A Multisurface Anisotropic Model for Quasi-brittle Materials

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A Multisurface Anisotropic Model for

Quasi-brittle Materials

Paulo B. Lourenço1 and Jan G. Rots2

Abstract. A novel yield criterion capable of modelling the

softening behaviour of anisotropic materials under plane

stress conditions is presented. Individual yield criteria are

considered for tension and compression, according to two

different failure mechanisms. The former is associated with

a localised fracture process, denoted by cracking of the

material, and, the latter, is associated with a more

distributed fracture process which is usually termed as

crushing of the material. The model is capable of

reproducing independent (in the sense of completely

diverse) elastic and inelastic behaviour along a prescribed

set of material axes. The energy-based regularisation

technique resorts then to four different fracture energies

(two in tension and two in compression).

1 INTRODUCTION

The difficulties in accurately modelling the behaviour

of orthotropic materials are, usually, quite strong. This

is due, not only, to the fact that comprehensive

experimental results (including pre- and post-peak

behaviour) are generally lacking, but also to intrinsic

difficulties in the formulation of orthotropic inelastic

behaviour. It is noted that a representation of an

orthotropic yield surface solely in terms of principal

stresses is not possible. For plane stress situations,

which is the case here, a graphical representation interms of the full stress vector in a predefined set of

material axes (σ x, σ

y, τ

xy) is necessary.

In this article, the theory of plasticity is utilised to

combine anisotropic elastic behaviour with

anisotropic inelastic behaviour. Even if many

anisotropic plasticity models have been proposed from

purely theoretical and experimental standpoints, only

a few numerical implementations and calculations

have actually been carried out. An example is given in

[1] where the implementation of an elastic-perfectly-

plastic Hill criterion is fully treated. In principle,

hardening behaviour could be simulated with the

fraction model [2] but not much effort has been done

in this direction. More recent attempts are given in

[3], where linear tensorial hardening is included in the

Hill criterion, and [4], where linear hardening is

included in a modified (pressure dependent) Von

Mises to fit either the uniaxial tensile or the com-

pressive behaviour.While problems can be acute in the implementation

of isotropic plasticity models, they can become even

more pronounced for anisotropic models where

algebraic simplifications are hardly possible. The

present article represents, thus, a step further in the

formulation of anisotropic plasticity models.

Individual yield criteria are considered for tension and

compression, according to different failure

mechanisms, one in tension and the other in

compression. This represents an extension of

conventional formulations for isotropic quasi-brittle

materials to describe orthotropic behaviour. The

proposed yield surface combines the advantages of

modern plasticity concepts with a powerful

representation of anisotropic material behaviour,

which includes different hardening/softening

behaviour along each material axis. The behaviour of

the model is demonstrated by means of single element

tests and a comparison between numerical results and

experimental results for the case of masonry shear

walls. The comparison shows good agreement both

for ductile and brittle failure modes.

2 PROPOSED YIELD SURFACE

Different approaches for the conception of a yieldsurface can be used. One approach is to describe the

material behaviour with a single yield criterion. The

Hoffman criterion is quite flexible and attractive to

use, see e.g. [5], but yields a non-acceptable

representation of quasi-brittle materials, see [6], with

very poor fit of the experimental values. A single

1 Delft University of Technology, Faculty of Civil

Engineering, The Netherlands. Presently back at the

Department of Civil Engineering, School of Engineering,

University of Minho, Azurém, P-4800 Guimarães,

Portugal2 Delft University of Technology, Faculty of Civil Engi-

neering, The Netherlands. Also at TNO Building andConstruction Research, P.O. Box 49, 2600 AA Delft, The

Netherlands

© 1996 P.B. Lourenço and J.G. Rots

ECCOMAS 96.

Published in 1996 by John Wiley & Sons, Ltd.



Section Title 2 P.B. Lourenço and J. G. Rots

criterion fit of experimental data would lead to an

extremely complex yield surface with a mixed harden-

ing/softening rule in order to describe properly the

inelastic behaviour. It is believed that this approach is

practically non-feasible. Thus, a different approach

will be adopted. Formulations of isotropic quasi-

brittle materials behaviour consider, generally, differ-

ent inelastic criteria for tension and compression. Inthe present study, an extension of [7], where this

approach is utilised for concrete with a Rankine and a

Drucker-Prager criterion, will be presented. In order

to model orthotropic material behaviour, a Hill type

criterion for compression and a Rankine type criterion

for tension, see Fig. 1, are proposed. Note that the

word type is used here because the yield surfaces

adopted are close to the original yield criteria.

Nevertheless, they represent solely a fit of experimen-

tal results.

Figure 1. Proposed composite yield surface with iso-shear

stress lines. Different strength values for tension and

compression along each material axis

3 FORMULATION OF THE MODEL

The most relevant aspects of the proposed yield

criteria are given next. For a complete description of

the model the reader is referred to [6].

3.1 A Rankine type criterion

An adequate formulation of the Rankine criterion is

given by a single function, which is governed by the

first principal stress and one yield value σ t which

describes the softening behaviour of the material as,

see [7],

)(22

2

2

1 t t xy

y x y x f κ σ τ

σ σ σ σ −+

−+

+=

(1)

where the scalar κt controls the amount of softening.

The assumption of isotropic softening is not

completely valid for quasi-brittle materials such as

concrete or masonry which can be loaded up to the

tensile strength even if in the perpendicular direction

damage has already occurred. A solution for this

problem seems quite complex, see e.g. [8]. Therefore,

the scalar κt measures the amount of softening

simultaneously in the two material axes, even though

the model still incorporates two different fracture

energies.

The expression for the Rankine criterion, cf.eq. (1), can be rewritten as

2

2

1

2

))(())((

2

))(())((

xy

t t yt t x

t t yt t x f

τ κ σ σ κ σ σ

κ σ σ κ σ σ

+

−−−

+−+−

=

(2)

where coupling exists between the stress components

and the yield value. Setting forth a Rankine type

criterion for an orthotropic material, with different

yield values σ κtx t ( ) and σ κty t ( ) along the x, y directions is now straightforward if eq. (2) is modified

to

2

2

1

2

))(())((

2

))(())((

xy

t ty yt tx x

t ty yt tx x f

τ α κ σ σ κ σ σ

κ σ σ κ σ σ

+

−−−

+−+−

=

(3)

where the parameter α, which controls the shear stress

contribution to failure, reads

ατ

= f f

tx ty

u t ,

2

(4)

Here, f tx, f

ty and τ

u,t are, respectively, the uniaxial

tensile strengths in the x, y directions and the pure

shear strength. Note that the material axes are now

fixed with respect to a specific frame of reference and

it shall be assumed that all stresses and strains for the

elastoplastic algorithm are given in the material

reference axes.

Eq. (3) can be recast in a matrix form as

f T

t

T

11

21

2

12= +( { } [ ]{ }) { } { }ξ ξ π ξP (5)

where the projection matrix [Pt ] reads

−

−=

200

0

0

][P2

12

1

21

21

t

(6)




the projection vector {π} reads

{ } { }π = 1 0 0T

(7)

the reduced stress vector {ξ} reads

{ } { } { }ξ σ η= − (8)

and the back stress vector {η} reads

{η} = { ( ) ( ) }σ κ σ κtx t ty t T 0 (9)

Exponential tensile softening is considered for both

equivalent stress-equivalent strain diagrams, with

different fracture energies (G fx

and G fy

) for each yield

value, which read

−=

−= t

fy

ty

tytyt

fx

tx

txtx k G

f h f k

G

f h f expexp σ σ (10)

where the standard equivalent length h is related to theelement size.

A non-associated plastic potential g1

g T

g

T

11

21

2

12= +( { [ ]{ )ξ} ξ} {π} {ξ}P

(11)

is considered, where the projection matrix [P g ]

represents the original Rankine plastic flow, i.e. α = 1

in eq. (6).

The inelastic behaviour is described by a strain

softening hypothesis given by the maximum principal

plastic strain ε1 p.

as

κ ε ε ε ε ε γt

p x

p

y

p

x

p

y

p

xy

p= = + + − +⋅ ⋅ ⋅⋅⋅⋅ ⋅

11

22 2

2( ) ( ) (12)

which reduces to the particularly simple expression

κ λt t

=⋅ ⋅ (13)

The behaviour of the model in uniaxial tension along

the material axes is given in Fig. 2. The values chosen

for the material parameters illustrate the fact thatcompletely different behaviour along the two material

axes can be reproduced. In the first example, the ma-

terial strength in the y direction degrades at a faster

rate than the material strength in the x direction. The

second example yields isotropic softening, which

means that degradation of strength in the x and y

Example 1

Example 2

Example 3

Figure 2. Possible behaviour of the model along the

material axes for three different sets of material

parameters.




directions occurs at the same rate. Finally, the third

example yields elastic-perfectly-plastic behaviour in

the y direction while softening is allowed to occur in

the x direction.

3.2 A Hill type criterion

The simplest yield surface that features differentcompressive strengths along the material axes is a

rotated centred ellipsoid in the full plane stress space.

The expression for such a quadric can be written as

f cy c

cx c

x x y

cx c

cy c

y

xy cx c cy c

2

2 2

20

= + + +

− =

σ κ

σ κσ βσ σ

σ κ

σ κσ

γ τ σ κ σ κ

( )

( )

( )

( )

( ) ( )

(14)

where σ κcx c

( ) and σ κcy c

( ) are, respectively, the

yield values along the material axes x and y. The β

and γ values are additional material parameters thatdetermine the shape of the yield surface. The

parameter β controls the coupling between the normal

stress values, i.e. rotates the yield surface around the

shear stress axis, and must be obtained from one

additional experimental test, e.g. biaxial compression

with a unit ratio between principal stresses. The

parameter γ, which controls the shear stress

contribution to failure, can be obtained from

γτ

= f f

cx cy

u c,

2

(15)

where f cx, f cy and τu,c are, respectively, the uniaxialcompressive strengths in the x, y directions and a

fictitious pure shear in compression.

The proposed yield surface can be rewritten in a

matrix form as

f T

c c c21

2

12= −( {σ} σ})[ P ]{ σ κ( ) (16)

where the projection matrix [Pc] reads

=

γ κ σ

κ σ β

β κ σ

κ σ

200

0

)(

)(2

0)(

)(2

][P

ccy

ccx

ccx

ccy

c

(17)

the yield value σc is given by

σ κ σ κ σ κc c cx c cy c( ) ( ) ( )= (18)

and the scalar κc controls the amount of hardening and

softening.

The inelastic law adopted comprehends parabolic

hardening followed by parabolic/exponential

softening for both equivalent stress-equivalent strain

diagrams, with different compressive fracture energies

(G fcx

and G fcy

) along the material axes. The problem of

mesh objectivity of the analyses with strain softeningmaterials is a well debated issue, at least for tensile

behaviour, and the stress-strain diagram must be

adjusted according to an equivalent length h to

provide an objective energy dissipation. The inelastic

law features hardening, softening and a residual

plateau of ideally plastic behaviour. The peak strength

value is assumed to be reached simultaneously on

both materials axes, i.e. isotropic hardening, followed

by anisotropic softening as determined by the

different fracture energies. A residual strength value is

considered to avoid a cumbersome code (precluding

the case when the compressive mode falls completely

inside the tension mode) and to achieve a more robustcode (precluding degeneration of the yield surface to a

point). For practical reasons, it is assumed that all the

stress values for the inelastic law are determined from

the peak value.

An associated flow rule and a work-like

hardening/softening hypothesis are considered. This

yields

κσ

σ ε λc

c

T pc= =⋅ ⋅ ⋅1

{ } { } (19)

The behaviour of the model in uniaxial

compression along the material axes is given in Fig. 3.

Again, the values chosen for the material parametersillustrate the fact that completely different behaviour

along the two material axes can be reproduced. In the

first example, the material strength in the x direction

degrades at a faster rate than the material strength in

the y direction. The second example yields elastic-

perfectly-plastic behaviour in the y direction while

softening is allowed to occur in the x direction.




Example 1

Example 2

Figure 3. Possible behaviour of the model along the

material axes for two different sets of material parameters.

4 EXAMPLES

The performance of the anisotropic continuum modelis validated next by a comparison with experimental

results in hollow clay brick masonry shear walls [9].

These experiments are well suited for the validation of

the model because most of the parameters necessary

to characterise the model are available from biaxial

tests.

Fig. 4 shows the geometry of the walls, which

consist of a masonry panel of 3600 × 2000 × 150 mm3

and two flanges of 150 × 2000 × 600 mm3. Additional

boundary conditions are given by two concrete slabs

placed in the top and bottom of the specimen.

Initially, the wall is subjected to a vertical load p

uniformly distributed over the length of the wall. Thisis followed by the application of a horizontal force F

on the top slab causing a horizontal displacement d . A

regular mesh of 24 × 15 4-noded quadrilaterals is used

for the panel and 2 × 15 cross diagonal patches of 3-

noded triangles are used for each flange. The analyses

are carried out with indirect displacement control with

line searches, whereas the snap-backs are traced with

COD control over the most active crack. It is noted

that the self-weight of the wall and the top slab is also

considered in the analyses.

Figure 4. Geometry and loads for masonry walls.

Two walls from the experiments, denoted by W1 and

W2, are analysed with the composite plasticity model.

The properties of the composite material are obtained

from [10], see Table 1 to 3.

Table 1. Elastic properties

E x E

y ν

xy G

xy

2460 N/mm2 5460 N/mm2 0.18 1130 N/mm2

Table 2. Rankine material parameters (α = 1.73)

f tx f

ty G

fx G

fy

0.28 N/mm2 0.05 N/mm2 0.02 N/mm 0.02 N/mm

Table 3. Hill material parameters (β = -1.05, γ = 1.20)

f cx

f cy

G fcx

G fcy

1.87 N/mm2 7.61 N/mm2 5.0 N/mm 10.0 N/mm

The first wall analysed (W1) is subjected to an

initial vertical load p of 0.61 N/mm2 and shows a very

ductile response with tensile and shear failure along

the diagonal stepped cracks [9]. The comparison

between numerical and experimental load-

displacement diagrams, for wall W1, is given in

Fig. 5. Good agreement is found. The low initial

vertical load combined with the confinement provided

by the flanges and the top concrete slab yields anextremely ductile behaviour. The unloading found at d

equal to 2.0 mm is due to the mode I crack opening of

the left flange. The behaviour of the wall is depicted

in Fig. 6 in terms of the deformed mesh at ultimate

stage, where the centre node of the crossed diagonal

patch of the flanges is not shown in order to obtain a




more legible picture. Even at the ultimate stage the

stress values are considerably below the maximum

compressive strength in the vertical direction which

confirms the fact that failure is exclusively governed

by the tension regime.

Figure 5. Wall W1. Load - displacement diagrams.

Figure 6. Wall W1. Total deformed mesh at a displacement

of 12.0 mm.

The second wall analysed (W2), is subjected to aninitial vertical load p of 1.91 N/mm2 and, initially,

shows a relatively ductile behaviour, followed by

brittle failure with explosive behaviour due to

crushing of the compressed zone [9]. The comparison

between numerical and experimental load-

displacement diagrams is given in Fig. 7. From a

qualitative perspective good agreement is found

because the same trend is observed in both diagrams.

Remarkably, the explosive type of failure observed in

the experiments at a displacement of approximately

8.0 mm is also predicted by the analysis. Less good

agreement is found with respect to the calculated

collapse load value, which is 20 % higher than theexperimental value. Even if the sharp reproduction of

the collapse load value is not the main issue here, it is

likely that the difference can be explained by the

variation of the material properties in compression

between the biaxial tests and the wall. In [6], good

agreement is found for other tests even in the case of

failure due to masonry crushing.

Figure 7. Wall W2. Load - displacement diagrams.

Figure 8. Wall W2. Incremental deformed mesh at a

displacement of 8.0 mm.

The behaviour of the wall W2, depicted in Fig. 8 in

terms of the deformed mesh at ultimate stage, is quite

different from the behaviour of wall W1. Theexplosive type of failure due to crushing in the

compressed toes, which is also observed in the

experiments, has been traced with arc-length control

over the nodes in the bottom row of elements of the

panel.

5 MESH SENSITIVITY

A crucial point in the analysis of strain softening

materials with standard continuum is the sensitivity of

the results with respect to the mesh size. The fracture

energy based regularisation which has been adopted in

this study is widely used in engineering practice toovercome this problem. It suffices to incorporate an

equivalent length in the material model which is

related to the area of an element. Fig. 9 shows the

comparison between the results of the analysis for

wall W1 with the original mesh and a mesh refined by




a factor two. It is observed that, for practical purposes,

the results can be considered mesh

insensitive.

Figure 9. Wall W1. Mesh sensitivity analysis.

6 CONCLUSIONS

An anisotropic composite continuum model capable

of reproducing independent inelastic behaviour along

two orthogonal axes has been formulated. It is

assumed that two failure mechanisms can be

distinguished, one associated with localised fracture

processes and one associated with a more distributed

fracture process which can be termed crushing of the

material. Orthotropic elasticity is combined with

orthotropic softening plasticity in a frame of reference

associated to a set of material axes. The model

includes a Rankine type criterion for tension and a

Hill type criterion for compression, which is flexibleenough to accommodate the behaviour of quasi-brittle

materials.

A comparison between numerical results and

experimental results for masonry shear walls shows

the good performance of the model.

ACKNOWLEDGEMENTS

The calculations have been carried out with DIANA

finite element code of TNO Building and

Construction Research. The research is supported

financially by the Netherlands Technology

Foundation (STW) under grant DCT 33.3052.

REFERENCES

[1] R. de Borst and P.H. Feenstra, Studies in anisotropic

plasticity with reference to the Hill criterion, Int. J.

Numer. Methods Engrg., 29, p. 315-336, 1990.

[2] J.F. Besseling, A theory of elastic, plastic and creep

deformations of an initially isotropic material showing

anisotropic strain-hardening, creep recovery andsecondary creep, J. Appl. Mech., 22, p. 529-536, 1958.

[3] C.C. Swan and A.S. Cakmak, A hardening orthotropic

plasticity model for non-frictional composites: Rate

formulation and integration algorithm, Int. J. Numer.

Methods Engrg., 37, p. 839-860, 1994.

[4] X. Li, P.G. Duxbury and P. Lyons, Considerations for

the application and numerical implementation of strain

hardening with the Hoffman yield criterion, Comp.

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[5] J.C.J. Schellekens and R. de Borst, The use of the

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1096, 1990.

[6] P.B. Lourenço, Computational strategies for masonry

structures, Dissertation, Delft University of Technology,

Delft, The Netherlands, 1996.

[7] P.H. Feenstra and R. de Borst, A composite plasticity

model for concrete, Int. J. Solids Structures, 33(5),

p. 707-730, 1996.

[8] P.B. Lourenço, J.G. Rots and P.H. Feenstra, A 'tensile'

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[9] H.R. Ganz and B. Thürlimann, Tests on masonry walls

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[10] H.R. Ganz and B. Thürlimann, Tests on the biaxial

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A Multisurface Anisotropic Model for Quasi-brittle Materials