1. Experimental methods E181101
EXM6
Material testing constitutive equations
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. 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 x y z s zz zy zx yz yy yx xz xy xx
Index of plane index of force component (cross section) (force acting upon the cross section i)

3. 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

4. Viscous Fluids (kinematics)
EXM6
Delvaux

5. 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
U=ux(H) y x
The only nonzero component of deformation rate tensor in case of simple shear flow
is called shear rate

6. 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. .

7. Viscous Fluids (rheology)
EXM6
General formulation for fully three-dimensional velocity field
Viscosity  is constant in Newtonian fluids and depends upon second invariant of deformation[1] 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.

8. 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)
Viscoelastic material – phase shift 0<<90
Viscous liquid- zero stress corresponds to zero strain rate (maximum ) =900 .

9. Rheograms (shear rate-shear stress)
EXM6
n=1 Newtonian fluid, n<1 pseudoplastic fluids (n is power law index)
n=1.5
n=1
n=0.8
n=0.5
Shear stress
Shear rate .

10. DMA Dynamic Material Analysis and Oscilograms
EXM6
storage modulus
loss modulus
polyoxymethylene
Viscous properties E’’
Elastic properties E’ .

11. Viscoelastic effects EXM6
Weissenberg effect (material climbing up on the rotating rod)
Barus effect (die swell)
Kaye effect .

12. Upper convective derivative
Viscoelastic models
EXM6
Oldroyd B model
Extra stress S
Deformation rate
Upper convective derivative .

13. Rheometers EXM6 Rheometry (identification of constitutive models).
Rotating cylinder
Plate-plate, or cone-plate
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. .

14. Capillary rheometer EXM6
1–glass 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 .

15. 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 .

16. Capillary rheometer
EXM6
DPtotal = DPres + DPe + DPcap .

17. Elastic solids
EXM6 .
Lempická

18. 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.
"Dogbone" sample .

19. Elastic solids Deformation tensor
EXM6
transforms a vector of a material segment from reference configuration to loaded configuration. Special case - thick wall cylinder R r r t z
reference configuration
loaded configuration
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 .

20. Elastic solids Cauchy Green tensor
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

21. Elastic solids Mooney Rivlin model
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 = z=1) and Ic=3, IIc=3, 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) t z .
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).

22. Evaluation of stretches and stresses
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 z h p r0
G-force t .

23. Elastic solids instruments
EXM6
Elastic solids instruments
Uniaxial testers
Sample in form af a rod, stripe, clamped at ends and stretched
Static test
Creep test
Relaxation test .

24. Elastic solids instruments
EXM6
Elastic solids instruments
Biaxial testers
Sample in form of a plate, clamped at 4 sides to actuators and stretched
Anisotropy
Homogeneous inflation .

25. CCD cameras of correlation system Q-450
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