An extremum principle is basically a mathematical concept that relies on some physical law. In mechanics, extremum principles such as the principle of minimum total potential energy and minimum total complementary energy form an important base of knowledge that has provided the means for obtaining approximate solutions to a variety of problems in engineering. This is particularly the case with the theory of elasticity. The celebrated principles of least work attributed to Alberto Castigliano, are also in the realm of extremum principles that have been used extensively in the solution of problems in classical structural mechanics dealing with elastic materials. In general, extremum principles and for that matter variational principles start with the basic premise that the solution to a problem can be represented as a class of functions that would satisfy some but not all of the equations governing the exact solution. It is then shown that a certain functional expression, usually composed of scalar quantities such as the total potential energy, strain energy, energy dissipation rate, etc., that have physical interpretations associated with them and are defined through the use of this class of functions, will yield an extremum (i. e. either a maximum or a minimum) for that function. Moreover, the extremum will satisfy the remaining equations required for the complete solution. For example, the principle of minimum total potential energy states that of all the kinematically admissible displacement fields in an elastic body, which also satisfy the governing constitutive equations, only those that satisfy the equations of equilibrium will give rise to a total potential energy that has a stationary value or an extremum. Furthermore, this stationary value will be a minimum for systems that are in stable equilibrium. The underlying power of extremum principles in elasticity is clearly indicated in their earlier applications to structural mechanics and the recent developments associated with numerical methods such as the Rayleigh-Ritz method, the precursor to and the mathematical basis of the finite-element method. An extremum principle is, however, a stronger principle than a variational one since it establishes the existence of an extremum by considering all admissible functions of a certain class and not restricting it to those that are infinitesimal in the proximity of the extremum. Also, in general, for a variational principle, the existence of even a local extremum is not a requirement; it is only sufficient that the functional satisfying the variational principle has a stationary value. Considering the success these principles have enjoyed in their applications to a wider class of problems in mechanics, it is therefore entirely natural to enquire whether extremum principles can indeed be developed to facilitate the development of solutions for materials that exhibit plasticity effects.
The study of extremum principles and indeed the general area of variational methods is quite a mathematically demanding subject. The purpose of this presentation is not to indulge in rigorous mathematical proofs applicable to all types of elasto-plastic materials, but to give a brief expose of the basic facets of extremum principles since they constitute the basic foundation upon which the theorems of limit analysis have been developed. We can appreciate the power of the upper and lower bound solutions when we begin to realize that the solution to a plasticity problem is provided with a set of ‘bounds’ without ever solving the complete set of partial differential equations governing the problem. This is a distinct advantage since these equations are generally non-linearpartial differential equations. Excellent accounts of the developments concerning extremum principles are given in the original articles by pioneers of this area of research, notably G. Colonetti, L. M. Kachanov, M. A. Sadowsky, G. H. Handelman, A. A. Markov, H. J. Greenberg, A. Nadai, R. Hill, W. Prager, D. C. Drucker and P. G. Hodge. The references to the articles by these researchers and more complete accounts of developments of extremum principles applicable to elastic-plastic media and those materials experiencing large-strain phenomena can be found in the bibliography cited at the end of this Appendix.
As a prelude to the discussion of extremum principles for elastic-plastic solids it is instructive to illustrate, as an example, the proof of the principle of minimum potential energy, bearing in mind that the principle is applicable only to elastic solids. In a typical boundary value problem in elasticity, displacements are usually prescribed on apart of the boundary and tractions are prescribed on the remainder. It is also possible to generalise this by considering a part of the boundary where in each of the three independent directions we specify either a component of the displacement or a component of traction. These are the so-called mixed-mixed boundary conditions. An example would be a body in smooth contact with a rigid plane where a single displacement is prescribed and two components of the traction are specified as zero. For the present purposes let us restrict our attention to the specification of the conventional displacement boundary conditions on Su in the form
Ui = Ui on Xi є Su (D.1)
and traction boundary conditions on the remainder of the boundary such that
ffijnj = Ti = Ti on Xi є St (D.2)
where ui and T are specified functions and ni are the direction cosines of the outward unit normal to ST. For the purposes of the discussions that follow, it is sufficient to assume that the region S = Su U ST, and during any deformation Su П ST = 0. Considering the elasticity problem, we assume that the solution to any well-posed boundary value problem can be expressed in terms of the stresses aij and strains eij, that are required to satisfy certain conditions. For example, any stress state aij that satisfies both the equations of internal equilibrium, which in the absence of body forces and dynamic effects reduce to
ajj = 0 on Xi є V (D.3)
and the traction boundary conditions
ajnj = Tj on Xi є S (D.4)
and where ni are the components of the outward unit normal to S, is considered to be a statically admissible stress state. Also Cauchy’s condition (D.4) ensures that at all boundary points where a vector Tj is specified, the internal stress field aij satisfies equilibrium between the applied tractions and the internal stresses.
The strain field eij, on the other hand, must be determined from a displacement vector ui, such that given eij, we should be able to determine ui, at least to within a set of rigid body displacements. If we now consider a displacement field uj, which satisfies all the boundary conditions applicable to the displacements (i. e. of the type (D.1)) and ej the corresponding strains, then these strains are considered to be kinematically admissible.
In elasticity, the statically admissible stresses aj and the corresponding strains ejj are related through Hooke’s law, as follows:
ejj = (D.5)
where Cijkl is the generalised elasticity matrix. The inversion of (D.5) is assured by the positive definiteness of the generalized elasticity matrix. Similarly, the kinematically admissible strains ejj and the stresses derived from these strains are also related to each other through Hooke’s law as follows:
ej = Qjkri (D.6)
In general, the strains ejj cannot be integrated to obtain the displacements uj and the stresses aj generally do not satisfy equilibrium.
Since we have a kinematically admissible set of displacements uj and a statically admissible set of stresses ajj applicable to the same region V with boundary S, we can apply the principle of virtual work to the region; combining (C.6) and (C.7) and setting bi = 0, we have
fffv ajetjdV = // Ti°ujdS (D.7)
The internal energy per unit volume associated with any kinematically admissible state is
U = f a* j de*j (D.8)
Since we are considering linear elastic behaviour (and isothermal or adiabatic deformations) we have from (D.6) and (D.8)
U = 2 Cijklai kajl (D.9)
Hence the total potential energy for the kinematically admissible state of deformation is
nj = 2fffv Cijkl*>]l dV – JJs Tiuj dS (D.10)
The equivalent expression for the total potential energy associated with the exact solution takes the form
П = 2 ЦjT Cijklffikffjl dV – fj TtutdS (D.11)
Theorem D1. The theorem of minimum total potential energy states that, of all the kinematically admissible states of deformation in an elastic body, only the true one will minimise the total potential energy.
with the equality sign applicable when uj = ui. Using (D.10) and (D.11) we have
АП = 2 j jj Cijki(a*kaji – OikOji) dV – j j Ti(u* – щ) dS (D.13)
Since ut and u* have to satisfy the same prescribed displacement boundary conditions on Su of the type (D.1), we must have
ff Ti (uj – Щ) dS = 0 (D.14)
J J Su
f f Ti(uj – ui)dS = Ї f Ti(uj – ui)dS = Ї f ffijnj (uj – ui)dS (D.15)
ST S S
and applying Green’s theorem to the above, we can show that since aij = Oji
/ / Ti (u* – ui) dS = ffij(sij – Bj) dV = Cijkiffik(ff *i – ffji) dV
ST V V
Combining (D.13) and (D.16) we have
ЛП = 2 j jCijki(ff*kffji – 2алоji + ffikffji) dV (Ш7)
Note that since Cijki is symmetric and, since the summation is carried out over the complete set of indices to provide a scalar result, we can interchange the suffixes without altering the final result. We can write (D.17) in the form
ЛП = fff Cijki(«*k – ffik)(ffji – Oji) dV (D.18)
Since Cijki is positive definite, the integrand of (D.17) is positive definite at each xi є V. Hence АП > 0 with the equality being applicable if and only if aj = aij. This latter condition implies that ej = eij and u* = ui to within a rigid body displacement. This proves the assertion that, of the kinematically admissible sets of displacement fields, the exact one, which also satisfies the equations of equilibrium, renders the total potential energy a minimum.
We can use the principle of minimum complementary energy to develop a similar proof for any staticaiiy admissibie stress fieid; i. e. of all the statically admissible stress fields only the stress state that will also give compatible strain fields will render the complementary energy a minimum. Both of these extremum principles and their mixed versions feature prominently in aspects related to the development of procedures for obtaining approximate computational solutions to problems in elasticity. These aspects are discussed in detail in works cited in the bibliography at the end of this Appendix.
Let us now focus attention on the discussion of the extremum principles that are applicable to elastic-plastic materials. First, in keeping with the developments consistent with the theory of plasticity, we will consider velocities, strain rates and stress rates as opposed to displacements, strains and stresses, with the understanding that the specification of the rate is to account for the incremental nature of the developments. Analogous to (D.1), we can define a region Sv on which velocities are prescribed: i. e.
Vi = Vi on Xi є Sv
Similarly, for a surface on which the traction rate is defined we have
ffijnj = Ti = Ti on Xi є St (D.20)
The class of boundary value problems to be solved usually assumes that at a certain time t, the displacements and stresses are known throughout V and the traction rates and velocities are prescribed on S in relation to (D.19) and/or (D.20). The objective here is to determine the stress rates and velocities within V. In keeping with the decomposition rule applicable to small-strain rates, we now assume that the total strain rate eij consists of the summation of the elastic and plastic strain rates ej and e jl respectively. We further assume that the elastic strain rates are derived from Hooke’s law and the plastic strain rates are obtained through the specification of a yield criterion and a flow rule. We shall restrict attention to only the class of materials that satisfy the associated flow rule. We also assume that, given a yield criterion, ej can be determined uniquely. This is, of course, not the case with yield functions with edge surfaces such as those encountered in the Tresca yield surface or even for that matter the vertex point in the Drucker-Prager conical yield surface. This restriction can be removed from the presentation that follows by adopting a discussion to include edges or points where, conventionally, the orientation and magnitude of the plastic strain rate is undetermined. These aspects can be further
We now define a statically admissible field of stress rates &£j such that they satisfy the equilibrium equations in V and boundary traction rates Ti on the surface St as defined through (D.20), and do not violate the plasticity conditions (D.22). The requirement concerning non-violation of the plasticity conditions (D.22) is automatically satisfied if f < k, but imposes the additional constraint that if
f = k then f0 < 0 (D.24)
Here, the superscript 0 refers to the quantity evaluated at the stress state corresponding to the statically admissible state. The strain rates corresponding to (D.21) applicable to the value of the statically admissible stress state are now given by
Є 0j = Cijkld^ + X0 dL (D.25)
Considering (D.23) and (D.24) , the above expression is subject to the following constraints:
X0 > 0 if f = k and f0 = 0, (D.26)
X0 = 0 if either f < k or if f = k f0 < 0 (D.27)
A point to note here is that we have chosen a statically admissible stress state that will specifically eXclude yield in the material, which should be present if plastic strain rates are to manifest. At the outset it would appear that the third condition of (D.26) implies that there may be plastic energy dissipation. However the specification of the additional
constraints (D.24) along with (D.26) and (D.27) safeguards the non-violation of the yield condition which is necessary for any stress state aj to be considered statically admissible (see e. g. Hodge (1958) and Koiter (I960)).
The analogous kinematically admissible velocity field v* is one that satisfies the velocity boundary conditions of the type (D.19) on Sv. The strain rates e * are derived directly from the velocity vector v*. The related stress rates are any solution satisfying
e*i j = Cijki&ki + X* (D.28)
with the constraints
if f = k and f * = j then X* > j (D.29)
if f < k or f * < j then X* = j (D.30)
In (D.24)-(D.3j), it should be noted that quantities such as f and df /doij depend only on the stress rather than the stress rate and are evaluated for the actual given stress state.
Since e*j represents any kinematically admissible strain rate derived from a velocity field that satisfies the velocity boundary conditions, the corresponding analogy to the energy per unit volume of the material is the energy production rate per unit volume of the material, which is given by
W * = f 6ij de *j = j (сто* d 6ji + aX (D.31)
With regard to the last term on the right-hand side of (D.31), the differential of df /doij depends solely on the stresses and not the stress rates. Also considering (D.28)- (D.3j) it follows that since either f * or X* is identically zero, we have
( -^—6*1 ) dX* = f * dX* = j dOij l]) J
W = 2 Cijki6ik 6ji
and the total energy rate is given by
A* = 2 fffv Cijki6*k6*i dV – ffs TiV* dS
We can now use this functional to develop the first of two extremum principles applicable to elastic-plastic materials.
Theorem D2. The first minimum principle states that, of all the kinematically admissible velocity fields in an elastic plastic material, the true velocity field will minimise A*.
Proof. The procedure for developing the proof is similar to that outlined in connection with the principle of minimum total potential energy for an elastic material. We consider the total energy rate associated with the exact result, which is given by
A = 1 //fv Cijkl&ik&jl dV – f£ TiVi dS (D.35)
and construct the difference between the total energy rate (D.34) associated with the assumed kinematically admissible velocity field v* and the result (D.35). This gives (after converting the resulting surface integral in the expression to a volume integral)
AA = A* – A = 2 уjj CijU(d*k – &ik)(a* – Oji) dV + j j j ^~оч(к – к*) dV
We need to prove that AA is positive definite. The integrand of the first integral in (D.36) is always positive in view of the fact that Cijkl is positive definite and the remaining term is in a quadratic form. Considering (D.34), the integrand in the second term can be written as f (к – к*) and, in view of (D.22), this term will vanish at every plastic point. If, on the other hand, the material is elastic it follows from (D.23) that к = 0 with the result that the integrand is equal to – f к. Now if f < k, then no finite stress rates can make the neighbourhood of a stress state immediately plastic, so that from (D.30) we have к* = 0; if, on the other hand, f = k, then we require from (D.23), f < 0 and from (D.29) we have к* > 0. Hence – f к is always positive and the integrand of the second integral is also positive. Consequently, AA is positive definite. Implicit in this positive definiteness assumption is the requirement that the material is elastic-perfectly plastic and is non-softening, in order to satisfy the constraint.
The analysis can be extended to the consideration of the total complementary energy rate defined by
A0 = 2 fffv Cijkrfktfi dV – УУ TS dS (D.37)
where the superscripts 0 are associated with the statically admissible stress states, to develop a second extremum principle.
Theorem D3. The second minimum principle states that among all the statically admissible states of stress rates, the true one will minimise A0.
Proof. Again by considering the difference between the integral expressions for the complementary energy rate applicable to a statically admissible state of stress rate and the complementary energy rate applicable to the exact solution we obtain
and we need to prove that AA0 is positive definite. Since the integrand of the first integral in (D.38) is positive definite, attention can be directed to proving that the integrand of the second integral is always positive definite. We can rewrite the second integrand as к( f – f 0). In the case of elastic behaviour, in view of (D.23), this quantity will be zero. For plastic behaviour, from (D.22), к > 0 and f = 0 and from (D.27) and (D.28),
f 0 < 0. As a consequence X( f — f0) > 0, and the integrand is positive definite, which proves the second extremum principle.
The two theorems presented here can be combined to give upper and lower bounds on either A or Ac as follows:
—A0 < —Ac = A < A* (D.39)
This represents the basis for the development of a number of important relationships associated with not only the upper and lower bound theorems but also to address the question of uniqueness of solution. Let us also not overlook the fact that the extremum principles for elastic-plastic materials, experiencing small strains presented here, have as their basis the requirement concerning the applicability of the associated flow rule for the determination of the plastic strain increments. This indirectly provides the proof for the necessity of the associated flow rule and the normality condition as minimum requirements for the valid application of limit analysis techniques in the development of approximate solutions to problems in soil plasticity.