Isotropic Hardening

8 Key excerpts on "Isotropic Hardening"

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  The Mechanics and Thermodynamics of Continua

  • Morton E. Gurtin, Eliot Fried, Lallit Anand(Authors)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  74.1 Isotropic and Kinematic Strain-Hardening As is clear from our brief discussion of the phenomenology of the stress-strain re-sponse, the deformation of metals beyond the elastic limit is quite complicated. In this subsection, we discuss some idealizations of strain-hardening that are frequently used in theories of plasticity. A simple idealization of actual material response, referred to as isotropic hard-ening , accounts for strain-hardening but approximates the hardening by a straight line, neglects the Bauschinger effect, and assumes that after reversal of deforma-tion from any level of strain in the plastic regime, the magnitude of the flow stress upon which reverse yielding begins has the same value as the flow stress from which the unloading was initiated. The stress-strain response corresponding to Isotropic Hardening is shown schematically in Figure 74.4 (a). Let the stress level from which reversed deformation is initiated be denoted by σ f , and denote the stress at which the stress-strain curve in compression begins to deviate from linearity by σ r < 0. The closed interval [ σ r , σ f ] is called the elastic range , and its end points σ r , σ f are called the yield set . Let Y , with initial value Y 0 , Figure 74.3. A stress-strain diagram obtained from a tension-compression experiment show-ing the Bauschinger effect. 420 Some Phenomenological Aspects of the Elastic-Plastic Figure 74.4. Idealized stress-strain response for an elastic-plastic material with (a) linear Isotropic Hardening; (b) linear kinematic hardening. denote the flow resistance (a material property) of the material. The initial elastic range then has σ r = − Y 0 , σ f = Y 0 , and, during subsequent plastic deformation along the hardening curve, the defor-mation resistance increases linearly from Y 0 to Y due to strain-hardening, and the new elastic range becomes σ r = − Y , σ f = Y .

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  Nonlinear Problems in Machine Design

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

    (Publisher)

  4.3.2 I SOTROPIC H ARDENING For materials that harden during plastic deformation, the yield surface under loading changes. This phenomenon is called strain hardening. The change of the yield surface follows a hardening rule. Here, we shall consider an Isotropic Hardening rule, proceeding with an incremental derivation of the stress-strain correlation. Isotropic Hardening assumes that the yield surface expands uniformly while maintaining a constant center and shape. See Figure 4.14. The Isotropic Hardening model is defined by the yield function, σ e 3 2 --s 1 2 s 2 2 s 3 2 + + ( ) 3 2 --s ·· s ( ) = = d ε e p 1 1 v + -----------3 2 --d ε 1 p ( ) 2 d ε 2 p ( ) 2 d ε 3 p ( ) 2 + + [ ] 2 3 --d ε p ·· d ε p ( ) = = d λ 3 2 --d ε e p σ e -------= d ε p 3 2 --d ε e p σ e -------s = d ε x p 3 2 --d ε e p σ e -------s x , d γ xy p 3 d ε e p σ e -------τ xy = = d ε y p 3 2 --d ε e p σ e -------s y , d γ yz p 3 d ε e p σ e -------τ yz = = d ε z p 3 2 --d ε e p σ e -------s z , d γ zx p 3 d ε e p σ e -------τ zx = = 126 Nonlinear Problems in Machine Design (4.72) where f ( σ ) reflects a yield criterion (either von Mises or Tresca) and Y ( κ ) is a monotonously increasing function of hardening parameter κ . In its simplest form, κ denotes the accumulated incremental plastic strain, (4.73) where denotes either the Euclidean norm (4.74) or the infinite norm (4.75) In the case of von Mises potential, is the Euclidean norm, (4.76) In an alternative form, κ is assumed to be the accumulated plastic work, (4.77) FIGURE 4.14 Isotropic Hardening of von Mises yield surface. F σ , κ ( ) f σ ( ) Y κ ( ) – 0 = = κ d ε p t ∫ = … x x ·· x = x max x ij = d ε p d ε p d ε p ·· d ε p 3 2 --d ε e p = = κ W p s ·· d ε p t ∫ s ·· ∂ f ∂σ ------t ∫ d λ = = =

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  Multiscale Modeling of Heterogenous Materials

  From Microstructure to Macro-Scale Properties

  • Oana Cazacu(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  [5.6] for an improved representation of the anisotropy (see [PLU 08]).

  5.4. Modeling anIsotropic Hardening due to texture evolution

  Regardless of the shape of the yield surface, strain hardening can be isotropic or anisotropic. The former corresponds to an expansion of the yield surface without distortion. Any other type of hardening is anisotropic and leads to different properties in different directions after deformation, even if the material is initially isotropic. Pure translation of the initial yield surface could be described by the classic linear kinematic hardening laws (see [CHA 86]). To model the smooth elastic-plastic transition upon reverse loading more accurately, multi-surface models as well as non-linear kinematic hardening models have been proposed. Such models, reviewed by [LI 03], may capture hardening anisotropy associated with cyclic loading (e.g. Bauschinger effects) but not anIsotropic Hardening under monotonic strain paths.

  Modeling hardening in hexagonal metals is even more challenging since the hardening rate is strongly dependent not only on the loading path but also on the sense of loading (see [LOU 07]). To describe with accuracy the evolution of the yield locus, it is imperative to account for the most important sources of anisotropy in the given material: slip and/or twinning activity, substructure evolution at the grain level and texture development during deformation. In crystal plasticity models (see e.g. [DAW 03]), the available slip/twinning systems, their critical shear stresses and the distribution of lattice orientations in a given polycrystal are taken into account explicitly. Therefore, lattice rotations and the associated anisotropy evolution result directly from the use of a suitable homogenization scheme (e.g. a Taylor model or a self-consistent model). Recently, the application of crystal plasticity models to hcp metals and their use in finite element (FE) analyses have received much attention. Models that account for both slip and twinning activity and employ Taylor (e.g. [WU 07]) or self-consistent ([e.g. [LEB 93]) averaging schemes to predict the aggregate behavior have been proposed. For example, [STA 03] neglects hardening but accounts for the transgranular effects on plastic deformation observed in magnesium alloys. [WU 07] focused on the characterization of the strain-hardening behavior of high-purity titanium at room temperature and the development of slip-hardening and twin-hardening functions. These laws were used to simulate the stress-strain response and texture evolution for monotonic loading (uniaxial compression and simple shear). However, no attempt was made to predict the final deformed shape of the specimens or to perform benchmark simulations of more complex monotonic loadings such as bending. In [KAS 01], a self-consistent viscoplastic model linked to the explicit FE code EPIC has been successfully used for describing the deformation of pure zirconium with a strong initial texture under quasi-static monotonic loading. A polycrystalline aggregate was associated with each FE integration point. Such FE calculations have the advantage of keeping track of texture evolution. However, they are computationally very intensive and, thus, their applicability is limited to problems that do not require a fine spatial resolution.

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  Fundamentals of the Theory of Plasticity

  • L. M. Kachanov(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  2 = 0.2%); then we find the “true” yield limit by extrapolating to “zero” tolerance. When additional points are available the accuracy of the extrapolation can be improved.

  If for some reason the construction of a yield surface cannot be reliably effected, it is probably inevitable that the concept of a yield surface has to be abandoned. In particular problems, of course, a limited use of the yield surface (front portion) is still possible.

  Isotropic Hardening. In the preceding section the equation of a fixed yield surface had the form ƒ(σij ) = K. If we assume that, with plastic deformation, hardening develops equally in all directions and is independent of the hydrostatic pressure σ, the equation of the loading surface can be written in the form

  (17.1)

  where the scalar q > 0 is some measure of Isotropic Hardening, and ϕ is an increasing function.

  One common measure of hardening q is the work of plastic deformation Ap , i.e.

  Another, less frequent, measure is characteristic of the accumulated plastic strain (Odquist’s parameter):

  Note that equation (17.1) can contain, in general, several measures of hardening q 1 , q 2 , q 3 , .... If the medium is isotropic the function must depend only on the invariants of the stress deviatoric. In particular, if we take into account only the quadratic invariant — the intensity of tangential stresses T — (which is quite sufficient in the first approximation), then equation (17.1) takes the form

  (17.2)

  A similar condition has been considered earlier (§ 12).

  According to (17.1) the loading surface expands uniformly (“isotropically”) and remains self-similar with increasing plastic deformation (fig. 27 ). It should be understood that the Bauschinger effect is not involved here, since the yield limits in the direct (OM+ ) and inverse (OM—

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  Theory of Plasticity

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

    (Publisher)

  ‡ This is confirmed by the results of an experimental investigation by W. M. Shepherd, Proc. Inst. Mech. Eng. , 159 : 95 (1948). See also C. Zener and J. H. Hollomon, J . Appl. Phys. , 17 : 2 (1946). foundations of plasticity 69 Figure 2.9 Correspondence between the strain-hardening characteristics in simple tension and pure torsion for an isotropic material. (ii) AnIsotropic Hardening We shall now consider hardening rules that account for anisotropy and Bauschinger effect exhibited by real materials. † It is assumed that the material is initially isotropic, having identical yield stresses in tension and compression. In the kinematic hardening rule, due to Prager, the yield surface is assumed to undergo translation in a nine-dimensional stress space. The initial yield surface is represented by the equation f ( σ i j ) = k 2 , where k is a constant. If the resultant displacement of the yield surface at any stage is denoted by a symmetric tensor α i j , the current yield surface is given by f ( σ i j − α i j ) = k 2 (19) Since α i j is not a scalar multiple of the isotropic tensor δ i j , which represents a hydrostatic change in stress, the material becomes anisotropic as a result of the hardening process. It is reasonable to suppose that the incremental translation of the yield surface is in the direction of the plastic strain increment d ε p i j , considered as a † Experimental investigations on subsequent yield surface have been carried out by P. M. Naghdi, F. Essenberg, and W. Koff, J . Appl. Mech. , 25 : 201 (1958); H. J. Ivy, J. Mech. Eng. Sci. , 3 : 15 (1961); W. M. Mair and H. Ll. D. Pugh, J. Mech. Eng. Sci. , 6 : 93 (1964); P. S. Theocaris and C. R. Hazell, J. Mech. Phys. Solids , 13 : 281 (1965); J. Rogan and A. Shelton, J. Strain Anal. , 4 : 138 (1969). 70 theory of plasticity Figure 2.10 Stress-space representation of Prager’s hardening rule.

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  Applied Mechanics of Solids

  • Allan F. Bower(Author)
  • 2009(Publication Date)
  • CRC Press

    (Publisher)

  If the state of stress lies on the surface of the cylinder, the material yields and deforms plastically. If the plastic deformation causes the material to strain harden, the radius of the cylinder increases. The stress state cannot lie outside the cylinder; this would lead to an infinite plastic strain. Because the yield criterion f ( σ ij ) = 0 defines a surface in stress space, it is often referred to as a yield surface . The yield surface is often drawn as it would appear when viewed down the • • 120 ◾ Applied Mechanics of Solids axis of the cylinder, as shown in Figure 3.31a. The Tresca yield criterion can also be plotted in this way. It looks like a cylinder with a hexagonal cross section, as shown in Figure 3.31b. 3.7.5 Strain Hardening Laws Experiments show that, if you plastically deform a solid, unload it, and then try to reload it so as to induce additional plastic flow, its resistance to plastic flow will have increased. This is known as strain hardening. Obviously, we can model strain hardening by relating the size and shape of the yield surface to plastic strain in some appropriate way. There are many ways to do this. Here we describe the two simplest approaches. 3.7.5.1 Isotropic Hardening Rather obviously, the easiest way to model strain hardening is to make the yield surface increase in size but remain the same shape, as a result of plastic straining, as shown in Figure 3.32. This means that we must devise some appropriate relationship between Y and the plastic strain. To get a suitable scalar measure of plastic strain, we define the accumulated plastic strain magnitude ε ε ε p ij p ij p d d 2 3 . ∫ (The factor of 2/3 is introduced so that ε ε p p = 11 in a uniaxial tensile test in which the speci-men is stretched parallel to the e 1 direction. To see this, note that plastic strains do not change volume, so that d ε 22 = d ε 33 = − d ε 11 /2 and substitute into the formula.) FIGURE 3.31 Axial views of yield surfaces.

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  Structural Analysis of Polymeric Composite Materials

  • Mark E. Tuttle(Author)
  • 2012(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  111 3 Material Properties 3.1 Material Properties of Anisotropic versus Isotropic Materials The phrase “ material property ” refers to a measurable constant which is characteristic of a particular material, and which can be used to relate two disparate quantities of interest. Material properties are defined that describe the ability of a material to conduct electricity, to transmit (or reflect) visible light, to transfer heat, or to support mechanical loading, to name but a few. Material properties of interest herein are those used by engineers during the design of load-bearing composite structures. Two specific examples are Young’s modulus , E , and Poisson’s ratio , ν . These familiar material properties, which will be reviewed and further discussed in the following section, are used to relate the stress and strain tensors. The adjectives “anisotropic” and “isotropic” indicate whether a material exhibits a single value for a given material property. More specifically, if the properties of a material are independent of direction within the material, then the material is said to be isotropic. Conversely, if the material properties vary with direction within the material, then the material is said to be anisotropic . To clarify this statement, suppose that three test specimens are machined from a large block at three different orientations, as shown in Figure 3.1. The geometry of the three specimens is assumed to be identical, so that the only difference between specimens is the original orientation of each specimen within the parent block. Now suppose that the axial stiffness (i.e., Young’s modulus, E ) is measured for each specimen. Young’s modulus measured using specimen 1 will be denoted E xx , that is, subscripts are used to indi-cate the original orientation of specimen 1 within the parent block. Similarly, Young’s modulus measured using specimens 2 and 3 will be denoted E yy and E zz , respectively.

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  Fundamental Issues and Applications of Shock-Wave and High-Strain-Rate Phenomena

  • K.P. Staudhammer, L.E. Murr, M.A. Meyers(Authors)
  • 2001(Publication Date)
  • Elsevier Science

    (Publisher)

  However, the specimen must achieve a measurable plastic strain after a loading step which will influence the flow behaviour at the following loading step. The material behaviour at the rotating loading path shows a great influence of Isotropic Hardening, induced by the previous loading steps (cross effect). ~0 1 ,i .,.- =.. 5 8 9 110 / tl ,, 5 , ~ I proportional limit !I1 ._4 6 2 back extrapolation (3! t[ / to elastic modulus i l! / 3 back extrapolation !tl / to ordinate I ! i I 4 plastic offset strain 1/]/ 5 upper yield point ] 6 lower yield point offset E; strain Figure 8. Definitions of yielding of strain hardening material and yield-point behaviour 3 a) b) c) 2/_ ---~~~-~2 02 4 10 Figure 9. Strain path techniques for yield loci determination; a) Method using many virgin specimens; b) One- specimen method, rotating path; c) One-specimen method, alternating path Therefore the alternating loading path with a small Isotropic Hardening is better adapted but leading to a Bauschinger-effect. Figure 10 shows some results of rotating loading pathes investigated by Besdo et al. [52]. The yield surfaces of AIMg3 specimens are measured with increasing offset. The yield surfaces are closed and have an elliptic shape up to the offset strain of 0.04 %. This induces significantly plastic deformation at the large offset strain of 0. 1%. Figure 11 compares the yield surfaces of the case hardening steel 20MoCrS4 based on many specimens and by alternating strain pathes of one specimen at the offset strain of 0.1%. The schematic structure of the measurement and control facilities is shown in Figure 12. A specimen is loaded in a strain controlled tension-torsion-machine with a steered constant ratio y/e and a strain rate of 0.01 s-~. The measured torque Mt, axial force Faxial, shear y and axial strain Caxial are converted on-line, even within one sample time step, to von Mises equivalent 17 stresses O'eq, total and equivalent strains e eq, total using equations (1, 2 and 4).

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