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 Chapter 3 
 Constitutive
  Models  
 
 
 3.7 Small Strain, Rate Independent Plasticity: Metals loaded beyond yield 
 For many design calculations, the elastic constitutive equations outlined in Section 3.1 are sufficient, since large plastic strains are by and large undesirable and will lead to failure. There are some applications, however, where it is of interest to predict the behavior of solids subjected to large loads, sufficient to cause permanent plastic strains.  Examples include: 
 
 
 
 Plasticity theory was developed to predict the behavior of metals under loads exceeding the plastic range, but the general framework of plasticity theory has since been adapted to other materials, including polymers and some types of soil (clay). Some concepts from metal plasticity are also used in modeling concrete and other brittle materials such as polycrystalline ceramics. 
 
 
 
 3.7.1 Features of the inelastic response of metals. 
 We
  begin by reviewing the results of a typical tension/compression test on an
  annealed, ductile, polycrystalline metal specimen (e.g. copper or Al).  Assume that the test is conducted at
  moderate temperature (less than  
 The results of such a test are 
 
 
 
 
 
 
 
 
 
 We also need to characterize the multi-axial response of an inelastic solid. This is a much more difficult experiment to do. Some of the nicest experiments were done by G.I. Taylor and collaborators in the early part of the last century. Their approach was to measure the response of thin-walled tubes under combined torsion, axial loading and hydrostatic pressure. 
 The main conclusions of these tests were 
 
 
 
 
 
 In this section, we will outline the simplest plastic constitutive equations that capture the most important features of metal plasticity. There are many different plastic constitutive laws, which are intended to be used in different applications.  There are two broad classes: 1.      
  Rate
  independent plasticity  2.      
  Rate
  dependent plasticity, or viscoplasticity  
 There are also various different models within these two broad categories.  The models generally differ in two respects (i) the yield criterion; (ii) the strain hardening law. There is no completely general model that describes all the features that were just listed, so in any application, you will need to decide which aspect of material behavior is most important, and then choose a model that accurately characterizes this behavior. 
 Key ideas in modeling metal plasticity 
 Five key concepts form the basis of almost all classical theories of plasticity. They are 1. The decomposition of strain into elastic and plastic parts; 2. Yield criteria, which predict whether the solid responds elastically or plastically; 3. Strain hardening rules, which control the way in which resistance to plastic flow increases with plastic straining; 4. The plastic flow rule, which determines the relationship between stress and plastic strain under multi-axial loading; 5. The elastic unloading criterion, which models the irreversible behavior of the solid. These concepts will be described in more detail in the sections below. 
 For simplicity, we will at this stage restrict attention to infinitesimal deformations. 
 Consequently, we adopt the infinitesimal strain tensor 
 as our deformation measure.  We have no need to distinguish between the
  various stress measures and will use  
 It is also important to note that the plastic strains in a solid depend on
  the load history.  This means
  that the stress-strain laws are not just simple equations relating stress to
  strain.  Instead, plastic strain laws
  must either relate the strain rate in the solid to the stress and stress rate, or else
  specify the relationship between a small increment of plastic strain  
 
 
 3.7.2. Decomposition of strain into elastic and plastic parts 
 
 Experiments show that under uniaxial loading, the strain at a given stress has two parts: a small recoverable elastic strain, and a large, irreversible plastic strain, as shown in the picture. In uniaxial tension, we would write 
 Experiments suggest that the reversible part is related to the stress through the usual linear elastic equations.  Plasticity theory is concerned with characterizing the irreversible part. 
 For multiaxial loading, we generalize this by
  decomposing a general strain increment  
 The elastic part of the strain is related to stress using the linear elastic equations (discussed in detail in 3.1) 
 
 
 
 The
  yield criterion is used to determine the critical stress required to cause
  permanent deformation in a material.  There
  are many different yield criteria  
 
 
 
 
 In
  both cases, the criteria are defined so that the material deforms elastically
  for  
 
 
 where 
 are the components of the `von Mises effective stress’ and `deviatoric stress tensor’ respectively. 
 These yield criteria are based largely on the following experimental observations: (1) A hydrostatic stress (all principal stresses equal) will never cause yield, no matter how large the stress; (2) Most polycrystalline metals are isotropic. Since the yield criterion depends only on the magnitudes of the principal stresses, and not their directions, the yield criteria predict isotropic behavior. 
 Tests suggest that von Mises yield criterion provides a slightly better fit to experiment than Tresca, but the difference between them is very small.  Sometimes it simplifies calculations to use Tresca’s criterion instead of von Mises. 
 
 
 3.7.4 Graphical representation of the yield surface. 
 
 Any arbitrary stress state can be plotted in `principal stress space,’ with the three principal stresses as axes. 
 The
  Von 
 If
  the state of stress falls within the cylinder, the material is below yield
  and responds elastically.  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  
 Because
  the yield criterion  
 
 
 
 
 Experiments show that if you plastically deform a solid, then unload it, and then try to re-load it so as to induce further 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. 
 
 
 
 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. 
 This means 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 
 (the factor of 2/3 is introduced so that  
 Then we make Y
  a function of  Perfectly plastic
  solid:   Linear strain hardening
  solid:  Power In
  these formulas,  
 
 
 Â Â Â Â Â 
 
 
 An isotropic hardening law is generally not useful in situations where components are subjected to cyclic loading. It does not account for the Bauschinger effect, and so predicts that after a few cycles the solid will just harden until it responds elastically. 
 To fix this, an alternative hardening law allows the yield surface to translate, without changing its shape. The idea is illustrated graphically in the picture. As you deform the material in tension, you drag the yield surface in the direction of increasing stress, thus modeling strain hardening. This softens the material in compression, however. So, this constitutive law can model cyclic plastic deformation. The stress-strain curves for isotropic and kinematic hardening materials are contrasted in the figure below. 
 
 
 To account for the fact that the center of the
  yield locus is at a position  
 
 Here, Y is
  now a constant, and hardening is modeled by the motion of the yield surface.
  To do so, we need to relate  
  This hardening law predicts that the stress-plastic strain curve is a straight line with slope c. This is known as linear kinematic hardening. 
 
 
 A more sophisticated approach is to set 
 where c and  
 There are many other kinematic type hardening laws. New ones are still being developed. 
 
 
 
 To complete the plastic stress-strain relations, we need a way to predict the plastic strains induced by stressing the material beyond the yield point. Specifically, given 1.      
  The current stress  2.      
  The current yield stress (characterized by  3.      
  A small increase in stress  we wish to determine the small change in plastic
  strain  
 The formulas are given below, for isotropic and kinematic hardening. These are just fits to experiment (specifically, to the Levy-Mises flow rule). The physical significance and reason for the structure of the equations will be discussed later. 
 The plastic strains are usually derived from the
  yield criterion f defined in 3.6.3,
  and so are slightly different for isotropic and kinematic hardening.  A material that has its plastic flow law
  derived from f is said to have an
  `associated’ flow law  
 
 Isotropic Hardening (Von-Mises yield criterion) 
 where 
 denotes the Von-Mises yield criterion, and  
 Here,  
 
 Linear Kinematic Hardening (Von-Mises yield criterion) 
 where the yield criterion is now 
 and as before  
 Recall that for linear kinematic hardening the hardening law is 
 Substituting
  into the  
 
 Comparison of flow law formulas with the Levy-Mises flow rule 
 The
  Levy-Mises flow law (based on experimental observations) states that
  principal values of the plastic strain increment  
 It
  is straightforward to show that this observation is consistent with the
  predictions of the flow law formulas given in this section. To see this,
  suppose that the principal axes of stress are parallel to the  
 
 
 The flow law 
 gives 
 Thus, we see that 
 with similar expressions for other components. Some trivial algebra then yields the Levy-Mises flow law. 
 
 Differentiating the yield criterion 
 Differentiating the yield criterion requires some sneaky index notation manipulations. Note that 
 Now, recall that 
 and further that 
 Hence 
 and 
 However, observe that 
 so that 
 and finally 
 
 
 
 3.7.7 The Elastic unloading condition 
 
 There is one final issue to consider.  Experiments show that plastic flow is
  irreversible, and always dissipates energy. 
  If the increment in stress  
 For an isotropically hardening solid, this unloading condition may be expressed as 
 For kinematic hardening, 
 In both cases, the solid deforms elastically (no plastic strain) if the condition is satisfied. 
 
 
 
 3.7.8 Complete incremental stress-strain relations for a rate independent elastic-plastic solid 
 We conclude by summarizing the complete elastic-plastic stress strain relations for an isotropic solid with Von-Mises yield surface. 
 
 Isotropically hardening elastic-plastic solid 
 
 The solid is characterized by its elastic
  constants  
 In this case we have that 
 with 
 
 where
   
 These may be combined to 
 It is sometimes necessary to invert these expressions. A straightforward but tedious series of index notation manipulations shows that 
 
 This constitutive law is the most commonly used model of inelastic deformation. It has the following properties: 
 
 
 
 It has the following limitations: 
 
 
 
 
 Linear Kinematically hardening solid 
 The solid is characterized by its elastic
  constants  
 with 
 
 where
   Finally, the evolution equation for  
 
 This constitutive equation is used primarily to model cyclic plastic deformation, or plastic flow under nonproportional loading (where principal axes of stress rotate significantly during plastic flow). It has the following limitations: 
 
 
 
 
 
 3.7.9 Typical values for yield stress of polycrystalline metals 
 Unlike
  elastic constants, the plastic properties of metals are highly variable, and
  are also very sensitive to alloying composition and microstructure (which can
  be influenced by heat treatment and mechanical working).  Consequently, it is impossible to give
  accurate values for yield stresses or hardening rates for materials.  The table below (again, taken from
  `Engineering Materials,’ by M.F. Ashby and D.R.H. Jones, Pergamon Press)
  lists rough values for yield stresses of common materials  
 
 
 
 
 
 3.7.10 Perspectives on plastic constitutive equations - The Principle of Maximum Plastic Resistance 
 The constitutive law outlined in the preceding section has an important property, known as the principle of maximum plastic resistance. 
 Statement of the principle: Let  Then 
 
 
 Interpretation: The Principle of Maximum Plastic Resistance is a mathematical statement of the following ideas: (i) The Mises yield surface is convex (ii) The plastic strain rate is normal to the yield surface. 
 
 It is best to illustrate
  these ideas graphically. In principal stress space, the product  
 is
  equivalent to the requirement that the angle between the vectors formed by  
 The Principle of Maximum Plastic Resistance is important because it is the basis for a number of very important theorems concerning plastic deformation in solids. For example, it can be shown that the stress field in a material that obeys the Principle is always unique. In addition, the principle leads to clever techniques to estimate collapse loads for elastic-plastic solids and structures. 
 Proof of the principle of maximum plastic resistance 
 Our
  goal is to prove that  1.      
  Recall the
  plastic flow rule   2. Multiply both sides by stress 
 where
  we have noted that  3.       Recall that  4. Now consider 
 5. Note that 
 6.       Now, recall again that  
 Thus
   
 
 
 3.7.11 Perspectives on plastic constitutive equations - Drucker’s Postulate 
 
 Constitutive models of inelastic behavior are based largely on experimental observations of plastic flow in laboratory specimens. Similar constitutive laws are used to describe very different materials, including metals, ceramics, glasses, soils and polymers. The mechanisms of deformation in these materials are very different, so it is surprising that their response is similar. 
 One
  perspective on the structure of constitutive laws for inelastic solids was
  developed by Drucker in the 1950s. 
  Drucker introduced the idea of a stable
  plastic material, as follows. Consider a deformable solid, subjected to
  boundary tractions  
 
 It can be shown that, for a plastic material to be stable in this sense, it must satisfy the following conditions: 
 
 
 
 Furthermore, a material that is stable in the sense of Drucker must satisfy the principle of maximum plastic resistance. 
 This does not really explain why the constitutive law should have this structure, but materials that do not satisfy the Drucker stability criterion tend to be difficult to work with in calculations, so there is a strong incentive for choosing a constitutive law that meets the condition. It is not surprising, then, that the people developing constitutive laws ended up with a form that satisfies Drucker stability. 
 
 
 3.7.12 Microscopic Perspectives on Plastic Flow in Metals 
 
 It is possible to obtain some insight into the structure of the constitutive laws for metals by considering the micromechanisms responsible for plastic flow. 
 Plastic
  flow in metals is caused by dislocation motion.  Dislocations are line defects in
  crystalline solids  
 
 When the crystal is subjected to stress, these defects move through the solid and rearrange the crystal lattice. For example, if the model crystal shown above is subjected to a shear stress, the atoms rearrange so that the top part of the crystal is shifted to the right relative to the bottom part. 
 Because the crystal lattice is distorted near a dislocation, only a modest shear stress is required to drive the dislocation through the solid, causing permanent plastic deformation. 
 Experiments
  and atomistic simulations suggest that dislocation motion obeys Schmidt’s
  law: a dislocation moves through a crystal if the shear stress on its glide
  plane exceeds a critical magnitude  
 It can be shown that a material which deforms by dislocation glide, and which obeys Schmidt’s law, will satisfy the principle of maximum plastic resistance. This, in turn, implies that the yield surface for the solid must be convex and the plastic strain rate must be normal to the yield surface. 
 
 The
  notion of a yield surface and convexity for a material which deforms by
  dislocation glide can be illustrated with a simple double-slip model.  Consider a single crystal, which contains
  two dislocation glide planes oriented at  
 The picture below shows side views of the two slip planes. As an exercise, you should verify that the shear stresses (tangential component of traction) acting on the two slip planes are 
 
 The
  solid reaches yield if  
 Observe that the yield surface is convex: this is a consequence of Schmidt’s law. 
 
 Now,
  suppose that slip is activated on one of the glide planes.  Let  
 The deformed line element is given by 
 or, in index notation 
 The deformation gradient follows as 
 and the Lagrange strain tensor is 
 For
  small  
 
 Now, suppose that the stress satisfies 
 
 as marked on the yield locus shown below. This activates slip as shown in the picture. The normal and tangent to the appropriate slip plane are 
 The strain rate therefore follows as 
 Thus,
   
 
 
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| (c) A.F. Bower, 2008 |