of Table 15.1. In Table 15.1 u,, is the stress at the yield point in the simple tension test, and ul, u2 and u3 are the three principal stresses in the three-dimensional complex stress system in order of magnitude. Thus in the case of the maximum shear stress theory u1 -u3 is the greatest numerical difference between two principal stresses taking into account signs and the fact that one principal stress may be zero. Each of the first five theories listed in Table 15.1 will be introduced in detail in the following text, as will a sixth theory, (f) Mobr's modified sbear stress theory. Whereas the previous theories (a) to (e) assume equal material strength in tension and compression, the Mohr's modified theory attempts to take into account the additional strength of brittle materials in compression. 15.1. Maximum principal stress theory This theory assumes that when the maximum principal stress in the complex stress system reaches the elastic limit stress in simple tension, failure occurs. The criterion of failure is thus Ul = by It should be noted, however, that failure could also occur in compression if the least principal stress u3 were compressive and its value reached the value of the yield stress in compression for the material concerned before the value of u,,, was reached in tension. An additional criterion is therefore uj = up (compressive) Whilst the theory can be shown to hold fairly well for brittle materials, there is considerable experimental evidence that the theory should not be applied for ductile materials. For example, even in the case of the pure tension test itself, failure for ductile materials takes place not because of the direct stresses applied but in shear on planes at 45" to the specimen axis. Also, truly homogeneous materials can withstand very high hydrostatic pressures without failing, thus indicating that maximum direct stresses alone do not constitute a valid failure criteria for all loading conditions
s15.2 Theories of Elastic Failure 403 15.2.Maximum shear stress theory This theory states that failure can be assumed to occur when the maximum shear stress in the complex stress system becomes equal to that at the yield point in the simple tensile test. Since the maximum shear stress is half the greatest difference between two principal stresses the criterion of failure becomes (a1-03)=(a,-0) 1.e. 0103=0y (15.1) the value of o3 being algebraically the smallest value,i.e.taking account of sign and the fact that one stress may be zero.This produces fairly accurate correlation with experimental results particularly for ductile materials,and is often used for ductile materials in machine design.The criterion is often referred to as the"Tresca"theory and is one of the widely used laws of plasticity. 15.3.Maximum principal strain theory This theory assumes that failure occurs when the maximum strain in the complex stress system equals that at the yield point in the tensile test, i.e. 01-V02-V03=0y (15.2) This theory is contradicted by the results obtained from tests on flat plates subjected to two mutually perpendicular tensions.The Poisson's ratio effect of each tension reduces the strain in the perpendicular direction so that according to this theory failure should occur at a higher load.This is not always the case.The theory holds reasonably well for cast iron but is not generally used in design procedures these days. 15.4.Maximum total strain energy per unit volume theory The theory assumes that failure occurs when the total strain energy in the complex stress system is equal to that at the yield point in the tensile test. From the work of $14.17 the criterion of failure is thus 2E[o子+i+i-2vo102+a203+031】= 2E i.e. +3+-2w(o102+0203+03c1)= (15.3) The theory gives fairly good results for ductile materials but is seldom used in preference to the theory below. 15.5.Maximum shear strain energy per unit volume (or distortion energy)theory Section 14.17 again indicates how the strain energy of a stressed component can be divided into volumetric strain energy and shear strain energy components,the former being
$15.2 Theories of Elastic Failure 403 15.2. Maximum shear stress theory This theory states that failure can be assumed to occur when the maximum shear stress in the complex stress system becomes equal to that at the yield point in the simple tensile test. Since the maximum shear stress is half the greatest difference between two principal stresses the criterion of failure becomes i.e. (15.1) the value of a3 being algebraically the smallest value, i.e. taking account of sign and the fact that one stress may be zero. This produces fairly accurate correlation with experimental results particularly for ductile materials, and is often used for ductile materials in machine design. The criterion is often referred to as the “Tresca” theory and is one of the widely used laws of plasticity. 15.3. Maximum principal strain theory This theory assumes that failure occurs when the maximum strain in the complex stress system equals that at the yield point in the tensile test, i.e. a1 --a2 -va3 = ap This theory is contradicted by the results obtained from tests on flat plates subjected to two mutually perpendicular tensions. The Poisson’s ratio effect of each tension reduces the strain in the perpendicular direction so that according to this theory failure should occur at a higher load. This is not always the case. The theory holds reasonably well for cast iron but is not generally used in design procedures these days. (15.2) 15.4. Maximum total strain energy per unit volume theory The theory assumes that failure occurs when the total strain energy in the complex stress system is equal to that at the yield point in the tensile test. From the work of $14.17 the criterion of failure is thus - 1 0: [a: + a: + a: - 2v(a,a, + a2a3 + a3a1)] = - 2E 2E i.e. uf + af +a: - 2v(a1 a2 + aza3 + 03~1) = a: (15.3) The theory gives fairly good results for ductile materials but is seldom used in preference to the theory below. 15.5. Maximum shear strain energy per unit volume (or distortion energy) theory Section 14.17 again indicates how the strain energy of a stressed component can be divided into volumetric strain energy and shear strain energy components, the former being
404 Mechanics of Materials §15.6 associated with volume change and no distortion,the latter producing distortion of the stressed elements.This theory states that failure occurs when the maximum shear strain energy component in the complex stress system is equal to that at the yield point in the tensile test, i.e. 【oi-o+o,-,P+a,-a门-装 1 (eqn.(14.23a) or 6G[o+i+号-a12+20+3)=6记 (c1-0z)2+(c2-c32+(c3-1)2=2a3 (15.4) This theory has received considerable verification in practice and is widely regarded as the most reliable basis for design,particularly when dealing with ductile materials.It is often referred to as the "von Mises"'or "Maxwell"criteria and is probably the best theory of the five.It is also sometimes referred to as the distortion energy or maximum octahedral shear stress theory. In the above theories it has been assumed that the properties of the material in tension and compression are similar.It is well known,however,that certain materials,notably concrete, cast iron,soils,etc.,exhibit vastly different properties depending on the nature of the applied stress.For brittle materials this has been explained by Griffith,t who has introduced the principle of surface energy at microscopic cracks and shown that an existing crack will propagate rapidly if the available elastic strain energy release is greater than the surface energy of the crack.In this way Griffith indicates the greater seriousness of tensile stresses compared with compressive ones with respect to failure,particularly in fatigue environments. A further theory has been introduced by Mohr to predict failure of materials whose strengths are considerably different in tension and shear;this is introduced below. 15.6.Mohr's modified shear stress theory for brittle materials (sometimes referred to as the internal friction theory) Brittle materials in general show little ability to deform plastically and hence will usually fracture at,or very near to,the elastic limit.Any of the so-called "yield criteria"introduced above,therefore,will normally imply fracture of a brittle material.It has been stated previously,however,that brittle materials are usually considerably stronger in compression than in tension and to allow for this Mohr has proposed a construction based on his stress circle in the application of the maximum shear stress theory.In Fig.15.1 the circle on diameter OA is that for pure tension,the circle on diameter OB that for pure compression and the circle centre O and diameter CD is that for pure shear.Each of these types of test can be performed to failure relatively easily in the laboratory.An envelope to these curves,shown dotted,then represents the failure envelope according to the Mohr theory.A failure condition is then indicated when the stress circle for a particular complex stress condition is found to cut the envelope. t A.A.Griffith,The phenomena of rupture and flow of solids,Phil.Trans.Royal Soc.,London,1920. J.F.Knott,Fundamentals of Fracture Mechanics (Butterworths,London),1973
404 Mechanics of Materials $15.6 associated with volume change and no distortion, the latter producing distortion of the stressed elements. This theory states that failure occurs when the maximum shear strain energy component in the complex stress system is equal to that at the yield point in the tensile test, i.e. or 1 u2 - 6G c0: + 0: + 0: - (ala2 + 0203 + O3O1) = 2 6G .. (a1 - a2)2 + (a2 - up)? + (a3 - 61 )2 = 24 (15.4) This theory has received considerable verification in practice and is widely regarded as the most reliable basis for design, particularly when dealing with ductile materials. It is often referred to as the “von Mises” or “Maxwell” criteria and is probably the best theory of the five. It is also sometimes referred to as the distortion energy or maximum octahedral shear stress theory. In the above theories it has been assumed that the properties of the material in tension and compression are similar. It is well known, however, that certain materials, notably concrete, cast iron, soils, etc., exhibit vastly different properties depending on the nature of the applied stress. For brittle materials this has been explained by Griffith,? who has introduced the principle of surface energy at microscopic cracks and shown that an existing crack will propagate rapidly if the available elastic strain energy release is greater than the surface energy of the crack.$ In this way Griffith indicates the greater seriousness of tensile stresses compared with compressive ones with respect to failure, particularly in fatigue environments. A further theory has been introduced by Mohr to predict failure of materials whose strengths are considerably different in tension and shear; this is introduced below. 15.6. Mohr’s modified shear stress theory for brittle materials (sometimes referred to as the internal friction theory) Brittle materials in general show little ability to deform plastically and hence will usually fracture at, or very near to, the elastic limit. Any of the so-called “yield criteria” introduced above, therefore, will normally imply fracture of a brittle material. It has been stated previously, however, that brittle materials are usually considerably stronger in compression than in tension and to allow for this Mohr has proposed a construction based on his stress circle in the application of the maximum shear stress theory. In Fig. 15.1 the circle on diameter OA is that for pure tension, the circle on diameter OB that for pure compression and the circle centre 0 and diameter CD is that for pure shear. Each of these types of test can be performed to failure relatively easily in the laboratory. An envelope to these curves, shown dotted, then represents the failure envelope according to the Mohr theory. A failure condition is then indicated when the stress circle for a particular complex stress condition is found to cut the envelope. t A. A. Griffith, The phenomena of rupture and flow of solids, Phil. Trans. Royal SOC., London, 1920. $ J. F. Knott, Fundamentals of Fracture Mechanics (Butterworths, London), 1973
§15.6 Theories of Elastic Failure 405 Pure shear to yield D Pure compression Pure tensior to yield to yield Fig.15.1.Mohr theory on o-t axes. As a close approximation to this procedure Mohr suggests that only the pure tension and pure compression failure circles need be drawn with OA and OBequal to the yield or fracture strengths of the brittle material.Common tangents to these circles may then be used as the failure envelope as shown in Fig.15.2.Circles drawn tangent to this envelope then represent the condition of failure at the point of tangency. Fig.15.2.Simplified Mohr theory on o-t axes. In order to develop a theoretical expression for the failure criterion,consider a general stress circle with principal stresses of a and o2.It is then possible to develop an expression relatingo,2,the principal stresses,andthe yield strengths of the brittle material in tension and compression respectively. From the geometry of Fig.15.3, KL JL KMMH Now,in terms of the stresses, KL=(a1+02)-01+0%=(c%-01+02) KM=o%+和y.=(o%+c) JL=(a1+02)-0y=(o1+02-0 MH=0%.-o%=(og-0y)
$15.6 Theories of Elastic Failure 405 Y I Fig. 15.1. Mohr theory on 0-T axes. As a close approximation to this procedure Mohr suggests that only the pure tension and pure compression failure circles need be drawn with OA and OB equal to the yield or fracture strengths of the brittle material. Common tangents to these circles may then be used as the failure envelope as shown in Fig. 15.2. Circles drawn tangent to this envelope then represent the condition of failure at the point of tangency. r Fig. 15.2. Simplified Mohr theory on g-7 axes. In order to develop a theoretical expression for the failure criterion, consider a general stress circle with principal stresses of o1 and 02. It is then possible to develop an expression relating ol, 02, the principal stresses, and o,,, o,,, the yield strengths of the brittle material in tension and compression respectively. From the geometry of Fig. 15.3, KL JL KM MH -=- Now, in terms of the stresses, KL =$(.I +o,)-oa, +$c~,=$(D~,-Q~ +a,) K M = $a,, + *oyc = f (oY, + on) JL = $(01+ 02) -$o,, = $(GI + 62 - oy,) MH='~ -Lo -o ) 2 Yc 2 Y, 2 Yc Y