1.3 Notation 5 Lemaitre and Chaboche (1990), Maugin (1992, 1999), Coussy (1995). In this con- text, what we call a hyperplastic material has much in common with what the French authors define as a “standard material”, although we believe we have adopted a rather stricter formalism for the necessary definitions. There are numerous other works in which the thermodynamics of continua are addressed. Many attempt great generality [e. g. the works by Eringen (1962), Truesdell (1977), and Holzapfel (2000)], describing the mechanics in terms of appropriate tensor notation for large strain analysis, and often developing a somewhat axiomatic approach in which thermodynamics is introduced at a very general level. The disadvantages of this approach are twofold. Firstly, these works are impenetrable to all but the highly specialised reader. Secondly, we have found that, despite their generality, their treatment of plasticity theory is often superficial. Plasticity theory does not sit easily within formulations oriented principally toward the treatment of materials in which the dissipation is viscous (rate dependent). Often the authors of more general texts revert to “conventional” analysis of plasticity problems and make little connection be- tween plasticity and thermodynamics. Our approach is different in that (a) thermodynamics and hyperplasticity are intimately connected and (b) we make our ideas more accessible by adopting just small strain analysis, which makes much of the presentation more straight- forward. 1.3 Notation This book makes much use of vectors and tensors, and for these we use the sub- script notation. Thus ij V is a shorthand for a second-order tensor which could be written out in full in matrix form as 11 12 13 21 22 23 31 32 33 VVV ªº «» VVV «» «» VVV ¬¼ . We adopt the summa- tion convention over a repeated index, so that, for instance, 11 22 33ii V{VV V . Although the subscript notation can become rather cum- bersome at times, and many authors prefer the boldface notation in which the above tensor would simply be written as V , it has the advantage that there is absolute clarity about products, particularly contracted products. If the boldface notation is used, one has to pay meticulous attention to the order of multiplica- tion, and also elaborate notation for different types of product is necessary. Note that in any equation, dimensional consistency requires that the subscripts must be balanced. Thus ij ij ab or ij ik kj abc are meaningful statements, whilst ij ik ab is not. A problem arises when we wish to indicate the arguments of a function within an equation. Suppose, for instance, that kl f is a function of 6 1 Introduction a tensor ij x . Then we could write  kl kl ij yfx . The subscripts within the list of arguments of the function are there merely to indicate a tensorial argument and play no other role in the equation. We shall usually just use ‘ij ’ for the subscripts in such cases. The unit tensor (Kronecker delta) is given by ij G , where 1 ij G for ij and 0 ij G for 1. The deviator of a tensor is indicated by a prime notation, thus 1 3 ij ij ij kk c V{VGV . Other details of the terminology we shall use for tensors are given in Appendix B. We denote the time differential by the dot notation; thus yy t{w w  and 22 yy t{w w  . Spatial differentiation is denoted by a comma notation, so that if i x are the coordinate directions, then ,ii yy x{w w . In Chapter 12, we use the notation y  for the material derivarive ,ii yyy v   , where ii vx  is the velocity of a material point. 1.4 Some Basic Continuum Mechanics 1.4.1 Small Deformations and Small Strains In this book, we shall restrict our attention (with the exception of Chapter 12) to problems of small strain. It has become usual in continuum mechanics papers and textbooks to present ideas in the very general mathematical forms that can be used for large strain problems. However, it is our view that this adds a level of complexity which can make the underlying ideas less clear in some cases. Whilst the ideas that are described in this book can certainly be extended to large strain (see Chapter 12), one should not underestimate the difficulties involved; some areas would need considerable attention to matters of detail. It is worth, however, reflecting at this stage on precisely what we mean by the small strain assumption. We consider a process of deformation in which the initial position  0t of a material point in a body is given by the Cartesian coordinates i X . At some later time t, the position of the same material point, measured in the same coordinate system, is i x . Furthermore, we define the dis- placement of the point as ii i uxX  . The deformation gradient tensor is defined as ij i j AxX w w , and the dis- placement gradient tensor is ijijij uX Aww G . We define a small displace- ment gradient process as one in which 1 ij uXww  , where ij a defines an appropriate norm of the tensor ij a . Possible choices of the norm are , max ij ij a , 1.4 Some Basic Continuum Mechanics 7 , ij ij a ¦ , or ij ij aa . Use of any of these definitions ensures that the condition 1 ij uXww  requires that each individual component of ij uXww is itself small. For any small deformation process, it follows that ij ij A |G , so that iiki i i kj kj j kj k k j uuxu u u A XxXx x x wwww w w |G wwww w w . It follows therefore that for such processes, it is not necessary to distinguish between differentiation with respect to the initial coordinates i X (a Lagrangian formulation) and the current coor- dinates i x (an Eulerian formulation). For convenience (except in Chapter 12), we shall use an Eulerian formulation. Now we turn our attention to definitions of strains. Many definitions are in general possible, but the strains most commonly defined for large displacement processes are the Green-Lagrange tensor 1 2 j ikk ij j iij u uuu E XXXX §· w www  ¨¸ ¨¸ wwww ©¹ and the Euler-Almansi tensor 1 2 j ikk ij j iij u uuu e xxxx §· w www  ¨¸ ¨¸ wwww ©¹ . Both of these reduce to zero for processes that involve pure translation and rotation. For small displacement gradient processes as defined above, the quadratic terms become of second order, and both of these tensors become approximately equal to the Cauchy small strain tensor 11 22 j j ii ij j iji uu uu xx XX §·§ · ww ww H  |  ¨¸¨ ¸ ¨¸¨ ¸ ww ww ©¹© ¹ and 1 ij H . Clearly, small displacement gradient processes involve small strains, but the reverse is not necessarily true. A distinction is necessary between small strain problems and small displace- ment problems. It is possible to have a process in which the displacements are large, but the strains themselves are small everywhere. Trivial examples involve rigid body rotations (zero strain), but others involve small non-zero strains but large displacements. For instance, a thin metal strip can be bent such that the displacements are certainly of the same order as the dimensions, and yet the strains can still be very small throughout the strip. Small strain means that, for every element of the continuum, the linear strain (fractional change of length) and shear strain (change of angle) are “small”. However, how small is small? The answer lies in the precision required of the theory, but for most practical pur- poses, 1% is certainly small, whilst 10% is approaching the magnitude at which the approximations in the theory may be important. In the following, we shall strictly adopt definitions that are applicable only to small deformations, which is a stricter criterion than small strains. However, the constitutive responses we shall describe are applicable to the wider class of 8 1 Introduction problems involving small strains, and need be restricted only to small strains, and not necessarily to small deformations. Small deformation, according to the definition above, means that the displace- ments of any part of the continuum are small compared to the dimensions, and therefore need not be considered in any way commensurate with them. A result is that an apparently self-contradictory assumption is made, so that although the continuum undergoes displacements, no changes of geometry need be considered. The issue of how small is small again arises, and again the answer depends on the problem. It is important to realize, however, that the magnitude of displacements that might be regarded large, as far as the serviceability of a structure are con- cerned, would be quite unrelated to the magnitude of the displacements at which the mathematics of small displacement theory would no longer be valid. For in- stance, the serviceability of a building may be considered compromised if deflec- tions are greater than about 1300 of its linear dimension (due to aesthetic con- cerns about damages to finishes, etc.), yet by any standard, displacements of only 1 300 of the dimensions are “small” in the mathematical sense. For the small strain analysis in this book, we shall employ the Cauchy stress ij V (force per unit area) and the linear or Cauchy strain ij H , as defined above. 1.4.2 Sign Convention Throughout the entire book, except for Chapters 9 and 10, we adopt the usual convention of tension positive from continuum mechanics. Unfortunately, in soil mechanics and geotechnical engineering, it is usual to adopt a compression positive convention. Only in Chapters 9 and 10, which deal with applications in geomechanics, we reverse the sign convention and use compression positive, to make these chapters more accessible to geotechnical engineers. 1.5 Equations of Continuum Mechanics In the solution to any problem in continuum mechanics, four issues must be addressed: x Equilibrium x Compatibility x Constitutive relationships x Initial and boundary conditions. The bulk of this book is concerned with the third of these topics for a particu- lar class of materials, but it is worthwhile devoting some attention to the other three, so that the context of the constitutive relationships is properly understood. 1.5 Equations of Continuum Mechanics 9 1.5.1 Equilibrium The stresses in a continuum cannot simply be specified arbitrarily, but must satisfy certain equilibrium relationships. These are well-known and are derived by examining the equilibrium of a small element in a varying stress field. Firstly, the moment equilibrium of the element establishes the symmetry of the stress tensor: ji ij V V (1.1) Secondly, the direct equilibrium equations give ij ii j Fü x wV  U w (1.2) or more compactly ,ij j i i FüV U , where i F is the body force per unit volume (usually equal to i gU , where i g is the gravitational acceleration vector), U is the density, and i u the displacement. In large displacement theory, the term i ü would be augmented to ,iijj üuu  , which is the convective derivative of the velocity and represents the total time rate of change of the velocity of the ele- ment currently at the coordinate position under consideration. In static (or quasi-static) problems, the right-hand side of Equation (1.2) is zero. 1.5.2 Compatibility Similarly, the strains cannot be arbitrarily specified, but have to satisfy certain compatibility relationships. These arise because (in three dimensions) six inde- pendent strain components are derived, by definition, from the gradients of only three displacements: 1 2 j i ij ji u u xx w §· w H  ¨¸ ¨¸ ww ©¹ (1.3) or more compactly  1 ,, 2 ij i j j i uuH  . This definition of the strains is a com- plete statement of the compatibility requirements, although some authors prefer to eliminate the displacements and derive relationships between higher order derivatives of the strains. 1.5.3 Initial and Boundary Conditions Initial values of the stresses and/or the displacements [and hence through Equa- tion (1.3) the strains] need to be specified. The stresses must, of course, obey the equilibrium Equations (1.1) and (1.2). It is common (but not essential) to treat 10 1 Introduction the displacements and strains as zero at the initial conditions. Later in this book, we shall use internal variables (see Chapter 4 and following), and these will also need to be specified as part of the initial conditions, because they define the initial state of the material. The boundary conditions fall into two categories. On some sections of the boundary, the displacements will be specified. This is straightforward. On other sections, there will be boundary conditions on the tractions. These are often loosely referred to as “stress” boundary conditions, but we prefer to make a careful distinction between stresses (which are second-order tensors) and tractions on a free surface, which are defined as forces per unit area on the sur- face. The traction i t is therefore a vector. It is the tractions that can be defined at a boundary, and there is then an equilibrium relationship between these trac- tions and the stresses in an element immediately within the boundary: ij i j ntV (1.4) where i n is the unit outward normal to the boundary. There is also the possibility of mixed boundary conditions, either in which some displacement components and some traction components are specified, or in which some relationship between displacement and traction is defined. The most common example of the first type is the case where the shear traction on a surface is defined, together with the normal displacement component. This method would be used to describe, for instance, a contact with a smooth rigid body. An example of the second type would be a case where contact with an elastic support was to be simulated by expressing some relationship between the increment in displacement and the increment of traction. There are also occasions where boundary conditions can be expressed in dif- ferential form. For instance, in analyzing a problem of steady flow past some obstruction, it may be convenient to express the boundary conditions far up- stream (and/or far downstream) by noting that the differentials of certain quan- tities must vanish asymptotically in the direction of flow. Since we are concerned here principally with constitutive behaviour, we do not pursue the issue of boundary conditions further, but note that these need to be specified with some care if the equations to be solved are to form a well-posed mathematical problem. Further discussion is outside the scope of this book. 1.5.4 Work Conjugacy A very important concept in continuum mechanics is that of work conjugacy of stresses and strains. A proper set of definitions satisfies the condition that the stresses and strains are work conjugate in the sense that the rate of work input to the material, per unit volume, is given by the product of the stresses with the strain increments: 1.5 Equations of Continuum Mechanics 11 ij ij W V H   (1.5) It is straightforward to show that the conventional small-strain definitions of the Cauchy stress and the linear strain satisfy this condition. 1.5.5 Numbers of Variables and Equations Within a three-dimensional continuum, a solution is required for 15 variables (6 stresses components, 6 strain components and 3 displacements). The equilib- rium Equation (1.2) provides three equations that involve only the stresses, and the strain definitions (1.3) provide six equations that involve the strains and displacements. For a solution to the problem, we are missing six equations, and this is where the constitutive relations come in. They provide six relationships between the stresses (or more usually their increments) and the strains (or strain increments), thus satisfying the need for six more equations, and also providing a link between stresses and strains. The constitutive relations are quite different in character from the equilib- rium and compatibility relationships. Firstly, they are algebraic rather than dif- ferential equations. Secondly, and more importantly, the equilibrium and com- patibility relationships are the same for all materials and (subject to the usual provisos about the applicability of continuum mechanics to a given problem) are universally “true”. Each of the constitutive equations is different for different materials – this statement is a tautology, because it is these very differences in the constitutive relations that define the differences in the characteristics of the materials. Furthermore, the constitutive relations are simply approximations to the behaviour of real materials; none of which will behave exactly according to the idealisations employed. Thus constitutive relations are never “true” for a real material; they can only provide solutions that approximate what happens in reality to a certain degree of precision. Of course, in certain branches of me- chanics (notably the elasticity of metals within a limited stress range), the preci- sion of the constitutive model is so good that, for all practical purposes, it can be treated as a “true” model. In this book, however, we treat much more complex materials, and for these a certain lack of precision is inevitable. Whoever uses the models should understand this and must try to assess its implications for the problem in hand. Chapter 2 Classical Elasticity and Plasticity 2.1 Elasticity Fung (1965) provides elegant definitions for the different forms of elasticity theory, and we follow his terminology here. A material is said to be elastic if the stress can be expressed as a single-valued function of the strain:  ij ij ij fV H (2.1) where  ij ij f H is a second-order tensor-valued function of the strains. If  ij ij f H is linear in H ij , then it can be written  ij ij ij ijkl kl fdV H H , where ijkl d is a fourth-order tensor constant usually called the stiffness matrix: this is the common case of linear elasticity. The incremental stress-strain relationship for an elastic material is written  ij ij ij kl kl fwH V H wH   (2.2) Alternatively, if the stress-strain relationship is originally expressed in incre- mental form, the material is described as hypoelastic:  , ij ijkl ij ij kl fV H V H   (2.3) where  , ijkl ij ij f HV is a fourth-order tensor valued function of the strains or of the stresses, or even (rarely) of both. Note that we distinguish between the fourth-order tensor function ijkl f and the second-order tensor function ij f . If  , ijkl ij ij f HV is a constant, then the material is linear elastic and  , ijkl ij ij ijkl fdHV . 14 2 Classical Elasticity and Plasticity Alternatively, if the stresses can be derived from a strain energy potential, then the material is said to be hyperelastic:  ij ij ij fwH V wH (2.4) in which  ij f H is a scalar valued function of the strains. (Again note that we distinguish the scalar function f from the two tensor functions ij f and ijkl f ). It follows that  2 ij ij kl ij kl fwH V H wH wH   (2.5) If  ij f H is a quadratic function of the strains, then the material is linear elas- tic, and  2 ij ijkl ij kl f d wH wH wH . Example 2.1 Linear Isotropic Elasticity The following strain energy function can be used to express linear isotropic elasticity in hyperelastic form: 32 62 ii jj ij ij fK G cc HH HH  (2.6) where K is the isothermal bulk modulus and G is the shear modulus. Differ- entiating the above 1 gives the elastic form,  2 ij ij ij kk ij ij ij f fK G w c V H HG H wH (2.7) Differentiating once more leads to the hyperelastic incremental form, 2 2 ij kl ijkl kl kk ij ij ij kl f dK G w c V H H HG H wH wH    (2.8) from which one can derive 2 2 3 ijkl ij kl ik jl G dK G §·  GGGG ¨¸ ©¹ (2.9) 1 See table B.1 in Appendix B for some differentials of tensor functions. 2.1 Elasticity 15 Comparison of Equations (2.1) and (2.4) reveals that every hyperelastic mate- rial is also elastic, with   ij ij ij ij f f wH H wH . Conversely, however, an elastic mate- rial is hyperelastic only if  ij ij f H is an integrable function of the strains. Similarly, comparison of Equations (2.2) and (2.3) reveals that all elastic ma- terials are also hypoelastic, with   , ij ij ijkl ij ij kl f f wH HV wH . Again, the converse is not true, and a hypoelastic material is elastic only if  , ijkl ij ij f HV can be ex- pressed as an integrable function of the strains only. Thus hypoelasticity is the most general form, followed by elasticity and hy- perelasticity. This hierarchy is illustrated in Figure 2.1. What therefore are the advantages of the more restrictive forms? The first advantage is the compactness of the formulation. A hyperelastic material requires solely the definition of a scalar function  ij f H for its complete specification. An elastic material re- quires the definition of a second-order tensor function, and a hypoelastic mate- rial requires a fourth-order tensor function. The second advantage relates to the Laws of Thermodynamics. It is quite pos- sible to specify an elastic or hypoelastic material so that, for a closed cycle of stress (or of strain), the material either creates or destroys energy in each cycle. This is clearly contrary to the First Law of Thermodynamics. Furthermore, for a hypoelastic case, it is possible to specify a material for which a closed cycle of stress does not necessarily result in a closed cycle of strain, thus contradicting the notion of elasticity in its sense that it implies that no irrecoverable strains occur over a cycle of stress. Thus there are sound reasons why hyperelasticity should always be the pre- ferred form of elasticity theory. In this book, we build on this concept to express models for plastic materials, i. e. those that do display irrecoverable behaviour in a way that is consistent with thermodynamics. Hypoelastic Elastic Hyperelastic Figure 2.1. Classes of elasticity theory [...]... K 2G 3 ij kl 2G G ik jl k2 ij kl (2. 24) 22 2 Classical Elasticity and Plasticity It is important to note that this stiffness matrix is singular, so that it cannot be inverted to give a compliance matrix This is a feature common to all perfect plasticity models 2. 3 .2 Hardening Plasticity Incremental Response for Strain Hardening For strain-hardening plasticity, Equations (2. 13), (2. 14), and (2. 16) take... This important case is called “associated flow” or “normality” (in the sense that the plastic strain increments are normal to the yield surface, and not in the sense that “normal” means that this represents the usual behaviour of materials) There are many advantages of theories that adopt associated flow, sufficient that many practitioners go to elaborate means to avoid using non-associated flow theories... not translate as plastic straining occurs, then this is said to be isotropic hardening (or softening) This is illustrated in Figure 2. 3a for a simple one-dimensional material that hardens linearly and isotropically with plastic strain The material yields at A at a stress c, and during plastic deformation AB, it hardens and the stress increases to c1 It is then unloaded, and reverse yielding occurs at...16 2 Classical Elasticity and Plasticity 2. 2 Basic Concepts of Plasticity Theory Before moving on to the new formulation of plasticity theory that is the main subject of this book, it is useful to present the conventional formulation of plasticity theory This serves as a basis for comparison for the approach presented in later chapters The first and fundamental assumption of plasticity theory is that... incremental constitutive behaviour can be obtained by applying a purely automatic process to obtain the incremental stress-strain relationship This is important because no further ad hoc assumptions are necessary Note also that the derivation involves solely matrix manipulation and differentiation Both processes can be readily carried out using symbolic manipulation packages Example 2. 3 Von Mises Plasticity... meaning of the variables clear whenever there is any danger of ambiguity The yield surface defines the possibility of plastic strain increments, and the plastic potential then defines the ratios between the plastic strain increments; but what remains to be defined is the magnitude of the plastic strains It is in this area that the greatest variety of ways of specifying plasticity models is found, and... cipal stress space, the surface is a cylinder, centred on the space diagonal, 2 2 3 2 2 k 0 In plane stress 3 1 2 0 , this 1 2 2 3 2 2 2 2 0 The strength in pure tension reduces to the ellipse 1 1 3 3 3k 0 is 1 3k 2 3 2. 2 Basic Concepts of Plasticity Theory 17 It is important to note that the form of the yield function is not uniquely defined The same surface as defined in Example 2. 2 could equally... Incremental Stiffness in Plasticity Models 27 2. 3.4 Kinematic Hardening On the other hand, if the yield surface translates, but does not change size, as plastic strain occurs, then this is said to be kinematic hardening Figure 2. 4a, which should be contrasted with Figure 2. 3a, shows the response of a simple kinematic hardening material that hardens linearly with plastic strain The loading curve OAB is... the back stress, is a function of the plastic strain A simple form would be ij h ' ij p , which would result (for a straight strain path) in linear hardenp ing This hardening relationship could also be written ij h ij , so that the translation of the yield surface is in the same direction as the direction of 28 2 Classical Elasticity and Plasticity the plastic strain This type of hardening is often... as Prager’s translation rule Alternatively, one could write ij is a ij ij , where scalar multiplier, so that the translation of the yield surface is in the same direction as ij ij This is often known as Ziegler’s translation rule For the special case of the von Mises type of yield surface with associated flow, the two translation rules are identical: ij h ij h g ij 2h ij ij 2. 3.5 Discussion of Hardening . for large strain analysis, and often developing a somewhat axiomatic approach in which thermodynamics is introduced at a very general level. The disadvantages of this approach are twofold. Firstly,. elastic mate- rial is hyperelastic only if  ij ij f H is an integrable function of the strains. Similarly, comparison of Equations (2. 2) and (2. 3) reveals that all elastic ma- terials are also. surface, and not in the sense that “normal” means that this represents the usual behav- iour of materials). There are many advantages of theories that adopt associated flow, sufficient that many