Technology & Engineering
Von Mises and Tresca Criteria
The Von Mises and Tresca Criteria are mathematical models used in material science and engineering to predict the yield and failure of materials under stress. The Von Mises Criterion is based on the concept of equivalent stress, while the Tresca Criterion focuses on the maximum shear stress. These criteria are essential for designing and analyzing the structural integrity of various engineering components.
Written by Perlego with AI-assistance
Related key terms
1 of 5
10 Key excerpts on "Von Mises and Tresca Criteria"
eBook - PDF- Roger T. Fenner, J.N. Reddy(Authors)
- 2012(Publication Date)
- CRC Press(Publisher)
In practice, ratios for most ductile materials are closer to 0.577 than to 0.5. Measured yield behavior under more complex states of stress also tends to confirm the von Mises criterion as the more accurate. The Tresca maximum shear stress criterion nevertheless has a useful role to play. It is somewhat easier to use than von Mises, and has the important merit of being moderately conservative. Figure 9.38 shows that the Tresca locus either lies within or just touches the von Mises locus, so that it tends to predict the onset of yielding at stress levels somewhat below the actual ones. The difference, however, is not more than the 15% for the case of pure shear. A strategy sometimes used is to locate points of possible yield in a member with the aid of Tresca, and then to find the equivalent stress more accurately using von Mises. Figure 9.38 514 TRANSFORMATIONS OF STRESS AND STRAIN It is instructive to extend the plotting of yield criteria to three-dimensional states of stress. Figure 9.39 does this for Tresca and von Mises, the two-dimensional loci becoming three-dimensional yield surfaces or envelopes. An envelope represents the interface between elastic states of stress inside and plastic outside, according to the particular criterion of yielding. The von Mises envelope is a circular cylinder with its geometric axis lying along the line σ 1 = σ 2 = σ 3 of equal principal stresses, which is equally inclined to the three principal stress axes. The Tresca surface has the same geometric axis but has a cross section in the form of a regular hexagon just touching the von Mises cylinder at six positions around its circumference. The loci formed at the intersections of these surfaces with the ( σ 1 , σ 2 ) plane remain as shown in Figure 9.38. The line σ 1 = σ 2 = σ 3 is of particular interest, because according to both criteria yielding never occurs, irrespective of the magnitude of the stresses.
- Robert King(Author)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
• Several criteria can be used to define failure. The Von Mises and Tresca Criteria are used with ductile materials. Mohr-Coulomb and normal stress are for brittle materials. • In the von Mises criterion, the maximum value of the distortion strain-energy per unit volume of a material from an FEA is compared with the distortion strain-energy per unit volume required to cause yield. von Mises FOS / yld vM s s 5 If the FOS is greater than 1, the vM stress did not exceed the yield strength of the material. • The Tresca, or maximum shear-stress theory, criterion compares 2 max t to the yield strength. y Tresca FOS / Tresca s s 5 The Tresca criterion is more conservative than the von Mises criterion except in uniaxial or equibiaxial loading states where they are equal. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. Chapter 7 Failure Criteria 244 • The maximum normal stress, or Coulomb, criterion for failure of brittle materials consid-ers that the fracture plane in a brittle material is normal to the first principal stress direc-tion. In this criterion, the first principal stress value is compared with the ultimate tensile strength of the material: Coloumb FOS / ut 1 s s 5 • The maximum normal stress theory applies to materials with equal strength in tension and compression. The Mohr-Coloumb criterion applies to brittle materials with different strengths in compression and tension. This criterion compares the first and third principal stresses to the tensile and compressive strengths. • FOS plots distribution with color contours or under and over regions.
eBook - PDFMechanics of Biomaterials
Fundamental Principles for Implant Design
- Lisa A. Pruitt, Ayyana M. Chakravartula(Authors)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
The Tresca yield criterion indicates that the shear stress needed for plastic deformation is one-half of the uniaxial yield strength, and von Mises yield criterion indicates that the required shear stress for permanent strain is approximately 58% of the uniaxial yield strength for the material. Figure 8.16 Illustration of (a) hydrostatic pressure or triaxial stress state showing (b) the Mohr’s circle, (c) Tresca yield criterion, and (d) von Mises yield criterion. But criteria predict no yield (both the Tresca stress and the von Mises effective stress are equal to zero). the yield criteria no longer serve as the best assessment for failure. Instead, the critical normal stresses become better predictors of failure. Furthermore, it should be noted that the presence of a defect in the material subjected to triaxial tensile stresses war- rants the use of fracture mechanics, rather than yield criteria, for the safe design of the component. 259 8.6 Predicting yield in multiaxial loading conditions Example 8.3 Multiaxial loading in the femoral stem (i) Consider a cemented metallic femoral stem that is isotropic and is subjected to a multiaxial stress state as shown in Figure 8.17(a). These multiaxial stresses arise from several sources. From the joint contact force on the head, there is a net tensile bending stress (48 MPa) on the lateral side of the stem, a shear force of 16 MPa due to the load transmitted to the bone through the bone cement, and a compressive force in the transverse direction due to press fitting the implant in the bone ( −32 MPa). Assume that the alloy used in this device is made of an isotropic alloy that yields through a ductile shear mechanism. The orthopedic alloy has a tensile yield strength of 900 MPa. Calculate the effective (von Mises) stress for this implant – how does this compare to the magnitude of the individual stress components? Determine whether this component is safely designed against yielding (calculate the factor of safety).
- Gregory Szuladzinski(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
The yield strength formulations are often extended to failure or fracture. The maximum shear stress theory associated with the name of Tresca, is probably the most popular, as far as hand calculations of metallic structures are concerned. It says that yielding takes place when the maximum shear stress attains its critical value. The Mohr cir-cle (Appendix A) tells us that in a simple tension of magnitude σ, the largest shear is σ /2 (at 45° to the direction of tension); consequently, a simple form of yield condition is to say that m y /2 F τ = τ = (6.1) If the state of stress is three-dimensional (3D), then we have three principal stresses: σ 1 , σ 2 , and σ 3 and three Mohr circles representing three mutually perpendicular cutting planes. According to Tresca’s theory, the material remains elastic if none of the principal stress difference exceeds F y . This is equivalent to writing 1 2 y 2 3 y 3 1 y ; ; and < F F F σ − σ < σ − σ < σ − σ (6.2) 216 Formulas for Mechanical and Structural Shock and Impact For a 2D state of stress σ 3 = 0, so the above equation is reduced to 1 2 y 2 y 1 y ; ; and < F F F σ − σ < σ < σ (6.3) Equation 6.2 takes precedence when the principal com-ponents are of opposite signs, because then the largest Mohr circle is the one built on σ 1 and σ 2 . In Figure 6.2 the envelope of yield points, according to this theory, is shown by the thinner lines for a 2D case. A more recent theory, dating back to the beginning of twentieth century, is that of distortion energy theory , also referred to as Huber–Mises (HM) hypothesis, as mentioned before. * According to this concept, the state of stress can be resolved into a volumetric or hydro-static component and a distortional or shear compo-nent.
eBook - ePub- Wei Jiang(Author)
- 2019(Publication Date)
- Wiley(Publisher)
It is also called von Mises or von Mises–Hencky theory, giving credit to R. von Mises and H. Hencky for the development of it. When applying this theory, it is convenient to use an equivalent stress σ e, or von Mises stress, to transform multiaxial stresses into an equivalent uniaxial stress. The equivalent stress is the value of a uniaxial tensile stress that would produce the same level of distortion energy as the actual stresses involved [ 6, 7 ]. By the distortion energy theory, the equivalent stress can be derived and expressed by principal stresses as 2.8 Thus, the yield strength criterion is expressed as 2.9 When using xyz components of the stress, the von Mises stress and yield strength criterion can be rewritten as 2.10 For pure shear in plane problem where σ x = σ y = 0, the yield strength can be obtained from Eq. (2.10) as 2.11 Thus, the shear yield strength predicted by the maximum distortion energy theory is 2.12 Both the maximum shear stress theory and the maximum distortion energy theory can be applied in the analysis and design of a machine element. The maximum shear stress theory gives a simple and moderately conservative approach; while the maximum distortion energy theory provides a more accurate prediction [7]. 2.3 Fatigue Strength Previous discussion is mainly about strength analysis of elements under static loads. More often than not, machine elements are subjected to fluctuating loads and the behaviour of an element under variable loads is entirely different from that under static loads. The failure mode under fluctuating loads is fatigue. The stress‐life, strain‐life and elastic fracture mechanics methods are currently used to analyse fatigue strength. These methods aim to predict the life in a number of cycles to failure at a specific level of loads. This section will only introduce the stress‐life method. The latter two methods will not be covered in this book
eBook - PDFFundamentals of Machine Elements
SI Version
- Steven R. Schmid, Bernard J. Hamrock, Bo. O. Jacobson(Authors)
- 2014(Publication Date)
- CRC Press(Publisher)
maximum-shear-stress theory von Mises criterion postulate that failure is caused by elas-tic energy associated with deformation; identical to distortion-energy theory von Mises stress effective stress based on von Mises crite-rion Summary of Equations Stress concentration factor: Definition: K c = Actual maximum stress Average stress Elliptical hole in plate loaded in tension: K c = 1+2 a b Fracture Toughness: K Ic = Y σ nom √ πa Failure Prediction for Uniaxial Stress State: n s = σ all σ d Failure Prediction for Multiaxial Stress State: Maximum shear stress theory (MSST, or Tresca): n s = S y σ 1 -σ 3 von Mises stress: σ e = 1 √ 2 ( σ 1 -σ 2 ) 2 + ( σ 3 -σ 1 ) 2 + ( σ 3 -σ 2 ) 2 1 / 2 Distortion energy theory (DET, or von Mises): n s = S y σ e Maximum normal stress theory (MNST): n s = S ut σ 1 or n s = S uc σ 3 , whichever is lower. Internal Friction Theory (IFT, or Coulomb-Mohr): If σ 1 > 0 and σ 3 < 0 , σ 1 S ut -σ 3 S uc = 1 n s If σ 3 > 0 , n s = S ut σ 1 If σ 1 < 0 , n s = S uc σ 3 Modified Mohr Theory (MMT): If σ 1 > 0 and σ 3 < -S ut , σ 1 -S ut σ 3 S uc -S ut = S uc S ut n s S uc -S ut If σ 3 > -S ut , n s = S ut σ 1 If σ 1 < 0 , σ 3 = S uc n s Recommended Readings Anderson, T.L. (2005), Fracture Mechanics — Fundamentals and Applications , 3rd ed., CRC Press. Budynas, R.G., and Nisbett, J.K. (2011), Shigley’s Mechanical Engineering Design , 9th ed., McGraw-Hill. Hill, R. (1950) The Mathematical Theory of Plasticity , Oxford. Dowling, N.E. (1993) Mechanical Behavior of Materials , Pear-son. Juvinall, R.C., and Marshek, K.M. (2012) Fundamentals of Ma-chine Component Design , 5th ed., Wiley. Mott, R. L. (2014) Machine Elements in Mechanical Design , 4th ed., Pearson. Norton, R.L. (2011) Machine Design , 4th ed., Pearson Educa-tion. Sun, C.T., and Jin, Z.-H. (2012) Fracture Mechanics , Elsevier. References ASM International (1989) Guide to Selecting Engineering Mate-rials , American Society for Metals.
eBook - PDFMetal Failures
Mechanisms, Analysis, Prevention
- Arthur J. McEvily, Jirapong Kasivitamnuay(Authors)
- 2013(Publication Date)
- Wiley-Interscience(Publisher)
Hence V = ε x + ε y + ε z , so V V = = ε x + ε y + ε z . CHAPTER 3 3-1. 1. Use the Mohr circle to determine the principal stresses σ 1 = σ x 2 + σ 2 x 4 + τ 2 xy 1/2 σ 2 = 0 σ 3 = σ x 2 − σ 2 x 4 + τ 2 xy 1/2 s s 3 s x t xy t s 3 For the von Mises criterion for yielding, substitute in Eq. 3-18 to obtain σ x σ 2 + 3 τ xy σ 2 = 1 (1) For the Tresca criterion for yielding, substitute in Eq. 3-23 to obtain (σ x 2 /4 + τ xy 2 ) 1/2 = k = σ/2 (σ x 2 /4 + τ xy 2 ) = ( σ/2) 2 (σ x 2 + 4τ xy 2 ) = ( σ) 2 CHAPTER 3 423 (σ x 2 / σ 2 + 4τ xy 2 /( σ) 2 ) = 1 σ x σ 2 + 4 τ xy σ 2 = 1 (2) (or from the Mohr circle, ( σ x 2 ) 2 + τ xy 2 = k 2 = ( σ 2 ) 2 Equations 1 and 2 each define an ellipse, as shown in the accompanying figure. Tresca: Maximum shear stress Von Mises: Distortion energy 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Distortion energy Maximum shear stress s x / s 0 t xy / s 0 3-2. σ kyd = k − 2kθ . θ in this case is equal to—( π 2 − 1 2 π 4 = 3 8 π ). Therefore, σ hyd = k + 2k 3 8 π = k + 3 4 π k, and σ 1 = k + k + 3 4 π k = 2k ( 1 + 3 8 π ) . From Eq. 3-34, σ 1 2k = 2.18 = 1 + ln 1 + x R , ln 1 + x R = 1.18, 1 + x R = 3.25, x R = 2.25, x = (0.25) 2.25 = 0.56 mm σ 1 = 2.18 (2k) k = σ y 2 Tresca; k = σ y √ 3 von Mises According to the Tresca criterion, σ 1 is 2.18 times the yield stress in simple tension. According to the von Mises criterion, σ 1 is 2.18 × 1.15 = 2.51 times the yield stress in simple tension. Note that when the yield stress in torsion (pure shear) is taken as the basis for the yield criterion rather than the yield stress in tension, the Tresca hexagon circumscribes the von Mises circle. For tension loading, according to the Tresca criterion, the tensile yield will equal 2k, but according to the von Mises criterion, the tensile yield stress will be equal to (2/ √ 3)k = 2.30k.
eBook - PDF- William F. Hosford(Author)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
C. Predict the value of σ x when yielding occurred according to von Mises. D. Predict the value of σ x when yielding occurred according to Tresca. 12. A new yield criterion has been proposed for isotropic materials. It states that yielding will occur when the diameter of Mohr’s largest circle plus half of the diam-eter of Mohr’s second largest circle equals a critical value. This criterion can be Plasticity Theory 91 expressed mathematically, following the convention that σ 1 ≥ σ 2 ≥ σ 3 , as ( σ 1 − σ 3 ) + 1 / 2( σ 1 − σ 2 ) = C if ( σ 1 − σ 2 ) ≥ ( σ 2 − σ 3 ) and ( σ 1 − σ 3 ) + 1 / 2( σ 2 − σ 3 ) = C if ( σ 2 − σ 3 ) ≥ ( σ 1 − σ 2 ) . A. Evaluate C in terms of the tensile (or compressive) yield strength, Y . B. Let x, y, and z be directions of principal stress, and let σ z = 0. Plot the σ y versus σ x yield locus. (That is, plot the values of σ y / Y and σ x / Y that will lead to yielding according to this criterion.) ( Hint: Consider different loading paths (ratios of σ y /σ x ), and for each, decide which stress ( σ 1 , σ 2 , or σ 3 ) corresponds to σ x , σ y or σ z = 0, then determine whether ( σ 1 − σ 2 ) ≥ ( σ 2 − σ 3 ), substitute s x , s y , and 0 into the appropriate expression, solve, and finally plot.) 13. The tensile yield strength of an aluminum alloy is 14,500 psi. A sheet of this alloy is loaded under plane–stress conditions ( σ 3 = 0) until it yields. On unloading, it is observed that ε 1 = 2 ε 2 and both ε 1 and ε 2 are positive. A. Assuming the von Mises yield criterion, determine the values of σ 1 and σ 2 at yielding. B. Sketch the yield locus, and show where the stress state is located on the locus. 14. Consider a capped thin-wall cylindrical pressure vessel, made from a material with planar isotropy and loaded to yielding under internal pressure. Predict the ratio of axial to hoop strains, ρ = ε a /ε h , as a function of R , using A. The Hill criterion and its flow rules ( equations 5.34 and 5.35 ); B.
eBook - PDFTheory of Structures
Fundamentals, Framed Structures, Plates and Shells
- Peter Marti(Author)
- 2013(Publication Date)
- Ernst & Sohn(Publisher)
Such circumstances will not be discussed any further here. The theory of plastic potential presented here for a three-dimensional element is discussed further in chapters 20 and 21 using generalised force and deformation variables . The theory remains valid for generalised variables if it is assumed to be correct for all the elements of a system. 7.3.3 Yield conditions Only isotropic materials will be considered below. Corresponding yield conditions can be presented as a function of the basic invariants of the stress tensor, i. e. Y s I , s II , s III ð Þ w 0 (7 : 19) The principal axes of the stresses and the strain increments coincide because isotropy is presumed. 7.3.3.1 VON MISES and TRESCA yield conditions When a material is incompressible , e . I = 0, and the resulting yield surfaces are cylin-ders Y s II , s III ð Þ w 0 (7 : 20) parallel with the hydrostatic axis, see (5.40). Examples of this are the VON MISES yield condition Y w s II s f 2 y 3 w 0 (7 : 21) or the TRESCA yield condition Y w 4 s 3 II s 27 s 2 III s 9 f 2 y s 2 II S 6 f 4 y s II s f 6 y w 0 (7 : 22) 85 7.3 Perfectly plastic behaviour which can also be written in the form Y w Max s 2 s s 3 j j , s 3 s s 1 j j , s 1 s s 2 j j ð Þ s f y w 0 (7 : 23) or, even simpler, as Y w t max s f y 2 w 0 (7 : 24) where, as before, f y denotes the uniaxial yield limit. According to (7.21) and (5.43), in the principal stress space the VON MISES yield condition corresponds to a cylinder of radius ffiffiffiffiffiffiffi 2 = 3 p f y about the hydrostatic axis, see Fig. 7.8(a). The TRESCA yield condition (assuming the same uniaxial yield limit f y ) corresponds to a regular hexagonal prism inscribed within the VON MISES cylinder. Fig. 7.8(b) shows the intersection between the cylinder / prism and the devi-atoric plane, and Fig. 7.8(c) shows the yield loci in the coplanar stress state ( s 3 = 0).
eBook - ePubMechanics of Materials Volume 1
An Introduction to the Mechanics of Elastic and Plastic Deformation of Solids and Structural Materials
- E.J. Hearn(Author)
- 1997(Publication Date)
- Butterworth-Heinemann(Publisher)
15.14 Three-dimensional yield locus for Tresca (maximum shear stress) theory. Points representing stress conditions plotted on the principal stress coordinate axes indicate safe conditions if they lie within the surface of the hexagonal cylinder. The two-dimensional yield locus of Fig. 15.6 is obtained as before by the intersection of the σ 1, σ 2 plane (σ 3 = 0) with this surface. 15.9.2 Brittle materials Failure of brittle materials has been shown previously to be governed by the maximum principal tensile stress present in the three-dimensional stress system. This is thought to be due to the microscopic cracks, flaws or discontinuities which are present in most brittle materials and which act as local stress raisers. These stress raisers, or stress concentrations, have a much greater adverse effect in tension and hence produce the characteristic weaker behaviour of brittle materials in tension than in compression. Thus if the greatest tensile principal stress exceeds the yield stress then failure occurs, and such a simple condition does not require a graphical representation. 15.10 Limitations of the failure theories It is important to remember that the theories introduced above are those of elastic failure, i.e. they relate to the “failure” which is assumed to occur under elastic loading conditions at an equivalent stage to that of yielding in a simple tensile test. If it is anticipated that loading conditions are such that the component may fail in service in a way which cannot easily be related to standard simple loading tests (e.g. under fatigue, creep, buckling, impact loading, etc.) then the above “classical” elastic failure theories should not be applied. A good example of this is the brittle fracture failure of steel under low temperature or very high strain rate (impact) conditions compared with simple ductile failure under normal ambient conditions
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
