Proveden

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Experimental methods E181101
EXM6
Material testingconstitutive equations
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
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Constitutive equations
EXM6

• Constitutive equations represent description of
  material properties
• Kinematics (deformation) stress (dynamic
  response to deformation)
• kinematics is described by deformation of a body
  In case of solids
• kinematics describes relative motion (rate af
  deformation) In case of fluids

Deformations and internal stresses are expressed
as tensors in 3D case. Example stress tensor
describes distribution of internal stresses at an
arbitrary cross section
Index of plane index of force component
(cross section) (force acting upon the
cross section i)
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Stress in solids/fluids
EXM6
Tensor of stresses is formally the same in solids
and fluids (in both cases this tensor expresses
forces acting to an arbitrary oriented plane at a
point x,y,z) , however physical nature of these
forces is different. Solids intermolecular
forces (of electrical nature) Fluids stresses
are caused by diffusional transfer of molecules
(momentum flux) between layers of fluid with
different velocities
Total stress pressure dynamic
stress
Viscous stresses affected by fluid flow. Stress
is in fact momentum flux due to molecular
diffusion
Unit tensor
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Viscous Fluids (kinematics)
EXM6
Delvaux
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Viscous Fluids (kinematics)
EXM6
Rate of deformation
Gradient of velocity is tensor with components
(in words rate of deformation is symmetric
part of gradient of velocity)
Special kinematics Simple shear flow ux(y)
(only one nonzero component of velocity,
dependent on only one variable). Example
parallel plates, one is fixed, the second moving
with velocity U
x
The only nonzero component of deformation rate
tensor in case of simple shear flow
is called shear rate
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Viscous Fluids (rheology)
EXM6
Constitutive equations expressed for special case
of simple shear flows Newtonian fluid
Model with one parameter dynamic viscosity
Pa.s
Power law fluid
Model has two parameters K-consistency, n-power
law index.
.
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Viscous Fluids (rheology)
EXM6
General formulation for fully three-dimensional
velocity field
Viscosity ? is constant in Newtonian fluids and
depends upon second invariant of deformation1
rate (more specifically upon three scalar
invariants I, II, III of this deformation rate
tensor, but usually only the second invariant II
is considered because the first invariant I is
zero for incompressible liquids). General
definition of second invariant II
(double dot product, give scalar value as a
result) The second invariant of rate of
deformation tensor can be expressed easily in
simple shear flows
Power law fluid
.
1 Invariant is a scalar value evaluated from 9
components of a tensor, and this value is
independent of the change (e.g. rotation) of
coordinate system (mention the fact that the
rotation changes all 9 components of tensor! but
invariant remains). Therefore invariant is an
objective characteristics of tensor, describing
for example measure of deformation rate. It can
be proved that the tensor of second order has 3
independent invariants.
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Viscous Fluids (rheology)
EXM6

• More complicated constitutive equations exist for
  fluids exhibiting
• yield stress (fluid flows only if stress exceeds
  a threshold),
• thixotropic fluids (viscosity depends upon the
  whole deformation history)
• viscoelastic fluids (exhibiting recovery of
  strains and relaxation of stresses).
• Examples of
• Newtonian fluids are water, air, oils.
• Power law, and viscoelastic fluids are polymer
  melts, foods.
• Thixotropic fluids are paints and plasters.
• Yield stress exhibit for example tooth paste,
  ketchup, youghurt.

Oscillating rheometer sinusoidaly applied stress
and measured strain (not rate of strain!)
Hookean solid-stress is in phase with strain
(phase shift ?0)
Viscous liquid- zero stress corresponds to zero
strain rate (maximum ?) ?900
Viscoelastic material phase shift 0lt?lt90
.
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Rheograms (shear rate-shear stress)
EXM6
n1 Newtonian fluid, nlt1 pseudoplastic fluids (n
is power law index)
Shear stress
Shear rate
.
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DMA Dynamic Material Analysis and Oscilograms
EXM6
storage modulus
loss modulus
polyoxymethylene
Viscous properties E
Elastic properties E
.
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Viscoelastic effects
EXM6
Weissenberg effect (material climbing up on the
rotating rod) Barus effect (die
swell) Kaye effect
.
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Viscoelastic models
EXM6
Oldroyd B model
Extra stress S
Deformation rate
Upper convective derivative
.
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Rheometers
EXM6
Rheometry (identification of constitutive
models). -Rotational rheometers use different
configurations of cylinders, plates, and cones.
Rheograms are evaluated from measured torque
(stress) and frequency of rotation (shear
rate). -Capillary rheometers evaluate
rheological equations from experimentally
determined relationship between flowrate and
pressure drop. Theory of capillary viscometers,
Rabinowitch equation, Bagley correction.
.
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Capillary rheometer
EXM6
1glass cylinder, 2-metallic piston, 3-pressure
transducer Kulite, 4-tested liquid, 5-plastic
holder of needle, 6-needle, 7-calibrated resistor
(electric current needle-tank), 8-calibrated
resistor (current flowing in tank), 9-AC source
(3-30V), 10-SS source for pressure transducer
(10V), 11-A/D converter, 12-procesor, 13-metallic
head, 14-push bar, 15-scale of volume
.
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Capillary rheometer
EXM6
Example Relationship between flowrate and
pressure drop for power law fluid
Consistency variables
Model parameters K,n are evaluated from diagram
of consistency variables
.
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Capillary rheometer
EXM6
DPtotal DPres DPe DPcap
.
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Elastic solids
EXM6
.
Lempická
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Elastic solids
EXM6
Constitutive equations are usually designed in a
different way for different materials one class
is represented by metals, crystals, where arrays
of atoms held together by interatomic forces
(elastic stretches can be of only few percents).
The second class are polymeric materials
(biomaterials) characterized by complicated 3D
networks of long-chain macromolecules with freely
rotating links interlocking is only at few
places (cross-links). In this case the stretches
can be much greater (of the order of tens or
hundreds percents) and their behavior is highly
nonlinear.
.
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Elastic solids Deformation tensor
EXM6
transforms a vector of a material segment from
reference configuration to loaded configuration.
Special case - thick wall cylinder
?t ?r and ?z are principal stretches (stretches
in the principal directions). There are always
three principal direction characterized by the
fact that a material segment is not rotated, but
only extended (by the stretch ratio). In this
specific case and when the pipe is loaded only by
inner pressure and by axial force, there is no
twist and the principal directions are identical
with directions of axis of cylindrical coordinate
system. In this case the deformation gradient has
simple diagonal form
.
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EXM6
Elastic solids Cauchy Green tensor
Disadvantage of deformation gradient F - it
includes a rigid body rotation. And this rotation
cannot effect the stress state (rotation is not a
deformation). The rotation is excluded in the
right Cauchy Green tensor C defined as
Deformed state can be expressed in terms of
Cauchy Green tensor. Each tensor of the second
order can be characterized by three scalars
independent of coordinate system (mention the
fact that if you change coordinate system all
matrices F,C will be changed). The first two
invariants (they characterize magnitudes of
tensor) are defined as
Material of blood vessel walls can be considered
incompressible, therefore the volume of a loaded
part is the same as the volume in the unloaded
reference state. Ratio of volumes can be
expressed in terms of stretches
(unit cube is transformed to the brick having
sides ?t ?r ?z)
.
Therefore only two stretches are independent and
invariants of C-tensor can be expressed only in
terms of these two independent (and easily
measurable) stretches
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EXM6
Elastic solids Mooney Rivlin model
Using invariants it is possible to suggest
several different models defining energy of
deformation W (energy related to unit volume
this energy has unit of stress, Pa)
Example Mooney Rivlin model of hyperelastic
material
Remark for an unloaded sample are all stretches
1 (?t ?r ?z1) and Ic3, IIc3, therefore
deformation energy is zero (as it should be).
Stresses are partial derivatives of deformation
energy W with respect stretches (please believe
it wihout proof)
.
These equations represent constitutive equation,
model calculating stresses for arbitrary
stretches and for given coefficients c1, c2. At
unloaded state with unit stretches, the stresses
are zero (they represent only elastic stresses
and an arbitrary isotropic hydrostatic pressure
can be added giving total stresses).
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EXM6
Evaluation of stretches and stresses
Only two stretches is to be evaluated from
measured outer radius after and before loading
ro, Ro, from initial wall thickness H, and
lengths of sample l after and L before loading.
This is quadratic equation
Therefore it is sufficient to determine Ro,H,L
before measurement and only outer radius ro and
length l after loading, so that the kinematics of
deformation will be fully described.
Corresponding stresses can be derived from
balance of forces acting upon annular and
transversal cross section of pipe
?t
.
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EXM6
Elastic solids instruments

• Uniaxial testers
• Sample in form af a rod, stripe, clamped at ends
  and stretched
• Static test
• Creep test
• Relaxation test

.
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EXM6
Elastic solids instruments

• Biaxial testers
• Sample in form of a plate, clamped at 4 sides to
  actuators and stretched
• Anisotropy
• Homogeneous inflation

.
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EXM6
Elastic solids instruments

• Inflation tests
• Tubular samples inflated by inner overpressure.
• Internal pressure load
• Axial load
• Torsion

CCD cameras of correlation system Q-450
Pressure transducer
Pressurized sample (latex tube)
Laser scanner
Axial loading (weight)
.
Confocal probe