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# 2D Nonlinear Finite Element Analysis of Masonry Bedjoint

Page 21
THEORY OF ELASTICITY 21
2.2 Strain-displacement relations
Strains can be expressed as functions of displacement components at a point in
the three Cartesian coordinate directions. If u,v and w (all functions of location of
point, represented by its x, y, z coordinates) represent the displacement components
along x, y and z directions, then
x
u
x

y
v
y

z
w
z

(2.01)
x
v
y
u
xy

y
w
z
v
yz

z
u
x
w
zx

2.3 Stress-strain relations
Stress-strain relations or the constitutive equations (according to the
generalised Hooke law
E
) for linearly elastic and isotropic material can be
expressed as
;
)
(
E
z
y
x
x
��
��

G
xy
xy

� �
;
)
(
E
z
y
x
y
��

��

G
yz
yz

� � (2.02)
;
)
(
E
z
y
x
z

��
��

G
zx
zx

� �
where )
1
(
2

E
G is the shear modulus.
These equations can also be written in terms of stresses as function of strains.
� �
� �;
)
2
1
)(
1
(
1

��
��

z
y
x
x
E
)
1
(
2

xy
xy
E
� �
� �;
)
2
1
)(
1
(
1

��

��

z
y
y
y
E
)
1
(
2

yz
yz
E (2.03)
� �
� �;
)
2
1
)(
1
(
1

��
��

z
y
x
z
E
)
1
(
2

zx
zx
E

Page 22
THEORY OF ELASTICITY 22
These equations are expressed in the form as
� � � �� �

D
(2.04)
2.4 Equilibrium equations
Stress at a point in a component is described by the stress tensor one normal
stress component and two shear components on each of six sides of a cube around
that point. To reach an equilibrium of a cube, next eighteen stress components must
satisfy the following equilibrium conditions. Forces acting on a cube are Fx, Fy and Fz
and they act along axes X,Y and Z.
;
x
xz
xy
x F
z
y
x

� �

yx
xy
� �
;
y
yz
y
yx F
z
y
x

� �

zy
yz
� � (2.05)
;
z
z
zy
zx F
z
y
x

xz
zx
� �

Page 23

DAMAGE MODELS 23

3 Damage Models

3.1 Isotropic damage model

An isotropic damage model expects that the stiffness degradation is isotropic.
It means that modulus of elasticity corresponding to different directions decreases
tensor is expressed as where is a scalar damage variable and is
D (1 ��)D

e

the elastic stiffness tensor. The damage evolution law is assumed in an explicit form,
connecting the damage variable to the largest previously reached equivalent strain
level.
The equivalent strain is a scalar measure related from the strain tensor. The
shape of the elastic domain in the strain space has got an influence to the choice of
the specific expression for the equivalent strain. There are many supported
equivalent strain definitions, for example: Mazars definition based on norm of
positive part of strain, Rankine criterion of maximum principal stress, energy norm
scaled by Young’s modulus, Modifies Mises definition. All these definitions are
based on the three-dimensional description of strain (and stress). If they are used in a
reduced problem, the strain components from the underlying assumptions are used
in the estimation of equivalent strain.
Attention should be paid to proper regularization, since the growth of damage
usually leads to softening and may cause localization of the dissipative process. The
damage law should be modified according to the element size, in the spirit of the
crack-band approach, if the model is kept local.

D

e

3.2 Anisotropic damage model

The concept of an isotropic damage is appropriate for materials weakened by
gaps and empty spaces, but if the physical source of damage is the initiation and
propagation of micro-cracks, isotropic stiffness degradation can be considered only
as a first imprecise approximation. More sophisticated damage models take into
account the highly oriented nature or cracking, which is obtained in the anisotropic
character of the damaged stiffness or compliance matrices.
There are many anisotropic damage formulations in the literature. In our case
we use a model based on the principle of energy equivalence and on the construction
of the inverse integrity tensor by integration of a scalar over all three-dimensional
directions. This model uses concepts from microplane theory and it is called the
microplane-based damage model MDM.
Microplane damage model is very advanced technique to capture the failure.
The damage evolution is calculated in each integration point (node) in 21 directions
(on 21 micro planes having a different orientation). Our model is theoretically
dealing only with the tensile stresses. The compressive strength turns out to depend
on the Poisson ratio and its value is too low compared to the tensile strength. The

Page 24

DAMAGE MODELS 24

MDM is a very advantageous to capture moment of a failure and is a perfect method
in M. Jirásek: Comments on microplane theory, Mechanics of Quasi-Brittle Materials
and Structures, Paris, 1999.

Page 25

DAMAGE LOCALIZATION 25

4 Damage localization

Many materials exhibit softening when subjected to excessive tensile stress. It
means, a stress resistance ability is decreasing with growing deformation. The
softening problem is caused by a progress of material defects and decreasing ability
to carry stresses through the effective area. A problem in a brittle materials is, that
the crack development is too fast and it causes immediate model failure. Any stress
transfer in material practically disappears. Instead of softening process we call it
brittle failure.

Fig. 4.1: Local stress-strain diagram with linear softening

For illustration, we will use a simple material model (Fig. 4.1) for uniaxial
tension. We assume that the material is linearly elastic only until reaching a
deformation

0

. After that, according to the diagram, a linear softening is initiated
and the stress transfer totally disappears after reaching the deformation . The

maximum stress f

t

, which is also the strength of material, expressed as
where E is the modulus of elasticity.

f

ft E

0

,

4.1 Numerical solution of tensile test with softening
material

A numerical solution of one-dimensional task by finite element method splits
a whole bar (Fig. 4.2) into N parts elements. A number of connections are ,
called nodes. Another two nodes are on the ends of the bar. As we know, the method
is based on approximation of a fields with displacement functions, which are
continuous on the whole bar and linear on each element. Every function is specified
by a node displacement values. An element deformation is calculated as a difference
of a node displacement divided by bar length. A deformation of a whole bar is
approximated from steps by constant function. The stresses are obtained from
deformations. An equivalent condition needs to be satisfied a stresses in every
element are equal.

N 1
1

Page 26

DAMAGE LOCALIZATION 26

Fig. 4.2: Tensioned bar

As we mentioned before, the material is assumed to be linearly elastic until
reaches a peak of the stress-strain diagram. After that, a behaviour solution is not
clear. In every element a two situations can be distinguished elastic unloading or
whole bar and combination in every element. We are looking for a failure situation
and because of this requirement we can eliminate an elastic unloading of the whole
bar.
by force or displacement. Force control is not appropriate for our model, it would be
impossible to find out a moment of failure. So, we will use displacement control and
imagine moving with a right side of the bar. In case, that a material characteristic of
elements differs (for example strength of element) and number of steps is adequate,
only one element starts to soften it will be the weakest element with smallest
strength. In the next step, the element is more softening, the stress in it is decreasing.
This leads to the stress decreasing in other elements, which are still in elastic state
and their deformation is also being reduced. The non-elastic deformation in one
element is localized. It is common in practise to weaken one element on purpose to
help localization. Nevertheless, usually, in a numerical solution, localization is
reached spontaneously. It is caused by a rounding mistakes, which leading to small
deviations from idealised assumptions.

2

N

4.2 Damage work

We will show how much energy (work) is needed to be expended for
complete material damage of unitary volume. During an uniaxial tension and
monotone loading a relation between a stress and deformation can be written

�� s(
) 1 g
E4.1

where equation 4.1 is divided from deformation law (dependence of damage parameter on
deformation → ω=g(ε) ) and Hook’s law → σ=Eε. The variable g is a function taken from
a deformation law. The area g

t

under the local stress-strain diagram is calculated as
g

t

0

s

d�� E

0

g

d

1 4.2

Page 27
DAMAGE LOCALIZATION 27
For example, in a model with linear softening (Fig. 4.1), the area under a local stressstrain
diagram is calculated as
t
f
t
t E
f
g

0
2
1
2
1
4.3
For a model with exponential softening (Fig. 4.3) an area t
g can be obtained as

f
f
f
f
f
t
E
E
E
E
E
d
E
d
E
g

0
0
0
2
0
0
0
0
2
0
0
0
2
1
2
1
exp
2
1
exp
0
0
0
0
4.4
Fig. 4.3: Local stress-strain diagram
In general, an area under the stress-strain diagram represents a dissipated
energy - the energy spent on an object damage.
4.3 Fracture energy
For explanation, we can imagine a bar tensile experiment with a constant
cross-section. A material has got defined his special characteristics and behaves
linearly elastic until reaching a force-displacement diagram peak. A softening is
concentrated in a damage localized zone and we can denote it as p
L . A total energy
dissipated during the experiment f
W is related to the area of the cross-section. Then,
the ratio A
W f / is dependent only on a material characteristic and symbolizes energy
needed to break the bar. The fracture energy in brittle materials is not dissipated only
in one crack, but is dissipated in many micro-cracks localized in a process zone.

Page 28

DAMAGE LOCALIZATION 28

If G

f

is the fracture energy then it holds that

G

f

W

f

A
4.5

The advantage is, that it is easier to measure the fracture energy then the width of a
localization zone .

Ls
L

p

The numerical testing also localizes damage into a certain zone, but a width
symbolizes the width of a finite element and that is why a true width zone

differs from the width Ls

. The dissipated volume is
stress-strain diagram for uniaxial tension is
calculations is

W

f

AG

f

, then

s f
g
AL

g

t

ALs
L

and the area under the
, then a total dissipation obtained with
, which needs to be equal with the true dissipated energy

p

L g

s

f

G

f

4.6

The energy on the right side

the left side g

f

G

f

is a material characteristic constant. The energy on

is related with a selected material model and Ls

is the width of a

simulated localization zone. If the condition 4.6 is not valid, the area under a
simulated force-displacement diagram differs with a true area. In case of making the
finite element mesh smoother, the width is changing but not the area .
L

s

g

f

Page 29

PART II:

CALCULATIONS

Page 30

CALCULATION WITH CRUSHED BRICK PARTICLES 30

5 Calculation with Crushed Brick Particles

Geometry and loading of the tested specimen is described in on Fig. 6.1. The
geometry represents a unit cell of masonry wall and consists of a bedjoint connecting
two halves of bricks. There were tested two load-cases: pure shear and shear with
precompression P equal to 0.5 MPa .

Code

Joint
thickness
Crushed brick

particles

Quality of
crushed
bricks

[mm] coarse/fine/none high/low

pure shear/
shear + compression

T20_P_S 20 - - pure shear
T20_P_SC 20 - - shear + compression
T30_P_S 30 - - pure shear
T30_P_SC 30 - - shear + compression
T40_P_S 40 - - pure shear
T40_P_SC 40 - - shear + compression
T40_C_HQ_S 40 coarse high pure shear
T40_C_HQ_SC 40 coarse high shear + compression
T40_C_LQ_S 40 coarse low pure shear
T40_C_LQ_SC 40 coarse low shear + compression
T40_F_HQ_S 40 fine high pure shear
T40_F_HQ_SC 40 fine high shear + compression
T40_F_LQ_S 40 fine low pure shear
T40_F_LQ_SC 40 fine low shear + compression

Tab. 6.1: Codes of the samples