Acta Polytechnica CTU Proceedings


doi:10.14311/APP.2020.26.0056
Acta Polytechnica CTU Proceedings 26:56–63, 2020 © Czech Technical University in Prague, 2020

available online at https://ojs.cvut.cz/ojs/index.php/app

LOCALIZATION ANALYSIS OF AN ORTHOTROPIC
MULTI-SURFACE PLASTICITY MODEL UNDER UNIAXIAL

STRESS

Claudio Pagani∗, Milan Jirásek, Martin Horák

Czech Technical University in Prague, Faculty of Civil Engineering, Department of Mechanics, Thákurova 7,
166 29 Prague 6, Czech Republic

∗ corresponding author: claudio.pagani@fsv.cvut.cz

Abstract. Numerical simulations of masonry structures are often based on continuum macro-
modelling approaches that need constitutive laws able to phenomenologically reproduce the behavior
of the material. To capture the deformation process up to failure, appropriate softening laws are
needed to take into account the contraction of the yield stress domain caused by cracking and crushing.
It is well known that softening may lead to localization of inelastic strain. This paper focuses on
localization analysis of an orthotropic macro-scale model in the framework of multi-surface plasticity,
which describes the in-plane behavior of masonry structures. Preliminary results reported in this
short paper are limited to uniaxial stress states. Analytical localization conditions are first derived for
uniaxial stress states with principal axes aligned with the material axes of orthotropy. Then, localization
analysis is extended to an arbitrary angle between the principal stress axes and the axes of orthotropy.

Keywords: Multi-surface plasticity, localization, weak discontinuity, masonry.

1. Introduction
Masonry structures are characterized by two different
failure modes, the in-plane and out-of-plane mecha-
nisms. Regarding the study of in-plane mechanisms,
different strategies could be adopted for numerical sim-
ulations of the behavior of masonry structures. Among
others, continuum macro-modeling is a frequently used
approach, in which masonry structures are treated as
homogenized continua and, in general, discretized by
the finite element method. This kind of modelling
needs the choice of a constitutive model, which could
be formulated within the framework of nonlinear elas-
ticity, plasticity, damage mechanics, smeared crack
models, or their combinations, and its aim is to phe-
nomenologically reproduce the average behavior of
masonry.

This paper is focused on a study of an orthotropic
model in the framework of multisurface plasticity, with
the yield function described as the composition of two
different surfaces for the tensile and the compressive
behavior [1, 2]. Both failure surfaces evolve according
to hardening/softening laws that may lead to local-
ization of plastic strain. Conditions for the onset of
localization depend on many factors, such as the stress
state, the destabilizing effects of the non-associated
flow rule in plasticity, or, in this particular case, the
orthotropy of the elastic stress-strain law.
In this paper, the localization analysis of the pre-

sented macro-model is carried out under the condition
of uniaxial stress in tension and in compression. Con-
ditions for the onset of localization are derived and
preferential directions of the emerging weak disconti-
nuity are evaluated.

2. Model description
This paper deals with an advanced model for the
in-plane failure mechanisms of masonry structures
proposed by Lourenço [1, 2], which takes into account
the anisotropic structure of masonry and is formulated
in the framework of multisurface plasticity.

The analysis can be reduced to two dimensions when
plane-stress conditions are assumed. In the following
description, the Cartesian axes x and y are considered
aligned with the material axes, which, for regular
periodic masonry, are the horizontal and vertical ones
(bed and head mortar joints, respectively).

In order to avoid pathological mesh sensitivity of
the results to the element size, model parameters
that affect the stress-strain diagrams in tension and
compression are adjusted by means of the equivalent
length le. In the original paper, the crack band size
was related to the area of the elements:

le = αh
√
Ae (1)

where αh is 1 or
√

2 for quadratic and linear elements,
respectively. This rule, which has no convincing justifi-
cation, may induce a large error for elongated elements
and even for square elements since the crack band size
depends also on the orientation with respect to the
element sides. In this paper, the projection method
is considered, i.e., the effective crack band size is es-
timated by projecting the element onto the direction
perpendicular to the estimated crack band direction
[3].
The model requires in total 16 parameters, which

are reported in Table 1, and are divided into groups
governing orthotropic elasticity, tensile and compres-

56

https://doi.org/10.14311/APP.2020.26.0056
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vol. 26/2020 Localization analysis of an orthotropic multi-surface plasticity model

sive plastic behavior. The meaning of the symbols is
described in the following sections.

2.1. Orthotropic elasticity
The linear elastic stress-strain relation under plane-
stress conditions can be written as

σ = Deε (2)

where stress and strain components are stored in col-
umn matrices using the Voigt notation:

σ =

 σx
σy
σxy

 , ε =

 εx
εy
γxy

 (3)

The elastic stiffness matrix for an orthotropic material
reads

De = 1
ζ

 Ex νyxEx 0
νxyEy Ey 0

0 0 ζGxy

 (4)

where ζ = 1− νxyνyx is an auxiliary parameter.
The elastic behavior is characterized by four pa-

rameters: Ex, Ey, νxy, and Gxy. The other in-plane
Poisson’s ratio is determined from symmetry and is
given by νyx = νxyEyy/Exx.
The plane stress condition (σz = 0) leads to the

following expression for the out-of-plane normal strain:

εz = −1
ζ

[(νxz + νyzνxy)εx + (νxzνyx + νyz)εy] (5)

Hence, additional two Poisson’s ratios (νxz and νyz)
have to be known if one is interested in computing
the out-of-plane strain.

2.2. Plastic behavior in tension
An orthotropic Rankine type yield surface is consid-
ered for the tensile behavior, with orthotropic soft-
ening controlled by a single scalar quantity. The
corresponding yield function is defined as

ft(σ, κt) = (σx − σt,x(κt)) + (σy − σt,y(κt))
2 +

+

√(
(σx − σt,x(κt))− (σy − σt,y(κt))

2

)2
+ ατ2

xy

(6)

Degradation of the yield stress is governed by a single
scalar internal variable κt. The softening laws

σt,i(κt) = ft,i exp
(
− left,i
Gft,i

κt

)
, i ∈ {x, y} (7)

describe the dependence of the equivalent tensile yield
stresses in directions x and y in the form of exponential
softening and can be collected into the column matrix

qt =

 σt,x(κt)
σt,y(κt)

0

 (8)

Parameters ft,i and Gft,i, i ∈ {x, y}, are the tensile
strength and the fracture energy in tension correspond-
ing to the i-axis, respectively. To avoid snap-back at
the constitutive level, the value of the tensile strength
is reduced if the elements are too large, according to
the rule

ft,i =
√
Gft,iEi
le

if le >
Gft,iEi
f2
t,i

(9)

Parameter
α = ft,xft,y

τ2
u,t

(10)

controls the shear stress contribution to tensile failure,
where τu,t is the shear strength in tension.

The flow rule is non-associated and is derived from
the plastic potential

gt(σ, κt) = (σx − σt,x(κt)) + (σy − σt,y(κt))
2 +

+

√(
(σx − σt,x(κt))− (σy − σt,y(κt))

2

)2
+ τ2

xy

(11)

The rate of the softening parameter is considered
equal to the rate of the maximum principal plastic
strain:

κ̇t = ε̇P1 =
ε̇Px + ε̇Py

2 + 1
2

√(
ε̇Px − ε̇Py

)
+
(
γ̇Pxy
)2 (12)

It is assumed that the hardening regimes in ten-
sion and compression are uncoupled, i.e., each of the
surfaces is associated with its own hardening variable
that grows only if the surface is active. Under this
assumption, the expression of κ̇t reduces to

κ̇t = λ̇t (13)

where λt is the plastic multiplier associated with the
tensile yield surface.

In conclusion, there are five model parameters con-
trolling the tensile behavior: two strengths, ft,x and
ft,y, two fracture energies, Gft,x and Gft,y and the
parameter α that governs the coupling of shear with
the normal stresses.

2.3. Plastic behavior in compression
The adopted failure surface in compression corre-
sponds to a Hill-type yield function,

fc = σ2
x

σ2
c,x(κc)

+
βσxσy + γτ2

xy

σc,x(κc)σc,y(κc)
+

σ2
y

σ2
c,y(κc)

−1 (14)

where β is a parameter that controls the coupling
between the normal stresses. Parameter γ controls
the shear stress contribution to failure and is given by

γ = fc,xfc,y
τ2
u,c

(15)

57



C. Pagani, M. Jirásek, M. Horák Acta Polytechnica CTU Proceedings

where fm,x and fm,y are the values of uniaxial com-
pression strength in directions x and y, and τu,c is the
shear strength in compression.

Hardening and softening are controlled by a strain-
like internal variable κc, defined by the rate equation

κ̇c = 1√
σc,x(κc)σc,y(κc)

σT ε̇p (16)

which, under the assumption of uncoupled hardening
regimes in tension and compression, reduces to

κ̇c = λ̇c (17)

where λc is the plastic multiplier associated with the
compressive yield surface.
The values of current compressive yield stress in

directions x and y, σc,x and σc,y, are evaluated from
κc using the hardening-softening law (Figure 1)

σc,i =



σi,i + (σp,i − σi,i)
√

2κc
κp
− κ2

c

κ2
p

κc ≤ κp

σp,i + (σm,i − σp,i)
(

κc−κp
κm,i−κp

)2
κp < κc ≤ κm,i

σr,i + (σm,i − σr,i) e
2m(κc−κm)
κm−κp κm,i < κc

(18)

where parameter

m = σm,i − σp
σm,i − σr,i

(19)

corresponds to parabolic hardening followed by
parabolic and later exponential softening.

This law contains strain-like parameters κp, which
represents the value of κc at the peak, i.e., at transi-
tion from hardening to softening, and κm,x and κm,y,
which correspond to the transition from parabolic to
exponential softening, and also stress-like parameters
σi,x and σi,y, which are the initial values of yield stress
in directions x and y, σp,x and σp,y, which are the
peak values of yield stress, σm,x and σm,y, which are
the values of yield stress at transition from parabolic
to exponential softening, and σr,x and σr,y, which are
the values of residual yield stress.

Figure 1. Hardening/softening law for the compres-
sive yield stress

In absence of specific experimental tests, the char-
acteristic yield stress values can be set to

σp,i = fc,i σi,i = 1
3fc,i (20)

σm,i = 1
2fc,i σr,i = 1

10fc,i (21)

The value of the equivalent plastic strain at peak, κp,
is considered to be an additional parameter of the
model. The value of κm,i depends on the estimated
crack band width le (related to the finite element
mesh) and is given by

κm,i = 75
67
Gfc,i
lefc,i

+ κp (22)

where Gfc,i is the fracture energy in compression
corresponding to the i-axis. In order to avoid snap-
back at the constitutive level, the compressive strength
is reduced if the elements are too large, according to
the formula

fc,i =
√

75
67
Gfc,iEi
le

if κm,i <
fc,i
Ei

+ κp (23)

The values of current compressive yield stress in
directions x and y could be cast into the column
matrix

qc =

 σc,x(κc)
σc,y(κc)

0

 (24)

An associated flow rule for compression regime is
used, which means that the rate of plastic strain is
given by

ε̇p = λ̇c
∂fc(σ, κc)

∂σ
(25)

In summary, six model parameters are used for
description of the inelastic behavior in compression:
compressive strengths fc,x and fc,y, fracture energies
Gfc,x and Gfc,y, coupling parameter between normal
stresses in two directions, β, and the equivalent plastic
strain value corresponding to the peak, κp.

3. Localization analysis in
plasticity

Suppose that plastic loading takes place with only one
yield surface activated, and let i be the index of the
active surface ( i ∈ {t, c}).
In the following derivation, a comma denotes a

partial derivative with respect to the variable follow-
ing the comma and no sum over repeated indices is
considered.

The basic equations characterizing the flow theory
of plasticity include the elastic-plastic split,

ε = εe + εp (26)

the stress-strain law for the elastic part (2), the yield
condition

fi(σ, qi) = 0 (27)

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vol. 26/2020 Localization analysis of an orthotropic multi-surface plasticity model

the flow rule,
ε̇p = λ̇igi,σ (28)

the definition of hardening/softening variable in the
rate form,

κ̇i = λ̇i (29)
the dependence of the yield strength on the hardening
variable

qi = hi(κi) (30)
and the loading/unloading conditions

fi ≤ 0, λ̇i ≥ 0, λ̇ifi = 0 (31)

During plastic flow, the yield function must remain
equal to zero, and so the rate of its change is also zero.
This consideration leads to the consistency condition,

λ̇iḟi = 0 (32)

from which the elasto-plastic stiffness matrix can be
computed. The resulting formula reads

Dep = De −
De : gi,σ ⊗ fi,σ : De

fi,σ : De : gi,σ − fi,qi ·Hi
=

= De −
1
h
De : gi,σ ⊗ fi,σ : De (33)

where
h = fi,σ : De : gi,σ − fi,qi ·Hi (34)

and
Hi = qi,κi (35)

The condition of incipient weak discontinuity, as in-
spired by the early works of Hadamard [4] and Hill
[5] and developed, among others, for plasticity [6, 7]
and for damage [8, 9], is given by

(n ·Dep · n) ·m = 0 (36)

where n is the normal to the discontinuity surface,
and

Qep = n ·Dep · n (37)
is the elasto-plastic acoustic tensor. According to (36),
this tensor is singular at the onset of localization. The
eigenvector m corresponding to the zero eigenvalue is
called the polarization vector.

The elasto-plastic acoustic tensor can be expressed
as

Qep = n ·Dep · n =

= n ·De · n−
1
h
n ·De : gi,σ ⊗ fi,σ : De · n (38)

which can be recast in a simpler form

Qep = Qe −
1
h
b⊗ a (39)

where

Qe = n ·De · n (40)
a = n ·De : fi,σ (41)
b = n ·De : gi,σ (42)

After some calculations, by imposing singularity of
the elastoplastic acoustic tensor, the condition of an
incipient weak discontinuity (36) becomes

− fi,qi ·Hi = a ·Q−1
e · b− fi,σ : De : gi,σ (43)

Since we are in plane-stress conditions and considering
the plane of symmetry perpendicular to the z-axis,
one can restrict attention to normals n for which

n =

 n1
n2
0

 (44)

under the normalizing constraint n2
1 +n2

2 = 1. Consid-
ering major and minor symmetries of the orthotropic
elastic stiffness tensor and the plane-stress condition,
the quantities entering in (43) can be simplified.

Namely, the orthotropic linear elastic acoustic ten-
sor is given by

Qe =
[

D1111n
2
1 +D1212n

2
2 (D1122 +D1212)n1n2

(D1122 +D1212)n1n2 D1212n
2
1 +D2222n

2
2

]
(45)

where

D1111 = Exx
1− νxyνyx

(46)

D2222 = Eyy
1− νxyνyx

(47)

D1122 = νyxExx
1− νxyνyx

(48)

D1212 = Gxy (49)

Other important quantities are evaluated as

fi,σ : De : gi,σ = D1111f
11
i,σg

11
i,σ+

+D1122(f11
i,σg

22
i,σ + f22

i,σg
11
i,σ)+

+ 4D1212f
12
i,σg

12
i,σ +D2222f

22
i,σg

22
i,σ (50)

a =
[

(D1111f
11
i,σ +D1122f

22
i,σ)n1 + 2D1212f

12
i,σn2

2D1212f
12
i,σn1 + (D1122f

11
i,σ +D2222f

22
i,σ)n2

]
(51)

b =
[

(D1111g
11
i,σ +D1122g

22
i,σ)n1 + 2D1212g

12
i,σn2

2D1212g
12
i,σn1 + (D1122g

11
i,σ +D2222g

22
i,σ)n2

]
(52)

where f iji,σ and giji,σ are the ij components of the
derivative of the yielding function and the plastic
potential, respectively, with respect to stress tensor.

4. Localization analysis for
uniaxial stress

Let us consider the case of uniaxial stress (Figure 2)
whose direction is inclined by an angle φ with respect
to the x-axis that, together with the y-axis, forms the
pair of orthotropy axes. Let θ be the angle that the
generic normal n to the crack surface forms with the
material x-axis, so that it can be expressed as

n =

 cos θ
sin θ

0

 (53)

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C. Pagani, M. Jirásek, M. Horák Acta Polytechnica CTU Proceedings

The stress tensor matrix in the Oxy reference frame
corresponding to uniaxial tension of magnitude σ̄ and
rotated by a generic angle φ with respect to the x-axis
is given by

σ(σ̄, φ) = Qσ̂QT (54)
where

σ̂ =
[
σ̄ 0
0 0

]
Q =

[
cosφ − sinφ
sinφ cosφ

]
(55)

The value σ̄ should be the one that triggers local-
ization for the given stress angle φ, so it is the value
corresponding to the onset of plastic flow and is de-
termined by the condition fi = 0 where fi is yield
function given by (6) or (14) for tension or compres-
sion, respectively.
Subsequently, one can substitute the determined

value of σ̄ for a given φ into the right-hand side of
(43) and obtain an expression depending only on the
angle θ. The localization angle θloc is determined as
the one that maximizes the right-hand side of (43)
and the corresponding value is the critical hardening
modulus (Hcrit = H(θloc)). The polarization vector
m, the eigenvector of the elastoplastic acoustic tensor
corresponding to the zero eigenvalue, characterizes the
failure mode, ranging from tensile splitting (mode I)
with m = n to shear slip (mode II) with m perpen-
dicular to n.

Figure 2. Localization (n) and polarization (m) vec-
tors for uniaxial stress σ̄ inclined by φ with respect
the material x-axis

In the case of uniaxial tension and compression
along the x or y-axis, the expressions of the critical
hardening moduli can be found in a closed form. First,
it is useful to define the quantity ξ, common to all the
expressions:

ξ(θ) = −
(

sin4 θ

Exx
+ cos4 θ

Eyy
+

+Exx − 2Gxyνxy
ExxGxy

sin2 θ cos2 θ

)−1
(56)

t In the isotropic case, the previous expression reduces
to

ξ = −E (57)

where E is the Young modulus.
Subsequently, indicating by subscript t or c for

tensile or compressive uniaxial stress and by subscript
x or y its direction, the expressions of the critical
hardening moduli are given by

Ht,x(θ) = ξ(θ) sin4 θ (58)
Ht,y(θ) = ξ(θ) cos4 θ (59)

Hc,x(θ) = ξ(θ)
(
βµ

2 cos2 θ + 1
µ

sin2 θ

)2
(60)

Hc,y(θ) = ξ(θ)
(
µ cos2 θ + β

2µ sin2 θ

)2
(61)

where µ =
√
fcx/fcy.

All the critical hardening moduli have the maxi-
mum value equal to zero, so localization starts at the
onset of plastic flow for tension (since, under tension,
abrupt softening occurs) and at the peak of the com-
pressive yield stress for compression. In the case of
uniaxial tension (Figures 3-4), normal n is aligned
with the stress, in fact θ = φ. In the case of uniax-
ial compression (Figures 5-6), for the chosen set of
mechanical parameters, the expressions of the critical
hardening moduli exhibit two peaks corresponding to
two different possible cracks that are symmetric with
respect to the compressive stress axis.

Figure 3. H(θ) for uniaxial tension in x-direction

Figure 4. H(θ) for uniaxial tension in y-direction

The localization analysis is then extended to the
generic case of uniaxial tension or compression inclined
by an angle φ with respect to the x-axis, with φ ∈
[0, π/2]. The mechanical parameters used are the same
chosen by the author of the model for elementary tests
under uniaxial stress [1, 2] and are reported in Table 1.

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vol. 26/2020 Localization analysis of an orthotropic multi-surface plasticity model

Elastic parameters Tensile parameters
Exx Eyy νxy Gxy ft,x ft,y Gt,x Gt,y α
[GPa] [GPa] [-] [GPa] [MPa] [MPa] [N/m] [N/m] [-]
10.0 5.0 0.20 3.0 1.0 0.5 20 10 1.0

Compressive parameters
fc,x fc,y Gc,x Gc,y β γ κp

[MPa] [MPa] [N/m] [N/m] [-] [-] [-]
10.0 5.0 5000 2500 -1.0 3.0 5·10−4

Table 1. Mechanical parameters considered for localization analysis

Figure 5. H(θ) for uniaxial compression in x-direction

Figure 6. H(θ) for uniaxial compression in y-direction

The critical hardening moduli, except for the partic-
ular case of uniaxial tension along the material axes,
always exhibit two maxima that correspond to the
formation of two potential weak discontinuity surfaces.
The solid and dashed lines represented in Figures 7
to 12 correspond to the aforementioned two solutions.
The dependence of the localization angles on the

uniaxial stress angle φ for tension is represented in
Figure 7, while the difference between the localization
angles and the tension angle is shown in Figure 8.
The variation of the inner product n ·m between
the normal to crack surface and the polarization vec-
tor is represented in Figure 9, which shows that for
φ = {0, π/2} its value is equal to 1 (tensile splitting),
whereas for intermediate values of φ the inner prod-
uct is lower, which corresponds to a misalignment
between n and m. It is worth noting that the two
graphs are coincident since the polarization vectorm1
corresponding to the normal n1 coincides with the
other normal n2 and vice versa.

Figure 7. Dependence of θloc on φ for uniaxial tension

Figure 8. Difference between θloc and φ as a function
of φ for uniaxial tension

Figure 9. Dependence of n · m on φ for uniaxial
tension

In the case of uniaxial compression, the dependence
of the localization angles on the stress angle is reported
in Figure 10). From Figure 11 one can see that the
differences between the localization angles and the
stress angle are almost constant and close to the values
π/4 and 3π/4. Figure 12 shows that the graphs of

61



C. Pagani, M. Jirásek, M. Horák Acta Polytechnica CTU Proceedings

Figure 10. Dependence of θloc on φ for uniaxial
compression

Figure 11. Difference between θloc and φ as a func-
tion of φ under uniaxial compression

the inner products n ·m corresponding to the two
crack surfaces coincide as for the tensile behavior
and that the polarization vector is orthogonal to the
corresponding normal (shear slip) for φ = 0, whereas
for different values of φ the inner product increases
up to its maximum value for φ = π/2.

5. Conclusions
Localization analysis under uniaxial tensile and com-
pressive stress has been carried out.

Analytical results in the particular cases of tension
or compression aligned with the material principal
axes have been obtained and it has been shown that,
for tensile stress, the weak discontinuity surface will
form in the direction orthogonal to the stress direc-
tion while, in the case of compression, two different
discontinuity surfaces, symmetric with respect to the
stress direction, could develop.

The analysis has been extended by varying the angle
that the stress forms with the material axes. It has
been shown that for both tension and compression two
localization angles can be found and some observations
related to the dependence of the localization angles
and the polarization vectors on the stress angle have
been presented.
For all the cases described here, the localization

condition is for the first time satisfied at the peak
of the stress-strain diagram, i.e., the critical plastic
modulus associated with the most favorable direction
of a potential discontinuity is zero. For this reason, the
results are not affected by the size of finite elements.
The adjustment of the softening laws dependent on

Figure 12. Dependence of n ·m on φ for uniaxial
compression

the element size according to (7) or (22) affects only
the post-localization stage of response.

List of symbols
α Parameter that controls the shear stress contribution

to tensile failure
β Parameter that couples the normal stresses in the

compressive failure surface
γ Parameter that controls the shear stress contribution

to compressive failure
le Crack bandwidth

H Hardening modulus
Hcrit Critical hardening modulus
κp Value of the compressive hardening parameter at peak

fi,σ Gradient of the i-th yield function with respect to
the stress tensor

gi,σ Gradient of the i-th plastic potential with respect to
the stress tensor

n Normal to the discontinuity surface
m Polarization vector
φ Uniaxial stress angle with respect to the x-axis
θ Angle that the generic normal to the discontinuity

surface forms with the x-axis
θloc Localization angle

Acknowledgements
This research has been financially supported by
the European Regional Development Fund through
the Center of Advanced Applied Sciences at the
Czech Technical University in Prague (project
No. CZ.02.1.01/0.0/0.0/16_019/0000778).

References
[1] P. Lourenço. An orthotropic continuum model for the
analysis of masonry structures. Tech. rep., University of
Delft, Delft, Holland, 1995.

[2] P. Lourenço. An anisotropic macro-model for masonry
plates and shells: implementation and validation. Ph.D.
thesis, Delft University of Technology, Faculty of Civil
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	Acta Polytechnica CTU Proceedings 26:56–63, 2020
	1 Introduction
	2 Model description
	2.1 Orthotropic elasticity
	2.2 Plastic behavior in tension
	2.3 Plastic behavior in compression

	3 Localization analysis in plasticity
	4 Localization analysis for uniaxial stress
	5 Conclusions
	List of symbols
	Acknowledgements
	References