microstructure-properties composites: part 4 absorbing shocks profs. a
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Microstructure-Properties Composites: part 4 Absorbing Shocks
Profs. A. D. Rollett, M. De Graef Microstructure Properties
Processing Performance Last modified: 18th Dec. 15 Lecture
Objectives: Composites
The main objective of this lecture is to introduce you to
microstructure-property relationships in composite materials. We
discuss the way in which cellular materials are useful for
absorbing shocks and how to quantify this property. This is an
example of the application of composite properties of foams, which
are cellular materials. Please acknowledge Carnegie Mellon if you
make public use of these slides Q&A for Part 4 What are the
main differences in the response of metal foams, ceramic foams and
polymer foams to compression (crushing)? One obvious difference is
that metals and ceramics typically crush in an irreversible fashion
whereas most polymer foams recover (visco-)elastically. (#6) How is
it that foams can absorb large amounts of energy? The most
important feature of their stress-strain curves is the large strain
at constant stress (the plateau region), which means that there is
a large area under the curve. (#8) How do we translate a
stress-strain curve into maps of Energy Absorbed versus Peak
Stress? Integrate the curve to get the area and adjust the units to
get energy; plot this value versus the max. stress. Why is it that
the Energy does not increase much beyond a certain point? Because
the foam starts to approach full density, which means that the
stress rises sharply with little increase in strain. How is it
possible for foams (esp. polymers) to deform at constant stress
over large strains? Through bending and deformation of the cell
walls. In the plots of energy absorbed versus normalized stress,
how do we obtain the envelopes at constant density? By identifying
the point on the energy-stress curves where the stress enters the
plateau region (#10). For a shock cushion, how do we calculate the
energy absorbed and the maximum stress? By computing (W*A*t), #14.
How does the drop height specify the energy that must be absorbed?
Via the kinetic energy, #11.What about the strain rate? From
velocity / thickness, #12. In the worked example, how do we get the
quantities such as energy absorbed and normalized stress in order
to use the graphs provided for specific materials? From the kinetic
energy (above) and the mass, specified max. deceleration and area,
#11. In the worked example for expanded polyurethane, how does the
iterative procedure work to arrive at a suitable thickness for the
shock cushion that satisfies the requirement for maximum stress and
for the strain rate? By varying the thickness of the foam pad and
re-computing the energies, strain ratesetc. until convergence is
obtained. Please acknowledge Carnegie Mellon if you make public use
of these slides Advertising Looks pretty good, right? How does it
work?
How can we check it out? With all the reporting on concussion of
athletes, shock protection is obviously important. Please
acknowledge Carnegie Mellon if you make public use of these slides
Examinable Cellular Materials This next section provides some basic
information on cellular materials. Why study cellular
materials?Answer: cellular materials provide a range of properties
that are not achievable in bulk materials.Especially when load
carrying capacity at very low densities is required, only cellular
materials can satisfy the requirements.Shock resistance is also a
vital characteristic of cellular materials. Cellular structures are
feasible (and used for engineering applications) with all materials
types.Metal honeycombs are used in transport applications.Ceramic
foams are used in insulation.Cellular structures are ubiquitous in
biomaterials (wood, bone, shells). A good basic reference is
Cellular Solids, Pergamon, L.J. Gibson and M.F. Ashby (1988), ISBN
Please acknowledge Carnegie Mellon if you make public use of these
slides Honeycombs: properties
Examinable Honeycombs: properties Note the contrast between tension
and compression (plateau present), 4.2a vs. 4.2b. Even brittle wall
materials exhibit progressive failure in compression, 4.2e. The
stress-strain curves are labeled by their characteristic stages.
Very important consequences for energy absorbing structures (see
later slides) [Gibson & Ashby: Cellular Materials] Please
acknowledge Carnegie Mellon if you make public use of these slides
Examinable Energy Absorption Why are foams useful?!One reason is
their capacity to absorb energy. Remember: energy absorbed = area
under the stress-strain curve. [Gibson] Please acknowledge Carnegie
Mellon if you make public use of these slides Energy Absorption: 2
Examinable
How do these two graphs connect? Each line on the 2nd graph
correspond to a locus of points from the 1st graph, for a
particular relative density.Note the turn-over in the curve of
energy versus stress: this is the most efficient use of the
material. Another way to explain the procedure is like this.For
each colored point in the LH graph, extract the stress and the area
under the curve associated with that point.The stress becomes the X
coordinate and the area the Y coordinate of each corresponding
point in the new graph on the RH side. [Gibson] Please acknowledge
Carnegie Mellon if you make public use of these slides Energy
Absorption: 3 Fully Densified Wall Buckling Elastic
Examinable Energy Absorption: 3 During wall buckling, densification
proceeds at a approximately constant external stress. Fully
Densified [Gibson] Wall Buckling Elastic Note that, once the foam
starts to densify (steep upturn in the stress-strain curve) then
the stress rises with little increase in energy absorbed. Please
acknowledge Carnegie Mellon if you make public use of these slides
Examinable Energy Absorption: 4 As seen before, the stress-strain
(8.4a) can be re-plotted as energy absorbed versus stress
(8.4b).Varying the density varies the maximum energy that can be
absorbed at the plateau stress. We can draw an envelope through the
points of maximum energy plateau stress. Variations in other
parameters such as strain rate can also be shown on such an
energy-stress diagram by plotting only these envelopes. [Gibson]
Please acknowledge Carnegie Mellon if you make public use of these
slides Examinable Shock Cushions Once one knows the energy-stress
characteristic of a material, it is possible to calculate the
optimum thickness. Given the kinetic energy to be absorbed, U, and
the area of contact between object and foam, A, the thickness, t,
is given by t = U / W A (Eq. 1)where W is the energy absorbed per
unit volume in the foam. Typically, the mass of the object, m, and
the peak deceleration, a, is also specified (as a multiple of
gravitational acceleration, g) which determines the maximum stress,
s, s = m a / A (Eq. 2) Please acknowledge Carnegie Mellon if you
make public use of these slides Examinable Shock Cushion: 2 In
addition, a drop height is specified which in turn sets the
velocity, v, and the energy, U, that must be absorbed;U = m v2 /
2.Thus the thickness, t, is given by t = m v2 / (2 W A) (Eq. 3)
This in turn specifies the strain rate, de/dt, in the foam which
affects the energy-stress relationship (see Fig. 8.4c): de/dt = v /
t (Eq. 4) A good place to start is to identify the maximum
allowable stress and read off the associated energy at a high
strain rate.The energy is, however, a function of both stress and
strain rate, so some iteration is required to identify a suitable
thickness that allows the required energy to be absorbed. Please
acknowledge Carnegie Mellon if you make public use of these slides
Examinable Shock Cushion: 3 Worked Example Problem specification
Mass of packaged object: 500 gms. Area of contact between object
and foam: A = 0.01 m2 Velocity of package on impact, v = 4.5 m/s
(drop height, h = 1 m) Energy to be absorbed, U = mv2/2 = 5 J Max.
allowable force on package (10g deceleration), F = ma = 50 N Max.
allowable peak stress (Eq. 2), sp = F/A = 5 kPa Solid modulus of
polyeurethane foam, Es = 50 MPa Max. allowable peak stress,
normalized = sp/Es = We use Gibson-Ashby, fig. 8.8 (next slide).
Gibson & Ashby: Table 8.2, p. 231 Please acknowledge Carnegie
Mellon if you make public use of these slides Shock Cushion: 4 To
see where to start, consider the stress limit (red, vertical
line).This intersects with the solid line for 0.01 relative
density.Therefore it is reasonable to assume that we are seeking a
solution that uses this density Intersections of the r=0.01 (heavy)
line with a specific strain rate (thin) line tell us the energy per
unit volume. The dashed green and blue lines are examples. Note
that the energy value varies (slightly) according to the strain
rate Please acknowledge Carnegie Mellon if you make public use of
these slides [Gibson] Examinable Shock Cushion: 5 The reason that
we have to iterate is that changing the thickness changes the
strain rate and therefore the W value. To start working on the
problem, we have to make some rather arbitrary choices of thickness
that bracket the likely result. Choice of thickness, t: 1 m m
Strain rate, de/dt=v/t (Eq 4): 4.5 s s-1 Energy/modulus (W/Es) at
sp/Es = : (Fig. 8.8) Energy absorbed/unit volume: 2.62 kJ/m kJ/m3
Energy absorbed (W*A*t): 26.2 J J To complete the problem, we have
to iterate on the thickness until we converge on a self-consistent
result and the energy absorbed is the required value, 5J. Please
acknowledge Carnegie Mellon if you make public use of these slides
[Gibson] Examinable Shock Cushion: 6 To continue with the problem,
we re-calculate the thicknesses from Eq. 1. Thickness, t = U/WA:
0.19 m 0.14 m Strain rate, de/dt=v/t (Eq 4): 24 s-1 32 s-1
Energy/modulus (W/Es) at sp/Es = : (Fig. 8.8) Energy absorbed/unit
volume: 3.30 kJ/m kJ/m3 Energy absorbed: 6.27 J 4.69 J Clearly we
have nearly converged, so we have to iterate on the thickness one
more time, using t = U/WA, which givest= 150 mm, W*A*t=5J, and an
optimum relative density = 0.01. Please acknowledge Carnegie Mellon
if you make public use of these slides Examinable Summary: Part 4
Foams or cellular materials are an example of composite materials.
We developed an example of how cellular materials are useful as
shock cushions. This lead to worked example of how calculate the
optimum thickness of such as shock cushion. Please acknowledge
Carnegie Mellon if you make public use of these slides Questions
Many of us commonly use Wikipedia to learn about new topics or to
check information.What do you think of this wiki page? Please
acknowledge Carnegie Mellon if you make public use of these slides
References Cellular Solids, Pergamon, L.J. Gibson and M.F. Ashby
(1988), ISBN L. Gong, S. Kyriakides, W.-Y. Jang, Compressive
response of open-cell foams; Part I: Morphology and elastic
properties, Intl. J. Solids Structures, 42 (2005) Materials
Principles & Practice, Butterworth Heinemann, edited by C.
Newey & G. Weaver. Mechanical Behavior of Materials, T. H.
Courtney (2000), Boston, McGraw-Hill. Mechanical Behavior of
Materials, N.E. Dowling (1999), Prentice-Hall. Structural
Materials, Butterworth Heinemann, edited by G. Weidmann, P. Lewis
and N. Reid. The New Science of Strong Materials, J. E. Gordon,
Princeton. An Introduction of Composite Products, Chapman &
Hall, K. Potter (1997), ISBN X. An Introduction to the Mechanical
Properties of Solid Polymers, Wiley, I.M. Ward and D.W. Hadley
(1993), ISBN Plasticity: A Treatise on Finite Deformation of
Heterogeneous Inelastic Materials, Cambridge University Press,
2009, S. Nemat-Nasser, ISBN The Theory of Composites, Cambridge
University Press, 2001, G. F. Milton, ISBN Please acknowledge
Carnegie Mellon if you make public use of these slides Supplemental
Slides The following slides contain supplemental material that will
be of interest to those who are curious to obtain more detail.
Please acknowledge Carnegie Mellon if you make public use of these
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