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

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)

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

Viscous Fluids (kinematics) EXM6 Delvaux

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

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. .

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.

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 .

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 .

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

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

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

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. .

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 .

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 .

Capillary rheometer EXM6 DPtotal = DPres + DPe + DPcap .

Elastic solids EXM6 . Lempická

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 .

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 .

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

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).

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 .

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 .

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 .

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