integration of tresca and mohr-coulomb constitutive …...mohr-coulomb solids to be considered in...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 33, 163-196 (1992)

INTEGRATION OF TRESCA AND MOHR-COULOMB CONSTITUTIVE RELATIONS IN PLANE STRAIN

ELASTOPLASTICITY

S. W. SLOAN

Department of Civil Engineering and Surveying, The University of Newcastle, N.S. W. 2308, Australia

J. R. BOOKER

School of Civil and Mining Engineering, University of Sydney, N.S. W. 2006, Australia

SUMMARY This paper considers the problem of integrating the constitutive relations for Tresca and Mohr-Coulomb materials under conditions of plane strain. In the case of a Tresca material, we show that the constitutive law may be integrated exactly by assuming the strain rates dE/dt to be constant. We also derive a semi-analytic method for integrating both types of constitutive law which assumes that the quantities d&/dl are constant. This approach is motivated by the fact that the exact variation of the strains during a time interval is unknown and leads to a single non-linear equation in a which can be solved efficiently to yield the unknown stresses. Finally, we compare the results from the analytic and semi-analytic methods with those from a variety of numerical integration schemes.

1. INTRODUCTION

The constitutive behaviour of an elastoplastic material may be characterized by an incremental stress-strain relation of the form

i = D,,i (1) where i denotes a vector of stress components, i is a vector of strain components and the superior dot represents a derivative with respect to time. For the perfectly plastic Tresca and Mohr-Coulomb solids to be considered in this paper, the elastoplastic constitutive matrix D,, is solely a function of the stresses.

When the displacement type of finite element method is used to analyse elastoplastic behaviour, the nodal forces are applied in increments and the corresponding nodal displacement increments are computed from the global tangent stiffness relations. Once these are known, the strain increments can be determined from the strain-displacement relations and it is then necessary to ascertain the increments of stress. If we assume the strain rates are constant, equation (1) constitutes a set of ordinary differential equations which can be used to find these stresses. The integration of this set of differential equations is not a trivial task, however, since in general the matrix D,, is a highly non-linear function of the stresses.

Although an exact integration scheme for a von Mises material has been given by Krieg and Krieg,' this step is usually performed numerically for Tresca and Mohr-Coulomb materials. One

0029-598 1/92/010163-34$17.00 0 1992 by John Wiley & Sons, Ltd.

Received 3 October 1989 Revised 12 December I990

164 S. W. SLOAN AND J. R. BOOKER

of the simplest numerical schemes, which has been used widely in finite element codes, is the first order Euler algorithm. This has the great advantage of being straightforward but also has the disadvantage of being accurate only for very small time steps. To avoid this shortcoming, it is usual to subdivide the particular time step into a number of smaller substeps and apply the Euler scheme to each of these.2*3 This partially overcomes the disadvantages of the Euler approach but usually leads to a set of stresses which do not lie precisely on the yield surface at the end of each time step. Since these errors are cumulative, and may lead to unacceptable results in subsequent computations, it is usual to apply some form of correction to the stresses to restore them to the yield surfaces4 Although various correction procedures have been proposed for use in finite element codes, most of these are difficult to justify theoretically and are usually founded on empirical testing.

An alternative numerical approach, which varies the size of each step automatically to control the error in the computed stresses, has been proposed by Sloan.’ This technique computes two estimates of the stresses for each substep, the first with a formula of order p and the second with a formula of order p + 1, and uses the difference between these to measure the local truncation error. If this error exceeds a specified tolerance, then the substep is abandoned and the integration is repeated with a new substep which is reduced in size by local extrapolation. Local extrapolation is also used to predict the size of a substep which follows a successful stress update. This technique is particularly robust and, provided a suitable tolerance is chosen, permits a very general class of elastoplastic constitutive laws to be integrated with negligible drift from the yield surface. The preliminary numerical tests reported in Sloan’ indicate that no form of stress correction is necessary, although Gens and Potts6 have elected to implement a version which does employ a correction in their critical state codes. It is possible to develop very efficient pairs of formulae which permit this type of error control and the order of these commonly ranges from first and second order through to fourth and fifth order. In general, the high order schemes are preferred for computations with tight error tolerances whilst the lower order schemes are more efficient for looser tolerances.

Yet another attractive method of numerical integration has been described by Wissmann and This technique also attempts to control the error in the computed stresses, but does so by

using Richardson extrapolation to select a number of fixed size substeps. The results of Wissmann and Hauck suggest that this method performs well for a von Mises constitutive law with isotropic hardening, and it is of interest to see how effectively it integrates Tresca and Mohr-Coulomb relations. Richardson extrapolation can be applied to all of the common integration schemes, from the simple first order Euler scheme up to the classical fourth order Runge-Kutta scheme, and assumes that the truncation errors for each substep are additive and do not multiply.

This paper shows that the constitutive relations for a Tresca material may be integrated exactly under conditions of plane strain, provided we assume the strain rates to be constant. The final form of the analytic solution is particularly simple and can be used in preference to a numerical integration scheme in a finite element code. We also develop a semi-analytic method for integrating the Tresca and Mohr-Coulomb constitutive laws which is based on the assumption that the quantities dc/dl are constant throughout an increment. This approach is justified on the grounds that the strain increments can be assumed to vary linearly with any quantity which increases monotonically with time, and leads to a single non-linear equation in the plastic multiplier. Solving this equation numerically is shown to be straightforward and permits the unknown stresses to be computed efficiently. To complete our study, we examine the accuracy and efficiency of a number of numerical integration schemes that are used widely in finite element programs. We also propose some new methods which have proved to be very effective.

RELATIONS IN PLANE STRAIN ELASTOPLASTICITY 165

2. INCREMENTAL CONSTITUTIVE RELATIONS FOR AN ELASTOPLASTIC SOLID

The theory of plasticity divides the strain rate into two components according to

(2) & = &" + &P

where Ee denotes the elastic strain rate and EP denotes the plastic strain rate. The elastic component is related to the stress rate i by Hooke's law, so that

(3) 2 = D-16

while the plastic component is derived from a flow rule

where Q is a plastic potential and i is a plastic multiplier rate, indicating that it is only the ratios of the plastic strain rates which are known and not their magnitudes. Combining equations (2), (3) and (4), we see that the stress rate is given by

& = i" - iDb (5 ) where i" = Di denotes the elastic stress rate and i remains to be determined.

plastic solid, may be written in the general form Plastic straining occurs once the stress state has reached the yield surface which, for a perfectly

F(a) = 0 Using the consistency condition

which holds during plastic flow, and substituting equation (5), we can eliminate i to obtain an expression for the plastic multiplier rate according ;o

. a*$ A=---- aTDb

Equations (5 ) and (7) permit the stress-strain relations to equation (1) where

(7)

be written in the alternative form of ~,

DbaTD aTDb

D,, = D - ~

Note that when the constitutive law is written in the form of equation (5), a negative value of indicates elastic unloading whilst a zero value for i corresponds to neutral loading. In both of these cases, the correct stress rate is given by i = i". If the gradient vectors a and b are parallel, the material is said to have an associated flow rule and the matrix D,, is symmetric. Otherwise, the material is said to have a non-associated flow rule and the matrix D,, is in general unsymmetric.

3. EXACT INTEGRATION OF THE TRESCA CONSTITUTIVE RELATIONS FOR A CONSTANT STRAIN RATE

We consider the behaviour of an isotropic Tresca material which is subjected to plane strain loading in the x-y plane. It is assumed, for the sake of simplicity, that the z direction is also the

166 S. W. SLOAN AND I. R. BOOKER

direction of the intermediate principal stress (this is precisely the situation when the material is incompressible and corresponds to the undrained deformation of a saturated clay). Under these conditions, the yield criterion may be written as

F = X 2 + P - ~ ’ = 0 (8) where c is the undrained shear strength of the material and

x = *(ax, - c y y )

Y = axy

with tensile stresses taken as positive. For the case of an associated flow rule, equations (2), (3) and (4) give

1 .jxy = 5 &xy + 2AOxy

where v* = v/(S - v)

E* = E/(1 - v’)

G = E/2(1 + V) = E*/2(1 + v*)

(9)

and v, E and G denote Poisson’s ratio, Young’s modulus and shear modulus respectively. Ayxy are known and vary linearly with

respect to time over a time interval At so that the strain rates ixx, iyy, j xy are constant. To simplify the integration process, it is convenient to introduce the Mohr circle representation of stress

Let us suppose that the strain increments A E ~ ~ ,

axx = z + c cos28

ayy = z - c cos 28 (1 1) oxy = c sin 26

where = *(axx + ayy) = +(a, + 6 3 )

and 8 denotes the angle between the principal stress direction and the x axis.

that Differentiating equations (1 1) with respect to time and combining with equations (10) we see

(13) A E ~ A E ~ ~ + A E ~ ~ - Z At At K * -- - --

(14) A E ~ cos 2cD AE,, - AE,, - 2cd sin 28 - - + 2ci cos 28 - -

At At G

A E ~ sin 2cD A?,, - 2cd cos 28 - + 2c;Z sin 28 At At G


where K * = E*/2(1 - v * ) is the plane strain bulk modulus and we have used the Mohr circle representation for the strain increments according to

AE, = + A E ~ = ,/( AE, - AE,,)~ + AyZy

AYxy A%

sin 2cD = __

Equation (13) can be integrated immediately to give

Z ( t ) = Z o + K * A & - ( i t t o ) where to denotes the time at the commencement of the load step, Z, = %(to), and to < t < to + At. Eliminating i from equations (14) and (15) we also obtain

2cd G At

- A E ~ sin2(8 - 0) - --

This can be integrated to give

1 tan 51 = tan R, exp

where

tan 80 - tan cD tanR, =

1 + tan8, tan@

YO tan8, = ~ x, + c

and sin 2cD

1 + cos 2@ tan@ =

can be evaluated from equations (16). All quantities denoted by the subscript zero are values which are known at time to. This completes the integration as the field quantities oxx, guy., oxy can now be recovered from equations (1 1) and (17) together with the trigonometric identities

cos 28 = cos 2R cos 2cD - sin 2Q sin 20

sin 28 = sin 2R cos 2@ + cos 2R sin 2cD

where 1 - tan2R 1 + tan2R

cos2R =


2 tanR 1 + tan2R

sin 2R =

can be evaluated from equation (18) and sin 2(0, cos 2(0 are given by equations (16). The values at the end of the time increment are, of course, found by setting t = to + At in equations (17) and (18).

4. SEMI-ANALYTIC INTEGRATION SCHEMES

In the previous section, we integrated the Tresca stress-strain relations exactly by assuming a constant strain rate. Since the exact variation of the strains during the time interval to < t < to + At is not known, there is no reason to suppose that this is the only valid assumption. In fact, provided the time increment is sufficiently small, there is no reason to prefer one type of monotonic variation of strain over any other. Bearing in mind that the parameter 1 is monotonic (i is strictly positive during plastic flow) it follows, therefore, that we are at liberty to postulate that the variation of strain is linear in 1.

4.1. Semi-analytic integration of the Tresca constitutive relations

We again assume that the z direction coincides with the direction of the intermediate principal stress and plane strain conditions. Supposing that the variation of strain is linear in 1, equations (10) become

dX AXE - + 2GX = __ d1 AA

dY A YE -+2GY=- d1 A1

dZ AZE -=- d1 A1

where AXE = G(A&, -

and X , Y and Z are again defined by equations (9) and (12) and A1 denotes the as yet unknown increment in 1 which occurs over the time interval to < t < to + At. This set of equations may be integrated to give

X F = X ( t o + At) = X0e-' + - (1 - e-a) ct

AYB Y , = Y ( t o + A t ) = roe-' + -(1 - e-a) ct

ZF = Z(t0 + A t ) = Zo + AZE

where the subscript zero denotes values which are known at time to, ct = 2GA1, and we have

RELATIONS IN PLANE STRAIN ELASTOPLASTICITY 1 69

assumed, without loss of generality, that A varies from zero to AA monotonically as t varies from to to to + At.

In order to determine AA, we need to employ the yield criterion of equation (8)

F = X $ + Y $ - c 2 = 0 (22) Equations (21) and (22) now define a single non-linear equation in A I which can be solved numerically to give the stresses at the end of the increment. Once X,, YF and 2, are known, the Cartesian stress components are given simply as

f lxx = 2, + x, flyy = ZF - XF

= yF

A variety of numerical schemes may be used to solve equation (21). Two algorithms which have proved to be very effective, however, are the well known Newton-Raphson and secant methods. With the former scheme, an initial estimate for AA is readily obtained from the incremental form of equation (7) with a and b evaluated at b0, the stresses at the start of the increment. For the secant scheme, which requires two initial values to start the iteration procedure, a second estimate of A I can be computed by applying equation (7) at the stress (F = (F, + Dep(a0)A&.

A particularly interesting feature of the semi-analytic method is that it depends primarily on the flow rule, since the yield condition is not invoked until the final step of the analysis. This permits a variety of different plasticity models to be handled with only minor adaptation. In the case of an isotropically hardening Tresca material, for example, the stress state would still be found by solving equations (21) and (22), except that the shear strength c would depend on the plastic strains and thus the parameter a.

4.2. Semi-analytic integration of the Mohr-Coulomb constitutive relations

We now consider a Mohr-Coulomb material which is subjected to plane strain loading in the x-y plane. We assume that the intermediate principal stress direction coincides with the z direction and remains fixed throughout an increment. Under these conditions, and denoting tensile stresses to be positive, the Mohr-Coulomb yield criterion may be written as

F = X2 + P - g2 sin2qb = 0

2 = f(., + OYY) - ccotcp = z - ccotcp

(23)

(24) - where

and X, Y, Z are given by equations (9) and (12). If we again assume that the variation of strain is linear in A, the strain relationships may be written as

dX AXE -+2GX=--- d I 41

-+2GY=- dY A YE

dA A I

- AZE d g dA Ail

2 sin2$K*Z = - ___-

where AXE, AYE, 62, are defined by equations (20). These equations are remarkably similar to


those developed for the Tresca criterion and can be integrated to obtain the solutions

XF = X ( t , + At) = X0e-. + -(1 AXE

Y, = Y(t, + At) = Yoe-" + -(1 A YE

- e-.)

- e-.)

a

a

AzE fl 2, = z(t, + At) = zoea + - (e - 1) D

where a = 2GA1

a sin2 #K * D =

Equations (25), combined with the yield criterion

F = X g + Yg-Zgsin2#=0 (26) define a single non-linear equation in A 1 which can be solved numerically to give the stresses at the end of the increment. Having obtained X,, YF and Z,, the unknown Cartesian stresses can be computed simply from -

0,- = Z F + XF + c cot# -

0 y y z, - XF + c cot 4 Oxy yF

5. NUMERICAL INTEGRATION SCHEMES

The exact and semi-analytic solutions described in the previous sections may be implemented in a displacement finite element program to furnish accurate solutions for elastoplastic analysis with the Tresca and Mohr-Coulomb yield criteria. Many existing finite element codes, however, integrate these stress-strain relations numerically and it is thus of interest to investigate the performance of some of the more widely used schemes.

The majority of numerical integration methods assume that the strain rates are constant and, hence, equation (1) becomes

A& At & = Dep-

where At is the time interval over which loading takes place and to < t < to + At. This equation defines a classical initial value problem since the 6, = &(to) and A& are known and the matrix D,, is a function of the stresses CT. We now briefly describe a variety of techniques that may be used to perform this integration numerically.

5.1. The Euler scheme

The Euler scheme is a simple and widely used method for updating the stresses in elastoplastic finite element codes. In this algorithm, the stresses at the end of a time step are computed from

CT =Z GO + D,,A& (28)


The Euler method is a first order scheme and hence the stresses are accurate only if the strains As are small. Since no attempt is made to control the error in the integration process, other than by choosing the load increments to be very small, the stresses at the end of the increment do not typically satisfy the yield criterion. Because any deviations from the yield surface are cumulative, and may result in unacceptable errors if left unchecked, it is usual to 'correct' the stresses so that the yield criterion is satisfied. There is no unique method for doing this, but the most common approach is to scale the stresses along a direction which is normal to the yield surface. For the most general type of perfectly plastic yield condition, the correction procedure may require the solution of a single non-linear equation. For the Tresca and Mohr-Coulomb criteria, however, it is possible to devise a very simple correction scheme which is exact and requires no iteration. In terms of the principal stresses, the Mohr-Coulomb yield condition may be written as

F = o , -cr3+(o, +a3)s in# -2ccos#=0 (29) where al > a3 and the Cartesian stresses are

a,, = +(., + a3) + +(., - 6 3 ) C O m

ayy = +(a, + a3) - +(a, - a3) cos2e (30) a,,, = +(a, - a3) sin28

If we correct the stresses along a direction which is normal to the yield surface, so that the principal stress direction 8 remains unchanged, we have

a:, - a ; y

a; - a; cos 28 = cos 28" =

2 G sin 28 = sin 28" = ____ a; -a;

and aF aa1

~1 = 0: + s- = cry + $1 + sin#)

aF a3 = 0; + s- = a; - s(l -

803 sin #)

where the superscript u denotes uncorrected values and s is a scalar which is yet to be determined. By substituting the above expressions for a1 and cr3 into equation (29), the value of s is obtained as

and hence

F(olf, a;)(l + sin#) 2(1 + sin2 #)

a1 = a'i -

F(a;, o;)(l - sin#) 2(1 + sin2 #)

cr3 = a; + Equations (30)-(32) now define the corrected stresses. The corrections appropriate for the Tresca yield condition are obtained merely by substituting # = 0. Although this procedure ensures that the stresses satisfy the yield condition exactly, it does not, of course, ensure that they are


sufficiently accurate. The role of the flow rule is clearly assumed to be subordinate to that of the yield criterion and, unless the strain increments are small, the computed stresses may still be in error.

5.2. The mod@ed Euler scheme

The modified Euler algorithm is a second order method and requires two evaluations of the matrix D,, to update the stresses. It first uses the Euler formula to compute the stress increment

A o 1 = D,,(oo)Ac:

and then uses this to compute a second stress increment

AG, = D,,(ao + Aa,)A&

The update is then carried out using the average of these two increments according to

= no + ~ ( A G , + Aa,) (33) As with the Euler scheme, these stresses may be scaled back to lie on the yield surface using equations (30)-( 32).

5.3. The classical Runge-Kutta scheme

The classical Runge-Kutta method is a fourth order scheme that employs four evaluations of the matrix D,, to update the stresses. Although a number of fourth order Runge-Kutta formulae can be developed, the best known and most commonly used one computes the increments

h, = Dep(a0)A&

An, = D,,(oo + &Aa,)A&

Aa3 = D,,(ao + +Aa,)A&

An, = D,,(a0 + Aa3)A&

before updating the stresses according to

a = a. + &Anl + 2Aa, + 2Aa3 + Aa,) (34) Again the stresses can be restored to the yield surface at the end of each step if required.

5.4. The Euler scheme with substeps of equal size

This is perhaps the most widely used method for updating the stresses in elastoplastic finite element codes. Since the Euler scheme may become insufficiently accurate for large increments in the strains A&, these are divided into M increments of equal size and the stresses are computed from the recurrence relation

a, = a,-, + Aa,; i = 1 , 2 , . . . , M (35) where

and Aai = D,,(a,-,)hA&

1 h = - M


At the end of the substepping process, the stresses may be restored to the yield surface using equations (30)-(32).

For the stresses to be sufficiently accurate it is clearly necessary to choose a suitable number of substeps M . In many existing codes, M is selected on the basis of an empirical rule which is formed by trial and error (see, for example, Nayak and Zienkiewicz’ and Owen and Hinton3). A more rigorous approach for estimating M , which has the important advantage of providing an estimate of the error in the computed stresses, has been proposed by Wissmann and H a ~ c k . ~ Their method is based on Richardson extrapolation which states that the error in the stresses for the Euler method can be estimated by a single and a double step calculation according to

e = 2 1 1 % - bl II where dl is determined from a single step of size h = 1 and (T’ is determined from two steps of size h = 1/2. This error estimate assumes that the propagation of the local truncation error is additive and ignores all effects of round-off error. Dividing both sides by 11 e2 I/, an estimate of the relative error in the stresses (rl can be obtained from

Noting that the local truncation error in the Euler scheme is O(h’), the relative error at the end of M substeps is thus approximated by

2l l%-~,II M II bz I1

If we insist that this quantity is less than some specified tolerance, 6, then the number of substeps required is

In practice, we compute M as the largest integer which is less than or equal to

2 1 / ~ , - ~ 1 I I +

6 I/ 6 2 II The tolerance, 6, controls the overall accuracy of the integration and it usually has a value in the range lo-’ to Ideally, the global relative error in the stresses computed is of the same order of magnitude as 6.

Richardson extrapolation can, of course, be applied to any of the common integration schemes. For a method of order p , the error is again estimated from a single step and a double step calculation according to

11% - el II (1 - 2-P) e x

Since the local truncation error is O(hP+’), the relative error in the stresses at the end of M substeps may be estimated from


For a specified relative error tolerance, 6, the number of substeps is chosen as the largest integer which is less than or equal to

Wissmann and Hawk7 found that Richardson extrapolation provided a simple and efficient means of integrating a von Mises constitutive law with isotropic hardening. Their results verify the effectiveness of this type of error control for a variety of schemes and, in particular, suggest the use of the fourth order Runge-Kutta procedure for optimum efficiency.

5.5. The modijed Euler scheme with substeps of equal size

Application of Richardson extrapolation to the modified Euler scheme is similar to that for the Euler method. The stresses G, and (r2 are again computed from a single step and a double step calculation, respectively, using equations (33). The number of substeps required is then set to the largest integer which is less than or equal to

G, - a, 11

The stresses for the specified strain increment A& are found from the recurrence relation (35) with

AG, = ~ ( A G , + Ac,)

and ACJ, = De,(ai- l ) h A ~

AG, = D e p ( ~ i - l + Aa,)hA&

5.6. The Runge-Kutta scheme with substeps of equal size

This is identical to the two previous schemes except that a fourth order formula is used to update the stresses. The number of substeps required is set to the largest integer which is less than or equal to

where equation (34) is employed to compute the single and double step stresses a, and az. The recurrence relation (35) is then used to give the stresses with

doi = $(AG, + 2Aa, + 2Aa3 + Aa,)

and AG, = Dep(ai-,)hA&

A G ~ = Dep(ai - l + $AG,)hA&

AG, = De,,(~i-.l + $AL\a2)hA&

AG, = D e p ( ~ i - l + Aa,)hA&


5.7. The rnod$ed Euler scheme with substeps of variable size

This scheme was originally proposed by Sloan5 and also uses extrapolation to control the growth in the local truncation error. Unlike the Richardson schemes, however, the size of each substep is adjusted throughout the integration process instead of being fixed a priori. During each stage we compute two estimates of the stresses, one using the first order Euler method and one using the second order modified Euler method, according to

= a0 + An, (36)

(37) Ci = c0 + $(A@, + Aa2)

where

Aha, = D,,(oo)hiAE

Aa2 = D,,(ao + AL\o1)hiA&

and hi is the size of the current substep with 0 < hi < 1. Noting that the local truncation errors in a and Ci are O ( h 2 ) an O(h3) , respectively, an estimate of the relative error in a can be obtained from

The current substep in accepted if this relative error is less than some specified tolerance, 6, and rejected otherwise. In both cases, the next substep size hi+ , is given by

hi+, = qhi where q > 0. The factor q is found by enforcing the condition E i + < 6, where Ei+ is the relative error for substep hi+ ,, and noting that E,+ x q2Ei . This gives

Because this estimate is based on local extrapolation, it is prudent to choose q to be a few per cent less than this upper bound. Moreover, it is wise to introduce constraints on q so that the extrapolation is not carried too far. Numerical experiments suggest that q should be computed using

- q = 0.9 /: (39)

and constrained to lie within the limits

0.01 < q < 2 (40) in order to minimize the number of rejected steps.

To start the integration procedure, we apply equations (36) and (37) with hi = 1. If a substep is rejected because E , > 6, its size is reduced and the substep is repeated without updating the stresses. Note, however, that in this case only one additional evaluation of the matrix D,, is required since the new value of da, is merely equal to its old value times q. If a substep is accepted, the stresses are updated using either equation (36) or equation (37) before computing the size of the next substep and continuing the integration. In general, it is better to use the second


order update of equation (37), as this compensates for the fact that the error control is based solely on approximate estimates of the local truncation error.

As with all of the schemes discussed so far, the stresses may be restored to the yield surface on completion of the integration process. Typical values for the relative error tolerance, 6, are again in the range SO-2-SO-5.

5.8. The Runge-Kuttu-England scheme with substeps of variable size

A number of different methods which employ local extrapolation to adjust the step size automatically can be formulated. Indeed, Sloan’ considered the use of a pair of fourth and fifth order Runge-Kutta formulae developed by England.8 Even though it requires six evaluations per step, this high order scheme rapidly becomes competitive with the modified Euler scheme as the error tolerance is tightened. The theory for implementing the Runge-Kutta-England formulae is identical to that for the Euler pair of formulae. During the integration procedure we again compute two estimates of the stresses for each substep, the first with a formula of order 4 and the second with a formula of order 5, and use the difference between these to measure the local truncation error. The two stress updates are calculated from

= a0 + Q(Aa1 + 4Aa3 + Aa4) (41)

(42) 6 = a. + &(S4Aa1 + 35Aa4 + S62Aa5 + 125Aa,)

where An1 = D,,(ao)hiA&

An2 = D,,(a0 + iAal )hiA&

Ac, = D,,(ao + $(Aa, + Aa2))hiA~

Ah04 = D,,(ao - An2 + 2Ao3)hiA~

Aa, = De,(ao + &(7AG1 + 10Aa2 + Aa4))hiA&

Aa, = D,,(ao + &&SAC, - 125Aa2 + 546Aa3 + 54Aa4 - 378Aa,))hiA&

to give the relative error estimate, E i , from equation (38). Since the local truncation error in the fourth order formula is O(hJ), equation (39) is replaced

by

but the constraints of equation (40) still apply. Again it is usually advantageous to update the stresses with the higher order formula following a successful step.

5.9. The Runge-Kutta-Fehlberg scheme with substeps of variable size

An alternative pair of fourth and fifth order integration formulae has been developed by Fehlberg.’ This scheme has gained widespread use in the mathematical literature and the error estimate is again obtained at the cost of six evaluations of the matrix Dep. The implementation of the Runge-Kutta-Fehlberg formulae is identical to that for the Runge-Kutta-England formulae


with equations (41) and (42) being replaced by

25 1408 2197 1 216 2565 4104 5

16 6656 28 561 9 2 135 12 825 50 55 56 430

D = GO + - A a 1 + - A G ~ + - Aa4 - - An5

Aa4 - - Aa5 + - AG, Ac3 + - i = 60 + -AG~ + -

Aa, + 7296 A a 3 ) hiA& 1932 7200 ( 2197 2197 2197 60, =I D,, GO + - A c ~ - -

4104 439 3680

513

3 544 1859

5.10. The Runge-Kutta-Dormand-Prince scheme with substeps of variable size

Yet another pair of fourth and fifth order integration formulae has been proposed by Dormand and Prince.l0 An interesting feature of their scheme is that the coefficients have been chosen to estimate and control the local truncation error as accurately as possible. Extensive testing of the method suggests that its performance is generally superior to that of the Runge-Kutta-Fehlberg scheme. The Runge-Kutta-Dormand-Prince formulae are implemented in the same manner as the Runge-Kutta-England formulae with equations (41) and (42) replaced by

31 190 145 351 1 540 297 108 220 20

19 1000 125 81 5 216 2079 216 88 56

a = a o + - A a l + - A a 3 - - - - A a 4 + - A a , + - A a ,

i = a , + - A ~ 1 + - A I s ~ - - A c T ~ + - A G ~ +-Ac,

where AG, = D,,(ao)hiA&

1 ACT, = D,, ( a. + AS,) hiAE

40


729 226 25 880

Ac, = D,, <rO + - A c ~ - - A c ~ + ~ Aa3 + ( 729 27 729

Ao, = D,, q0 - ~ 60, + - Ac2 - __ A c ~ - ( 270 2 297 181 5 266 91

6. TEST PROCEDURE

In the previous sections, we have described various schemes for integrating Tresca and Mohr-Coulomb constitutive laws. We now describe a numerical procedure for testing the accuracy and efficiency of these algorithms over a broad range of strain paths.

In terms of the quantities X , Y and Z, defined by equations (9) and (12), the Mohr-Coulomb yield criterion is given by

This plots as a cone as shown in Figure 1. For the first of two different types of tests, we consider loadings which cause the elastic trial

stress increments to lie in the plane Z = 0. The increments are generated in a fan, centred on a state of pure shear, to give a broad range of paths as shown in Figure 2. Each stress increment vector has a magnitude r and inclination 8 to the X axis, where 0 < r < c cos S$ and 6 is restricted to lie in the range 0 d 6 d 7c/2 (owing to symmetry in the yield function). To generate the fan of elastic stress increment vectors, r and 8 are computed from

X 2 + Y2-(Zs in# -ccos$ )2=0

r = i Ar;

8 = ( j - 1)Ae; i = 1,2, . . . , n,

j = 1,2,. . . ,no + 1 where the number of radial and angular increments, n, and n,, are specified and

c cos # Ar = ~

nr

(43)

(44)

Figure 1. Mohr-Coulomb yield criterion under plane strain conditions

RELATIONS IN PLANE STRAIN ELASTOPLASTICITY

Y

point o f pure shear with *

u = c XY

179

Figure 2. Stress fan for testing the MohrCoulomb criterion in the plane 2 = 0

For a given r and 0, the elastic stress increments are

Ao:, = r cos 8

Ao;, = - AoZ, (45) Ao;, = r sin0

where it is assumed that AZe = $(Aot;, + Ao;,,) = 0. These define the corresponding strains since

1 E

AE,, = (A@:. - v*Ao;,)

. "I Because of the condition AZ" = 0, it follows that each of the strain paths defined by equations


(43)-(46) is incompressible. Numerical experiments suggest that a total of 100 trajectories is sufficient to measure the average performance of a particular integration scheme accurately, and we thus choose n, = 10 and n, = 9. As we impose the constraints 0 < r < ccos 4 and 0 < 13 < 4 2 , it also follows that the elastic stress increments lie within the following bounds:

0 < Aczx < c cos+

- C C O S ~ < Ac;y < 0

0 < Ac;, < C C O S ~

It is perhaps worth remarking that quite large stress changes may occur in displacement finite element analysis, particularly in collapse studies or under displacement controlled loading, and these variations are not unreasonably large.

In the second type of test, we generate a fan of elastic stress increment vectors in the tangent plane Y + Z sin+ - c cos+ = 0 which passes through a point of pure shear. When viewed normal to this plane, the Mohr-Coulomb yield criterion appears as shown in Figure 3. From this

point o f pure shear with

Figure 3. Stress fan for testing the Mohr-Coulomb criterion in the plane Y + Z sin 4 - c cos 4 = 0


figure, we see that each stress increment vector of magnitude r has an inclination p to the axis Z , , / m . Owing to symmetry in the yield surface, f i is restricted to lie in the range 0 < p < n whilst r is again chosen such that 0 < r < c cos 4. The fan of elastic stress increment vectors is generated by computing r and j? from

r = i Ar;

f l = ( j - 1)Ap;

i = 4 2 , . . . , n ,

j = 1,2, . . . , nS + 1

where

and n, and nS are prescribed. Given r and p, the elastic trial stress increments are

cos p ACE,. = r (,,/- + sinp

1 + sin2 4 Ac;,, = r (~~ cos p - sinp

1 + sin2 4

(47)

(49)

- r cosp sin4 ACL= Jm

and the corresponding strain increments are completely specified by direct substitution into equations (46). Since the elastic stress increments lie in the tangent plane Y + Z sin4 - c cos 4 = 0, we have A Ye + AZe sin 4 = 0 and each strain path must satisfy GAy, + K *(A&,.. + AeY,)sin 4 = 0. To test the various integration schemes, a total of 190 trajectories are generated by choosing n, = 10 and nP = 18 in equations (46)-(49).

For the Tresca constitutive law, the cone of Figure 1 is replaced by a cylinder whose equation is given by equation (8). Since this criterion is independent of the stress quantity Z , it is necessary only to consider stress paths which cause changes in X and Y. The first type of test procedure is thus identical to that for the Mohr-Coulomb material (with 4 = 0) whilst the second type of test procedure is not required.

To assess the accuracy of each numerical scheme over each strain path, the errors in the elastoplastic stresses are computed using the measure

II * - *exact II m

II *exact II m E =

where Q are the approximate stresses, cexact are the stresses computed from the exact solution and 11 . I / co indicates the max norm. For the Mohr-Coulomb constitutive relation, no exact solution is available so oexact is computed by a highly accurate numerical scheme (fourth order Runge-Kutta with 100 substeps of equal size). The above quantity is a useful error measure since it gives a rough indication as to the number of significant figures in the stresses. We choose to work with the max norm, rather than the more usual Euclidean norm, as it is cheaper to compute and simpler to implement. To be consistent, we have also used this norm in the error control mechanisms of the


various substepping schemes. The average and worst case performances for a particular integration method are of primary interest and these are measured using

and

Em,, = max ( E i ) l d i Q N

where E , is the error computed from equation (50) for each of the N strain paths. In subsequent discussions, these quantities will be referred to as the average and maximum relative errors.

7. RESULTS

We now compare the performance of various schemes for integrating the Tresca and Mohr-Coulomb stress-strain relations.

7.1. Tresca yield criterion

As described in the previous section, the various schemes for integrating the Tresca constitutive law are tested by generating a fan of elastic trial stress increments in the plane 2 = 0. The initial stresses are chosen to lie on the yield surface in a state of pure shear with ox, = o,,,, = 0 and ox,, = c. To provide a benchmark for comparing the accuracy of alternative methods, we assume a constant strain rate and compute the exact stresses from the analytic solution of Section 3. All tests are conducted with the material parameters v = 0.49 and G/c = 100, but the results are independent of these quantities and are thus valid for all values of v and G/c.

The maximum and average relative errors, expressed as percentages, for the Euler, modified Euler and Runge-Kutta single step schemes are shown in Table I. When used without the stress correction, the average and maximum relative errors for the Euler scheme are, respectively, 9.91 and 46-2 per cent. As expected, the use of the higher order modified Euler and Runge-Kutta methods reduces these errors substantially, with the latter yielding average and maximum relative

Table I. Errors for integration of Tresca criterion with various single step schemes (elastic stress increments in plane Z=O)

Maximum error Average error Emax Eaw

Method Order (%.) (”/)

Euler 1 46.2 9.9 1 (1 3.5) (3.40)

Modified Euler 2 13.7 2.32 (7.83) (1.96)

Runge-Kutta 4 0.94 0.13 (0.50) (0.07)

Notes: Tests conducted with initial stress state of pure shear Values denoted by ( ) indicate errors in stresses after correction normal to yield surface Em,, and E,,, defined by equations (51) and (52)


errors of only 0.13 and 0.94 per cent. In all cases, scaling the stresses back to the yield surface improves the accuracy of the integration process. This improvement is most marked for the Euler scheme, where the maximum relative error of 13.5 per cent is a reduction of some 70 per cent on the uncorrected values. Error contours for the Euler method without the stress correction are shown in Figure 4. Not surprisingly, the largest errors occur when the direction of the elastic trial stress increment is tangent to the yield surface. Similar contours for the corrected Euler scheme, shown in Figure 5, display markedly different characteristics. This procedure clearly performs at its worst when the direction of the trial stress increment is inclined at roughly 45" to the normal of the yield surface.

The semi-analytic method, described in Section 4.1, was implemented using the secant iteration scheme. By choosing a suitable convergence tolerance when solving the non-linear equation in AL, the resulting stresses can be made to satisfy the yield criterion with high precision and no stress correction is necessary. (Our implementation assumes convergence once the percentage

1

0.5 A l e -- C

0 0 0.5

- A x e C

1: 5.00 2: 10.00 3: 15.00 4: 20.00 5: 25.00 6: 30.00 7: 35.00 8 : 40.00 3: 15.00

1

Figure 4. Error contours for single step Euler integration of Tresca material (no stress correction, elastic trial stress increments in plane Z = 0)


1

a a

f: t.80 1: 2.50 3: 5.04 I: 7.50 5: 10,OO 6: 12.50

0.5 1

Figure 5. Error contours for single step Euier integration of' Tresca material (stress correction, elastic trial stress increments in plane Z = 0)

change in A1 is less than between successive iterations.) Although there is no reason to assume that the stresses obtained from this scheme are less accurate than those obtained from the analytic constant strain rate scheme, we suppose the latter to be exact for the purposes of comparison. The error contours for the semi-analytic method, shown in Figure 6, indicate that the maximum relative error of 4-67 per cent occurs when the direction of the elastic trial stress increment is tangent to the yield surface. For many of the stress paths, however, the error is quite small and averages only 049 per cent. The secant scheme proved to be very efficient in solving the governing non-linear equation and, even with our stringent convergence tolerance, required a maximum of only 6 iterations.

The results for various numerical integration schemes, which assume a constant strain rate and use substepping with error control, are shown in Table11. The Euler, modified Euler and Runge-Kutta formulae were implemented with Richardson extrapolation and the resulting relative errors are normalized with respect to the error tolerance 6. Noting that the maximum

Tabl

e 11

. R

esul

ts fo

r in

tegr

atio

n of

Tre

sca

crite

rion

with

var

ious

sub

step

ping

sch

emes

(ela

stic

stre

ss in

crem

ents

in p

lane

z=o

) M

axim

um n

umbe

r of

ste

ps

Rel

ativ

e C

PU ti

me

Em*x

/6

E,V,

/6 M

etho

d O

rder

6=

10-3

6=

10-J

6=

.10-

3 6=

10-4

6=

10-’

6=

10-4

6=

:10-

3 6

=w

4 6

=iO

-’

Eule

r with

1

0.99

1 .o

o 1 .o

o 0.

66

0.67

0.

67

445

4445

44

445

10.0

68

0 47

9 su

bste

ps o

f (0

.81)

(0

.90)

(0

.90)

(0

36)

(0.3

8)

(0.3

8)

equa

l siz

e

Mod

ified

Eul

er

2 1.

13

1.21

1.

26

0.53

0.

64

0.72

12

37

11

5 1.

46

2.14

3.

86

with

sub

step

s (1

.19)

(1

.23)

(1

.26)

(0

.54)

(0

.64)

(0

.72)

of

equ

al s

ize

Run

ge-K

utta

4

0.70

1.

28

1.42

0.

09

024

037

2 4

6 1.

41

1.19

1.

09

with

sub

step

s (0

.35)

(0

74)

(0.8

8)

(0.0

5)

(0.1

5)

(0.2

5)

of e

qual

siz

e

Mod

ified

Eul

er

with

sub

step

s of

varia

ble

size

R

unge

-Kut

ta-

Engl

and

with

su

bste

ps o

f va

riabl

e siz

e

Run

ge-K

utta

- Fe

hlbe

rg w

ith

subs

teps

of

varia

ble

size

Run

ge-K

utta

- D

orm

and-

Prin

ce

with

sub

step

s of

var

iabl

e si

ze

1-2

0.20

(0

19)

4-5

0.51

(0

.26)

4-5

090

(077

)

4-5

033

(016

)

020

0.20

0.

10

(0.2

0)

(0.2

0)

(010

)

0.46

0.

54

0.12

(0

19)

(0.1

6)

(0.0

5)

1.96

1.

78

011

(1.9

4)

(1.1

2)

(007

)

0.32

0.

32

0.04

(0

.17)

(0

.19)

(0

03)

0.11

01

1 26

77

23

9 2.

72

5.30

11

.26

(011

) (0

.11)

0.14

01

5 3

4 6

1.39

1.

36

1.30

(0

06)

(006

)

0.34

0.

40

1 3

5 1.

00

1 .oo

1-00

(0

.21)

(0

.29)

006

007

3 4

5 1.

17

1.19

1.

14

(0.0

4)

(0.0

5)

Not

es:

Test

s co

nduc

ted

with

initi

al s

tress

sta

te o

f pur

e sh

ear

Val

ues d

enot

ed b

y ( )

indi

cate

err

ors

in s

tress

es af

ter

corr

ectio

n no

rmal

to

yiel

d su

rfac

e Em,, an

d E,

,, de

fined

by

equa

tions

(51)

and

(52)


1

nye 0.5 C

0

I: 0.io 2: 0.50 1: 1.00 I : 1.50 5: 2.00 6: 2.50 7: 5.00 8: 3.50 9: 1.00

Figure 6. Error contours for semi-analytic integration of Tresca material ( elastic trial stress increments in plane 2 = 0)

values for these relative errors should be equal to unity, all three algorithms clearly perform well without the use of stress correction. The first order Euler scheme, however, requires an excessive number of substeps for small error tolerances. The control of the average relative error with each of the schemes is also good although, as expected, this tends to be more conservative than the control of the maximum relative error. For paths where the strain increment vectors are of small magnitude, the high order methods inevitably yield additional accuracy when they are used with permissive error tolerances. Generally speaking, the accuracy of the Richardson extrapolation substepping schemes is further improved by correcting the stresses back to the yield surface. A notable exception to this rule, however, is the modified Euler method whose accuracy is largely unaffected by stress correction. Also shown in Table I1 are results for various schemes which permit the size of each substep to vary throughout the integration process. All of the procedures described in Section 5 are considered and the relative errors are again normalized with respect to the error tolerance 6. The error control mechanisms for the modified Euler, Runge-Kutta-England, Runge-Kutta-Fehlberg and Runge-Kutta-Dormand-Prince schemes


are clearly effective, although their maximum relative errors tend to be more conservative than those measured for the Richardson extrapolation schemes. A notable feature of the results is the small average error produced by the fourth and fifth order pair of Dormand and Prince." This supports the claim that their fifth order formula for updating the stresses is particularly accurate. With the exception of the modified Euler method, the accuracy of each of the adaptive integration schemes is again enhanced by scaling the stresses back to the yield surface. Considering all of the substepping integration schemes, the relative CPU times shown in Table I1 suggest that the most efficient algorithms are the R unge-Kut t a-Fehlber g met hod, the Runge-Kut t a-Dormand-Prince method and the classical Runge-Kutta method with Richardson extrapolation. Although it is not based on the same assumption of a constant strain rate, the semi-analytic algorithm of Section 4.1 also proved to be very efficient. Results are not shown for this scheme, but timing runs indicate that it is typically twice as fast as the best of the substepping schemes for error tolerances in the range i O - 3 - i O - 5 .

1.2. Mohr-Coulomb yield criterion

In the first of two different tests for the Mohr-Coulomb criterion, we again generate a fan of elastic trial stress increments in the plane Z = 0. The initial stresses are set to a state of pure shear with a,, = ayy = 0 and aXy = c cos 4. In all tests for the Mohr-Coulomb constitutive law, material parameters of v = 0.3, G/c = 100 and 4 = 30" are employed. Note that the results presented are valid for all values of G/c once v and 4 are fixed. Since no analytic solution is available for integrating the Mohr-Coulomb relations exactly, a constant strain rate is assumed and the exact stresses are computed by using the classical Runge-Kutta scheme with 100 substeps of equal size.

Under these conditions, the average and maximum relative errors for the Euler, modified Euler and Runge-Kutta single step methods are shown in TableIII. As expected, the average and maximum errors are greatest with the first order Euler scheme (8.94 and 39.4 per cent) and lowest with the fourth order Runge-Kutta scheme (0.04 and 0-32 per cent). In contrast to the results for the Tresca condition, scaling the stresses back to the yield surface is beneficial only in the case of Euler's method where the maximum relative error is reduced by roughly 50 per cent. Error

Table 111. Errors for integration of Mohr-Coulomb criterion with various single step schemes (elastic stress increments in

plane Z=O)

Maximum error Average error Em,, E w e

Method Order (%) (Yo)

Euler 1 39.4 8.94 (215) (4.11)

Modified Euler 2 11.7 1.75 (12.1) (1.76)

Runge-Kutta 4 032 0.04 (034) (0.04)



contours for this scheme, without and with stress correction, are respectively shown in Figures 7 and 8. In both cases the maximum error for a single Euler step occurs when the direction of the elastic trial stress increment is tangent to the yield surface.

Error contours for the semi-analytic method, shown in Figure 9, indicate that the integration error for this technique is again largest when the elastic trial stress path is tangent to the yield surface. The average and maximum relative errors, although not shown, were measured to be 0.48 and 3-58 per cent respectively and are thus similar in magnitude to those for the Tresca criterion. For all the paths considered, the secant iteration scheme required a maximum of 6 iterations to solve the governing non-linear equation in AA.

Results for various substepping schemes, with and without stress correction, are shown in Table IV. Since these algorithms incorporate error control, their average and maximum relative errors are again normalized with respect to the error tolerance 6. The Euler, modified Euler and Runge-Kutta methods, when used with Richardson extrapolation to determine a number of

1

0.5 CCOSQ

0 0

\ 7 \

1

0.5

& CCOSQ

1

1: 1.00 2: 5.00 3: 10.00 I : 15.00 5: 20.00 6: 25.00 I : 50.00 I: 35.00

Figure 7. Error contours for single step Euler integration of Mohr-Coulomb material (no stress correction, elastic trial stress increments in plane Z = 0)

1

0.5 ccosg

C 0


\

0.5

& ccosg

1

1: 2.50 2: 5.00 5 : 7.50 I : 10.00 5: 12.50 6: 15.00 7: 17.50 I: 20.00

Figure 8. Error contours for single step Euler integration of Mohr-Colomb material (stress correction, elastic trial stress increments in plane Z = 0)

substeps of fixed size, control the error in the stresses with good accuracy. Except for the Runge-Kutta scheme with the largest error tolerance, all of the normalized maximum errors are sufficiently close to the desired value of unity. Owing to their additional accuracy when integrating over ‘short’ strain paths, the higher order methods give average errors which are substantially less than unity. Also shown in Table IV are results for the modified Euler, Runge-Kutta-England, Runge-Kutta-Fehlberg and Runge-Kutta-Dormand-Prince schemes which permit the size of each substep to vary. The error control for these methods is again impressive, although they tend to be more conservative than the Richardson extrapolation algorithms in their choice of step size. Comparing the relative CPU times shown in Table IV, we see that the Runge-Kutta-Fehlberg and Runge-Kutta-Dormand-Prince substepping algorithms are again the most efficient methods for integrating with a variable step size. The fourth order Runge-Kutta procedure with Richardson extrapolation is also competitive with these two methods, although it may be expensive when used with a large error tolerance. Timing results for

Tabl

e IV

. R

esul

ts fo

r in

tegr

atio

n of

Moh

r-C

oulo

mb

crite

rion

with

var

ious

sub

step

ping

sch

emes

(ela

stic

stre

ss in

crem

ents

in p

lane

2 =

0)

Max

imum

num

ber

of s

teps

R

elat

ive

CP

U ti

me

Em

ax

/6

Ea

veI6

M

etho

d 6=

;10-

3 6=

10-4

6=

10-5

6=

10-3

6=

10-4

6=

:10-

5 6=

10-3

6=

10-4

6

=1

0-~

6

=1

0-~

6

=w

4 6

=1

0-~

Eule

r with

1.

11

1.12

1.1

2 0.

71

0.73

0.

74

425

4243

42

425

9.30

83

.0

526

subs

teps

of

(0.7

2)

(0.7

8)

(0.8

3)

(042

) (0

.44)

(0

.44)

eq

ual s

ize

Mod

ified

Eul

er

0.90

08

4 0.8

8 0.

40

05

1

0.59

12

36

11

2 1.

34

252

4.02

w

ith s

ubst

eps

(0.8

9)

(084

) (0

87)

(0.4

0)

(0.5

1)

(0.5

9)

of e

qual

siz

e

Run

ge-K

utta

0.2

2 1.

01

1.16

00

3 0.1

6 0.3

3 2

3 5

1.42

1 4

.4

1.16

w

ith s

ubst

eps

(0.2

3)

(1.0

3)

(1.1

8)

(003

) (0

.16)

(0

.32)

of

equa

l siz

e

Mod

ified

Eul

er

with

sub

step

s of

varia

ble

size

R

unge

-Kut

ta-

Engl

and

with

su

bste

ps o

f va

riabl

e siz

e R

unge

Kut

ta-

Fehl

berg

with

su

bste

ps o

f va

riabl

e siz

e R

unge

-Kut

ta-

Dor

man

d-Pr

ince

w

ith s

ubst

eps

of va

riabl

e siz

e

0.15

0.

14

014

(0.1

5)

(0.1

4)

(0.1

4)

030

027

0.25

(0.1

9)

(0.1

4)

(0.1

3)

0.38

2.0

7 1.

41

(0.3

8)

(2.1

0)

(1.4

1)

0.15

0.

26

0.36

(0

17)

(025

) (0

.32)

006

0.06

(0

06)

(006

)

0.05

00

6 (0

03)

(0.0

4)

004

0.28

(0

03)

(025

)

002

0.05

(0

.02)

(0

.05)

0.06

26

78

24

3 2.

72

6.89

13

.1

(0.0

6)

0.07

3

5 6

1.11

1.

52

1.42

(0

.05)

0.39

1 3

5 1.0

0 1-

00

1.00

(0

.36)

0.07

3

4 6

1.02

1.39

1.25

(0.0

7)

Not

es:

Test

s con

duct

ed w

ith in

itial

stre

ss st

ate o

f pur

e sh

ear

Val

ues d

enot

ed b

y ( )

indi

cate

erro

rs in

str

esse

s afte

r co

rrec

tion

norm

al to

yiel

d su

rfac

e Em

,, an

d E,

,, de

fined

by

equa

tions

(51)

and

(52)


I

-!f-. 0.5 CCOS@

0

- .. .......

....... ...... ........ .............

....

........

1: 0.10 2: 0.50 3: 1.00 I: 1.50 5: 2.00 6: 2.50 7: 3.00

Figure 9. Error contours for semi-analytic integration of Mohr-Coulomb material (elastic trial stress increments in plane z = 0)

the semi-analytic scheme are not shown, but these indicate that this is the most efficient of all the techniques since it is typically 20 per cent faster than the Runge-Kutta-Fehlberg algorithm.

In the second type of test for the Mohr-Coulomb criterion, we generate a fan of elastic trial stress increments in the tangent plane Y + 2 sin (b - c cos (b = 0 with initial stresses of a,., = a,,,, = 0 and a,.,, = c cos (b.

The average and maximum relative errors for the Euler, modified Euler and Runge-Kutta single step methods are shown in TableV. When used without the stress correction, Euler’s scheme gives a maximum error of 59 per cent and an average error of 9 per cent. Scaling the stresses back to the yield surface is clearly worthwhile in this case as it reduces both of these errors by roughly 45 per cent. Error contours for the single step Euler method, without and with stress correction, are shown in Figures 10 and 11 respectively. From the results of Table V, we again see

192 S . W. SLOAN AND J. R. BOOKER

Table V. Errors for integration of Mohr-Coulomb criterion with various single step schemes (elastic stress increments in

plane Y + Z sin4 -c cos 4=0)

Method

Maximum error Average error

Euler 1 59.0 9.00 (32.3) (4.99)

Modified Euler 2 19.2 2.04 (20.0) (2.07)

Runge-Kutta 4 0.3 1 0.03 (0.25) (0.03)


that the accuracy of the higher order modified Euler and Runge-Kutta schemes is largely unaffected by the use of a stress correction.

Figure 12 shows a plot of the errors for the semi-analytic scheme. The average and maximum relative errors for this procedure were measured to be 0.75 and 6-79 per cent respectively. Even though the semi-analytic method assumes the strains to vary linearly with I , instead of linearly with time, the small average error indicates that there is little to choose between these two assumptions. In order to solve the governing non-linear equation in A i , the secant iteration scheme again required a maximum of only 6 iterations.

1

C C O S @

0

.............................

........ ...................... .............................. .............. ................. *-.*.I.. ............ ................

-1 0 1

1: 1,00 2: 5.00 3: 10.06 I : 20.00 5: 50.00 6: 40.00 7: 50.00 8 : 5 5 . 0 0

Figure 10. Error contours for single step Euler integration of Mohr-Coulomb material (no stress correction, elastic trial stress increments in plane Y + 2 sin g5 - c cos g5 = 0)


1

ccosl$

0 -1

1: 1.00 2: 5.00 3: 10.00 I: 15.00 5: 20.00 I: 25.00 7: 30.00

0 1

ccoso

Figure 11 . Error contours for single step Euler integration of Mohr-Coulomb material (stress correction, elastic trial stress increments in plane Y + Z sin # - c cos # = 0)

aX" ccosl$

-1 0 1

1: 0.10 2: 0.50 3: 1.00 I : 2.00

6: j.00 7: 5.00 8 : 6.00

Figure 12. Error contours for semi-analytic integration of Mohr-Coulomb material (elastic trial stress increments in plane Y + 2 sin4 - c cos4 = 0)

Results for the Euler, modified Euler and Runge-Kutta schemes, with fixed size substeps obtained by Richardson extrapolation, are shown in Table VI. Although their normalized maximum relative errors are mostly greater than the desired value of unity, the error control of all these methods is, on average, certainly adequate for engineering computations. Also shown in

Tab

le V

I. R

esul

ts fo

r in

tegr

atio

n of

Moh

r-C

oulo

mb

crite

rion

with

var

ious

sub

step

ping

sch

emes

(ela

stic

stre

ss in

crem

ents

in p

lane

Y+

Z si

n 4 -

cos 6, =

0)

Met

hod

Eule

r w

ith

subs

teps

of

equa

l siz

e

Mod

ified

Eul

er

with

sub

step

s of

equa

l siz

e R

unge

-Kut

ta

with

sub

step

s of

equa

l siz

e

Mod

ified

Eul

er

with

sub

step

s of

var

iabl

e siz

e

Run

ge-K

utta

- En

glan

d w

ith

subs

teps

of

varia

ble

size

Run

ge-K

utta

- Fe

hlbe

rg w

ith

subs

teps

of

varia

ble

size

Run

ge-K

utta

- D

orm

and-

Prin

ce

with

ste

ps

of v

aria

ble

size

6=10

-3

2.61

(1

.26)

1.04

(1

.04)

0.23

(0

.27)

0.2 1

(0

.21)

0.94

(0

.81)

056

(0.5

6)

0.18

(0

.19)

Em

ax/d

6=

10-

4 ._

__

__

_

2.74

(1

.30)

1.42

(1

.42)

1.15

(1

.21)

020

(020

)

0.57

(0

48)

1.75

(1

.69)

0.24

(0

.23)

~-

2.76

(1

.31)

1.42

(1

.42)

1.73

(1

.91)

0.98

1.

04

1.05

50

8 50

76

5075

7 89

0 65

.8

(0.4

6)

(0.4

9)

(049

)

0.40

0.55

0.66

15

46

14

4 1.

35

2.3 1

(0

.40)

(0

55)

(066

)

0.02

0.

14

0.3 1

2

3 5

1.42

1.

29

(0.0

2)

(0.1

4)

(0.3

2)

020

(0.2

0)

0.53

(0

41)

1.94

(1

.95)

0.18

(0

.19)

0.07

(0

.07)

0.07

(0

.05)

0.04

(0

.04)

0.02

(0

.02)

0.07

00

7 28

86

27

0 2-

63

6.2 1

(0

07)

(0.0

7)

010

0.10

3

5 7

1.05

1.

3 1

(0.0

6)

(0.0

7)

0.16

0.

32

1 3

5 1.

00

1 .oo

(0.1

4)

(0.2

6)

0.03

0.

04

3 5

6 1.

07

1.28

(0

.03)

(0

.04)

468

415

1.14

13.4

1.36

1 .oo

1.29

Not

es:

Test

s co

nduc

ted

with

initi

al s

tress

sta

te o

f pu

re s

hear

V

alue

s den

oted

by

( ) i

ndic

ate

erro

rs in

stre

sses

afte

r cor

rect

ion

norm

al t

o yi

eld

surf

ace

Em,,

and

E,,,

defin

ed b

y eq

uatio

ns (51) a

nd (

52)


Table VI are results for the modified Euler, Runge-Kutta-England, Runge-Kutta-Fehlberg and Runge-Kutta-Dormand-Prince algorithms which permit the size of each substep to vary adaptively throughout the integration process. With the exception of the Runge-Kutta-Fehlberg method, these procedures are again more conservative than the Richardson extrapolation schemes in their control of error. As in previous tests, the Runge-Kutta-Dormand-Prince scheme is notable for its accurate and efficient error control. Comparing results for all of the methods shown in Table VI, it is apparent that the stress-strain relations are integrated efficiently by the Runge-Kutta-Fehlberg and Runge-Kutta-Dormand-Prince schemes. The Runge-Kutta algorithm with Richardson extrapolation is also competitive, although its efficiency is less good for large error tolerances. Timing runs for the semi-analytic scheme, which are not shown, indicate that this is again the fastest of all the procedures by a margin of roughly 20 per cent.

8. CONCLUSION

By assuming a constant strain rate throughout an increment, the Tresca stress-strain relations may be integrated exactly under conditions of plane strain. The analytic solution, described in Section 3, is simple to implement in a finite element code and ensures that both the yield criterion and flow rule are satisfied precisely, regardless of the size of the strain increments.

The semi-analytic schemes, described in Section 4, assume that the strains vary linearly with the plastic multiplier and require the solution of a single non-linear equation. This equation may be solved numerically, using a Newton-Raphson or secant algorithm, to give AA and the stresses can be made to lie arbitrarily close to the yield surface by choosing a suitably small convergence tolerance. Because this procedure assumes that the strains vary linearly with A, it is difficult to compare its results directly with those from the other techniques which all assume a constant strain rate. Nonetheless, it would appear to give good estimates of the stresses since, on average, these differ by less than 1 per cent from those obtained by the best constant strain rate schemes. The semi-analytic algorithm can be modified to deal with von Mises and Drucker-Prager yield criteria and can also be extended to incorporate isotropic hardening models. Neither of these generalizations has been pursued here, however. When implemented with a secant iteration scheme, the semi-analytic procedure proved to be an efficient means of integrating the Tresca and Mohr-Coulomb stress-strain relations. For the test cases considered, it was typically 20-50 per cent faster than the best of the numerical substepping schemes.

The various substepping schemes, described in Section 5, assume a constant strain rate during plastic flow and are based on mathematical techniques developed for integrating systems of ordinary differential equations. All of these methods attempt to control the relative error in the stresses by using an estimate of the local truncation error to divide the strain increment into substeps of appropriate size. Richardson extrapolation, which assumes the substeps to be of equal size, computes the number of substeps required by a single step and a double step calculation. Wissmann and who considered the integration of a von Mises material with isotropic hardening, concluded that Richardson extrapolation is most efficient when used with the classical fourth order Runge-Kutta scheme. Our results support this observation, although the Runge-Kutta algorithm is less efficient if the strains are small or the error tolerance is not stringent. In contrast to the techniques based on Richardson extrapolation, the adaptive substepping schemes permit the size of each substep to vary during the integration process. Our results suggest that the fourth and fifth order schemes of Fehlberg9 and Dormand and Prince," which both require six evaluations of the constitutive law per substep, are the most efficient methods for this type of integration. The modified Euler and Runge-Kutta-England methods, originally investigated by S l ~ a n , ~ are generally less competitive than either of these two


algorithms. Not surprisingly, the modified Euler scheme can become particularly inefficient if it is used with a small error tolerance. Our results for the Tresca and Mohr-Coulomb materials suggest that the error control of the Dormand-Prince method is more conservative than that of the Fehlberg method. Since it gives a substantial gain in accuracy with only a small increase in computational effort (roughly 20 per cent overall), the Dormand-Prince scheme is thus the preferred option for implementation in a finite element code. The choice of error tolerance for a substepping scheme is dependent upon the size of the load increments employed and the type of calculation undertaken. The authors have used a value of in collapse studies with good results, and this value is generally recommended.

ACKNOWLEDGEMENTS

Part of this work was funded by the Australian Research Council and the authors are grateful for this support.

REFERENCES

1. R. D. Krieg and D. B. Krieg, ‘Accuracies of numerical solution methods for the elastic perfectly plastic model’, J .

2. G. C. Nayak and 0. C. Zienkiewicz, ‘Elastoplastic stress analysis: A generalisation for various constitutive laws

3. D. R. J. Owen and E. Hinton, Finite Elements in Plasticity: Theory and Practice, Pineridge Press, Swansea, U.K., 1980. 4. D. M. Potts and A. Gens, ‘A critical assessment of methods of correcting for drift from the yield surface in elastoplastic

5. S. W. Sioan, ‘Substepping schemes for the numerical integration of elastoplastic stress-strain relations’, Int. j . numer.

6. A Gens and D. M. Potts, ‘Critical state models in computational geomechanics’, Eng. Computations, 5, 178-197

7. J. W. Wissmann and C. Hauck, ‘Efficient elastic-plastic finite element analysis with higher order stress-point

8. R. England, ‘Error estimates for Runge-Kutta type solutions to systems of ordinary differential equations’, Comp. J.,

9. E. Fehlberg, ‘Klassische Runge-Kutta Formeln Vierter und Niedrigerer Ordnung mit Schrittweiten-kontrolle und

10. J. R. Dormand and P. J. Prince, ‘A family of embedded Runge-Kutta formulae’, J. Comp. Appl. Math., 6,19-26 (1980).

Press. Vessel Technol. ASME, 99, 510-515 (1977).

including strain softening’, Int. j . numer. methods eng., 5, 113-135 (1972).

finite element analysis’, Int . j . numer. anal. methods geomech., 9, 149-159 (1985).

methods eng., 24, 893-911 (1987).

(1988).

algorithms’, Comp. Struct., 17, 89-95 (1983).

12, 166-170 (1969).

Ihre Anwendung auf Warmeleitungs-probleme’, Computing, 6, 61-71 (1970).

integration of tresca and mohr-coulomb constitutive …...mohr-coulomb solids to be considered in...

Documents