Category Plasticity and Geomechanics

I. F. Collins, A note on the interpretation of Coulomb’s analysis of the thrust on a rough

retaining wall in terms of the limit theorems of plasticity theory, Geotechnique, 24, 442-447 (1973).

A. Drescher and E. Detournay, Limit load in translational failure mechanisms for associative and non-associative materials, Geotechnique, 43, 443-456 (1993).

J. Heyman, Coulomb’s Memoir on Statics – an Essay in the History of Civil

Engineering, Cambridge University Press, Cambridge, 1972.

[1] Complete references to cited works are given at the end of the chapter where they first appear.

[2] We use the symbol V to denote the del operator ■lx i + dy j + dz k where i, j, к denote the triad of unit base vectors.

[3] In solid mechanics the shear strain represents the decrease in the right angle. We have the increase because of the assumption that compression is positive and our sign convention for displacements.

[4] It is the scalar quantity defined by V • u = (df i + dyi + dz j) • (uxі + Uy j + uz k) =

■ duz + d z.

[5] Why are only six numbers needed to describe the three principal stresses and three principal directions? See Exercise 1.5.

[6] Note that there is no need to let в run to 2n since a rotation of only n radians brings us back to our starting surface.

[7] If we were to make tensile stress positive, the negative sign in (1.32) would become positive. This is one of the rare occasions where the usual equations of solid mechanics differ from ours.

[8] Often the Lame constant is represented by the lowercase X. We will use X extensively later for other purposes.

[9] Students of geomechanics will be aware that if drainage happens slowly, for example in the case of a clay soil, then the strain in (2.26) will also develop slowly. The theory of one-dimensional consolidation would be applicable in that case.

[10] Of course the logarithmic scale means that void ratio and effective stress are not linearly related, and this tells us that any assumption we might make concerning the use of the linear theory of elasticity may be slightly flawed. A non-linear elastic relation of the form deldp = к/p, where к is constant, is sometimes used to replace equation (2.7) and thereby more accurately represent this part of the soil response.

[11] There have been other names associated with the п-plane. These include Pi-plane, octahedral plane, deviatoric plane and Haigh-Westergaard plane.

[12] Note that Wp = ||v||||F|| cos Q, where Q is the angle between the two vectors. The maximum value of cos Q for Q € (0, n/2) occurs when Q = 0, that is when the vectors are aligned. Also note that if Q € (0, n), then negative work is done for Q € (п/2, n), which is a violation of the laws of thermodynamics. This aspect of normality was exploited by D. C. Drucker in a celebrated paper discussing the requirements of normality for non-negative work during plastic deformation; see Appendices E and F.

[13] In some works X appears with a raised dot, X, to emphasise the fact that the dimensions of (4.12) must be homogeneous. We merely note that our X has dimensions of 1/time.

[14] Negative signs associated with displacements in (4.54) and (4.55) as well as some equations that follow all result from the sign convention adopted in Chapter 1.

[15] Here ‘loads’ is used in a general sense. We may be interested in boundary tractions or body forces in the form of gravity loads, or both, depending upon the particular problem being considered.

[16] This follows from the observation that, at collapse, the actual stress field in a non-associative soil will be statically admissible. Therefore, by the lower bound theorem, the collapse load for a non-associative material cannot exceed that for a corresponding material with the associated flow rule.

[17] We can assume a в to be approximately equal to the in situ stress in the rock at the depth of the tunnel.

[18] Of course if the x-axis happens to coincide with the а-direction, then we would have Z = 0 and (6.47) would show that єPx = 0. The same would apply to the в-line.

[19] Referring to equation (6.49), if we replace n with Z the result is zero as shown in (6.50). Also sXx = dvx/дx with similar expressions for the other strain components. We see that the terms in (6.53) are exactly the same as those in (6.49), with n = Z, and therefore the sum is zero as stated above.

[20] See Appendix E. Drucker’s postulate is based on the idea that for a yielding body in equilibrium under a system of loads any small change in loading should not cause the body to do work on its surroundings. The postulate can be used to derive the normality condition.

[21] Here we imply the effective stress wherever appropriate.

[22] Other geologic processes such as transport may tend to sort the particles into more uniform gradations but, for a static soil evolving under increasing ambient stress, we can expect to find a wide range of particle sizes.

[23] In fact, (7.39) is not identical to the result obtained by McDowell and Bolton. Note that the work equation (7.33) assumes all dissipation occurs through particle fracture. McDowell and Bolton include another term in their work equation to account for frictional dissipation as well. This leads to a small difference in the final void ratio-stress relationship, but the essential aspects of their argument are as presented.

[24] d

+ — i zT – (Ur i r + Ug I g + Uzl z )

hz d z

Performing the operation, we obtain

‘ dUr / dUg / dUz

[28] u = 1 H7)* r* r + ( d7)* r* g +1 1* r* z

[29]

.

Limit analysis and limiting equilibrium

Coulomb’s retaining wall analysis was based on equilibrium of forces acting upon a wedge of soil isolated behind the retaining wall. His method is generally referred to as a limiting equilibrium analysis. It gives exactly the same result as the energy balance method used in the upper bound theorem for any translational collapse mechanism.* Moreover the equivalence of the two methods holds regardless of what material model we choose, particularly the choice of flow rule. To see why this is so consider the rigid triangular soil element with area A shown in Figure I.1.

The soil element in Figure I.1 could have more than three sides, but a triangle will be the most simple geometry for our purposes. The element itself is numbered 1 while the surrounding elements are numbered 2, 3 and 4. The velocities of each element with respect to some common stationary point 0 are denoted v0k, k = 1, , 4. Relative

velocities are shown on the hodograph and are denoted v1k, k = 2, 3, 4. Let the three sides of element 1 be numbered L 12, L 13, L 14 and let the traction vectors which act on those sides be T12, T13, T14. Then we can write out the external rate of work R associated with our element as follows:

Here b denotes the body force vector. Equation (I.1) would apply to a unit thickness of soil measured perpendicular to the plane of the Figure I.1. We can also write out the corresponding expression for the rate of dissipation D.

When we use the upper bound theorem we set D equal to R.

Note from the velocity hodograph in Figure I.1 how the relative velocity vectors are associated with the element velocities.

v02 = v01 + v12, v03 = v01 + v13, v04 = v01 + vu (I.3)

* By a translational collapse mechanism we mean a plane system of rigid soil blocks separated by thin shear bands exactly as used throughout Chapter 5.

Figure I.1. A triangular material element with velocities of surrounding elements. Using these relationships in (I.1) and setting R = D shows that (I.1) reduces to

J J b dA (I.4)

Note that the vector v01 can be factored from this expression. Then since v01 is arbitrary we conclude that

(I.5)

Equation (I.5) is a statement of equilibrium of forces for the triangular element. Thus we see that, for a translational collapse mechanism, the energy balance equation from the upper bound theorem is equivalent to the equations of equilibrium. Note that no reference has made to the material that composes the triangular element.

Theorems of limit analysis

the interested reader is encouraged to consult the bibliography presented at the end of this Appendix.

The development of the proofs of the limit theorems is facilitated by first considering the following Lemma:

In the case of elastic—perfectly plastic materials, upon attainment of the limit load, the stress state experiences no change and the increments of strain are only due to the development of plastic strains in the medium. As a result, the application of elastic—perfectly plastic relations is formally similar to the application of rigid perfectly plastic relations. With continued plastic deformations, the elastic deformations can be neglected from the analysis.

To prove the lemma, let us consider a finite region V of an elastic-perfectly plastic material with surface S. Also, ST and Sv are, respectively, subsets of S on which tractions and velocities can be prescribed. The tractions T act on the surface ST and body forces act in V and satisfy the conditions for static equilibrium

the comma denotes the partial derivative with respect to the spatial variable and are the direction cosines to the outward unit normal to S. In addition to this statically admissible set of stresses, v* represents the kinematically admissible velocity field in the region V and e*j the corresponding strain rates that satisfy the kinematic relations

2®o = vi, j +v h

Neither the equilibrium set T, b and Oij nor the kinematically admissible set v* and e*j need be the actual state nor in any way related to each other. We shall proceed to develop the rate form of the virtual work equation where we have

 bi vidV + f f Ti vidS + f f Ti vidS = ff f &ij&* j dV (H.6)

The result (H.5) can also be extended to cover the rate of work of internal forces associ­ated with discontinuities that are encountered in plastically deforming materials; this is not, however, central to the discussion that pertains to the proofs of the limit theorems. The rate form of the equation of virtual work gives

 bi vidV + f f Ti vidS + f f Ti vidS = f f f &ijS *jdV (H.7)

This result is applicable to any equilibrium set of tractions, body forces and stresses. Therefore we can choose the set as increments of tractions, body forces and stresses. Substituting the increments or rates of the equilibrium set we obtain from (H.6)

of the body forces, surface tractions and stresses and vf and efj the corresponding velocity and strain rates associated with a collapse state, such that, analogously to (H.7), we obtain

and the superscript ( )c is intended to signify the fact that all the quantities are associated with the collapse state. Let us note that in the development of the proof of this Lemma, we will be considering only elastic-perfectly plastic materials. For such a material, at limiting or collapse conditions, the peak values of all the forces, tractions (either applied over ST or induced at Sv) and stresses have been reached giving rise to the requirements

bf = 0; xi є V (H.9)

rTlc = 0; xi є S (H.10)

In view of (H.9) and (H.10), the entire left-hand side of equation (H.8) is zero. If we further assume that the strain rates efj admit an additive decomposition of the form

eij — eij + eij

where the superscripts (el) and ( pl) refer, respectively, to the elastic and plastic strain rates, (H.8) gives

fffv (j + \$pl)) &Cj dV — 0 (H.12)

From considerations of the stability postulate of Drucker (Appendix E) that utilises the concepts of the convexity of the yield surface, the associated flow rule (Appendix F) and the normality condition, it can be shown that at collapse the increment of the plastic energy dissipation rate is zero: i. e. the plastic strain increment vector is orthogonal to the stress increment vector. This gives

fffv ecjpl)&c dV — 0 (H.13)

which reduces (H.12) to

fffe Cj^fjdV — 0 (H.14)

Since, for any elastic material, the integrand of (H.14) is positive definite for dc — 0, the vanishing of the integral in (H.14) implies that

acj — 0; xi є V (H.15)

This leads to the conclusion that, at collapse, there is no incremental change in the stress and accordingly there is no incremental change in the elastic strain during deformations occurring at the collapse load. In other words, the elastic deformations play no role in defining the collapse load. With this important result we can proceed to provide proofs of the limit theorems as originally postulated in the classic papers by D. C. Drucker, W. Prager and H. J. Greenberg.

Theorem H1. Consider a region V of a perfectly plastic material with surface S. If an equilibrium distribution of stress denoted by oE applicable to V can be found such that

 0Ecnc = T for x, Є ST (H.16) and is everywhere below yield, i. e. f (4c)
 of satisfies the traction boundary conditions

 then collapse will not occur under the action of the loads.

Proof. The basic approach to proving this theorem involves a proof by contradiction, which makes the assumption that the theorem is false and that such an assumption will lead to a contradiction. Let us consider the finite region V with surface S that is subjected to tractions Ti on ST a subset of S and body forces bi in V. At some values of these loads we assume that collapse of the body occurs, resulting in an actual state of stress of, strain rates eE and corresponding velocities vf. This state of collapse will obviously correspond to tractions

ofn – = Ti on x, Є St (H.18)

and body forces bi that satisfy

of j – b, = 0 in x, Є V (H.19)

but with the requirement that, at collapse, the region under consideration is suitably constrained to eliminate any undetermined rigid body movement.

There are two equilibrium states; one corresponding to {T,, b,, of} and the second corresponding to the assumed state of collapse {T,, b,, of}. We can apply the virtual rate of work equation to each of these states. Using (H.6) we can write

Subtracting (H.21) from (H.20) and noting that at collapse the influence of elastic de­formations can be neglected, we obtain the result

From assumptions of convexity of the yield surface the normality of the plastic strain increment vector to the yield surface implied by the associated flow rule, we require

,c ~ – v о— – (H.23)

and the resulting sum of positive terms cannot be zero. As a result (H.22) must be false and the lower bound theorem is proved. That is, if

then the body may be in an imminent state of collapse. The result (H.25) can then be used to compute the external loads that are necessary just to initiate collapse in the body, which provides the lower bound of the capacity of the body. In terms of the extrema, this is the absolute minimum carrying capacity of the body, and the real collapse load is expected to be greater than the lower bound.

We can now proceed to provide, in a similar manner, a theorem concerning the upper limit of the carrying capacity of an elastic-perfectly plastic body.

Theorem H2. Collapse must occur for any compatible mechanism of plastic deformation for which the rate of working of the external forces either equals or exceeds the rate of internal energy dissipation. The collapse load which is obtained by considering the balance between the rate of working of the external forces and the rate of internal energy dissipation will either be greater than or equal to the true collapse load.

Proof. The procedure for the proof is again to assume that the theorem is false and to show that the assumption leads to a contradiction. Consider the region V of an elastic- rigid plastic material with surface S, which is under the action of surface tractions Ti acting on ST and body forces bi in the volume. Suppose the loads computed by equating the rate of working of the tractions and body forces to the rate of internal dissipation are less than the actual collapse load. Then the body will not experience collapse at this load and we can obtain an equilibrium state of stress oE such that

and

where VF U VE = V. At this point we should clarify the conditions implied by (H.28) and (H.29) in the light of comments made both at the beginning of this Appendix and in the introductory paragraphs of Chapter 5. It is implicitly assumed that failure may occur in restricted regions within V, without attainment of collapse conditions in the body.

Let us now consider a compatible mechanism of plastic deformation corresponding to a collapse mechanism, which is defined by a plastic component of the velocity vc(pl) in the region Sv, a resulting plastic strain rate ejpl) and a corresponding stress state ojpl) in the region V. Considering the stress state at collapse and the velocity field at collapse, the rate form of the virtual work equation (H.6) can be written as

(H.30)

Similarly, considering the equilibrium stress state and the velocity field at collapse we can write

Considering (H.28) and (H.29), we can rewrite (H.32) as

fffv ^jPl) – °E) j dV + fffv (4pl) – 4) j dV = 0 (H.33)

We are aware from Appendices E and F that, for a convex yield surface and an associated flow rule obeying the normality condition, the quantity (aCpl) — aE )gjjpl) can be no smaller than zero. Moreover, if aE is not a yield state (as is the case in the region VE), then (ajpl) — aE )eff[29] must be strictly positive. Therefore the first integral in (H.33) is non-negative and the second integral is positive definite. This contradicts our initial assumption and we conclude that the theorem must be true. The upper bound theorem assures us of the fact that the true collapse load must either be less than or at most equal to the collapse load obtained by equating the rate working of the external loading with the rate of internal energy dissipation for a compatible mechanism of plastic deformation.

In the preceding sections we have presented the two basic limit theorems that provide distinct upper and lower limits to the actual collapse loads, which could be obtained by solving the complete set of partial differential equations governing a problem in the theory of perfectly plastic solids. Furthermore, if the upper and lower bounds coincide then this would also correspond to the exact solution of the problem. There are also several corollaries that arise from the classical lower bound theorem, since the original state of stress is also applicable in the modified situation. These extensions are described in detail by Drucker et al. (1952) and in the text by Chen (1975).

It is also worth re-iterating the important role that requirements such as convexity of the yield surface and the associated flow rule play in the developments of the theorems of limit equilibrium. Indeed the theorems cannot be proved in a general sense without the aid of the convexity and associativity arguments. Despite these advantages, it is worth recognising the fact that the flow laws for many geomaterials point specifically to their non-associated character. The uniqueness of solutions and the validity of the upper and lower bound theorems do not, in a general sense, extend to such materials. Several investigators have examined the conditions under which the bounding techniques can be applied to geomaterials, which obey non-associated flow rules. Examples of these are given, among others, by A. D. Cox, Z. Mroz, G. de Josselin de Jong, A. C. Palmer, G. Maier, E. H. Davis, T. Hueckel, J. L. Dais, I. F. Collins and J. Salencon. Other approximate procedures for the calculation of limit loads for geomaterials exhibiting non-associated flow laws have also been proposed recently by A. Drescher and E. Detournay and R. L. Michalowski.

Theorems of limit analysis

A uniqueness theorem for elastic-plastic. deformation

The concept of the uniqueness of a solution is an essential requirement to the well-posed nature of a boundary value problem. A uniqueness theorem assures us that there is only one solution possible for the governing set of equations subject to appropriate boundary conditions. In EG we have discussed a uniqueness theorem in the context of the linear theory of elasticity. With linear theories in mechanics and physics, the development of a proof of uniqueness of solutions to boundary value problems and initial boundary value problems is well established. Comprehensive discussions of these topics are given in many texts on mathematical physics and on the theory of partial differential equations and also discussed in recent volumes by Selvadurai (2000a, b). The question that arises in the context of plasticity focuses on the development of a uniqueness theorem for what is basically a non-linear problem. This is not a straightforward issue, even with regard to certain situations involving non-linear behaviour of linear elastic materials. Examples that illustrate the concept of non-uniqueness of elasticity solutions can be readily found in problems dealing with elastic buckling of structural elements such as beam-columns and shallow shells under lateral loads. In these categories of problem the structure can exhibit multiple equilibrium states corresponding to the same level of loading. The purpose of the discussion given below is then to address the basic question of what constraints should be imposed, specifically regarding plastic stress – strain relations, in order that the solution to a particular boundary value problem is unique.

Theorem G1. In keeping with the formulation of many approaches to proof of unique­ness, let us consider a finite region Vof an elastic-plastic material that is bounded by a surface S. The region is subjected to tractions, displacements and body forces as follows:

 II о xi € st (G.1) Ui = u0; Xi € Su (G.2) II о Xi € V (G.3)

where ST and Su are complementary subsets of S; i. e. ST U Su = S and ST П Su = 0,

and T0, u0 and b0 are prescribed functions (Figure G.1a). The state of stress and strain in
the medium due to the application of the prescribed tractions, displacements and body

Figure G.1. Initial and final states of stress and strain in an elasto-plastic body.

forces in the respective regions is given by

 aij — j xi Є V (G.4) &ij — &ij; xi Є V (G.5) We now alter the boundary tractions, boundary displacements and body forces by their corresponding incremental values as follows: Ti — T0 + dTi; X Є St (G.6) Ui — u0 + dui; Xi Є Su (G.7) bi — b0 + dbi; Xi Є V (G.8) We assume that the corresponding states of stress and strain can be expressed in the forms in the body (Figure G.1b) aij — ai + d ; Xi Є V (G.9) Sij — + dSij; Xi Є V (G.10)

where the incremental values of the stresses and strains are implied. The examination of the question of uniqueness of the solution reduces to the assessment of whether the stress increments daij and the strain increments deij are uniquely determined by the increments of change in the surface tractions dTi, the increment in the surface displacements dui and the increment in the body forces dbi. In this connection we hope to prove that the associated flow rule is both necessary and sufficient for the condition for uniqueness of the stress increments daij and the strain increments deij.

states of stress and strain in the following forms:

a

ч

and

ffiJ — 4

eij — e?. + df

Now we make use of the principle of virtual work described in Appendix C. Assuming that the displacement field is continuous throughout V, such that incremental tractions df are defined through equilibrium considerations and incremental displacements dui satisfy kinematic or compatibility constraints we have

There is, of course, no requirement for the two solutions mentioned above to be in any way related. The difference between the two states given by (G.11)-(G14) gives

 A(daij) — da j — da^; x( є V (G.16) A( de,-j) — dej — de(j}; x( є V (G.17) and with df — dTt(2) — dT1a) — 0; Xi є St (G.18) dui — duf) — du21 — 0; x( є Su (G.19) Substituting (G.16)-(G.19) in (G.15) we obtain the result JJ j A(daij) A(deij) dV — 0 (G.20) Since V is finite we obtain, from the Dubois-Reymond lemma, (G.20) must vanish everywhere in V, i. e. that the integrand of dl — A(d aij)A(d ej = 0 (G.21) If we could now show that, for a particular elastic-plastic constitutive relation, the quantity dl is positive definite, then we obtain the contradiction we seek. We could then satisfy (G.21) if and only if A(dei;) were identically zero. So how can we proceed to show that A(daij )A(dє(. ) is positive definite? To begin, assume that the difference in the incremental strains can be represented as a linear combination of the elastic and plastic components as follows: A(deij) — A( dej + A(dej (G.22) Using (G.22) we can rewrite (G.21) in the form dl — A(daij) [A(deej) + A(dep)] = 0 (G.23)

Considering the elastic behaviour of the material we have for every state of incremental stresses

Therefore the problem is reduced to the examination of the conditions under which the scalar product A(doij) A(dej) is positive definite. In order to examine this we need to consider separately threepossible states, relating to loading and unloading (Figure G.2) associated with dofl, doj and A(doij).

Case 1. Consider the case when both increments doj and doff * correspond to loading

(1) (f)

paths. In this instance, both doj and doj lie on the tangent plane at the stress point on the failure surface and by virtue of (G. l6), A(doij) also lies on the tangent plane. It is evident that for the scalar product A(doj A(dej) to be positive definite for all vector increments A(doj that are tangent to the failure surface, the plastic strain vectors

d j and d j, and consequently A(defj), must be normal to the failure surface. So the associated flow rule ensures uniqueness in this case.

Case 2. Consider the situation where both loading increments correspond to unloading. In this case by definition (see also Appendix F) A(dej) = 0. Hence by virtue of (G.24), dl is positive definite and uniqueness is a trivial consequence.

Owing to this assumption we have depr> = 0 and the part of dl whose positive defi­niteness needs to be investigated is

(2)

Since daf corresponds to a loading path, this stress increment is located on the tangent plane through the stress point. By virtue of the associated flow rule the plastic strain increment vector dep2> is orthogonal to dap Hence the scalar product daf dep2) is identically equal to zero.

Let us now focus on the second term on the right-hand side of (G.25). By defini­tion daff is an unloading path and the vector is directed to the interior of the failure surface (Figure G.2c), commencing from the stress point. The plastic strain increment vector def>, for the loading path, is on the other hand directed away from the failure surface normal to the tangent plane at the stress point. This means that so long as the increment datf is an unloading path and the increment daf is a loading path, the in-

j (1) p(2) j

cluded angle between daj and def will be an obtuse angle. Hence the scalar product, – daP-1 d efwill always be positive definite, for any convex failure surface. Therefore the positive definiteness of the integral is assured and we can conclude that uniqueness of the boundary value problem is assured, provided the failure surface is convex and the plastic strains are determined through the associated flow rule.

The associated flow rule

In Appendix E we have examined Drucker’s postulate for the stability of the material undergoing plastic deformations. To develop the plastic constitutive equations or the associated flow rule it is necessary to assume that a yield function exists, i. e.

f (ац) = k

As discussed in Chapter 3, when referred to the multi-dimensional stress space, the convex yieldfunction with a unique normal at each point identifies the boundary between elastic states in the material for which f (ai.) < k and plastic states for which f (ai.) = k. For the present purposes we shall restrict attention to non-strain hardening materials. Consider the inequalities given by (E.7) in relation to a vector space consisting of the stress tensor and the strain rate vector. The expression related to plastic energy dissipation rate can be visualised as the scalar product of two vectors (ai. — a-j) and eP. In order for the first inequality of (E.7) to be satisfied, the included angle between the vectors (aij — a.) and the plastic strain rate vector eP. should be acute. This condition will hold for any aj. located either within the yield surface or on the yield surface itself.

Consider the point B in Figure F.1, which is located on the yield surface f (aij) = k, and assume that the associated flow rule with the governing normality condition gives the plastic strain rate vector, which will therefore be normal to the yield surface. Now consider the tangent plane to the yield surface at this point. We are assured, by the convexity of the yield surface, that any stress point aij will lie to one side of this tangent plane. The line of action of the vector (ai. — a.) must therefore make an acute angle with eP.. Therefore for the rate of plastic energy dissipation to be positive definite we must have

Figure F.1. Geometrical representation of the stability postulate.

Other arguments for this form of the associated flow rule are also given by R. Hill and T. Y. Thomas.

The second inequality of (E.7) dealing with the requirement for material stability can be written as

p df

= XT—- – 0 (R4)

d&ij

In the case of neutral loading, the loading path follows the yield surface itself and the unloading process results in no plastic deformation. The condition (F.4) implies that for plastic deformations to occur the scalar multiplier must be non-negative; i. e.

X – 0 (F.6)

If the stress state satisfies the yield condition but with X = 0, then there is no plastic deformation. Also for a perfectly plastic material, there is no essential difference between the processes of loading and neutral loading since in the stress space (df /doij)dij = 0, whenever dij lies on the yield surface.

In the preceding we have focused on the application of Drucker’s stability postulate to the development of the associated flow rule for failure surfaces that have a unique normal at each point on the surface. Let us now focus attention on situations where the failure surface can have either edges or corners along which the orientation of the unit normal is not determined uniquely. Examples of such failure surfaces can include the Tresca and Coulomb failure criteria. We can extend the definition of the associated flow rule to cover such non-singular boundaries (Figure F.2). When considering non-singular failure surfaces, the associative flow rule should be modified to include several (say n) intersecting surfaces at a point. In such a case, the associated flow rule can be written as

where the derivatives dfa/doij are linearly independent in view of the fact that the failure

Figure F.2. Convex failure surface with non-singular points.

 p df1 df2 Sf: = M— + Л2— = к 1 1 d*ij dffij

surfaces themselves are independent. The complete plastic strain rate is found according to (F.7) and the resultant of these plastic strain rates will be contained within the region obtained by surfaces spanning the unit normals to the segments of the yield surfaces, which intersect at the singular points. For example, referring to Figure F.2, where n = 2, the plastic strain rate at the corner C is given by

where 0 < ф < 1. It should be noted that although the direction of the plastic strain rate is not unique, the energy dissipation rate is uniquely determined, since for a given plastic strain rate the corresponding stress is unique. For example, considering a perfectly plastic material, which obeys the associated flow rule, the energy dissipation rate is given by

D = j (F.9)

If we realize that, at failure, the stresses are uniquely determined by the failure criterion, then the dissipation function can be expressed solely in terms of the plastic strain rate, i. e. D = D(sf ), and such a representation can be used to present an inverse form of the associated flow rule (F.3), where now the stresses can be expressed in terms of the strain rates.

Drucker’s stability postulate

An approach to the development of the constitutive equations of plasticity involves the consideration of plastic energy dissipation in an irreversible process. This is somewhat analogous to the determination of the constitutive equations for an elastic material by considering the energy stored during deformation. The notion of material stability is an important aspect of the development of any self-consistent theory of plasticity, which not only includes the constitutive equations governing plastic behaviour but also appropriate uniqueness theorems and procedures for the solution of boundary value problems.

The concept of material stability implies the existence of a one-to-one correspondence in the constitutive equations in the range of small strains. The notion of material stability in the small can be illustrated by appeal to the behaviour of a material in uniaxial straining. Figure E.1 shows non-linear stress-strain behaviour where a is the uniaxial Cauchy stress, e is the corresponding small strain and e is the strain rate. Let us consider the situation where the specimen is subjected to an arbitrary stress a and Aa is the increment in stress, which produces a corresponding increment in strain Ae. The material is said to stable if

Aa Ae > 0 (E.1)

The result (E.1) implies that in a stable material, strain increments result in positive work from the stresses. We can generalize (E.1) to the following form involving all components of the strain tensor and the stress tensor to give the following requirement for a stable material:

AaijAeij > 0 (E.2)

The notion of material stability and plastic energy dissipation during yielding is central to Drucker’s stability postulate, which applies to geomaterials that exhibit strain hardening phenomena and as a special case can also apply to perfectly plastic materials.

The plastic energy dissipation during a closed cycle of loading can be demonstrated by appeal to the closed path shown in Figure E.2. If the loading that induces a strain increment Aeij commences from a reference stress state ajj, the inequality (E.2) can be written as

(ajj – a°j)Aefj> 0 (E.3)

and the incremental irreversible plastic strains Aeipj are those that occur subsequent to the application of the reference stress ajj.

Drucker’s postulate hinges on this concept of a stable material. Let us consider a stable state in a material, which can experience plastic energy dissipation. We subject the body

Figure E.1. The uniaxial stress-strain behaviour for a stable material.

to tractions, body forces and displacements, which result in the stress state аЯ. We now slowly alter the tractions, body forces, etc., such that the new stress state is а, і. Finally, we return slowly to the original reference stress state аЯ. If plastic strains develop during this stress cycle then the work done by the stresses is non-negative. Drucker’s postulate therefore states that, for any stable material, the rate of work done by the stresses during plastic deformation at a point in the medium, over a closed cycle involving loading and unloading, is non-negative. If we denote this plastic work rate by WP, we have

wp = f (ар – 0°)ёц dt > 0 (E.4)

0

We can use a schematic geometric representation shown in Figure E.3 to illustrate stress cycling in relation to the yield surface f (ai;) = 0. Considering the closed stress cycle (i. e. the stress cycle commences from a0 and returns to a0), the rate of work done by the stresses on the elastic strain rates e C is fully recoverable. Therefore we can focus on the representation of the stability postulate in terms of the total work of the stresses done on the plastic strain rates, Wp, over the history of the stress cycle where t є (t1 , t2), and for which the stresses satisfy the yield condition. Therefore in terms of the plastic

Figure E.3. Closed stress cycle 0-1-2-0 in a generalised stress space.

strain rate, (E.4) can be written as

Wp = fa – efjXj dt > 0 (E.5)

ti

Since plastic strains materialise only at t = t1, we can expand Wp as a Taylor series about the neighbourhood of oij (t1); this gives

Then, considering the leading terms on the right-hand side of (E.6), if the plastic work rate is to satisfy Drucker’s postulate we must require (since (t – t1) > 0)

(E.7)

The first requirement of (E.7) is associated with the energy dissipation and the second requirement refers to the stability of the material since, if oijej > 0 and eP = 0, material stability is assured. The stability postulate is a key feature in the development of associated flow rules in the theory of plasticity and in the development of uniqueness theorems for perfectly plastic behaviour. Several investigators including H. Ziegler, A. E. Green and P. M. Naghdi have discussed the stability postulate in the context of thermodynamics of continua. Ziegler’s work shows that Drucker’s stability postulate is a special case of maximum entropy production. Green and Naghdi’s work has shown that the postulate implies constraints on the flow rule that do not necessarily follow from laws of thermodynamics. This makes the concept of stability as constitutive assumption valid for certain classes of materials.

Extremum principles

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)

1

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

Hence

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

(D.16)

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)

dffij

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)

j d6j

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 (D.32) Therefore 1 W = 2 Cijki6ik 6ji (D.33) and the total energy rate is given by A* = 2 fffv Cijki6*k6*i dV – ffs TiV* dS (D.34)

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

(D.36)

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.

Principles of virtual work

The principles of virtual work, which bring together the concepts of equilibrium and compatibility, or kinematics, are an important development in the mechanics of solids and in applied mechanics in general. The fact that the principles do not rely on the constitutive behaviour that pertains to the material is a major advantage in their applicability to elastic as well as inelastic materials and to problems that deal with dynamic and stability effects. There are, of course, various versions of the principle of virtual work, the forms of which will depend on the manner in which mechanical and kinematic variables are defined and presented. In the following we shall present a general statement of the principle of virtual work, which is of particular relevance to applications to elastic as well as plastic continua.

Let us consider a continuum region of finite extent V, which is bounded by the surface S. The region is restrained against rigid motions by suitable boundary constraints (Figure C.1). We now apply prescribed values of tractions and displacements T* and u* respectively, which act over separate regions of the boundary S. The displacement and traction boundary conditions applicable to the boundary value problem can be written as

Ui = u*; Xi Є Su (C.1)

Ti = ffijUj = T*; Xi Є St (C.2)

where ni are the components of the outward unit normal to S. In general we assume that S = ST U Su. It must be noted that this presupposes that there are no regions where both traction and displacement boundary conditions are prescribed simultane­ously. There are situations where this is possible, an example being that of the contact between a rigid footing with a smooth frictionless base and the surface of a contin­uum region such as a halfspace. Here, the component of the normal displacement and the tangential component of the tractions at the contact region are specified. In this case the regions Su and ST will have to overlap to account for the mixed nature of the boundary conditions applicable to the same region. For the purposes of the present discussion it is sufficient to assume that the boundary conditions correspond to (C.1) and (C.2), with the understanding that the discussion that follows can equally well be extended to cover this class of mixed-mixed boundary conditions. In addition to these prescribed displacements and tractions, we assume that the region V is also subjected to a body force field defined by the vector bi and ignore the effects of dynamics, thereby reducing the problem to a boundary value problem rather than an initial boundary value problem. Let us assume that the applications of these bound­ary displacements, boundary tractions and body forces gives rise to a kinematically

Figure C.1. Undeformed and equilibrium configurations and the virtual displacement field.

admissible set of displacements ui and stresses aij that satisfy boundary conditions (C.1) and (C.2) and the equations of equilibrium

ffijj – bi = 0 (C.3)

where the subscript comma implies partial differentiation with respect to the appropri­ate spatial variable. By definition, a kinematically admissible displacement field is one that satisfies any external constraints, as defined by (C.1), any internal constraints (such as either material incompressibility or inextensibility), and is continuous and piece­wise continuously differentiable in the region V, which includes S. Also, we define a stress field aij that satisfies the traction boundary conditions (C.2) and the equations of equilibrium (C.3) as being statically admissible.

Let us now consider a virtual displacement field as defined by the difference between neighbouring kinematically admissible displacement fields. The term ‘neighbouring’ immediately introduces the notion of the infinitesimal into the definition. If we consider ui to be one displacement state and ui + S ui as the neighbouring displacement field then, by definition, S ui is also a kinematically admissible field with small deformation gradients (i. e. |9 (S ui )/dxj | ^ 1), and the qualifier S is intended to signify an incremental

difference. We can also define the virtual strain field Seij associated with the virtual displacements Sui as

1

Seij = 2^SUi, j + Suji) (C.4)

If we consider the variation of Sui on S then by virtue of (C.1),

S ui = 0; Xi e Su (C.5)

which satisfies the requirement of kinematic admissibility for the second displacement state. We can now compute the work done by this virtual displacement field by considering the body forces and the tractions Ti*. We can define this ‘external work’, AWext ast

AWm = – ff f bi Sut dV + [[ T*Sut dS (C.6)

J J Jv J Jst

We can also consider the internal work AWnt, associated with the virtual strains, result­ing from the virtual displacements, operating on the stresses aij. We have

We note that, in the absence of body couples, the stress tensor is symmetric, i. e. aij = Oji. Using this fact, we can write

OijSeij = Oij(Sui),j = [OijSui],j – ffjjSui (C.8)

Using (C.8) in (C.7) and making use of Green’s theorem we have

A~W{nt = J*^ OijnjSui dS Ш oij, jSui dV (C.9)

In view of (C.5) the surface integral in (C.9) can be restricted to ST rather than S. Combining (C.6) and (C.9) we have

AWint – AWext = – f f f [Oij, j – bi ]Sui dV + [[ [Oijnj – Ti*]Sui dS (C.10)

J J J V J Jst

The integrals in (C.10) will vanish for every choice of the set of virtual displacements Sui, if and only if the terms in the bracketed quantities reduce exactly to zero. This is ensured by the equations of equilibrium (C.3) and the traction boundary conditions (C.2). Hence, a body is in equilibrium under the application of a system of applied forces if and only if the ‘principle of virtual work, defined by

AWeXt = AWint

is satisfied identically.

At this point, a comment regarding the expression for the internal work AWint, as defined by (C.7), is in order. In the expression for AWjnt, we have not specified the nature of the internal work nor the agencies that would be responsible for generating this internal work. It is only sufficient that such a measure exists. The only appar­ent requirement is that the body should experience ‘virtual straining’, under the im­posed virtual deformation. We have not even specified whether such internal work is

t Although it might appear that the negative sign preceding the first integral in (C.6) is in error, that is not the case. The sign results from our convention that positive displacements act in negative coordinate directions while positive body forces act in positive coordinate directions.

either conservative or dissipative. This leaves room for choosing the measure of internal work to conform to the dominant internal process associated with the generation of in­ternal work. For example, with elastic materials, this internal work could be associated with the elastic energy that is stored in the material during the virtual deformation, with the assumption that for an elastic material the stored energy is indeed fully recoverable. Alternatively, we can assume that the medium under consideration is an ideally plastic solid, in which case the internal work AWint, as defined by (C.7) will now correspond to the plastic energy dissipation resulting from the virtual plastic straining, resulting from the application of the virtual displacements Sui. This ability to interpret the internal work in a manner appropriate to the continuum under consideration makes virtual work principles a powerful tool in mechanics.

Now we proceed to define the second principle of virtual work. The principle of complementary virtual work is based on the concept of a virtual stress field as opposed to a virtual displacement field. Again we assume two statically admissible stress fields, such that their difference gives the symmetric virtual stress field Saij. From (C.2) and (C.3) we note that this virtual stress field should satisfy, respectively,

Saij nj = 0; xi є ST (C.12)

Sffijj = 0; xi є V (C.13)

In connection with the derivation of the above equations, let us note that the body force field bi and the applied external tractions T* are exactly the same for two neighbour­ing states. When we interpret a virtual stress field as the difference between the two neighbouring states, these terms will naturally disappear from the equations governing internal equilibrium in the region V and on the boundary S. We can now define the external complementary virtual work AWext as

AW^ =ff u*SaunjdS (C.14)

J JSu

Similarly, we can define an internal complementary virtual work AWCint as

AWCnt = fffsijS0ijdV (C.15)

 (ui — u*)Saij nj dS +

Again we can take the difference between these two measures and apply Green’s theorem as well as (C.12) and (C.13) to arrive at the following the result:

(C.16)

Again, the integrals occurring in (C.16) will vanish identically for every choice of the virtual stress field Saij provided the terms within the brackets vanish identically. For this to be satisfied, the strain field eij should be compatible with the kinematically admissible displacement field ui. The principle of complementary virtual work thus gives

AWCnt = AWX (C.17)

Here again, the comments made earlier in relation to the definition of the inter­nal virtual work AWint, are also applicable to the definition of the complementary internal virtual work AWint.

When dealing with the application of the principles of virtual work to problems arising from the theory of plasticity, we are dealing with quantities such as virtual velocities Svi and virtual stress rates S&ij, as opposed to virtual displacements and virtual stresses, to
indicate primarily the incremental nature of the problem formulation. In this case, the definitions of the internal and external virtual work and their complementary counterparts have to be identified as rates of virtual work. The result (C.11) for the principle of virtual work can be restated as a principle of the rate of virtual work in the form

AWeXt = AWim (C.18)

and the result (C.17) for the principle of complementary virtual work can be restated as a principle of the rate of complementary virtual work in the form

AW-nt = AWCext (C.19)

Both principles are used quite extensively in the development of governing equations, specific solutions and the development of computational schemes for the numerical solu­tion of problems in mechanics. In the context of the theory of plasticity of geomaterials, the principle of the rate of virtual work is used quite extensively for the development of proofs of the upper and lower bound theorems in plasticity and in the development of associated solutions.

Mohr circles in three dimensions

We can extend Mohr circle construction to three dimensions. As with the two­dimensional case the procedure is most conveniently demonstrated using the principal stress state where a1 is the maximum principal stress, a2 is the intermediate principal stress and a3 is the minimum principal stress with the result, o1 > o2 > a3. The result (B.14) concerning the normal stress acting on an oblique plane and referred to a gen­eralized state of stress is equally valid for the principal stress state. If we consider the principal stress state shown in Figure B.10 and consider the obliquely oriented plane with a unit normal having components

n = n1 i 1 + n2 i 2 + n3 i 3 (B.30)

where the unit base vectors in the principal directions are implied, the normal stress acting on the oblique plane is given by

Onn = n?0-1 + n2o2 + n23ff3 (B.31)

Figure B.10. Traction vectors on an oblique plane referred to the principal stress space.

We can also use the reduced version of the result (B.16) to define the shear stress that acts on the oblique plane. This gives

alt = (И10ї)2 + (Я202)2 + (Я3СТ3)2 – (я^ + я^ + n23ff31 Using both (B.31) and (B.32) we can obtain two equations

ann + 0nt = (n1a1)2 + (n2a2)2 + (n3a3)2

(ann)2 = (nfo + n2&2 + ^03)

which can be combined with the consistency condition for the direction cosines,

n2 + n2 + n? = 1

to give a set of equations for the squares of the three direction cosines. The set of equations (B.33)-(B.35) has a non-trivial solution, which can be evaluated using sym­bolic mathematical manipulation programs such as MATHEMATICA®, MACSYMA® or MAPLE®. Imposing the constraints

n"2 > 0; n2 > 0; n2 > 0

The relevant solutions are

Since the principal stresses are in a ranked order, the equations (B.37)-(B.39) are equiv­alent to

fflt + (0nn – 02)(0nn – 03) > 0 (B.40)

0lt + (0nn – 01)(0nn – 03) < 0 (B.41)

fflt + (0nn – 01)(0nn – 02) > 0 (B.42)

and the different directions of the inequalities should be noted. We can rewrite

 & nt Figure B. ll. Mohr circle in three dimensions.

(B.40)-(B.42) in the following forms:

(B.43)

(B.44)

(B.45)

The similarity between equation (B.29) and equations (B.43)-(B.45) is abundantly ev­ident, the only difference being the inequalities that appear in the latter equations. How can we interpret the result? We can do so graphically and assign ann as the axis corresponding to 0XX and ant as the axis corresponding to aXY used in the Mohr circle plot shown in Figure B.8. For the sake of convenience let us restrict our attention only to the region where both ann and ant are considered to be positive. We can plot the circular boundaries defined by the three Mohr circles (B.43)-(B.45) and identify the region to which the inequalities apply. This is shown in Figure B. ll. Every combination of ann and ant shown in the shaded area is an admissible state of stress in a general­ized sense. This is a real bonus when it comes to identifying states of stress that are responsible for failure of materials. For example, if failure of the material is governed by the maximum shear stress, the three-dimensional stress state can be converted to its principal components and the maximum shear stress is given by

l

Tmax = ^(ff1 – °3) (B.46)

The relationship between the principal stresses can also be expressed in terms of Lode’s parameter

(a2 — a3)

X = 2 2 ^ – 1 (B.47)

(o-i – CT3)

which determines the position of the intermediate principal stress o2 in relation to the other principal stresses as x varies from -1 to +1,

for pure compression: o1 > 0; o2 = o3 = 0 and x = ~1

for pure tension: o1 = o2 = 0; o3 < 0 and x = +1

for pure shear: o1 > 0; o2 = 0; o3 = —o1 and x = 0

 nn

 nn

 Figure B.12. Alternative admissible stress states.

compressive; this is not a requirement for the identification of the domain of admissible stress states. Figure B.12 shows alternative representations.

This Appendix summarizes some basic attributes of the Mohr circle. Other features associated with its development are covered within the context of the chapters in this volume, which utilise such features extensively. Further valuable discussions can also be found in the suggested reading.

Mohr circles

The graphical construction for the representation of the state of stress at a point within a continuum region is generally attributed to the German engineer Otto Christian Mohr. Although the use of graphical techniques in structural and solid mechanics has been an important area of activity both for engineering calculations and stress analysis, par­ticularly in the eighteenth and nineteenth centuries (see, e. g., Todhunter and Pearson (1886, 1893) and Timoshenko (1953)), the contributions of Karl Culmann and Otto Mohr to the development of this area are regarded as being particularly significant. Despite the passage of time these graphical constructions have continued to serve as efficient educational tools for the visualisation of difficult concepts related to the representation of three-dimensional states of stress, particularly in relation to the de­scription of failure states in materials. The fact that the techniques developed in rela­tion to the stress state at a point that can be represented in terms of a stress matrix of rank two or a second-order tensor implies that the procedures are equally appli­cable to the description of other properties and states in continua, which can be de­scribed in a similar manner. Examples include the description of moments of inertia of solids, flexural characteristics of plates and the hydraulic conductivity characteristics of porous media, etc. The purpose of this Appendix is to present a brief outline of the significant features of Mohr circles and to develop the basic equations applicable to the three-dimensional graphical representation of the stress state at a point. The nam­ing of the graphical procedures for the representation of the state of stress at a point, in honour of Otto Mohr is very much in recognition of his formal development of the procedures through archival publications. There are earlier references to techniques re­sembling a graphical method in the work of Karl Culmann, although they are in a form that is perhaps less well developed than the presentations of Mohr.

As a prelude to the development of the relevant equations, we first consider Cauchy’s relationship, which deals with the stress state at a point within the medium and the traction vectors that act on an arbitrary plane either through or located at an infinitesi­mal distance from the point. Before doing this it is worthwhile making some remarks with regard to sign conventions that are used to identify particular stress states. From an engineer’s perspective, sign conventions are crucial to identifying the ‘sense’ of stress components accurately. This is clearly not the case if we were to treat the stress state as a ‘matrix’ or a ‘tensor’, which is amenable to purely mathematical opera­tions. In this context we are not concerned as to whether the normal stresses are compressive or tensile or whether the shear stresses are positive or negative. These are simply elements of a matrix or a tensor; we can transform the matrix, calculate its eigenvalues and perform all the legitimate operations of linear algebra without ever
worrying about the physical significance of the manipulations. Also, for example when a stress transformation rule such as

[acp] = [Hf [a][H] (B.1)

which involves the transformation of the stress matrix from the Cartesian to the cylin­drical polar equivalent, the sign convention adopted in the definition of the stress matrix referred to the rectangular Cartesian coordinate system, [a], simply translates to the definition of the sign convention for the stress matrix |_acp referred to the cylindrical polar coordinate system. There are of course ‘bonuses’ that arise from these mathemat­ical operations, such as the fact that the eigenvalues of a symmetric matrix must always be real, which straight away translates to the deduction that the principal stresses must always be real, but this is a secondary issue. From an engineering perspective, sign conventions are crucial to the proper physical understanding of the ‘mechanics’ of the manipulations.

Sign conventions for the description of the stresses are many and varied and they are, at the same time, a vexation to expert and novice alike. There are many possible sign conventions that are found in the literature. The fair advice is to suggest that if a particular sign convention works for you, by all means use it. The purpose of this commentary is to outline the limitations of some commonly adopted sign conventions and to suggest a sign convention that will be user-friendly in most circumstances. First, let us consider the sign convention normally associated with the axial stresses. In solid mechanics in general, tensile stresses are usually considered to be positive whereas in geomechanics, and in this text in particular, compressive stresses are considered to be positive. This is not a major area of concern since we can associate some differences in the physical actions that will result from the applications of either a tensile stress or a compressive stress. Line elements can either extend or shorten depending on the nature of the axial stress. What about the shear stresses? The usual procedure is to consider first the shear stress acting on a surface of interest and to select a point just outside the region. If the shear stresses cause clockwise moments then, in solid mechanics, the shear stresses are considered to be positive. In geomechanics counterclockwise replaces clockwise in their definition (Figure B.1a). (In doing this we have also, by deduction, introduced the definition of a negative shear stress.) Performing a simple operation, however, unravels this definition. Let us draw Figure B.1(a) on an acetate transparency and look at the figure from the opposite side. The view will correspond to that shown in Figure B.1(b). We now have the same shear stress but it appears to have a negative magnitude. Regrettably, this definition of the shear stress becomes dependent on the point of view of the observer. The previous definition is perfectly acceptable so long as you do not move outside of the plane of the paper. This is obviously somewhat restrictive when three-dimensional

b a

Figure B.1. Sign convention for shear stresses based on a clockwise and a counterclock­wise sense of the moment induced by the shear stress about an exterior point.

Figure B.2. Sign convention for normal stresses and shear stresses in association with coordinate directions and alignment of unit normals.

states of stress are encountered, and the dependence on a change in the perspective only compounds the problem. Naturally, there are means of overcoming this deficiency and the simplest is to attach a frame of reference to the definition of the sign convention. For example, consider the definition of tensile stresses as being positive in the context of solid mechanics. The appropriate definition of a positive tensile stress is one in which the traction vector acts in a positive (or negative) coordinate direction and on planes the outward normal of which is also oriented in a positive (or negative) coordinate direction. A similar definition can be adopted when defining a positive shear stress consistent with this definition of a positive tensile stress. Referring to Figure B.2(a), all the stresses shown there are positive stresses in the context of solid mechanics. We can draw Figure B.2(a) on an acetate transparency and look through from the reverse side and still all the stresses will be positive according to our definition. We have eliminated the observer dependence by attaching a system of coordinates to the element, the positive directions of which will remain observer invariant. In the same way, we can now proceed to define a consistent set of definitions to account for the geomechanics convention of considering compressive stresses as being positive. So, the definition of a positive compressive or shear stress is one where the traction vector acts in the negative (or positive) coordinate direction and on planes the outward normal to which acts in the positive (or negative) direction. Referring to Figure B.2(b), all the stresses shown there are considered to be positive stresses in the context of geomechanics. As has been demonstrated, a certain consistency is necessary in assigning a sign convention for the stress components defining the state of stress at a point. Several such possibilities exist and the prudent option is to select one with the minimum number of limitations.

With the above comments in mind let us restrict our attention to a system of rectangular Cartesian coordinates and a stress state defined by

Figure B.4. Traction vectors on a tetrahedral element located at an infinitesimal distance from P.

where ix, іy and iz are the unit vectors in the x-, y – and z-directions, respectively. Let us now consider an oblique plane S located at an infinitesimal distance from point P (Figure B.4) such that the unit normal to the plane ft is defined by

ft — nx і x + n y і y + n z і z (B.6)

where the components of n in the x-, y – and z-directions are implied. We can consider equilibrium of forces acting on the tetrahedral element with infinitesimal dimensions and show that (see, e. g., Davis and Selvadurai 1996)

Tn = fix T x + n y T y + nz T z (B.7)

Upon substituting (B.3)-(B.5) in (B.7) we have

Tn = (n x @xx + n y &yx + n z ffzx) i x + (n x @xy + n y &yy + n z ffzy) i y

+ (n x &xz + n y &yz + n z °’zz) i z (B.8)

Note that, although the stress matrix is symmetric, we will retain the designations for the components of the stress matrix defined by (B.2) primarily to illustrate the development of a sequence. We can now define the stress vector Tn on the oblique plane in terms of projections along the x, y and z axes such that

Tn @’nxix + yi y + @’nziz

where the following scalar definitions apply:

&nx = Пх &xx + n y <Tyx + n z ffzx = n a& ax (B.10)

&ny = n x & yx + n y & yy + n z &yz = na&ay (B.11)

&nz = n x&xz + n y&yz + nz&zz = na&az (B.12)

In these equations a summation takes place over the repeated a indices. Equations

(B.10)-(B.12) now define the components of the stress at the point P on an oblique plane S passing through P, the normal of which is defined by ft, using the six components of the symmetric stress matrix [t]. It is evident that when the plane S is located directly at the point P, the relationships (B.10)-(B.12) are, in fact, the traction boundary conditions given by

T = п[т] or, using the summation convention, T’i = njffij (B.13)

There are two other results that can be deduced from expressions (B.10)-(B.12); we can express the components of the traction as a resultant of a stress normal to the oblique plane onn and a shear stress tangent the oblique plane, o„~t. The normal stress to the plane is given by

Similarly, we can evaluate the shear component of the resultant shear traction on the oblique plane &nt by considering its relationship between the Euclidean norm of the vector T n and ann, which takes the form

2 2 2

I Tn II — + & n

Evaluating (B.15) we have

2 2 2 2 2

& nt = & nx + & ny + & nz & nn

We can now proceed to discuss the graphical representations associated with Mohr circles of stress. While several aspects of these graphical representations can be discussed we shall restrict attention to the following: the first deals with the use of Mohr circles as a graphical interpretation of the transformation rule applicable to stresses, strains and other dependent variables encountered in geomechanics and the second deals with the use of Mohr circles as a means of identifying admissible states of stress acting on arbitrary planes located through a point at which the stress matrix is defined in terms of the principal components. For the discussion of the first aspect of Mohr circles, it is convenient to further restrict one’s attention to a two-dimensional state of stress characterized by a state of plane stress defined by

 xx xy 0 [t] = yx yy 0 0 0 0

We consider this particular state of stress, which may now be referred to a new set of rectangular Cartesian coordinates (X, Y, Z), obtained by rotating the (x, y, z) coordi­nate system about the z-axis by an angle в in the anticlockwise direction (Figure B.5). Following developments given in Appendix A, the transformation matrix [H] is given by

Figure B.5. State of stress and rotation of the reference coordinate system.

Oyy

Figure B.6. Stresses on an oblique plane.

The stress matrix [E] referred to the rotated coordinate system is given by

(B.19)

The non-zero components of (B.19) are given by

oxx = oxx cos2 0 + oyy sin2 0 + 2oxy sin0 cos 0 (B.20)

oyy = axx sin2 0 + ayy cos2 0 — 2ffxy sin0 cos 0 (B.21)

aXY = (ayy — a"xx)sin0 cos0 + (cos2 0 — sin2 0)axy (B.22)

We can, for example, consider these expressions when 0 = 0, which gives

oxx = Oxx’; oyy = Oyy oxy = Oxy (B.23)

Similarly when 0 = n/2, (B.20)-(B.22) give

OXX = Oyy; OYY = Oxx; OXY = —°xy (B.24)

the negative sign of the shear stress indicating that the positive shear stress acts in a direction opposite to that indicated by oxy (see, e. g., Figure B.6).

Since the description of the state of stress referred to a coordinate system is dependent on the orientation of that coordinate system, we can choose an orientation in the (x, y, z) configuration such that the matrix [a] corresponds to a principal state of stress with

 to1 0 0 [a] = 0 to2 0 0 0 0

where the convention that to > to2 is used and both stresses to and a2 are assumed to be compressive. Using this result in (B.20)-(B.22) we obtain

1 1

toXX = 2to + to) + ^to – 02)cos26> 11

&YY = ^to + to) – ^to – 02)cos26> 1

OXY = “to – CT2)sin26>

If we interpret these transformed stress components in relation to the new set of axes (X, Y, Z), we note that the stress aXX is the normal stress acting along the X-direction and toXY is the positive shear stress acting on the same plane. Similar interpretations can be given to the stresses aYY and aXY (Figure B.7). The negative sign for the shear stress toXY is consistent with the fact that since a1 > a2, the actual direction of aXY will be opposite to that indicated by the positive sign convention. The magnitudes of the stresses toXX, toYY and aXY will thus vary with the choice of the angle of orientation в. Therefore this variation in the magnitudes of the components of the stress matrix in the transformed configuration can be illustrated graphically by constructing a diagram in which either the set aXX and aXY or aYY and aXY are taken as coordinates. To obtain such a relationship we square (B.26) and (B.28); the addition of these gives

2

This represents the equation of a circle in the aXX vs. aXY plane with its centre at (2(to + to2), 0) and radius 2(to – to). This circle is referred to as the Mohr circle (Figure B.8).

At this point we need to talk about sign conventions once again. There is no difficulty with the normal stresses aXX and aYY. Compressive normal stresses are taken as positive in geomechanics. But another question arises concerning the shear stress aXY. In the

context of our user-friendly sign convention, aXY is positive when it acts in the negative Y-direction on the surface whose outward normal points in the positive X-direction, as shown in Figure B.6. The exact same stress acts on the surface with normal pointing in the Y-direction as shown in the same figure. Nevertheless, when we look at the Mohr circle in Figure B.8 it appears that the two shear stresses have opposite signs. This seems to pose a serious contradiction. In fact, it is not serious; it is simply the result of the fact that aXY enters (B.29) as a squared quantity. The equation cannot distinguish between positive and negative shear stress and the resulting Mohr diagram cannot either.

It will still be useful, however, to be able to interpret the sign of the shear stress strictly within the context of Mohr circles. We will see where the utility arises in a moment, but we must first return to our original sign convention. Only for the purpose of Mohr circles, we will interpret the sign of the shear stress as follows. If the shear stress induces a counterclockwise moment about the point P in Figure B.1(a), then we will plot the stress on the Mohr diagram as positive. If a clockwise moment is indicated then we plot the stress as negative. Looking at Figure B.7(a) we see that aXY there would be plotted as positive on a Mohr diagram. In Figure B.7(b), aXY would be plotted as negative. This convention gives exactly the result shown in Figure B.8. Note that this applies only for interpretation of the Mohr diagram.

In one sense this is all a storm in a teacup. The sign of the shear stress does not have a physical significance similar to compressive and tensile normal stress. If a material fails due to excessive shear stress it makes little difference whether that stress was positive of negative; failure is just as inconvenient in either case. But there is one more feature we can associate with the Mohr diagram that makes it useful for us to introduce our special sign convention. That is the existence of a unique point on the circumference of any Mohr circle called the pole. The concept of the pole, sometimes called the origin of planes, is especially useful in developing a graphical understanding of any stress state, and we must use a special sign convention to make it work.

To understand the concept of the pole we begin by noting from (B.26) and (B.28) that the stress point (aXX, aXY) makes a central angle of 20 with the horizontal axis in Figure B.8. If we alter 0 the stress point moves around the circumference of the circle to some new point. Let (aXX, aXY) and (aXX, aXY) be two stress states acting on two surfaces. If the orientations of the surfaces differ by an angle a, then the central angle on the Mohr circle between two stress points will be 2a as shown in Figure B.9. This doubling of angle applies for any surfaces we wish to consider. Next, suppose we draw on the Mohr diagram two lines: one line through the point (aXX, aXY) oriented parallel to the surface

on which those stresses act and the other through (aXx, &xy) oriented parallel to the surface on which the second stress state acts. It is a fact that the two lines will always intersect at a point on the circumference of the circle. This follows from a theorem of geometry stating that the central angle between any two points on a circle will always be twice the corresponding inscribed angle. Note that the angle between our two lines must be a since they were drawn parallel to the two surfaces. Then on the Mohr diagram the two lines must intersect at some point on the circumference of the circle. We call this point the pole, denoted by OP. These ideas are all summarised in Figure B.9.

Note that the pole is a unique point. The two surfaces used in the discussion above were totally arbitrary and therefore every line drawn through the stress point corresponding to any surface will intersect the circle at the pole. Conversely, any line drawn through the pole must intersect the circle at the stress state which acts on the surface parallel to that line. This is an extremely powerful tool for visualisation of the stresses associated with any surface. Once the location of the pole is determined, the stress on any surface is found by simply drawing a line parallel to that surface. We can find the pole so long as we know the stresses acting on a surface of known orientation. In all of this the special sign convention concerning shear stress applies. Stress points on the upper half of any Mohr circle imply a positive shear stress producing a counterclockwise moment about the point nearby the surface.