Yield Stress

10 Key excerpts on "Yield Stress"

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  eBook - ePub

  Mechanical Properties of Solid Polymers

  • Ian M. Ward, John Sweeney(Authors)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Figure 12.8 (a)). Because this stress is achieved at a comparatively low elongation of the sample, it is often adequate to use the engineering definition of the Yield Stress as the maximum observed load divided by the initial cross-sectional area.

  FIGURE 12.8 (a) The Yield Stress is defined as the load divided by the cross-sectional area at the point A. (b) The Yield Stress is defined as the stress at the point B. (c) The Yield Stress is defined as the stress at the point C.

  In some cases, there is no observed load drop and another definition of Yield Stress is required. One approach is to determine the stress where the two tangents to the initial and final parts of the load–elongation curve intersect (Figure 12.8 (b)).

  An alternative is to attempt to define an initial linear slope on the stress–strain curve and then to draw a line parallel to this, which is offset by a specified strain, say 2%. The interception of this line with the stress–strain curve then defines the offset or proof stress, which is considered to be the Yield Stress (Figure 12.8 (c)).

  12.2 Ideal Plastic Behaviour 12.2.1 The Yield Criterion: General Considerations

  The simplest theories of plasticity exclude time as a variable and ignore any feature of the behaviour, which takes place below the yield point. In other words, we assume a rigid-plastic material whose stress–strain relationship in tension is shown in Figure 12.9 . For stresses below the Yield Stress there is no deformation. Yield can be produced by a wide range of stress states, not just simple tension. In general, it must therefore be assumed that the yield condition depends on a function of the three-dimensional stress field. In a Cartesian axis set, this is defined by the six components of stress, σ 11 , σ 22 , σ 33 , σ 12 , σ 23 and σ 31

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  Introduction to Polymers

  • Robert J. Young, Peter A. Lovell(Authors)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  531 22 Yield and Crazing 22.1 INTRODUCTION The determination of the stress–strain characteristics of a material up to failure is particularly use-ful as it gives information concerning important mechanical properties such as Young’s modulus, yield strength and brittleness. These parameters are vital in design considerations when the material is used in a practical situation. Stress–strain curves can be readily obtained for polymers by subject-ing a specimen to a tensile force applied at a constant rate of testing. The stress–strain behaviour of polymers is not fundamentally different from that of conventional materials, the main difference being that polymers show a marked time or rate dependence (i.e. viscoelasticity). The situation can be simplified greatly by making measurements at a fixed testing rate, but it is important to bear in mind the underlying time dependence that still exists. 22.2 PHENOMENOLOGY OF YIELD When a material undergoes elastic deformation, as described in detail in Chapter 19, there is com-plete recovery of strain on the removal of stress. Yield, on the other hand, can be defined as the onset of permanent plastic deformation in a material which is not recovered when the stress is removed. It will be considered first of all from a phenomenological viewpoint. 22.2.1 D EFINITIONS Figure 22.1 shows an idealized stress–strain curve for a ductile polymer sample. In this case, the nominal stress σ n is plotted against strain e . The change in the cross-section of a parallel-sided specimen also is sketched schematically at different stages of the deformation. Initially the stress is proportional to the strain and Hooke’s law is obeyed. The tensile modulus can be obtained from the initial slope. As the strain increases, the curve decreases in slope until it reaches a maximum. This is conventionally known as the yield point and the Yield Stress σ y and yield strain e y are indicated on the curve.

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  Engineering Plasticity

  Theory and Applications in Metal Forming

  • Z. R. Wang, Weilong Hu, S. J. Yuan, Xiaosong Wang(Authors)
  • 2018(Publication Date)
  • Wiley

    (Publisher)

  Among a great number of different suggestions, Mises yield criterion, proposed by Mises in 1913 [8], emerged as the most widely used one. Mises yield criterion is expressed as 1.97 where the flow stress can also be determined through a uniaxial tension test, i.e.,. Actually, Tresca and Mises yield criteria are more likely geared to strength theory in mechanics of materials with (initial yield state). To distinguish the application of yield criterion in the plastic deformation from that in the strength theory, it is of critical importance to construct a proper yield criterion. The strength theory deals with the stress condition within the range of elastic deformation. Applying yield criterion to plastic deformation means that the equation is just the offset of plastic strain taking place in a stressed material element (i.e.,), which means that the stresses element starts to enter a plastic deformation process. In other words, is the basis to judge whether the material element has reached the state of plastic deformation. Application of yield criterion to plastic deformation mainly means to take the state into consideration. 1.3.4 Twin Shear Stress Yield Criterion Another yield criterion worthy of being mentioned is twin shear stress yield criterion proposed by Yu in 1961 [9, 10]. Similar to Tresca yield criterion though, it is linked to strength theory and mostly used in mechanics of materials

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  Nonlinear Problems in Machine Design

  • Eliahu Zahavi, David M. Barlam(Authors)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  This point is called yield strength or yield point and is designated as S yp . A rising curve up to point C indicates material’s hardening with an increase of the load. This point is called ultimate strength and is designated as S u . Beyond this point, the curve drops, ending abruptly, which indicates material’s final failure. High-carbon steels and nonferrous metals do not have a marked yield point. They exhibit a gradual transition from linear elastic to nonlinear plastic behavior (see Figure 4.2b). For such cases, the yield point is defined by drawing a parallel line to the elastic line at an offset of 0.2%. The cross point of the offset line with the stress-strain line is assumed to be the yield point S yp . A ′ A ′ 108 Nonlinear Problems in Machine Design In both cases, if the specimen is unloaded before reaching final failure, the stress-strain line follows path DE, which is nearly parallel with the elastic path (see Fig. 4.2). Point E at zero load indicates a permanent deformation of the specimen. Upon reloading from point E , the specimen follows Hooke’s law along the elastic path ED . Stress and Strain Definitions Let us consider a typical tensional-test diagram. To describe this test, one uses the engineering stress and strain, which are defined as follows: the engineering stress is the stress relative to the original cross-section area A 0 , (4.1) while the engineering strain is the ratio of elongation versus original length l 0 , (4.2) See Figure 4.1. The engineering stress and strain defined by Equations (4.1) and (4.2) are applicable to small deformations only. When large plastic deformation takes place, a more functional definition becomes necessary, referred to as true stress and true stain . The true stress is defined as the load divided by the current (or deformed) cross-sectional area, (4.3) FIGURE 4.1 Tension test. S P A 0 -----= e l l 0 – l 0 ----------= σ P A ---=

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  Engineering Plasticity

  The Commonwealth and International Library: Structures and Solid Body Mechanics Division

  • C. R. Calladine, B. G. Neal(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  2.7. The Yield Locus We have already seen that in pure tension and compression tests of ductile metals we find that at any stage in a history of 30 ENGINEERING PLASTICITY loading the material has an elastic range of stress, and that if we are prepared to make the idealisations indicated in Figs. 2.1(b) and (c) the extent of the elastic range is defined precisely, in a way which is suitable for formulation in mathematical symbols as follows: For 0 < σ < G y the material is elastic. In this statement ,, is the current Yield Stress, which may be either an initial Yield Stress for a previously unstressed specimen or a subsequent Yield Stress for a specimen which has already undergone some plastic deformation. By neglecting the Bauschinger effect we extend the elastic range thus: For — σ ν < σ < a y the material is elastic. (2.8) Note that we can define an elastic range in terms of limits on stress, and without any reference to the reversible relationship between changes of stress and strain which is valid within that elastic range. In general, to develop a satisfactory theory of plasticity we shall need to be able to deal with more complicated, three-dimensional, states of stress. It will therefore be advantageous if we can generalise our ideas of elastic range and yield point to more complex stress systems. To proceed in relatively small steps, let us consider next a bi-axial, i.e. two-dimensional, state of stress. Suppose we devise a testing machine which is capable of applying tension and com-pression independently in two perpendicular directions parallel to the plane of a thin sheet of material, as indicated in Fig. 2.10(a). Also suppose that by plastic straining in the 1-direction we have established an elastic range (no Bauschinger effect) in the 1-direction, bounded by two yield points, as shown on the σ χ axis of the stress-space of Fig.

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  Rheology V1

  Theory and Applications

  • Frederick Eirich(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  STRESS-STRAIN RELATIONS IN PLASTIC RANGE 99 may be ignored. History, or path of loading, is all important because plastic deformation is an irreversible process. There can be no unique relation between stress and strain. The simplest relation which can exist is between the strain increments and the existing state of stress and increments of stress. In general, it is found that the relation between the increments of stress and strain depends upon the complete prior history. A brief discussion will also be given of the apparent similarity between frictional and plastic behavior and the fundamental difference will be brought out. An analysis of existing experimental results in terms of mathematical theory will complete the survey of the relationship between stress and strain. II. Experiments on Yielding 1. A SIMPLE AND USEFUL TEST SPECIMEN The onset of plastic action, termed yielding, is to a certain extent a mat-ter of definition. For the usual materials with curved stress-strain diagrams in simple tension, Fig. 1, it is customary to use an offset yield strength which does not involve a strong element of judgment. A permanent tensile strain of 0.002, designated by AC, is generally chosen and defines the yield strength, point B. The assumption will be made in the further discussion, however, that the limit of elasticity is well defined and onset of yielding will refer to it. FIG. 1. Stress-strain curve for simple stress. The total strain 6 produced by stress p is composed of an elastic part €« and a plastic part e* FIG. 2. Combined stress in a thin-walled cylinder. Internal pressure, axial force P, and twisting moment T produce known stress components p x , p y , p xy or equiva-lently principal stresses pi, pt . 100 D. C. DRUCKER Knowledge of the behavior of metal under uniaxial stress is not sufficient to predict what will happen under more complicated states of stress.

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  Formulas for Mechanical and Structural Shock and Impact

  • Gregory Szuladzinski(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  215 6 Yield and Failure Criteria THEORETICAL OUTLINE B ASIC T HEORIES OF C OMBINED S TRESS Consider a two-dimensional (2D) case of a plane stress, as described in Appendix A. Stress at a point is a tensor, which means that three components are needed to define it. Having those three components, σ x , σ y , and τ in some coordinate system is not always meaningful. For this reason, one often seeks the extreme values or three stress invariants , σ min , σ max , and τ m . Having these three clarifies the state of stress, but does not answer the essential ques-tion: Will the material suffer permanent deformation or damage because of being stressed to that level? Typically, materials are tested in a uniaxial fashion: steel in tension, concrete in com-pression, etc. Structural members, which are made of those materials, are usually subjected to more than one component of stress. To answer the questions related to the effect of a prescribed stress field, a designer must refer to combined stress theories . The oldest of those is the maximum stress theory . It simply says that in deciding on the effects of the stress field one calculates the maximum direct stress and compares it with the allowable value. The above criterion was devised having brittle materials in mind. It is usually extended to accommodate the fact that tensile and compressive strengths are different. The resulting damage envelope, based on uniaxial compressive strength F c and tensile strength F t , is shown in Figure 6.1. This theory works well for brittle materials, such as concrete, rock, and often soil. For those materials, damage usually means fracture. For ductile materials, the tensile yield strength F y is the main material property, as far as yielding is concerned. 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.

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  Continuum Mechanics and Plasticity

  • Han-Chin Wu(Author)
  • 2004(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  In this case, the yield function is f ( J 2 ) = k 2 (6.5) Mises [13] used a simple form of (6.5) which is − J 2 = k 2 or 1 2 σ ij σ ij = k 2 (6.6) 270 Continuum Mechanics and Plasticity Equation (6.6) is known as the Mises yield criterion and is widely used. A phys-ical interpretation of the Mises yield criterion is that yielding begins when the elastic distortion energy reaches a critical value. This interpretation is reasonable for isotropic materials. Using (6.6), an explicit yield function may be obtained for each specified loading condition. In the case of plane stress condition subjected to principal stresses, the stress and deviatoric stress components are [ σ ] =   σ 1 0 0 0 σ 2 0 0 0 0   , [ σ ] =       2 σ 1 − σ 2 3 0 0 0 2 σ 2 − σ 1 3 0 0 0 − (σ 1 + σ 2 ) 3       (6.7) Then, (6.6) is reduced to 1 2 2 σ 1 − σ 2 3 2 + 2 σ 2 − σ 1 3 2 + σ 1 + σ 2 3 2 = k 2 (6.8) or σ 2 1 + σ 2 2 − σ 1 σ 2 = 3 k 2 (6.9) In the case of combined axial–torsion of a thin-walled tube, the stresses are [ σ ] =   σ τ 0 τ 0 0 0 0 0   , [ σ ] =       2 σ 3 τ 0 τ − σ 3 0 0 0 − σ 3       (6.10) and (6.6) reduces to σ 2 + 3 τ 2 = 3 k 2 (6.11) From (6.11) we see that the Mises yield criterion leads to an ellipse, and it is a circle when plotted in the ( √ 3 τ , σ) space. This result agrees with experimental findings as previously mentioned. Using (6.11), the relationship between the Yield Stress in tension and that in shear may be determined. Setting σ = 0, (6.11) gives τ = k , which is the shear Yield Stress. On the other hand, by setting τ = 0, (6.11) gives σ y = Y = √ 3 k . Thus, according to the Mises yield criterion, Y and k are related by Y = √ 3 k . Some alternative forms of (6.6) are (σ 1 − σ 2 ) 2 + (σ 2 − σ 3 ) 2 + (σ 3 − σ 1 ) 2 = 6 k 2 (6.12) The Flow Theory of Plasticity 271 or (σ 11 − σ 22 ) 2 + (σ 22 − σ 33 ) 2 + (σ 33 − σ 11 ) 2 + 6 (σ 2 12 + σ 2 23 + σ 2 31 ) = 6 k 2 (6.13) Another well-known yield criterion is the Tresca yield criterion .

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  Continuum Mechanics

  Elasticity, Plasticity, Viscoelasticity

  • Ellis H. Dill(Author)
  • 2006(Publication Date)
  • CRC Press

    (Publisher)

  159 4 Plasticity 4.1 CLASSICAL THEORY OF PLASTICITY We review here the classical theory of plasticity for small displacements, a theory that describes the relation between load and deformation for many metals at higher stresses. Temperature changes will not be considered in this section. The constitutive relations are rate-independent. Time is used to keep track of events but the relations are independent of the time scale. Whenever components are introduced, a normal basis is implied. In very qualitative terms, if one performs a tensile test on metals (Fig. 4.1.1) by applying a uniaxial stress τ 11 = σ ( t ), apparent elastic behavior occurs until a limiting stress σ = Y , called the Yield Stress, is exceeded. After loading beyond the initial Yield Stress to σ = σ b , unloading to zero stress leaves a residual strain ε 11 = ε P called the plastic strain. The response to subsequent loading is incrementally elastic as long as the previous maximum stress is not exceeded, σ – σ b < 0, but yielding again occurs at σ – σ b = 0. Furthermore, the measured stress and strain are independent of the rate of loading. For any stress, ε = ε P + ε e = ε P + σ /E. If the equation of the curve a – b is ε = f ( σ ), then ε P = f ( σ ) – σ / E . So, ε P is a function of σ , or inversely σ = H ( ε P ). For plastic loading from a to b , , where is the plastic modulus. In summary, there is a yield criterion , and the material is incremen-tally elastic ( and ) if or if and . Incre-mental plastic strain ( ) occurs if and . During plastic loading, σ max increases so that we always have the yield criterion σ – σ max = 0. FIGURE 4.1.1 Tensile test for elastic-plastic material. σ ε = > E P P 0 E H dH d P P = ′ = / ε σ σ − ≤ max 0 ε P = 0 σ ε = E σ σ − < max 0 σ σ − = max 0 σ < 0 ε σ P P = / E σ σ − = max 0 σ > 0 o a b E E Y 1 1 ε (t) σ (t) 1 E T ε p Elastic regime Plastic loading Initial Yield Stress Initial elastic regime Plastic strain Elastic strain ε e σ max Reload yield

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  Theory of Plasticity

  • Jagabanduhu Chakrabarty(Author)
  • 2012(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  When the existing state of stress in the rigid/plastic body is regarded as given, the prescribed velocity distribution will be consistent if the corresponding strain rate vector is normal to the yield surface at the given stress point. In general, the existing stress would change discontinuously so as to become consistent with the strain rate at all points. The material is assumed to be isotropic, so that the unit normal n i j to the yield sur-face (which is regular) has principal axes coinciding with those of the current stress σ i j . Since σ ˚ i j n i j = ˙ σ i j n i j , the constitutive equations for a work-hardening rigid/plastic † Detailed investigations of the bifurcation in uniaxial tension have been carried out by S. Y. Cheng, S. T. Ariaratnam, and R. N. Dubey, Q . Appl. Math. , 29 : 41 (1971); A Needleman, J. Mech. Phys. Solids , 20 : 111 (1972); J. W. Hutchinson and J. P. Miles, ibid., 22 : 61 (1974); L. G. Chen, Int. J. Mech. Sci. , 25 : 47 (1983). The bifurcation problem in plane strain tension has been treated by S. T. Ariaratnam and R. N. Dubey, Q . Appl. Math. , 27 : 349 (1969); R. Hill and J. W. Hutchinson, J. Mech. Phys. Solids , 23 : 239 (1975). For axisymmetric bifurcation, see also D. Durban and P. Papanastasion, J. Appl. Mech. , 67 : 550 (2000). 110 theory of plasticity solid become ˙ ε i j = (3 / 2 H ) ˙ σ kl n kl n i j ˙ σ kl n kl 0 ˙ ε i j = 0, ˙ σ kl n kl 0 (84) whenever an element is stressed to the yield point. Taking scalar products of the first relation of (84) with ˙ σ i j and ˙ ε i j in turn, and remembering that n i j n i j = 1, it is easily shown that ˙ σ i j ˙ ε i j = 3 2 H ( ˙ σ i j n i j ) 2 = 2 3 H ˙ ε i j ˙ ε i j (85) The last expression also holds in a plastic element that instantaneously unloads, as well as in an element that is currently nonplastic, the strain rate being identically zero in both cases.

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