6_6. Elasto-Plastic Behaviour

6. Elasto-plastic behaviour

6.1 SynopsisThis chapter introduces the framework and assumptions of elasto-plastic materialbehaviour. Elastic perfectly plastic and elastic strain hardening/softening plasticbehaviour is defined. The concepts of yield and plastic potential surfaces and ofhardening and softening rules are introduced. It is shown that the framework cansimulate many of the facets of behaviour exhibited by real soils. The chapterfinishes by deriving the theoretical equations necessary for construction of theelasto-plastic constitutive matrix.

6.2 IntroductionIn the previous chapter some elastic constitutive models were presented. Whilesuch models are relatively simple, they cannot simulate many of the importantcharacteristics of real soil behaviour. Improvements can be made by extendingthese models using the theory of plasticity. The objective of this chapter is todescribe the concepts of elasto-plastic behaviour and how it may be formulated foruse in finite element analysis.

The chapter begins by considering the uniaxial behaviour of a linear elastic-plastic material. By using this simple problem as an example, the ideas of plasticyield, hardening and softening are introduced. It is then shown that many of thefacets of real soil behaviour are consistent with such concepts and, therefore, thatthe framework has considerable potential. While the uniaxial problem provides anideal introduction, the concepts of plasticity must be expressed in general stressand strain space, if they are to be used to describe soil behaviour under multi-stressconditions. The basic ingredients for a general elasto-plastic model are thereforepresented. These are then used to examine the behaviour of an elasto-plasticmaterial subjected to a biaxial stress state. The chapter finishes by deriving thetheoretical equations necessary for construction of the elasto-plastic constitutivematrix for use in finite element analysis. Examples of specific elasto-plastic modelsare given in Chapters 7 and 8.

Elasto-plastic behaviour / 1 33

6.3 Uniaxial behaviour of a linear elastic perfectly plasticmaterial

Figure 6.1 shows a uniaxially loaded bar of an ideal linear elasto-plastic material,which is loaded by applying a compressive axial strain, s. This figure also showsthe stress-strain curve for this bar. On first straining the bar behaves elastically andits stress-strain response travels along the line AB. As the material is linear elastic,the gradient of the line AB is given by the Young's modulus, E. If the strainingprocess is stopped before the stress in the bar reaches point B and the bar isunstrained (i.e. unloaded), the stress-strain response moves back down the line BA.Consequently, as long as the straining does not cause the stress to reach point B,the bar behaves in a linear elastic manner and, when unloaded, returns to itsoriginal undeformed state, with no permanent strains. If the bar is strained beyond£B, to point C, the stress strain curve passes through point B. At B the yield stress,aY, is reached and the bar becomes plastic. There is no longer a linear relationshipbetween stress and strain and the stress in the bar remains constant and equal to oY.If the bar is now unloaded it becomes elastic and the stress strain curve follows thepath CD, which is parallel to the path BA. When unloaded such that the axial stressis zero, i.e. point D, there is still a strain in the bar. This strain is equivalent to theplastic strain experienced when straining along the path BC and is given by sc

p =cc - 8B. Thus the bar does not return to its original shape, but suffers a permanentshortening. If the bar is now reloaded, the stress strain curve re-traces the path DCuntil point C is reached, at which point the axial stress equals the yield stress andthe bar becomes plastic again. It then moves along path CF. Clearly, if the strainingis not sufficient to take the stress to point C, the bar behaves linear elastically.

1 Appliedstrain e

a

P P£c

Figure 6.1: Uniaxial loading of linearelastic perfectly plastic material

Behaviour is reversible, and therefore elastic, on the inclined paths AB and DC,whereas on the path BCF behaviour is not reversible. For example, it is impossibleto retrace the path CB. If, instead of subjecting the bar to strain controlled loading,the bar is loaded by an increasing stress, it is impossible to apply a stress greaterthan the yield stress oY. Any attempt to do so would result in infinite strains. Amaterial conforming to the idealised behaviour shown in Figure 6.1 is called linearelastic perfectly (or ideal) plastic.

134 / Finite element analysis in geotechnical engineering: Theory

6.4 Uniaxial behaviour of a linear elastic strain hardeningplastic material

Figure 6.2 shows another idealiseduniaxial stress-strain curve. Behaviourof the bar when strained is similar tothat described above, but with onesubtle difference. Initially, if the bar isstrained it behaves elastically alongpath AB. If it is strained beyond pointB to point C the initial yield stress om ™ *" SB ^ £c °is exceeded, but instead of the stressremaining constant and equal to am, Figure 6.2: Uniaxial loading of linearas in Figure 6.1, it increases to oYC • elastic strain hardening plasticOn unloading from point C the bar materialbecomes elastic again and follows thelinear path CD which is parallel to BA. As before, there is a permanent (plastic)strain left in the bar when it is fully unloaded to point D, i.e. o = 0. On reloadingthe bar initially behaves elastically, path DC, until point C is reached, where itbecomes plastic. The yield stress at point C, oyc, is greater than that at B and hasincreased as a result of the plastic straining from B to C. Eventually, if the bar isstrained to point F the stress strain curve becomes horizontal and the stress in thebar remains constant. A material which behaves in this manner is called a linearelastic strain (or work) hardening plastic.

j~J~Appliedstrain £

6.5 Uniaxial behaviour of a linear elastic strain softeningplastic material

A third class of plastic material cannow be identified where the yieldstress, instead of increasing duringplastic straining, decreases. Thebehaviour of such a material whensubjected to uniaxial straining isshown in Figure 6.3. Behaviour isagain linear elastic on first loading &and during any unload - reload loops.However, during plastic straining, r. ~ o ## • - , , -#• * ,•i \u ™^r \.u • u 4. Figure 6.3: Uniaxial loading of linearalong path BCF, the yield stress , +- + • f+ • # *-, & F

o . . : . „ . elastic strain softening plasticreduces. Such a material is called a + • ,, . ,^ materiallinear elastic strain (or work)softening material.

From an engineering point of view, material that behaves in this brittle manneris of particular concern because, if strained beyond its initial yield point (i.e. pointB in Figure 6.3), its yield stress reduces and its capacity to resist load diminishes.


6.6 Relevance to geotechnical engineeringAlthough the above examples are forsimple uniaxial conditions, theyhighlight the difference between thethree main forms of elasto-plasticbehaviour. It is interesting to comparethe behaviour described above, withsome of the more common features ofsoil behaviour. For example, if thestress-strain curve given in Figure 6.2is re-plotted with the stress axishorizontal and the strain axis vertical,

Figure 6.4a, it bares some

F

ex

a) Uniaxial hardening

e

\ Virgin\ consolidation

\ l ine

Swelling \line \

log o v 'b) Oedometer test

seeresemblance to the behaviourobserved in an oedometer test, seeFigure 6.4b. Soil behaviour on aswelling line is often assumed to bereversible and therefore is akin tobehaviour on an elastic unload-reloadloop (i.e. path CDC in Figure 6.4a).Behaviour on the virgin consolidationline is irreversible (i.e. it is onlypossible to travel down the line, notup it) and results in permanent strains.It is therefore similar to behaviouralong the strain hardening path BCFof Figure 6.4a.

Similarity also exists between the

Figure 6.4: Similarity betweenuniaxiai hardening and an

oedometer test

a) Uniaxial softening b) Shear box test

Figure 6.5: Similarity betweenuniaxial softening and direct simple

shear test

strain softening behaviour given in Figure 6.3 and reproduced in Figure 6.5a, andthe shear stress - shear strain behaviour observed in a direct or simple shear test ondense sand, shown in Figure 6.5b.

6.7 Extension to general stress and strain spaceFor the above concepts of elasto-plastic materials to be of general use, behaviourmust be formulated in general multi-axial stress and strain space. Because there aresix independent components of stress and six of strain, this presents a formidabletask. However, if the material can be assumed to be isotropic (i.e. propertiesindependent of orientation) and as yield is essentially dependent on stressmagnitude, simplifications can be achieved by working with invariants of stressand strain. As noted in Chapter 5, three stress invariants are required to determinethe magnitude of stress. These invariants can be the principal stresses, or somecombination of these values. For plasticity it is usual to express behaviour in anincremental form and to use accumulated stress and incremental strain invariants.


Clearly, working in terms of invariants considerably reduces the complexity ofthe task of formulating elasto-plastic behaviour, because the number of stress andstrain parameters reduces from six to three. However, it comes at the cost ofassuming the material to be isotropic. To model anisotropic behaviour theformulation must be in terms of six independent stresses and six strains.

6.8 Basic conceptsTo formulate an elasto-plastic constitutive model requires the following fouressential ingredients.

6.8.1 Coincidence of axesThe principal directions of accumulated stress and incremental plastic strain areassumed to coincide. This differs from elastic behaviour where the principaldirections of incremental stress and incremental strain coincide.

6.8.2 A yield functionIn the uniaxial situations described in Sections 6.3 to 6.5 the yield stress, oy,indicates the onset of plastic straining. In the multi-axial situation it is not sensibleto talk about a yield stress, as there are now several non- zero components of stress.Instead, a yield function, F, is defined, which is a scalar function of stress(expressed in terms of either the stress components or stress invariants) and stateparameters, {A:}:

F({cr},{k}) = 0 (6.1)This function separates purely elastic from elasto-plastic behaviour. In general,

the surface is a function of the stress state {a} and its size also changes as afunction of the state parameters {£}, which can be related to hardening/softeningparameters. For perfect plasticity {k} is constant and represents the magnitude ofthe stresses at yield. It is analogous to oY in Figure 6.1. For hardening and softeningplasticity {A;} varies with plastic straining to represent how the magnitude of thestress state at yield changes. It is analogous to the curves BCF in Figures 6.2(hardening) and 6.3 (softening). If the hardening or softening is related to themagnitude of the plastic strains, the model is known as strain hardening/softening.Alternatively, if it is related to the magnitude of plastic work, the model is knownas work hardening/softening.

The value of the yield function F is used to identify the type of materialbehaviour. Purely elastic behaviour occurs if F({(T} , {k}) < 0, and plastic (or elasto-plastic) behaviour occurs if F({(r},{k}) = 0. F({<j},{k}) > 0 signifies an impossiblesituation. Equation (6.1) plots as a surface in a stress space. For example, ifEquation (6.1) is expressed in terms of the principal stresses and a2 = 0, the yieldfunction can be plotted as shown in Figure 6.6a. Such a plot of the yield functionis called a yield curve. If a2 is not set to zero but is allowed to vary, the yield

Elasto-plastic behaviour / 137

function has to be plotted in threedimensional ox-o2-(j3 space where itforms &yieldsurface, see Figure 6.6b.The space enclosed by this surface iscalled the elastic domain. Theadvantage of assuming isotropicbehaviour and therefore expressingthe yield function in terms of stressinvariants should now be apparent. Ifsuch an assumption is not made, theyield function has to be expressed interms of six stress components and it

a) Yield curve b) Segment of yield surface

Figure 6.6: Yield functionpresentation

therefore forms a surface in six dimensional space. Clearly, it is not possible todraw such a space and therefore visualisation of such a surface is difficult!

6.8.3 A plastic potential functionIn the uniaxial examples considered in Sections 6.3 to 6.5 it is implicitly assumedthat the plastic strains take place in the same direction as the imposed stress. Forthe uniaxial case this is self evident. However, in the multi-axial case the situationis more complex as there are potentially six components of both stress and strain.It is therefore necessary to have some means of specifying the direction of plasticstraining at every stress state. This is done by means of a flow rule which can beexpressed as follows:

0<J;(6.2)

where def represents the six components of incremental plastic strain, P is theplastic potential function and A is a scalar multiplier. The plastic potential functionis of the form:

(6.3)

where {m} is essentially a vector of state parameters the values of which areimmaterial, because only the differentials of P with respect to the stresscomponents are needed in the flow rule, see Equation (6.2).

Equation (6.2) is shown graphically in Figure 6.7. Here a segment of a plasticpotential surface is plotted in principal stress space. Because of the assumption ofcoincidence of principal directions of accumulated stress and incremental plasticstrain, it is possible to plot incremental principal strains and accumulated principalstresses on the same axes. The outward vector normal to the plastic potentialsurface at the current stress state has components which provide the relativemagnitudes of the plastic strain increment components. This is more easily shownin Figure 6.7b, where it is assumed that a2 = 0 and the plastic potential function isplotted in two dimensional ox - <r3 space. It should be noted that the normal vector


only provides an indication of therelative sizes of the straincomponents. The value of the scalarparameter A in Equation (6.2)controls their magnitude. A isdependent on the hardening/softeningrule which is discussed later. Ingeneral, the plastic potential can be afunction of the six independent stresscomponents and has a correspondingsurface in six dimensional stressspace, to which the components of avector normal to the surface at the

a) Segment of the plasticpotential surface b) Plastic potential curve

Figure 6.7: Plastic potentialpresentation

current stress state represent the relative magnitudes of the incremental straincomponents.

Sometimes a further simplification is introduced by assuming the plasticpotential function to be the same as the yield function (i.e. P({(r},{m}) =F({a},{k})). In this case the flow rule is said to be associated. The incrementalplastic strain vector is then normal to the yield surface and the normality conditionis said to apply. In the general case in which the yield and plastic potentialfunctions differ (i.e. P({(r},{m}) * F({(r},{k})), the flow rule is said to be non-associated.

Flow rules are of great importance in constitutive modelling because theygovern dilatancy effects which in turn have a significant influence on volumechanges and on strength, see Chapter 4. Whether or not the flow rule is associatedor non-associated also has a cost implication in finite element analysis. As shownsubsequently in this chapter, if the flow rule is associated, the constitutive matrixis symmetric and so is the global stiffness matrix. On the other hand, if the flowrule is non-associated both the constitutive matrix and the global stiffness matrixbecome non-symmetric. The inversion of non-symmetric matrices is much morecostly, both in terms of storage and computer time.

6.8.4 The hardening/softening rulesThe hardening/softening rules prescribe how the state parameters {A;} vary withplastic straining. This enables the scalar parameter, A, in Equation (6.2) to bequantified. If the material is perfectly plastic, no hardening or softening occurs andthe state parameters {k} are constant. Consequently, no hardening or softeningrules are required. In such materials A is undefined. This follows from the fact thatonce the stress state reaches, and is maintained at, yield the material strainsindefinitely. However, for materials which harden and/or soften during plasticstraining, rules are required to specify how the yield function changes.

For example, in the uniaxial compression of a strain hardening materialdiscussed in Section 6.4 and shown on Figure 6.2, the yield stress, aY, increases

with plastic straining along the pathBCF. At any point along this path the Y

strains can be separated into elasticand plastic components. It is thenpossible to plot how the yield stress,oY, varies with plastic strain, sp, asshown in Figure 6.8. A relationship ofthis type is called a hardening rule.For the strain softening uniaxialexample discussed in Section 6.5 andshown in Figure 6.3, the yield stressreduces and again it is possible to plothow oY varies with plastic strain ep.


hardening

softening

Figure 6.8: Examples ofhardening/softening rules

This is also shown in Figure 6.8 and such a relationship is called a softening rule.In multi-axial situations it is common to relate changes in size of the yield

surface to the components (or invariants) of the accumulated plastic strain. Suchhardening/softening rules are then called strain hardening/softening. Alternatively,but less commonly, the change in size of the yield surface can be related to theincrease in plastic work, W = j{a}T{Aep}. Such hardening/softening rules arecalled work hardening/softening.

So in general, having accepted coincidence of principal directions ofaccumulated stress and incremental plastic strain, three further pieces ofinformation are required to formulate an elasto-plastic model. A yield functionwhich signals when the material becomes plastic, and a plastic potential functionwhich determines the direction of plastic straining, are compulsory ingredients. Ifthe material hardens or softens, a hardening/softening rule is required. Specificexamples of such models are given in Chapters 7 and 8.

To indicate how the above concepts work in multi-axis stress conditions, a twodimensional stress situation is now considered. For simplicity, associatedconditions are assumed, with the yield and plastic potential surfaces being givenby the same function.

6.9 Two dimensional behaviour of a linear elasticperfectly plastic material

For this material the yield surface is fixed in stress space and does not changeposition when loading takes place. If the stress state remains below the yieldsurface behaviour is entirely elastic. If the stress state reaches the yield surfaceplastic straining occurs. It is not possible for the stress state to extend beyond theyield surface.

Consider an element of soil acted upon by a two dimensional system of stressesox, oy, as shown in Figure 6.9. Initially the sample is stress free at point ' o \ Thestress component ax is then increased while keeping ay=0 until point 'a' is reached.Because the stress state remains below (inside) the yield surface (curve), behaviour


is entirely elastic. Although oy does not change, there is a strain ey due to thePoisson's ratio effect (sy is likely to be negative if 0 < ju < Vi). The strain ey canonly be zero if Poisson's ratio, ju, is zero. The stress ax is then kept constant and oyincreased until it reaches the yield surface at point ' b \ While the stress remainsbelow the yield surface, behaviour is again elastic and the strain ey is controlled bythe elastic moduli. Once the yield surface is reached (point 'b'), it is not possibleto increase oy any further. Plastic straining occurs. If the stress state is maintainedat point 'b ' , the plastic strains will keep increasing indefinitely. However, the ratiobetween plastic strain components dex

p and dsf is fixed by the gradient of the yieldsurface (equal to the plastic potential in this case) at point 'b'. The element of soilhas failed.

Figure 6.9: Two dimensional behaviour of alinear elastic perfectly plastic material

If the element of soil considered above is part of the soil mass in a boundaryvalue problem, for example under the edge of a footing, it may be surrounded bysoil that is behaving elastically, with stresses below yield. In such a condition theplastic straining of the element is restricted. Only when sufficient elements of soilare failing so that a collapse mechanism develops, can the strains increaseindefinitely.

6.10 Two dimensional behaviour of a linear elastichardening plastic material

In this case the position and/or size of the yield surface changes as plastic strainingoccurs. If the yield surface changes size, but is always centred about the sameposition, hardening is often called isotropic hardening, see Figure 6.10. If, on theother hand, the yield surface does not change in size, but changes position in stressspace, it is called kinematic hardening. In general, hardening can includecomponents of both isotropic and kinematic hardening.

Following the same loading sequence on an element of soil as described in theprevious section, results in the same behaviour until the initial yield surface isencountered at 'b ' , see Figure 6.11. With further increase in oy plastic strainingoccurs and the yield surface expands (isotropic hardening) according to thehardening law. As it is now possible to increase oy , elastic strains develop


simultaneously with the plastic strains (i.e. soil behaviour is elasto-plastic). Withfurther loading the gradient of the yield surface, and therefore the ratio of theplastic strain components def and de/, is likely to change. Eventually, the yieldsurface stops hardening and failure occurs, similar to the perfect plasticity casediscussed above.

Stress path

Yield surfaces

lsotropic hardening Kinematic hardening

Figure 6.10: Types of hardening

K<JX

Yieldsurfaces ,'

dej.cr,

Figure 6.11: Two dimensional behaviour of alinear elastic hardening plastic material

If at stage 'd' stress oy is removed such that the stress path 'd-a' is followed, thestress strain curve follows the path 'd-e\ Behaviour is elastic on initial unloadingand remains so if the model is of the isotropic hardening type. If the model is of thekinematic hardening type unloading may be sufficient to invoke plastic behaviouragain, but such complications are not considered here. After complete unloadingthere is a permanent strain, ey, which is equal to the plastic strain ey

p generatedduring loading from 'b' to 'd \ If oy is then increased again, behaviour remainselastic until the stress state reaches point 'd', when elasto-plastic behaviour is re-initiated.

6.11 Two dimensional behaviour of a linear elasticsoftening plastic material

In this case behaviour is similar to that for strain hardening, except that the size ofthe yield surface reduces with increase in plastic straining.


ri

ds;

o

Yield

a

/de"

Figure 6.12: Two dimensional behaviour of alinear elastic softening plastic material

Following the same loading sequence as described in Sections 6.9 and 6.10 forperfect and strain or work hardening plasticity, results in the same behaviour untilthe initial yield surface is encountered at 'b' , see Figure 6.12. Once the initial yieldsurface is encountered, plastic straining occurs and the size of the yield surfacereduces (isotropic softening). It is therefore not possible for oy to remain at thevalue associated with point ' b \ Any attempt to do so results in indefinite plasticstrains. If, instead of controlling oy attention is switched to controlling sy (i.e.switch from stress to strain boundary conditions), the strain softening stress-straincurve shown above can be followed. If at some stage during plastic straining ey isreduced, the soil behaves elastically (i.e. path 'c-d'). If ey is then increased again,the unloading path ('c-d') is retraced until the soil yields again (at point V ) . Theyield stress is now lower than it was initially.

cr

6.12 Comparison with real soil behaviourTo simulate the behaviour of real soil it is necessary to have a model that involvesboth strain hardening and softening, seeFigure 6.13.

It should be noted that one of thefundamental differences between elastic andelasto-plastic behaviour is that in the formerstrain increments are proportional to stressincrements, whereas in the latter strainincrements are a function of the current stressstate and therefore the strain increments arenot likely to be in the same directions as theapplied stress increments.

In the above discussion it is implicitlyassumed that elastic behaviour is linear.However, this is not realistic and it ispossible to combine nonlinear elasticity with Figure 6.13: Real soilthe elasto-plastic framework. Due to the behaviour involving hardeningcomplex nature of soil it has not been ancj softening

Elastic Hardening Softening


possible, to date, to develop an elasto-plastic model that can capture all the facetsof real soil behaviour and be defined by a limited set of input parameters that canbe readily obtained from simple laboratory tests. There are therefore many suchmodels currently in the literature. These range from simple to extremelycomplicated models. Some are formulated in terms of total and others in terms ofeffective stress. Some simple models will be discussed in Chapter 7, while morecomplicated ones are the subject of Chapter 8.

6.13 Formulation of the elasto-plastic constitutive matrixHaving defined the basic ingredients of an elasto-plastic constitutive model, it isnow necessary to obtain a relationship between incremental stresses andincremental strains in the form of Equation (2.17):

{Acr} = [Dep]{Ae} (6.4)where [Dep] is used to distinguish that the constitutive matrix is elasto-plastic, asopposed to purely elastic. The matrix [D] is retained to represent the purely elasticmatrix.

The incremental total strains, {As}, can be split into elastic, {A^}, and plastic,{Aep}, components to give:

{A£} = {Ase} + {Asp} (6-5)

The incremental stresses, {Ac}, are related to the incremental elastic strains,{Aee}, by the elastic constitutive matrix, [/)], in the form:

{} {} (6.6)or alternatively:

{Ase} = [D]-]{Acr} (6.7)

Combining Equations (6.5) and (6.6) gives:

ep}) (6-8)The incremental plastic strains, {A^}, are related to the plastic potential

function, P({<r},{m}) = 0, via the flow rule given by Equation (6.2). This can bewritten as:

|Mi|LWi| (6.9)

where A is a scalar. Substituting Equation (6.9) into Equation (6.8) gives:

{A a} =

When the material is plastic the stress state must satisfy the yield functionF({(r},{k}) = 0. Consequently, dF({<r},{k}) = 0 which, on using the chain rule ofdifferentiation, gives:


dF({(T}, {k}) = (^(W.W)} {A a}+j^W,W)l {A*} = o (6.11)[ da J [ d£ J

This equation is known as the consistency equation (or condition). It can berearranged to give:

f^(WW)|T

_ (6.12)dFUa},{k})

\ So-Combining Equations (6.10) and (6.12) gives:

A= 1 ^ 1 (6.13)[ 8 ( { } { } ) 1 f a ( { } { } ) |[ ( { } { } ) 1

where{ { A A } (6.14)

Substituting Equation (6.13) into (6.10) gives:

= [D]{Ae} 1 ^ l i ^ J (6.15)[aF({o- } , { * } ) ] r f g t W W ) ]1 r

Comparison of Equations (6.4) and (6.15) gives the elasto-plastic constitutivematrix [Dep] as:

r ,

vn-m- ^sr7"7sr[ (*•>«\dF({a},{k})\ \dP({a},{m})} , A[ Da J [ da J

The form of the parameter 4̂ given by Equation (6.14) depends on the type ofplasticity (e.g. perfect plasticity, strain hardening/softening plasticity or workhardening/softening plasticity).

Perfect plasticityIn this case the state parameters, {£}, are constants and consequently

1 * Jwhich in turn gives A = 0.


Strain hardening/softening plasticityIn this case the state parameters, {k}, are related to the accumulated plastic strains,{ep}. Consequently, Equation (6.14) can be written as:

If there is a linear relationship between {k} and {sp} so that

= a constant (i.e. independent of {sp}) (6.19)d{sp)

then on substitution into Equation (6.18), along with the flow rule given byEquation (6.9), the unknown scalar, A, cancels and^4 becomes determinant.

If there is not a linear relationship between {k} and {ep}, the differential ratioon the left hand side of Equation (6.19) is a function of the plastic strains andtherefore a function of A. When substituted into Equation (6.18), along with theflow rule given by Equation (6.9), the A's do not cancel and A becomesindeterminant. It is then not possible to evaluate the constitutive matrix [Dep].

In practice all strain hardening/softening models assume a linear relationshipbetween the state parameters {k} and the plastic strains {ep}.

Work hardening/softening plasticityIn this type of plasticity the state parameters, {£}, are related to the accumulatedplastic work, W, which is dependent on the plastic strains. It can be shown,following a similar argument to that presented above for strain hardening/softeningplasticity, that as long as there is a linear relationship between the state parameters{k} and the plastic work, W, the parameter ,4 defined by Equation (6.14) becomesindependent of the unknown scalar, A, and therefore is determinant. If therelationship between {k} and W is not linear, A becomes a function of A and it isnot possible to evaluate the constitutive matrix [Dep].

If the elastic constitutive matrix, [D], is symmetric, which it is for isotropic andcross anisotropic elasticity, then the elasto-plastic constitutive matrix, [Dep], givenby Equation (6.16), is also symmetric if its numerator, given by:

(w^})[Z)] ( 6 2 0 )

[ da Jis symmetric. It is a simple matter to show that this only occurs if:

\dP({cr},{m})} = \dF({<T},{k})}\ da J 1 da J

which implies that the yield and plastic potential functions must be identical. Asnoted earlier, this occurs in a special class of plasticity in which the flow rule issaid to be associated. Substitution of a symmetric [Dep] for all elements in a finiteelement mesh, into the assembly process, results in a symmetric global stiffnessmatrix.


For the general case in which the flow rule is non-associated and the yield andplastic potential functions differ, the constitutive matrix [Dep] is non-symmetric.When assembled into the finite element equations this results in a non-symmetricglobal stiffness matrix. The inversion of such a matrix is more complex andrequires more computing resources, both memory and time, than a symmetricmatrix.

Some commercial programs are unable to deal with non-symmetric globalstiffness matrices and, consequently, restrict the type of plastic models that can beaccommodated to those which have an associated flow rule.

6.14 Summary1. Elasto-plastic theory provides probably the best framework available in which

to formulate constitutive models that can realistically simulate real soilbehaviour. Three types of plastic behaviour are identified: perfect plasticity andstrain (or work) hardening and softening plasticity. These models assumeelastic behaviour prior to yield and can therefore utilise the benefits of bothelastic and plastic behaviour.

2. The elasto-plastic framework can incorporate both linear and nonlinear elasticbehaviour. Consequently, all the models described in Chapter 5 can beincorporated.

3. Elasto-plastic models are based on the assumption that the principal directionsof accumulated stress and incremental plastic strain coincide. They require twoessential pieces and one optional piece of information for their definition. Theessential ingredients are a yield function, which separates purely elastic fromelasto-plastic behaviour, and a plastic potential (or flow rule) which prescribesthe direction of plastic straining. The optional ingredient is a set ofhardening/softening rules which describe how the state parameters (e.g.strength) vary with plastic strain (or plastic work).

4. If the yield and plastic potential surfaces coincide, the model is said to beassociated (or to satisfy the normality condition). This results in a symmetricconstitutive matrix and consequently a symmetric global finite element stiffnessmatrix. If such a condition does not hold, both matrices are non-symmetric.This results in the use of greater computer resources, both time and memory,for finite element analyses.

6_6. Elasto-Plastic Behaviour

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