⁰The results presented in this paper were obtained in the course of research conducted under Contract Nonr 562(10) between the Office of Naval Research and Brown University.

1. Introduction

The fundamental definitions of work hardening and perfect plasticity have been shown to have strong implications with respect to uniqueness of solution for elastic-plastic bodies. It is not surprising, therefore, to find that they lead rather directly to the variational principles as well. Perfect-plasticity theory and both the incrementally linear and the incrementally nonlinear theories for work-hardening materials are considered. The several counterparts of the minimum-potential-energy and the minimum-complementary-energy theorems are derived in a unified manner for stress-strain relations of great generality. Absolute-minimum principles rather than relative ones are established.

There are any number of approaches to the establishment of variational principles. One is to state the principles directly and then proceed to prove them. Although clear and precise statements can be made, the motivation for the original inspiration does not appear. A newcomer to the field then frequently will be unable to appreciate the development and generally will not see how to produce appropriate theorems or modifications of his own. The approach to be followed here does not suppose the result to be known in advance. It is synthetic in a sense, because the basic theorems have been stated and proved for a number of special materials [2, 3, 4, 5, 6, 7]. Nevertheless, it is a procedure which arises logically from fundamental postulates in elasticity and in plasticity theory, and it is systematic.

In the theory of elasticity, whether linear or nonlinear, the steps are reasonably straightforward. The equation of virtual work is written first under the implicit assumption of continuity of displacement and what may be termed equilibrium continuity of the stresses (surface tractions must be continuous across any surface, but the normal stress components parallel to the surface may be discontinuous). In a common notation, repeated subscripts indicate summation:

(1)

The starred quantities are related through equilibrium, and the unstarred are compatible. There need be no relation between the two sets of quantities. For convenience, the surface area A is divided into the region A_T , on which the surface tractions T_i are specified, and the region A_u, over which displacements u_i are given. The true and unique solution (no buckling, no initial stress) to the boundary-value problem with given body forces F_i thus satisfies

(2)

If approximate solutions are sought, two procedures suggest themselves immediately. One is to choose a compatible strain-displacement field 𝜖_ij^c,u _i^c and satisfy the boundary conditions on A _u. The other is to select an equilibrium stress field σ_ij^E which satisfies the surface-traction boundary conditions on A_T . More elaborate mixed schemes may be devised, but they cannot be classed as obvious [4].

The value of an approximation procedure, or of a guess, must be determined by comparison of the approximate solution with the unknown true answer. The real difficulty and the intuitive heart of the problem lie in the decision on what should be compared. The equation of virtual work will be satisfied if the natural strains and displacements are replaced by any chosen set satisfying compatibility and the boundary conditions on A_u. Therefore

	∫ V σijt𝜖 ijc dV −∫ ATTiuic dA −∫ V Fiuic dV
	= ∫ V σijt𝜖 ijt dV −∫ ATTiuit dA −∫ V Fiuit dV.	(3)

Transposing and calling the difference between the true and assumed solution Δ𝜖_ij, Δu_i,

(4)

This form suggests strongly a consideration of the elastic-strain energy density written as a function of strain alone,

(5)

because dW = (∂W∕∂𝜖_ij)d𝜖_ij = σ_ijd𝜖_ij. Equation (4) can then be restated as

(6)

where δ_𝜖 is the first variation of the expression in brackets (the potential energy) as u_i and 𝜖_ij are varied in accordance with compatibility and the boundary conditions on u_i.

A variational principle is not necessarily very helpful in solving problems. The assumed state may not be close to the true one. What is required instead is an absolute-maximum or -minimum principle, a comparison of the value of the potential energy for the assumed state with that of the true state, without restriction on the magnitude of the difference between the states. The presentation here is, however, within the framework of small-displacement theory.

A comparison may be made with the aid of the identity

	∫ V W(𝜖ijc) dV −∫ ATTiuic dA −∫ V Fiuic dV
	= ∫ V W(𝜖ijt) dV −∫ ATTiuit dA −∫ V Fiuit dV
	+ ∫ V [W(𝜖ijc) − W(𝜖 ijt)] dV −∫ ATTiΔui dA −∫ V FiΔui dV.	(7)

In view of (4), therefore, the potential energy of any admissible compatible state is algebraically more than the potential energy of the true state by

(8)

The integrand may be rewritten as

(9)

The rectangles in Fig. 1 symbolize σ_ij^tΔ𝜖 _ij. The shaded triangles represent (9), the integrand of (8). In Fig. 1(a)–(c) the triangles are on the positive-strain-energy side of the symbolic stress-strain curves for any magnitude Δ𝜖_ij.

The potential energy is an absolute minimum for a linear or for a stable nonlinear elastic material. For an unstable material [Fig. 1(d) and (e)], the shaded triangles are on the negative side for some Δ𝜖_ij, and the potential energy is not an absolute minimum.

A similar set of steps leads to the principle of minimum complementary energy. Equation (2) is satisfied if the σ_ij,T_i,F_i system is replaced by any other in equilibrium. For any state of stress σ_ij^E which satisfies the boundary conditions on A _T and is in equilibrium with F_i,

(10)

where σ_ij^E − σ _ij^t = Δσ _ij and ΔT_i is the corresponding change in surface traction on A_u.

The complementary-energy density as a function of stress alone is suggested by the first integral:

(11)

because dΩ = (∂Ω∕∂σ_ij)dσ_ij = 𝜖_ijdσ_ij. Equation (10) can then be restated as

(12)

where δ_σ is the first variation of the expression in brackets (complementary energy) as σ_ij and T_i are varied in accordance with equilibrium with F_i and the boundary conditions on A_T .

Figure 2 is symbolic of the fact that the complementary energy is an absolute minimum for a stable material. Corresponding to (8) and (9),

(13)

where the integrand may be rewritten as

(14)

The symbolic representation of the stress and strain tensors by one dimension each in Figs. 1 and 2 and the terms stable and unstable can be given general meaning and made precise.

2. The fundamental postulate for elasticity and plasticity

A basic postulate has been formulated for both elastic and plastic media without time effects [5]. It is essentially a definition of a stable material and may be stated as follows:

No work can be extracted from the material and the system of forces acting upon it.

A more useful statement is in terms of an external agency which applies a set of additional forces to the body under a given load and then removes the added forces. The external agency must do positive work in the application of force. Over the cycle of application and removal the work done by the external agency must be positive if plastic deformation occurs in work-hardening material and will be zero if only elastic changes take place. For a perfectly plastic material, the work done by the external agency may also be zero when plastic deformation takes place, although generally it will be positive.

The basic postulate may be applied to a homogeneous material under homogeneous stress σ_ij^a and strain 𝜖_ij^a. Suppose that the external agency changes the state of stress by Δσ _ij to σ_ij^b. The strain will change by Δ𝜖_ij to 𝜖_ij^b. Then the postulate requires

(15)

The value of the integral is strongly path-dependent in the plastic range but is, of course, independent of path for any elastic material.

3. Absolute-minimum principles in elasticity

The inequality (15) is a formal expression of the requirement that the shaded triangles of Fig. 1 be on the positive-strain-energy side of the stress-strain curve. Although for a nonlinear elastic material 𝜖_ij^b depends upon 𝜖 _ij^a as well as upon Δσ _ij, the integral is path-independent. Choosing a straight-line path in stress space from σ_ij^a to σ _ij^b, it is obvious that inequality (15) may be continued as

(16)

Also, from

(σijb − σ ija)(𝜖 ijb − 𝜖 ija)	= ∫ abd[(σ ij − σija)(𝜖 ij − 𝜖ija)]
	= ∫ 𝜖ija𝜖ijb (σij − σija) d𝜖 ij + ∫ σijaσijb (𝜖ij − 𝜖ija) dσ ij,	(17)

we obtain

(18)

Inequality (18) expresses the requirement that the shaded triangles be as shown in Fig. 2(a)–(c) and not as in Fig. 2(d) and (e).

Materials of the type of Figs. 1(d),(e) and 2(d),(e) are thus excluded from our consideration, although not necessarily from physical reality. For elastic materials which follow the postulated behavior, comparison of (16) with (9) and of (18) with (14) proves that the potential energy and the complementary energy of the true state are both an absolute minimum:

(19)

with equality only in the trivial case of the admissible state c or E coinciding with the true state.

4. Deformation or total theories of plasticity

If no distinction is made between loading and unloading, or if each point of the body is assumed to be at the maximum load intensity in its history, deformation theories postulate a unique relation between stress and total strain. Although physically unacceptable, in general, because plastic deformation is path-dependent and irreversible, such theories do in some instances lead to very useful results. Under the assumptions mentioned, there is no need to consider deformation theory further, as the theory is indistinguishable from nonlinear elasticity. No matter how elaborate the stress-strain relation, if the material postulated is stable, the principles of minimum potential and minimum complementary energy apply without any change.

If, on the other hand, loading is taken to be nonlinear whereas unloading is assumed to follow a linear elastic relation, the inconsistency of deformation theory becomes of primary importance. The mathematical and physical meaning of solutions then becomes quite obscure.

5. Work-hardening relations involving increments of stress and strain

The fundamental postulate of positive work by an external agency has very far-reaching implications. As shown in Fig. 3, the plastic-strain increment or strain-rate vector 𝜖_ij^′p must be normal to the yield or loading surface at a smooth point and between normals to adjacent points at a corner. At a smooth point

(20)

where C_ijklσ_kl′ is the elastic response and G and f are functions of the state of the material, which may include strain and the history of loading as well as the existing state of stress. G may, in addition, be a homogeneous function of order zero in the stress rate σ_ij′. In pictorial terms (Fig. 3), G may depend upon the direction of σ_ij′, but doubling σ_ij′ doubles 𝜖_ij′. In all stress-strain relations in use in the paper, G is taken as completely independent of σ_ij′, so that in the form

(21)

the H_ijkl likewise are independent of σ_kl′. The coefficients H_ijkl, which appear similar to the C_ijkl of linear anisotropic elasticity, may be complicated functions of the present state and prior history.

If a corner is considered to be a set of intersecting loading surfaces [8, 9, 10, 11], each of which makes its independent contribution to the plastic-strain rate (Fig. 4), then

𝜖ij′ =	Cijklσkl′ + (1)H ijklσkl′ + (2)H ijklσkl′ + + (m)H ijklσkl′ +
=	(Cijkl + Bijkl)σkl′.	(22)

It is important to keep in mind that the coefficients ^(m)H _ijkl are to be taken as zero unless σ_ij′ has an outward-pointing normal component, just as in (20), where the plastic term must be chosen as zero if unloading takes place. Stress-rate vectors having different directions often will activate different loading surfaces, so that, despite the apparent linearity of (22), it is not true in general that the strain rate produced by two stress rates acting simultaneously is the sum of the strain rates for each individually.

In fact, at a corner the combination of two stress rates, each of which individually produces plastic action, may result in an unloading (Fig. 5). Even when the ^(m)H _ijkl are independent of σ_ij′, the B_ijkl will be functions of the stress rate (order zero).

The basic postulate requires the initial rate of work by the external agency to be positive; therefore

(23)

and because the elastic component is recoverable,

(24)

for a work-hardening material, unless 𝜖_ij^′p = 0.

In the demonstration of the uniqueness theorem for stress and strain rates [8, 9, 10, 11], the entire point lies in the proof of

(25)

where a and b are two assumed solutions for the rates from the stress point σ_ij^L. If an infinitesimal time, arbitrarily chosen as unity, is permitted to elapse, the two stress states are σ_ij^L + ^aσ _ij′ and σ_ij^L + ^bσ _ij′ (Fig. 6).

At a smooth point of the loading surface it is possible to go from stress point b to stress point a [Fig. 6(a),(b)] or from stress point a to b [Fig. 6(a),(c)] and change the strain by ^a𝜖 _ij′−^b𝜖 _ij′ or ^b𝜖 _ij′−^a𝜖 _ij′, respectively, in accord with (20) and (23). The work postulate for the b-to-a case then gives [see (9)]

(26)

The result (25) therefore is established [8], but the value of the integral itself is of interest here:

(27)

so that, as for (18),

(28)

For the a-to-b path, (27) and (28) merely interchange, and the result is therefore unaffected. When there are two or more loading surfaces at a corner [Fig. 6(d)], the permissible path may be from b to a for one set of plastic-strain rates and from a to b for another. As both (27) and (28) apply to each path, with the elastic-strain rates being counted only once, it is clear that (27) and (28) apply just as they are, even for this complicated case.

6. Two minimum theorems for incremental work-hardening theories

In general, theorems which hold in the elastic range cannot be expected to apply in the plastic. As a consequence of irreversibility, the uniqueness theorem for work-hardening theories of plasticity is in terms of the increments or rates of stress and strain and not of the stresses and strains themselves. The equivalent variational or minimum principles likewise will be in terms of rates. Following the procedure established previously, the principle of virtual work is written for the rates

(29)

Equation (29) applies to the actual rates, as it does for any compatible strain-rate distribution and any stress-rate field in equilibrium.

When the true stress rates are varied under the restriction that they satisfy equilibrium and the boundary conditions on A_T (T_i′ = 0), the analogue to (10) is

(30)

The analogue to complementary-energy density (11) is simply

(31)

corresponding to linear elasticity, because the stress-strain relations are time-independent, as exhibited by (21). The form suggested for a variational principle similar to (12) is

(32)

This complementary rate principle will be valid whenever the H_ijkl of (21) are independent of stress rate. All forms (20) for smooth loading surfaces considered here are in this category. At a corner, however, as previously explained, the H_ijkl of (22) which are nonzero depend upon the direction of σ_ij′. Therefore, δ_σ′H_ijkl≠0 for some or all directions of loading, and (32) is not valid.

A complementary rate minimum principle in a form equivalent to (19) would be much more valuable:

(33)

The right-hand side is algebraically larger than the left by

	∫ V [ω(σij′E) − ω(σ ij′t)]dV −∫ Au(Ti′E − T i′t)u i′dA
	= ∫ V dV,	(34)

where 𝜖_ij^′E is computed from σ _ij^′E [equation (22)]. The integrand of (34) may be rewritten as

(35)

Comparison with (28) shows that the fundamental work postulate requires (35) to be positive and thus guarantees the minimum principle (33), although (32) does not apply.

If now the strain rates are varied in the virtual-work expression for the true rate state while the condition u_i′ = 0 on A_u is satisfied, the equivalent of (4) is

(36)

The analogue to strain-energy density (5) would be

(37)

if (22) can be inverted as

(38)

Such inversion is not possible if the material is incompressible in either the elastic or the elastic-plastic range. This difficulty can be circumvented for such materials by solving for the stress-rate deviation

(39)

and writing

(40)

It can be shown, then, that a minimum-potential rate principle holds:

	∫ V w(𝜖ij′t) dV −∫ ATTi′ui′t dA −∫ V Fi′ui′t dV
	≤∫ V w(𝜖ij′c) dV −∫ ATTi′ui′c dA −∫ V Fi′ui′c dV.	(41)

PhysicsLibrary conversion note. In the original printing of (41), the two body-force-rate integrals over V end with dA. Since their domain is the volume V and the corresponding terms in (29) and (36) use dV , the volume element dV is used here; this correction is made explicitly rather than silently.

The right-hand side exceeds the left by

(42)

when the surface-traction-rate and body-force-rate integrals are replaced by volume integrals of stress rates. The sum of the integrands is

(43)

Comparison of (43) with (27) proves (41) and once again demonstrates that the minimum principles follow from the fundamental postulate or definition of work hardening.

Incrementally nonlinear stress-strain relations have not been studied in any detail. If G in (20) or H_ijkl in (21) depends on the direction of σ_ij′, the usual proof of the uniqueness theorem breaks down. On the other hand, if uniqueness is assured, the basic postulate in the form of (26) or the first inequality (28) will ensure the validity of the absolute-minimum principles (33) and (41), corresponding to complementary and potential energy. A simple assumption which leads to uniqueness is that the plastic-strain rate is a monotonically increasing function of the normal component of σ_ij′. Although apparently a very reasonable postulate, as a consequence of the proportionality between rates of stress and strain for a given direction of σ_ij′, it is equivalent, unfortunately, to a reduction to linearity.

7. Restricted minimum theorems

The complete parallelism of the developments of the minimum principles for the elastic and for the elastic-plastic cases may have blurred the basic approach. There are but a few independent combinations possible for stress, stress rate, strain, and strain rate, and it is worthwhile running through some of them. Since uniqueness of rates only is guaranteed, it is not likely that any new theorems of true generality will result for work-hardening materials.

Suppose a theorem is desired for the plastic range which contains the stresses and the strains themselves. Equations (2)–(4) and (10) are written exactly as for elastic bodies. Again (5) and (11) would be suggested by the form of (4) and (10). Now, however, the path of loading is important, and it is not true in general that ∫ ₀^𝜖_ijσ _ij d𝜖_ij = W is a function of final strain only, nor is∫ ₀^σ_ij𝜖 _ij dσ_ij a function of stress only. Nevertheless, if at each point of the material there has been no unloading from any of the loading surfaces, the irreversibility is not apparent to the material. In this very limited sense, minimum-complementary-energy and minimum-potential-energy theorems hold for a very restricted and yet possibly useful class of alternative admissible states [12].

Next suppose that a theorem is desired for strain rates and stress. Virtual work is then written in the form

(44)

Varying the strain-rate system from the true state without changing the displacement-rate boundary conditions leads to

(45)

The variational principle suggested is

(46)

where

(47)

Such a principle can have meaning only if 𝜖_ij′ determines σ_ij or at least σ_ij𝜖_ij′. As this will not be true in general, (46) can be valid for a restricted set of loading paths or special materials at most.

Varying the equilibrium system in the familiar manner results in

(48)

which has meaning only if σ_ij determines 𝜖_ij′ or at least σ_ij𝜖_ij′ so that

(49)

Therefore, (48) cannot apply to a work-hardening material.

Another possible set of theorems relates stress rates and strain. The corresponding equations then are found by interchanging primes and no primes in (44)–(49), and the end results are

(50)

where (and only if 𝜖_ij determines σ_ij′)

(51)

and

(52)

where (and only if σ_ij′ determines 𝜖_ij)

(53)

Such theorems as these are therefore inappropriate for work-hardening materials as defined. They will have limited validity at least for materials whose state is described by surfaces in strain space rather than in stress space [13, 14].

8. Elastic-perfectly plastic material

A perfectly plastic material may be defined directly or may equally well be considered as the limiting case of a work-hardening material for which all subsequent loading surfaces coincide with the initial yield surface, f = k, bounding purely elastic action. Unlimited plastic deformation may occur at yield. As no stress increment is required for flow at yield, the work done by an external agency may be zero when plastic deformation takes place. The plastic-strain-rate vector is normal to the yield surface in the extended sense (Fig. 3).

The stress-strain relation at a smooth point on the yield surface is

(54)

and at a corner

(55)

where each fixed intersecting yield surface makes its own contribution. The λ are homogeneous of degree −1 in time, because the stress-strain relations are independent of time. Each λ is to be taken as zero unless the stress point is on its yield surface and remains there in the interval under consideration. They are otherwise indeterminate for a homogeneous state of stress.

As a consequence of a fixed yield surface and normality,

(56)

for all permissible σ_ij′. Therefore

(57)

Also, the total-strain-rate vector is resolved uniquely into an elastic and a plastic component at each stress point on the yield surface. At an interior point, the strain rate is, of course, purely elastic. Therefore, when the existing stress is known and the strain rate is given, the stress rate is determined except, again, for an incompressible material, in which the stress deviation rate is determined [as in (38) and (39)] by

(58)

All the equations (26)–(43) are valid, therefore, with ω_p replacing ω and w_p replacing w. The discussion of what happens at a corner is simplified for a perfectly plastic material. If the strain-rate vector points outward from the yield surface and lies between the normals to adjacent points (see Fig. 3), it is purely plastic, and the stress rate is identically zero. If the strain-rate vector points outward but has a component tangent to the adjacent surface, that portion of the surface governs, and the stress point is then effectively at a smooth point on the yield surface.

It is not surprising that a minimum-complementary rate principle and a minimum-potential rate principle apply. As has been stated, a perfectly plastic material is a limiting case of a work-hardening one as all successive loading surfaces approach the initial yield surface. The lack of limitation on 𝜖_ij^′p for a homogeneous state of stress does not matter because, at a given stress point, σ_ij′𝜖_ij^′p is zero when the strain rate and the stress rate are related and is zero or negative when they represent two independent states.

Additional theorems of some generality would be expected for a perfectly plastic material, because the yield surface does not depend upon loading. The plastic-strain-rate vector is normal to the surface and so determines the stress point itself, or at worst a straight line or plane of the surface (Fig. 7). Actually, 𝜖_ij^′p determines the rate of dissipation, σ _ij𝜖_ij^′p, uniquely. The total-strain-rate vector by itself does not provide any such information for an elastic-plastic material. However, for a plastic-rigid material or for an elastic-plastic material at the limit load [15], the elastic-strain rates are identically zero. The total-strain rates are then plastic only and do determine the dissipation and the stress to a considerable extent.

Under the restriction of zero elastic-strain rate, following steps (44)–(47), 𝜖_ij′ does determine σ_ij𝜖_ij′ = φ, and the principle (46) is established. Following steps (48) and (49), σ_ij′𝜖_ij′ is now zero, so that Ψ = 0 and (48) applies.

Absolute-minimum principles can be established which are generalizations of the Markov [16] and Hill [17] principles and are in fact equivalent to the limit theorems [15]. The minimum principle for (46) is

	∫ V φ(𝜖ij′t) dV −∫ ATTiui′t dA −∫ V Fiui′t dV
	≤∫ V φ(𝜖ij′c) dV −∫ ATTiui′c dA −∫ V Fiui′c dV,	(59)

where 𝜖_ij^′c,u _i^′c is any compatible system taken as plastic only and where φ = [σ _ij(𝜖_ij′)]𝜖_ij′ is the dissipation function. The right-hand side is algebraically greater than the left by the volume integral of

(60)

which is positive or zero in accordance with the basic work postulate [5], or equally well from the convexity of the yield surface and the normality of the plastic-strain-rate vector, which themselves are consequences of the postulate.

The upper-bound limit theorem [15] may be obtained from (59) by observing that, if u_i^′t vanishes on A_u, the left-hand side is zero from virtual work, and

(61)

The minimum principle corresponding to (48) may be written as a maximum principle by multiplying through by −1:

(62)

Proof follows directly from virtual work, as the left-hand side exceeds the right by the volume integral of

(63)

which is positive or zero just as (60) was. It is (63) which is the key in the lower-bound theorem of limit analysis [15].

The yield value and the yield criterion as well need not be the same at each point of the material, since nowhere in the proofs do such restrictions appear. Therefore (59) and (62) apply equally well to a rigid work-hardening material at each stage of loading.

As mentioned in the previous section, theorems involving total strain are not appropriate for the types of materials postulated in the paper [14]; hence the list of simple minimum principles seems exhausted.

9. Conclusion

A systematic procedure is presented for establishing variational and minimum principles. The virtual-work expression is written in terms of the quantities for which a theorem is sought, and a variation is tried which suggests a possible principle. Use is then made of a basic postulate for stable materials without time effects, which had been formulated previously [5]: in the very strictest sense, work cannot be extracted from the stressed material and the system of forces acting upon it. Substitution of the relation between the quantities which is given directly by the fundamental postulate provides immediate proof of the valid absolute-minimum principles.

Minimum-potential-energy and minimum-complementary-energy theorems (19) are established for linear and nonlinear elastic bodies and for deformation theories of plasticity by virtual work (2), the appropriate variation (4) or (10), and inequalities (16) or (18), as given by the postulate.

Corresponding theorems [3, 6] in which rates replace total quantities, (41) and (33), are established for work-hardening and for perfectly plastic materials by virtual work (29), the appropriate variation (36) or (30), and inequalities (27) or (28) given by the postulate. All incrementally linear and the most complicated combinations of incrementally linear forms (Fig. 4) are included.

Extended theorems (59) and (62) involving stress quantities and strain and displacement rates [16, 17] are established for rigid-perfectly plastic materials or elastic-perfectly plastic materials at limit loading or collapse. The steps are virtual work (44), the variations (45) or (48), and inequalities given by the postulate, as indicated following (60) or (63).

Brown University, Providence, R.I.

Source note

This PhysicsLibrary entry is a conversion of the public-domain material in Reference [1].

References