ORIGINAL ARTICLE

Mechanical and damage behaviour of mortar–rubberaggregates mixtures: experiments and simulations

M. Turki Æ I. Ben Naceur Æ M. Makni ÆJ. Rouis Æ K. Saı

Received: 10 July 2008 / Accepted: 3 November 2008

� RILEM 2008

Abstract The reuse of rubber wastes of worn tires

in aggregate form, to serve as a building material, is

appreciated to preserve environment. This study aims

to examine the mechanical behaviour of a mortar–

rubber aggregates material. A multi-phase model

called 2M2C (2 Mechanisms and 2 Criteria) which

take the volume fraction substitution of rubber into

account is investigated with the help of stress–strain

curves. The proposed model is based on the locali-

zation of the stress on the phases level (rubber and

mortar, respectively) and the homogenization of the

local plastic strains. The model has also incorporated

an isotropic damage variable to describe the loss of

compressive strength. The experimental tests are well

simulated by the model. Also, the simulations provide

local informations such as damage evolution and

local plastic strains.

Keywords Mortar � Rubber aggregates �Mechanical behaviour � Damage �Multi-mechanism model

1 Introduction

Waste management is one of the principal axes of

sustainable development in many countries in recent

years. It counts today among the main priorities with

regards to environmental protection. The conven-

tional storage of worn tires has an adverse effect on

the environment such: fire danger, rodents and

mosquitoes proliferation. Due to these problems,

several states recycle the worn tires in order to

introduce them into new products. In addition, the

rubber particles of worn tire reveal many qualities in

the mortar–rubber aggregate mixtures such as flexi-

bility and resistance [1–3].

Several studies have been interested in different

kind of waste valorization. This includes the work of

[4–7] who have used a lightweight aggregate concrete

as composite materials to study their mechanical

behaviour in order to combine the durability phe-

nomenon and microstructure. Kaufman et al. [8] have

demonstrated the improvement of the material duc-

tility containing Portland cement by addition of metal

fibres.

Other researches have focused on the waste of

worn tires valorization by the addition of rubber

M. Turki � J. Rouis

LGEMC, Ecole Nationale d’Ingenieurs de Sfax, BP 3038,

Sfax, Tunisia

I. B. Naceur � K. Saı (&)

LGPMM, Ecole Nationale d’Ingenieurs de Sfax, BP 3038,

Sfax, Tunisia

e-mail: kacemsai@yahoo.fr

M. Makni

Institut Superieur des Etudes Technologiques de Sfax,

Sfax, Tunisia

Materials and Structures

DOI 10.1617/s11527-008-9451-1

aggregates in mortar mixture. Benazzouk et al. [2]

and Siddique et al. [9] have studied the mechanical,

acoustic, thermal, and hydral behaviours of cemen-

titious materials by incorporating rubber aggregates.

These materials are also known to be used to design

durable structure in aggressive environments.

The use of mortar–rubber aggregates for special

building is a motivation to study their mechanical

behaviour. Rubberized mortar, encountered in several

civil engineering structures, would be used in area

games, subbases for highway pavements, highway

medians, sound barriers, and other transportation

structure. Turki et al. [10] have established link

between microstructure and mechanical properties in

order to improve the elastic characteristics of a

mortar–rubber aggregates mixtures thanks to a self-

consistent approach. Some analytical works related to

the stress–strain distribution within a single inclusion

(individual aggregate particle) are available [11–15].

A micro-mechanical modelling of a similar material

(thermoplastic polymer made of polypropylene and

rubber waste) has been proposed in the work of [16].

These authors have established constitutive equations

within a self consistent formalism. Recently, Topcu

and Saidemir [17] have used a neural network and

fuzzy logic to predict flexural and compressive

strength of waste rubberized aggregates mortar type.

A non exhaustive list of studies dealing with the

properties of rubberized concretes includes the works

of [18–22].

The main goal of the present paper is to provide a

model to simulate the mechanical behaviour of a

mortar–rubber aggregate composite which:

– takes the volume fraction substitution of rubber

aggregates into account,

– accounts for the damage behaviour of the mortar

phase,

– provides local information such as, the plastic

strains and stresses on the phases level.

Multi-phase model type is a good candidate to

describe this behaviour. To the authors’ knowledge,

the application of such a model for mortar–rubber

aggregates material has not yet been attempted.

This paper is organized in the following manner:

experimental data on mortar–rubber aggregates is

provided in Sect. 2. In that section, the general trends

observed on compressive tests are discussed. In

Sect. 3, the multi-mechanism model formulation

devoted to account for the material behaviour is

presented in details. Based on the experimental

results presented in Sect. 2, the identification of the

model parameters is shown in Sect. 4 followed by the

presentation of the numerical results. The local

contribution of stress–strain state at each phase level

is also commented in Sect. 4 in order to check if the

model is appropriate for the mortar–rubber

aggregates.

2 Experimental procedure

2.1 Investigated mortar–rubber aggregates

material

Rubber particles were introduced in the mortar

mixture by partial substitution of sand. The rubber-

free material has been taken as reference material.

The dry apparent bulk density of rubber is 1200 kg/

m3 and the dry apparent bulk density of the sand used

in the composites is 2650 kg/m3: the bulk density of

sand or rubber without intergranular void is measured

using an air pycnometer. The gap-grading analysis of

rubber-aggregates and sand shows that rubber parti-

cles has 1–4 mm size grading (the maximum grain

size is of 3.15 mm) and that the sand has 0–2 mm

size grading (Fig. 1). The used hydraulic binder is an

artificial Portland cement CPA CEM I 42.5 (EN 196-

1) with a bulk density of 3100 kg/m3. Water should

be added in different contents for each rubber

aggregates substitution at same volume. The inhibitor

factor for water of rubber aggregate (the water-to-

cement ratio W/C) is equal to 0.5 (Table 1).

2.2 Experimental tests

The static Young’s modulus (Es) is determined from

the stress–strain response of cylindrical samples

(50 mm 9 100 mm) in the uniaxial compression test

with rubber substitution by volume of 10%, 30% and

50% after curing for 28 days at 20�C and 98% of

relative humidity. The static modulus Es corresponds

to the initial slope of the stress–strain curves given in

Fig. 2. The effect of adding rubber on the ultra sonic

velocity in composite is established by applying a

longitudinal ultrasonic vibration (Tester type E0641):

the ultra sonic velocity is an indicator of sound and

vibration behaviour. Each value is the average of

Materials and Structures

three tests and can be calculated the dynamic

Young’s modulus (Ed) determined as follows:

Ed ¼ð1þ mÞð1� 2mÞð1� mÞ qCL

2 ð1Þ

where q and m denotes the dry bulk density and the

Poisson’s ratio, respectively. q, expressed in (kg/m3),

is given by the ratio (M/V). M and V denotes the mass

and the apparent volume of the specimen, respec-

tively. CL, expressed in (m/s), is the velocity of the

ultra-sonic wave calculated by: CL = DL/Dt. DL is

the distance between two sensors fixed on the

specimen and Dt is the elongation duration.

The Poisson’s ratio of cementitious materials is

usually assumed to be in the range of 0.2–0.3. The

dynamic Young’s modulus is then delimited by two

bounds:

0:74 qCL2 \ Ed \ 0:9 qCL

2 ð2ÞThe average value of dynamic modulus is almost

equal to 0.82 qCL2. The upper and the lower dynamic

modulus are shown together with the average

dynamic modulus and the static modulus in Table 2

and in Fig. 3. The same decrease is noted between the

static and the dynamic modulus. The scatter among

Es and Ed can be attributed to the unknown Poisson’s

ratio (for the dynamic modulus) and the small linear

domain (for the static modulus).

The stress–strain curves of cementitious mate-

rial containing rubber aggregates are shown in

Fig. 2. The reference material reveals an impor-

tant elastic phase. Progressively, when the rate of

substitution of rubber increases, a larger plastic

phase takes progressively place and reduces the

brittleness of the material. The addition of rubber

aggregates allows large deformation before failure

[23–26].

The variation of the dry bulk density, the velocity,

the different Young’s modulus with the rubber

substitution percentage are shown in Table 2. Fur-

thermore, the global volume of the void spaces

around rubber aggregates may affect the overall

Young’s modulus of mortar–rubber aggregates mix-

tures. The progressive reduction of the static and the

dynamic Young’s modulus, when the percentage of

0

10

20

30

40

50

60

70

80

90

100

0.01 0.1 1 10

% P

assi

ng

Sieve size (mm)

SandRubber aggregates

Fig. 1 Gap-grading

analysis of sand and rubber

aggregates

Table 1 Mass composition of prismatic specimen

(40 mm 9 40 mm 9 160 mm)

Substitution

percentage

Cement

(kg/m3)

Sand

(kg/m3)

Water

(kg/m3)

Rubber

aggregates

(kg/m3)

0 1,350 0

10 450 1,215 225 61.13

30 945 183.40

50 675 305.66

Materials and Structures

substitution of sand with the rubber aggregates rises,

is shown in Fig. 3.

3 Mechanical modelling of the material behaviour

The following notations are adopted for the descrip-

tion of the constitutive equations of the material

behaviour:

– x denotes a single scalar,

– x�

denotes a second order rank tensor,

– x��

denotes a fourth order rank tensor,

– _x is the derivative of x with respect to the time,

– x�

: y�

denotes the tensor product contracted over

one indices of x�

and y�

.

The proposed model belongs to a large class of

models known as multi-mechanism model type.

These models are intermediate between:

– the so-called unified models in which all the

deformation sources are considered together and

only their mean effect is taken into account

through a single inelastic strain,

– the micro-mechanical models in which physical

ingredients are represented to describe local

information.

The multi-mechanism model type was initiated by

Cailletaud and Saı [27] and improved later by Saı and

Cailletaud [28]. These authors have named their

models respectively 2M1C (2 Mechanisms and 1

Criterion) and 2M2C (2 Mechanisms and 2 Criteria).

These models are especially intended to describe the

behaviour of multi-phase materials [29, 30]. Since

mortar–rubber aggregates consist of mortar phase and

rubber phase, the 2M1C and 2M2C are good

candidates to describe their behaviour as a composite

material. The purpose of this paper is to modelize the

mechanical behaviour of the mortar–rubber aggre-

gates composite taking into account the influence of

the rubber volume fraction in mortar phase.

The 2M2C model is selected in this work to

preserve the opportunity to separate the two different

behaviours in the constitutive equations. The model is

developed in the thermodynamic frame of non-

associated plasticity with the Continuum Damage

Mechanics (CMD) theory. This framework allows to

0

2

4

6

8

10

12

14

16

18

20

0 1 2 3 4 5 6 7 8

Axi

al s

tres

s (M

Pa)

Axial deformation (mm/m)

Reference mortar

10 %

30 %

50 %

Fig. 2 Experimental

stress–strain curves for

different rubber aggregates

substitution

Table 2 Static and dynamic elastic Young’s modulus values

of mortar–rubber aggregates for different rubber substitution

Substitution percentage 0% 10% 30% 50%

Es (MPa) 22,500 9,000 4,500 2,100

Ed (MPa) 28,001 24,303 12,182 9,116

q (kg/m3) 2,086 2,010 1,746 1,651

CL (m/s) 4,046 3,840 2,917 2,595

Materials and Structures

take into account observations and experimental

results and avoid behaviour incompatibilities. Under

the assumption of infinitesimal elasto-plasticity, the

total strain can be decomposed into an elastic part and

a plastic part:

e�¼ e�

e þ e�

p ð3Þ

The plastic strain e�

p is considered here as the volume

average of the plastic strain of the rubber phase e�

r

and the plastic strain of the mortar phase e�

m.

e�

p ¼ z e�

r þ ð1� zÞ e�

m ð4Þ

where z denotes the volume fraction of the rubber

phase in the composite. It is well known that

rubber is an elastic material at small strains. From

a physical point of view it is abstract to treat

rubber particles as having plastic deformation.

However Chaboche [31] has demonstrated that it

is possible to establish a link between different

theories (for instance viscoplasticity and viscoelas-

ticity). This author has made possible bridges

between the two kind of theories. In our paper,

the viscoplasticity of the rubber phase is just an

‘‘artifice’’. Moreover, the viscoplastic behaviour of

the second mechanism is not the intrinsic behaviour

of the rubber phase. It is a behaviour obtain within

the composite. The interaction between elasticity

and plasticity is neglected, so that the free energy

can be split into two parts: the elastic free energy

qwe and the plastic free energy qwp. On the other

hand, the CDM theory supposes that the crack

initiation is preceded by a progressive internal

deterioration of the material (i.e. micro cracks,

micro defects) which induce a loss of compressive

strength. The isotropic damage evolution is quan-

tified by means of a macroscopic scalar variable D

varying between 0 and 1 [32].

The influence of damage is introduced into the

linear elasticity behaviour:

qwe ¼ 1

2ð1� DÞ e

�e : K��

: e�

e ð5Þ

where K��

is the fourth-order tensor of Hook’s

elasticity. The proposed model aims to consider the

individual behaviour of each phase. The list of

variables is reported in Table 3. In the plastic free

energy qwp, it is supposed that the state of coupling

between hardening variables is restricted to the

kinematic–kinematic coupling through the coefficient

Crm, which is known to produce the most interesting

0

5

10

15

20

25

30

35

0 10 20 30 40 50

You

ng’s

Mod

ulus

(G

Pa)

Rubber aggregates substitution (%)

Static Young’s modulusUpper dynamic Young’s modulus

Average dynamic Young’s modulusLower dynamic Young’s modulus

Fig. 3 Elastic Young’s

modulus curves of mortar–

rubber aggregates for

different rubber substitution

Materials and Structures

mechanical effect [27]. On the other hand, only the

mortar phase is affected by the damage. So that:

qwp ¼ 1

3Crr a�

r : a�

r þ 1

3Crm a

�r : a�

m þ 1

3Cmm a

�m : a

�m

þ 1

2QrðrrÞ2 þ 1

2ð1� DÞQmðrmÞ2

ð6Þ

where the material parameters Crr, Cmm and Crm

characterize the kinematic hardening whereas Qm and

Qr are isotropic material parameters. The overall

stress tensor and the two local stresses simply write:

r�¼ q

owo e�

r�

m ¼ qowo e�

m ¼ ð1� zÞ r�

r�

r ¼ qowo e�

r¼ z r

�

8>>>>>>>><

>>>>>>>>:

ð7Þ

This ‘‘multiplicative’’ transition rule (too drastic for

the local stress modification) to compute the local

stresses may be changed by a localization rule

inspired from the self consistent formalism:

r�

m ¼ r�þ A �

�p � �

�m

� �

r�

r ¼ r�þ A �

�p � �

�r

� �

8<

:ð8Þ

where A is an accommodation material parameter.

The local strains in Eq. 8 may be replaced by

phenomenological variables to better capture the

plastic accommodation:

r�

m ¼ r�þ A b

�� b�

m

� �

r�

r ¼ r�þ A b

�� b�

r

� �

b�¼ zb

�r þ 1� zð Þb

�m

8>>>>><

>>>>>:

ð9Þ

The rubber phase (soft phase) exhibits a plastic

deformation ��

r higher than the average deformation

��

p. Thus, the stress level r�

r supported by the rubber

phase will be less pronounced than the macroscopic

stress r�

. Inversely, the mortar phase (hard phase) will

be subjected to a higher stress level r�

m comparing to

r�

. Indeed, the plastic deformation ��

m is less than the

average deformation ��

p. The evolution laws of these

variables are generated by two plastic potentials Fr

and Fm, using the generalized normally rule:

Fm ¼ f m þ 3Dm

4Cmmð1� DÞX�m : X

�m

þ bm

2Qmð1� DÞðRmÞ2 þ 1

ð1� DÞbA

cþ 1

Y

A

� �cþ1

Fr ¼ f r þ 3Dr

4CrrX�

r : X�

r þ br

2QrðRrÞ2

8>>>>>>><

>>>>>>>:

ð10Þ

The formulation of these two plastic potentials is

obtained thanks to the multi-mechanism approach

[28] for the hardening and the CDM theory for the

damage [32]. Each local stress (r�

r and r�

m respec-

tively) is involved in a different yield function (fr and

fm respectively), defining two different criteria:

f m ¼ Jðr�

m � X�

mÞ � Rm � Rm0

f r ¼ Jðr�

r � X�

rÞ � Rr � Rr0

(

ð11Þ

where R0m and R0

r are the size of the initial elastic

domains. Rm and Rr describe their evolution. X�

m and

X�

r denotes respectively the center of the elastic

domain of the mortar phase and the rubber phase.

Jðr�� X�Þ represents a distance associated with the

von Mises invariant in the stress space:

Jðr�� X�Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3

2ðS�� X�Þ : ðS

�� X�Þ

r

ð12Þ

where S�

and X�

denotes respectively the deviators

associated with r�

and X�

.

Table 3 List of the variables for the proposed model

State variables Associated forces

Observable

variables

Internal

variables

��

(total strain) r�

Overall stress

��

r r�

r Inelastic stress

��

m r�

m Inelastic stress

a�

r X�

r Kinematic hardening

a�

m X�

m Kinematic hardening

Qr Rr Isotropic hardening

Qm Rm Isotropic hardening

D Y Damage variable

Materials and Structures

The evolution of the state variables is then given

by the following expressions:

_km and _kr are the plastic multipliers obtained thanks

to the consistency conditions. n�

m and n�

r are the

normal tensor to the two elastic domains.

The evolution of the kinematic variables a�

and the

accommodation variable b�

is obtained as the result of

the competition between a linear term ( _e�

m or _e�

r) and

a fading memory term depending on the actual value

of X�

:

_a�

m ¼ � _kmoFm

oX�

m ¼ _��

m � 3Dm

2CmmX�

m _km

_a�

r ¼ � _kr oFr

oX�

r ¼ _��

r � 3Dr

2CrrX�

r _kr

_b�

m ¼ _���m � dm b

�m _km

_b�

r ¼ _��

r � dr b�

br _kr

8>>>>>>>>>>><

>>>>>>>>>>>:

ð14Þ

For the isotropic hardening variables, non linearities

are also given as fading memory terms:

_rm ¼ � _kmoFm

oRm¼ _km 1

ffiffiffiffiffiffiffiffiffiffiffiffi1� Dp � bmQm

� �

_rr ¼ � _kroFr

oRr¼ _krð1� brQrÞ

8>><

>>:

ð15Þ

Finally the isotropic ductile evolution is defined by

the damage [32] rate given by:

_D ¼ � _km oFm

oY¼ _km Y

A

� �c1

ð1� DÞbð16Þ

Table 4 summarizes the constitutive equations of the

proposed model in both 3D and 1D case. The 2M2C

model is implemented into the FE code Zebulon [33]

using a h—method solved by an implicit Newton

scheme for the local integration.

4 Results and discussions

In the previous section, Eqs. 3–15 display some

material coefficients that have to be determined. The

identification of these parameters is performed with

the help of Zset FE code [34] provided with an

optimizer routine [33]. The Young’s modulus E is

supposed to be known and equivalent to Es (see

Sect. 2). From the compressive tests (Fig. 2), the

apparent Young’s modulus are checked by calculat-

ing the initial slope of the stress–strain curve.

The parameters used for the simulations of the

numerical results (Table 5) are found by best fit to the

experimental compressive tests as given in Fig. 4.

These parameters are unique for all the rubber phase

level except the initial size of the elastic domain R0

and Young’s modulus E. Good agreement is encoun-

tered by both simulations and experimental curve for

the whole range of the rubber phase. Figure 5 shows

the evolution of the simulated damage variable D

with deformation. It can be seen that, for a fixed

strain level, the damage decreases when the percent-

age of rubber substitution increase. As a matter of

fact, brittleness is reduced by the addition of rubber

particles. This result is confirmed by the photographs

of each specimen (0%, 10%, 30%, 50%) after

compressive tests.

On the other hand, the plastic strain is more

pronounced (as expected) for the ‘‘soft’’ rubber phase

than for the ‘‘hard’’ mortar phase (Fig. 6). This

difference is more important for the low rubber

substitution.

For higher percentage of rubber, the two plastic

strains become closer thanks to the coupling param-

eter Crm. Note that these two different behaviours are

depicted when the two phases interact in the aggre-

gates (they are not their ‘‘intrinsic’’ behaviours).

It is to be mentioned that this number of param-

eters is rather great if the purpose is just to simulate

the uniaxial stress–strain curves. In this work, it is

assumed that all parameters are activated even if

actually no relevant tests have been carried out in

order to address individually each coefficient. Some

of these parameters may be set to zero which is not

performed here. Additional experiments such as

_e�

m ¼ � _km of m

or�

m¼ _km n

�m with n

�m ¼ 3

2ffiffiffiffiffiffiffiffiffiffiffiffi1� Dp Sm � Xm

Jðr�

m � X�

mÞ

_e�

r ¼ � _kr of r

or�

r¼ _kr n

�r with n

�r ¼ 3

2

Sr � Xr

Jðr�

r � X�

rÞ

8>>><

>>>:

ð13Þ

Materials and Structures

Table 4 Summary of the constitutive equations of the proposed model

3D constitutive equations 1D constitutive equations

e�

p ¼ z e�

m þ ð1� zÞ e�

r �p = z �m ? (1-z)�r

r�¼ ð1� DÞ K

��: e�� e�

P� �

r = (1-D)E(�-�p)

r�

m ¼ r�þ A b

��b�

m

� �rm = r ? A(b-bm)

r�

r ¼ r�þ A b

�� b�

r

� �rr = r ? A(b-br)

b�¼ zb

�r þ ð1� zÞb

�m b = zbr ? (1-z)bm

X�

m ¼ 2

3Cmmð1� DÞa

�m þ Cmr a

�r

� �Xm = Cmm(1-D)a1 ? Cmra

r

X�

r ¼ 2

3Cmr a

�m þ Crr a

�r

� �Xr = Cmra

m ? Crrar

Rm = Q1(1-D)rm

Rr = Q2rr

Y ¼ 1

2ð e�� e�

pÞ : K��

: ð e�� e�

pÞ þ 1

3Cmmða�

mÞ : a�

m þ 1

2QmðrmÞ2 Y ¼ 1

2Eð�� �pÞ2 þ 1

2Cmmam2 þ 1

2QmðrmÞ2

f m ¼Jðr�

m � X�

mÞ � Rm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� DÞ

p � Rm0

f m ¼ jrm � Xmj � Rm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1� DÞ

p � Rm0

f r ¼ Jðr�

r � X�

rÞ � Rr � Rr0 f r ¼ jrr � Xrj � Rr � Rr

0

_e�

m ¼ _km n�

m _�m ¼ signðrm � XmÞ _km

_e�

r ¼ _kr n�

r _�r ¼ signðrr � XrÞ _kr

_a�

m ¼ _e�

m � 3Dm

2CmmX�

m _km _am ¼ _�m � Dm

CmmXm _km

_a�

r ¼ _e�

r � 3Dr

2CrrX�

r _kr _ar ¼ _�r � Dr

CrrXr _kr

_b�

m ¼ _e�

m � dm b�

m _km _bm ¼ _�m � dmbm _km

_b�

r ¼ _e�

r � dr b�

r _kr _br ¼ _�r � drbr _kr

_rm ¼ _km 1ffiffiffiffiffiffiffiffiffiffiffiffi1� Dp � bmQm

� �

_rr ¼ _krð1� brQrÞ

_D ¼ _km Y

A

� �c1

ð1� DÞb

Materials and Structures

loading unloading tests (under study) are then needed

to provide more consistent parameter set. Readers can

refer, for instance, to the ratchetting tests (necessary

to activate kinematic hardening parameters) in

Fig. 12 of the work of [2]. Moreover, the model

being developed in a three dimensional framework,

multi directional tests can be used to fix some

material parameters.

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Axi

al s

tres

s (M

Pa)

Axial deformation (%)

Reference mortar

10 %

30 %

50 %

Fig. 4 Experimental

(symbols) and modelling

results (dashed lines) of

stress–strain curves for

different rubber aggregates

substitution

Table 5 Identified parameters of the proposed model

Parameters Values Units

Elasticity z 0 0.1 0.3 0.5

E 22,500 9,000 4,500 2,100 MPa

m 0.3

Mortar

phase

km 90 MPa s

nm 2

R0m 11.0 3.5 2.4 1.1 MPa

Qm 31.4 MPa

bm 20.3

Cmm 1800 MPa

Dm 37

Rubber

phase

kr 90 MPa s

nr 2

Table 5 continued

Parameters Values Units

R0r 14 5.5 3.1 1.3 MPa

Qr 21.4 MPa

br 16.9

Crr 1400 MPa

Dr 51

Interaction A 25 MPa

Cmr 800 MPa

dm 5

dr 7

Damage B 0.0045

c 5.8

b 0.005

Materials and Structures

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80

Pla

stic

str

ain

(%)

Time (s)

Rubber phase

Mortar phase

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80

Pla

stic

str

ain

(%)

Time (s)

Rubber phase

Mortar phase

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 10 20 30 40 50 60 70 80

Pla

stic

str

ain

(%)

Time (s)

Rubber phase

Mortar phase

(a) (b)

(c)

Fig. 6 Evolution of the

plastic strain for the mortar

and rubber phases. a 10% of

rubber. b 30% of rubber.

c 50% of rubber

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Dam

age

Axial deformation (%)

Reference mortar 10 % 30 %

50 %

Fig. 5 Evolution of the

damage in compressive

tests

Materials and Structures

5 Conclusion

The purpose of this paper is to test the capabilities of

a multi-mechanism model to represent the mechan-

ical behaviour of the mortar–rubber aggregate

composites with different percentage of rubber par-

ticles (0%, 10%, 30% and 50%). The comparison

between experimental data and simulated results

shows good accordance. The local contributions of

the plastic strain at each level are then correctly

estimated. This model is implemented in the FE code

Zebulon. The constitutive equations of the model can

be used to analyze inelastic behaviour of structures

made of mortar–rubber aggregates in multi-direc-

tional framework. The next step of this work is to

validate the model for a successive loading-unloading

cycles tests with different percentage of substitution.

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