Michael Nieves. We consider a transition wave that propagates inside a discrete periodic structure, composed of massless beams and periodically placed masses, as a result of energy supplied by a remote point load. This scenario may represent the collapse of a civil engineering structure, such as a rooftop or a bridge, due to some unwanted vibrations. A summary of analytical results, obtained using the Wiener-Hopf technique, concerning the dynamic behaviour of the structure during the collapse is given. The structure’s dispersive nature is then used to reveal the steady-state collapse speeds observed in numerical simulations. We show that average speed of collapse propagation is a continuous function of the load amplitude and identify intervals when steady-state propagation occurs. Outside these intervals, the collapse propagates non-steadily, and can oscillate rapidly. Here the collapse can occur in small bursts or clusters and for large load amplitudes it possesses what was recently discovered as a forerunning propagation. Michael Nieves gratefully acknowledges the support of the EU H2020 grant MSCA-IF-2016-747334-CAT-FFLAP.

1. Progressive collapse of flexural systems
M.J. Nieves∗
Co-authors: G.S. Mishuris (Aberystwyth University),
L.I. Slepyan (Tel Aviv University).
∗Marie Curie Fellow
University of Cagliari, Department of Mechanical,
Chemical and Material Engineering, Cagliari, Italy,
Keele University, School of Computing and Mathematics,
Keele, ST5 5BG, UK,
m.nieves@keele.ac.uk
10th November 2017, DCEE 2017, University of Cagliari
Acknowledgement: M.N. gratefully acknowledges the support of the EU H2020 grant
MSCA-IF-2016-747334-CAT-FFLAP.
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 1 / 12

2. Outline
1 Background and Motivation
2 Failure of 1D discrete flexural structure
3 Wiener-Hopf method and related techniques
4 Summary of results
5 Conclusions and future work
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 2 / 12

3. Background: Dynamic failure of discrete periodic media
Here we are concerned with modelling dynamic failure of frame-like structures, which is a major aspect of metamaterials
research. Such models allow one to:
a) incorporate effects due to the material geometry
b) consider different lattices eg. utilising periodic distributions of mass or different connectors (beams/springs)
c) analyse various types of dynamic fracture mechanisms (eg. semi-infinite cracks, bridge cracks)
d) trace various fracture regimes.
Most of the research has been carried out for mass-spring systems.
0.3 Dynamic fault in a two-dimensional inhomogeneous lattice
structure
The ideas of the previous sections will be extended to the case of
an inhomogeneous lattice which occupies an infinite plane. Non-uniformity
within the lattice is introduced in the same way as before, i.e. by assigning
different masses to the joints of the lattice. The problem corresponds to the
anti-plane deformation, and the elementary cell of the periodic lattice has a
rectangular shape, as shown in Fig. 4a. In this diagram, the particles repre-
sented by black (or white) discs are assumed to have the mass m1 (or m2).
The rows of joints of the same mass are aligned with the horizontal axis,
and with our choice of distribution of mass, an elementary cell of the doubly
periodic structure contains three particles, with two particles of mass m1
and one particle of mass m2.
(a) (b)
n = −2
n = −1
n = 0
n = 1
Figure 4. Nonuniform lattice structures: (a) Undamaged lattice, (b) Lat-
tice with a crack. The elementary cell is shown as a shaded rectangle. The
horizontal and vertical coordinates of the cell are denoted by m and n,
respectively.
Similar to the previous text, we introduce the normalization in such
a way that the stiffness of the bonds connecting neighbouring particles,
the lattice spacing between neighbouring particles, and the averaged mass
density within the elementary cell are equal to unity. This implies that
(2m1 +m2)/3 = 1, and that the low-frequency wave speed is equal to c = 1.
The notation r = m1/m2 is used for the contrast parameter of the lattice.
n = 0
1
2
3
4
1
2
3
4
5
6m = 7
A"homogeneous"square"la/ce""
(Slepyan"(2001),"(1))""
A"homo
(S
An"inhomogeneous"square"la/ce"with"
contras=ng"masses"(Mishuris"et"al."(2007))"
480 L.I. Slepyan / J. Mech. Phys. Solids 49 (2001) 469–511
Fig. 4. Square-cell lattice.
to the force velocity, v, is directed towards the force. Thus, the power of the force is
now
N =
V[(q0 − q1)2
− q2
0]
2(V2 − 1)
: (20)
As the amplitude of the initial wave, q0, increases the power of the force, N, de-
creases becoming zero when q0 =q1=2 and then negative when q0 ¿ q1=2. In particular,
if q0 = q1, only the wave ahead the force remains and its energy ows to the force
which plays the role of an energy absorber. Such a wave can be called the feeding
wave.
Note that vg = v=2 for a wave on the deep-sea surface and it can represent another
example of the feeding wave of such a kind.
3. Dynamic crack growth in a square-cell elastic lattice
Consider an inÿnite lattice consisting of point particles of mass M. Each particle
is connected with four neighbors by the same linearly elastic bonds each of length a
(Fig. 4). For this lattice mode III crack propagation is studied. A semi-inÿnite crack
is assumed to propagate to the right with constant speed v; that is, the time-interval
between the breaking of neighboring bonds on the crack path, a=v, is constant. In this
‘steady-state’ process, one or several feeding waves can deliver energy to the crack
front. A part of the energy is spent on the bond disintegration on the crack path and
the rest is radiated by dissipative waves away from the crack front. The number of
these waves and their location depend on the crack speed. In outline, the plan of the
solution is as follows.
First, the Fourier transformation of the steady-state dynamic equations for an un-
Damage"p
"""""""""""""""""
3 Dynamic fault in a two-dimensional inhomogeneous lattice
structure
The ideas of the previous sections will be extended to the case of
inhomogeneous lattice which occupies an infinite plane. Non-uniformity
thin the lattice is introduced in the same way as before, i.e. by assigning
fferent masses to the joints of the lattice. The problem corresponds to the
ti-plane deformation, and the elementary cell of the periodic lattice has a
tangular shape, as shown in Fig. 4a. In this diagram, the particles repre-
nted by black (or white) discs are assumed to have the mass m1 (or m2).
e rows of joints of the same mass are aligned with the horizontal axis,
d with our choice of distribution of mass, an elementary cell of the doubly
riodic structure contains three particles, with two particles of mass m1
d one particle of mass m2.
a) (b)
n = −2
n = −1
n = 0
n = 1
n = 0
1
2
3
4
1
2
3
4
5
x
y
0 11 2456m = 7 3
A"homogeneous"square"la/ce""
(Slepyan"(2001),"(1))""
A"homogeneous"triangular"la/ce"
(Slepyan"(2001),"(2))""
An"inhomogeneous"square"la/ce"with"
contras=ng"masses"(Mishuris"et"al."(2007))"
480 L.I. Slepyan / J. Mech. Phys. Solids 49 (2001) 469–511
Fig. 4. Square-cell lattice.
to the force velocity, v, is directed towards the force. Thus, the power of the force is
now
N =
V[(q0 − q1)2
− q2
0]
2(V2 − 1)
: (20)
As the amplitude of the initial wave, q0, increases the power of the force, N, de-
creases becoming zero when q0 =q1=2 and then negative when q0 ¿ q1=2. In particular,
if q0 = q1, only the wave ahead the force remains and its energy ows to the force
which plays the role of an energy absorber. Such a wave can be called the feeding
wave.
Note that vg = v=2 for a wave on the deep-sea surface and it can represent another
example of the feeding wave of such a kind.
Damage"propaga=ng"through"a"bridge"
"""""""""""""""""(Brun"et"al."(2013))"
Dynamic fault in a two-dimensional inhomogeneous lattice
tructure
The ideas of the previous sections will be extended to the case of
omogeneous lattice which occupies an infinite plane. Non-uniformity
the lattice is introduced in the same way as before, i.e. by assigning
nt masses to the joints of the lattice. The problem corresponds to the
ane deformation, and the elementary cell of the periodic lattice has a
ular shape, as shown in Fig. 4a. In this diagram, the particles repre-
by black (or white) discs are assumed to have the mass m1 (or m2).
ws of joints of the same mass are aligned with the horizontal axis,
th our choice of distribution of mass, an elementary cell of the doubly
c structure contains three particles, with two particles of mass m1
e particle of mass m2.
(b)
n = −2
n = −1
n = 0
n = 1
n = 0
1
2
3
4
1
2
3
4
5
x
y
0 11 2456m = 7 3
A"homogeneous"square"la/ce""
(Slepyan"(2001),"(1))""
A"homogeneous"triangular"la/ce"
(Slepyan"(2001),"(2))""
An"inhomogeneous"square"la/ce"with"
contras=ng"masses"(Mishuris"et"al."(2007))"
480 L.I. Slepyan / J. Mech. Phys. Solids 49 (2001) 469–511
Fig. 4. Square-cell lattice.
to the force velocity, v, is directed towards the force. Thus, the power of the force is
now
N =
V[(q0 − q1)2
− q2
0]
2(V2 − 1)
: (20)
As the amplitude of the initial wave, q0, increases the power of the force, N, de-
creases becoming zero when q0 =q1=2 and then negative when q0 ¿ q1=2. In particular,
if q0 = q1, only the wave ahead the force remains and its energy ows to the force
which plays the role of an energy absorber. Such a wave can be called the feeding
wave.
Note that vg = v=2 for a wave on the deep-sea surface and it can represent another
example of the feeding wave of such a kind.
Damage"propaga=ng"through"a"bridge"
"""""""""""""""""(Brun"et"al."(2013))"
Homogeneous square lattice
[Slepyan (2001a)]
Homogeneous triangular lattice
[Slepyan (2001b)]
Inhomogeneous square lattice
with contrasting masses
[Mishuris et al. (2007)]
1 L.I. Slepyan: Feeding and dissipative waves in fracture and phase transition. I. Some 1D structures and a square-cell
lattice, J. Mech. Phys. Solids 49, 469-511. (2001a).
2 L.I. Slepyan: Feeding and dissipative waves in fracture and phase transition. III. Triangular-cell lattice, J. Mech. Phys.
Solids 49, 2839-2875, (2001b).
3 G.S. Mishuris, A.B. Movchan, L.I. Slepyan: Waves and fracture in an inhomogeneous lattice structure, Waves in Random
and Complex Media, 17, 409-428, (2007).
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 3 / 12

4. Dynamic fracture in beam structures
Mass-beam systems are more commonly found in applications such as civil engineering (eg. buildings, bridges, pipeline systems).
Above: Collapse of a rooftop, Tottenham, 2014.
ice. The problem corresponds to the
tary cell of the periodic lattice has a
In this diagram, the particles repre-
sumed to have the mass m1 (or m2).
re aligned with the horizontal axis,
ass, an elementary cell of the doubly
cles, with two particles of mass m1
)
n = −2
n = −1
n = 0
n = 1
es: (a) Undamaged lattice, (b) Lat-
is shown as a shaded rectangle. The
the cell are denoted by m and n,
introduce the normalization in such
connecting neighbouring particles,
ng particles, and the averaged mass
e equal to unity. This implies that
quency wave speed is equal to c = 1.
he contrast parameter of the lattice.
ritten for three particles within the
re, and the displacement of a node
n = 0, ±1, ±2, ..., is the multi-index
me(1)
+ 3ne(2)
, and the remaining
3
4
5
mogeneous"square"la/ce"with"
ng"masses"(Mishuris"et"al."(2007))"
Fig. 4. Square-cell lattice.
rected towards the force. Thus, the power of the force is
]
: (20)
nitial wave, q0, increases the power of the force, N, de-
q0 =q1=2 and then negative when q0 ¿ q1=2. In particular,
ead the force remains and its energy ows to the force
nergy absorber. Such a wave can be called the feeding
ave on the deep-sea surface and it can represent another
of such a kind.
a square-cell elastic lattice
consisting of point particles of mass M. Each particle
bors by the same linearly elastic bonds each of length a
e III crack propagation is studied. A semi-inÿnite crack
he right with constant speed v; that is, the time-interval
hboring bonds on the crack path, a=v, is constant. In this
several feeding waves can deliver energy to the crack
s spent on the bond disintegration on the crack path and
ative waves away from the crack front. The number of
n depend on the crack speed. In outline, the plan of the
mation of the steady-state dynamic equations for an un-
ormed and a general solution is derived. Then a dynamic
Damage"propaga=ng"through"a"bridge"
"""""""""""""""""(Brun"et"al."(2013))"
Below: 1D structure, mass-beam chain supported by
Springs [Brun et al. 2013]
Above: Collapse of the San Saba Railway bridge,
Texas, in 2013, [Brun et al. 2014]
Above: Progressive collapse of the Puente Viejo
bridge, Chile, 2010, [Brun et al. 2013]
Above: Dynamic failure of mass-beam structure within an
interface [Nieves et al. 2016]
1 Brun, M., Giaccu, G.F., Movchan, A.B., Slepyan, L.I.: Transition wave in the collapse of the San Saba Bridge, Front. Mater., (2014),
http://dx.doi.org/10.3389/fmats.2014.00012.
2 M. Brun, A.B. Movchan, L.I. Slepyan: Transition wave in a supported heavy beam, J. Mech. Phys. Solids 61, 2067-2085, (2013).
3 M.J. Nieves, G.S. Mishuris, L.I. Slepyan: Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98,
699-713, (2016).
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 4 / 12

5. Recent work: Dynamic fracture of 1D mass-beam system
Oscilla'ng*point**
force*at*infinity:*
Intact*region*
⌘ = 0
⌘ < 0 ⌘ 0
Damaged*region*
Euler:Bernoulli*beams:*
Young’s*modulus*
Second*moment*of*area****
Springs:*
s'ffness*
Point*masses*
with*mass*
Cri'cal*elonga'on*of*springs**wc
Moving*coordinate:**
M
E
I
{
transi'on*front*at*
Springs*and*beams*have*length**a
⌘ = m vt/a
P cos(!t + )
Oscilla'ng*point**
force*at*infinity:*
⌘ = 0
Euler:Bernoulli*beams:*
Young’s*modulus*
Second*moment*of*area****Point*masses*
with*mass*
Moving*coordinate:**
M
E
Itransi'on*front*at*
⌘ = m vt/a
P cos(!t + )
Oscilla'ng*point**
force*at*infinity:*
Intact*region*
⌘ = 0
⌘ < 0 ⌘ 0
Damaged*region*
Euler:Bernoulli*beams:*
Young’s*modulus*
Second*moment*of*area****
Springs:*
s'ffness*
Point*masses*
with*mass*
Cri'cal*elonga'on*of*springs**wc
Moving*coordinate:**
M
E
I
{
transi'on*front*at*
Springs*and*beams*have*length**a
⌘ = m vt/a
P cos(!t + )
Fracture of the springs is assumed to propagate steadily through the structure with a constant speed v.
1 M.J. Nieves, G.S. Mishuris, L.I. Slepyan: Analysis of dynamic damage propagation in discrete beam structures, Int. J. Solids Struct. 97-98,
699-713, (2016).
2 M.J. Nieves, G.S. Mishuris, L.I. Slepyan: Transient wave in a transformable periodic flexural structure, Int. J. Solids Struct. (to appear), (2017).
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 5 / 12

6. Wiener-Hopf method and related techniques
Oscilla'ng*point**
force*at*infinity:*
Intact*region*
⌘ = 0
⌘ < 0 ⌘ 0
Damaged*region*
Euler:Bernoulli*beams:*
Young’s*modulus*
Second*moment*of*area****
Springs:*
s'ffness*
Point*masses*
with*mass*
Cri'cal*elonga'on*of*springs**wc
Moving*coordinate:**
M
E
I
{
transi'on*front*at*
Springs*and*beams*have*length**a
⌘ = m vt/a
P cos(!t + )
1 Obtain equations of motion for each mass in the structure.
2 Take the continuous Fourier transform with respect to η
g1(k)w+ + g2(k)w− = Φ(k) = Φ+(k) + Φ−(k)
3 Solving Wiener-Hopf equation for w+ and w−. Use poles of these functions to determine dynamic
behaviour of structure.
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 6 / 12

7. Dynamic features of the beam structure
Behaviour of the Wiener-Hopf solution at singular points determines dynamic features of the
structure.
1 Behind the front: feeding and reflected waves, the slope of the beam.
2 for particular frequencies of load, ahead of the front there are transmitted waves.
-20 -15 -10 -5 0 5 10 15 20
0
20
40
60
80
100
120
140
160
180
200
0 5 10 15 20
-1
-0.5
0
0.5
1
1.5
2
Feeding
and
reflected
waves
along
the
slope
transi4on
front
transmi7ed
wave
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 7 / 12

8. The case of no transmission
-20 -15 -10 -5 0 5 10 15 20
0
10
20
30
40
50
60
Feeding
and
reflected
waves
along
the
slope
0 2 4 6 8 10
-1
-0.5
0
0.5
1
1.5
2
transi4on
front
No
wave
transmi8ed
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 8 / 12

9. Summary of results
Analytically predicted features:
(i) ranges of the parameters where the transmitted waves into the intact part of the
structure is possible and when all the waves are only reflected from the transition
front.
(ii) ranges of the parameters where the steady state regime is possible i.e. when the
front propagates with constant velocity.
(iii) non-zero slope of the damaged part of the structure
(iv) energy distribution in the system and others
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 9 / 12

10. Summary of results: Numerical simulations for long structures
timetime
30 40 50 60 70 80 90 100 110 120
position
0
10
20
30
40
50
60
load amplitude increasing
cluster propagation: size 2
clusterpropagation:size7
forerunningpropagation
v=1.7214
v = 0.3738
v = 0.2097
fractureposition
time
0 10 20 30 40 50 60
position
0
0.5
1
1.5
2
2.5
position of fractureinstantaneousfracturevelocity
load amplitude increasing
cluster propagation: size 7
v = 1.7214
v = 0.3738
v = 0.2097
positionoffracture
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 10 / 12

11. Summary of results: Numerical simulations for long structures
time
0 10 20 30 40 50 60
position
0
0.5
1
1.5
2
2.5
position of fracture
instantaneousfracturevelocity
load amplitude increasing
cluster propagation: size 7
v = 1.7214
v = 0.3738
v = 0.2097
0 10 20 30 40 50
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Pa/wc
averagetransitionfrontspeed
v = 0.2097
v = 0.3738
v = 1.7214
⇤ = 25.62
MATLAB&
COMSOL&
timetime
30 40 50 60 70 80 90 100 110 120
position
0
10
20
30
40
50
60
load amplitude increasing
cluster propagation: size 2
clusterpropagation:size7
forerunningpropagation
v=1.7214
v = 0.3738
v = 0.2097
fractureposition
`
load amplitude
cluster
regime
(intermediate
regime
between
steady
states)
stable
cluster
regimes
load amplitude increasing
positionoffracture
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 11 / 12

12. Conclusions and future work
1 The problem of failure propagation in a 1D flexural structure has been completely analysed
through analytical and transient analyses.
2 The Wiener-Hopf technique has been used to reveal all information about the dynamic
failure process and the phenomena associated with this particular toy problem.
3 The Wiener-Hopf technique provides tools for tackling more realistic and more challenging
problems related to the periodic collapse of framelike structures subjected to loads.
4 Numerical codes can be adapted to model collapse of other structures!
5 Some relevant civil engineering problems we hope to tackle analytically, numerically and
experimentally include:
Above: Crack propagating in a grillage
structure composed of heavy beams
connecting periodically placed masses
Above: Inhomogeneous
structures with contrasting
material parameters
Above: Net type
structures with diagonal
links
Your collaboration in exploring this new and rich area would be most welcome!
(e-mail: m.nieves@keele.ac.uk)
M.J.Nieves et al. (2017) Progressive collapse of flexural systems 10th November 2017 12 / 12