CH2 The Meaning of the Constitutive Equation

1. CH2 The Meaning of the Constitutive Equation Prof. M.-S. Ju Dept. of Mechanical Eng. National Cheng Kung University

2. 2.1 Introduction • Living system at cellular, tissue, organ and organism level sufficient to take Newton’s laws of motion as axiom • The smallest volume considered contains a very large no. of atoms and molecules. The materials can be considered as a continuum • Isomorphism between real number and material particles. Between any two material particles there is another material particle.

3. DV P DM Continuum: in Euclidean space, between any two material particles there is another material particle, each material particle has a mass. Mass density of a continuum at a point P is defined as:

4. Modification of definition Observation of living organisms at various levels of size: e.g., naked eye, optical microscope, electrical microscope, scanning tunnel microscope or atomic force microscope

5. Blood example • Whole blood – continuum at scale of heart, large arteries, large veins • Two-phase fluid (plasma & blood cells) at capillary blood vessels, arterioles and venules. • At smaller scale: red cell membrane as continuum and red cell content as another continuum

6. 2.2 Stress • Stress expresses the interaction of the material in one part of the body on another. • Unit: 1 Pa = 1 N/m2 • 1 psi = 6.894 kPa • 1 dyn/cm2 = 0.1 Pa • 1 atmosphere = 1.013 x 105 N/m2 = 1.013bar

7. DF n x3 DS S B x2 o x1 • Assume that asDS tends to zero the ratio DF/DS tends to a definite limit, dF/ds, and moment of the force acting on the surface DS tends to zero. Stress 

8. Let vector n benormal of DS Positive side: surface pointed by n Positive side exerts force DF on the negative side DF depends on location and size of DS and orientation of n When DS→0 moment of DF exerts on DS →0 force per unit area acting on the surface

9. t33 t32 x3 t31 t23 n t22 t13 t12 t21 t11 ds x2 o x1 Components of stress

10. Note: knowing components of a stress tensor one can write down the stress vector acting on any surface with unit outer normal vector n

11. (2) For a body in equilibrium we have Xi: components of body force (per unit volume) along ith axis Due to equilibrium of moments

12. (4) change of coordinate system • t’km components of stress tensor in the new coordinate system • tij components of stress tensor in theold coordinate system

13. 2.3 Strain • Deformation of a solid described by strain • Take one-dimensional problem as an example: elongation of a string, initial length L0

14. Other definitions: • Note: above strain measures are equal for infinitesimal elongation

15. M a • Another is Shear Strain，consider the twist of a circular cylindrical shaft. tana or tana/2 is defined as the shear strain M Constitutive Equation: a relationship between stress and strain

16. a3 ,x3 P’ P” Q” Q’ u P Q a2, x2 S (a1, a2, a3) o (x1, x2, x3) a1, x1 Deformation of living system is more complicated and require a general method. Let a body occupy a space S，let coordinate of a particle before deformation be (a1,a2,a3)，coordinate after deformation(x1,x2,x3)， Q: stretching and distortion of the body?

17. After the deformation p and p’ displaced to Q and Q’，QQ’ distance dS

19. Definition of strain tensors: Show that Eij and eij are symmetric

20. When ui is small eij reduces to Cauchy’s infinitesimal strain tensor: Cauchy strain tensor

21. or In the infinitesimal deformation, no distinction between Lagrangian and Eulerian strain tensors.

22. y y y x x x Geometric meaning y u u x x

23. 2.4 Strain rate • For fluid motion, consider velocity field and rate of strain. At point (x,y,z) velocity vector For continuous flow, Vi :continuous and differentiable

24. Define strain rate tensor Vij Define vorticity tensor Wij

25. 2.5 Constitutive equations • Properties of materials are specified by constitutive equations • Non-viscous fluid, Newtonian viscous fluid and Hookean elastic solid are most widely models for engineering materials • Most biological materials can not be described by above equations • Constitutive equations are independent of any particular set of coordinates. A constitutive equation must be a tensor equation: every term in it be a tensor of same rank.

26. 2.6 The Nonviscous Fluid • Stress tensor: • D Kronecker delta, p: pressure (scalar) • For ideal gas: equation of state • For real gas: f (p, r, T) = 0 • Incompressible fluid: r = constant

27. 2.7 Newtonian Viscous Fluid • Shear stress is proportional to strain rate • Stress-strain rate relationship sij: stress tensor, Vkl: strain rate tensor, p: static pressure • p= p(r, T) equation of state • Elements of Dijkl depend on temperature but stress or strain rate • Isotropic materials: a tensor has same array of components when frame of reference is rotated or reflected (isotropic tensor)

29. Coefficient of viscosity m • Newton proposed • Units: • 1 poise= dyne. s /cm2 = 0.1 Ns/m2 • Viscosity of air: 1.8 x 10-4 poise • Water: 0.01 poise at 1 Atm. 20 deg C • Glycerin: 8.7 poise

30. 2.8 Hookean Elastic Solid • Hooke’s law: stress tensor linearly proportional to strain tensor Note: elastic moduli are independent of stress or strain

31. Isotropic materials

32. Inverted form

33. E: Young’s modulus, G: shear modulus, n: Poisson’s ratio

34. 2.9 Effect of Temperature • The constitutive equations are stated at a given temperature T0 • Dijkl, Cijkl, m depend on temperature • If temperature is variable: Duhamel-Neumann form

35. For isotropic material a: linear expansion coefficient

36. 2.10 Materials with more complex mechanical behavior • In limited ranges of temperature, stress and strain, some real materials may follow above constitutive equations • Real materials have more complex behavior: • Non-Newtonian fluids: blood, paints and varnish, wet clay and mud, colloidal solutions • Hookean elastic solid: structural material within elastic limit, disobey Hooke’s law for yielding & fracture • Few biological tissues obey Hooke’s law

37. 2.11 Viscoelasticity • Features: hysteresis, relaxation, creeping • Stress Relaxation • Body is suddenly strained and maintained constant, the corresponding stress decreases with time • Creep • Body is suddenly stressed and maintained constant, the body continues to deform • Hysteresis • Body subjected to cyclic loading, the stress-strain relationship is different between loading cycle and unloading cycle.

38. Mechanical Models of Viscoelastic Materials • Maxwell model (series) • Voigt model (parallel) • Kelvin model (standard linear solid) (series + parallel) Lumped mass model consisted of linear springs and dashpots spring constant: m viscous coefficient of dashpot: h

39. h m F F u2 u1 u Maxwell model

40. h F1 m F F2 u Voigt model

41. h1 m1 F1 u1 u2 F F m0 F0 u Kelvin model relaxation time for constant strain relaxation time for constant stress

42. Creep function • When F(t) is unit-step function, solutions of (1)(2)(3) unit-step function

43. Creep function Maxwell Voigt Kelvin u u u 1/ER 1/m 1/h 1 t t t F F F 1 1 1 t t t

44. Relaxation function • When u(t) is a unit-step function, F(t)=k(t) d 1

45. Relaxation function F Kelvin F Voigt Maxwell hd(t-t0) m m ER t u u u 1 1 1 t t0 t0

46. General linear viscoelastic model by Boltzmann • Lumped mass  continuum (Boltzmann model) F(t) F(t) simple bar model u(t) u Dt t t F Dt t t

47. Assumptions • F(t) continuous & differentiable • In dt, increment of F(t) = (dF/dt)dt • Increment of u(t) due to F(t): du(t), t > t Relationship between du(t) and F’(t)dt creep function convolution integral

48. relaxation function Similarly, we can define the relaxation function • Notes: • Maxwell, Voigt & Kelvin models are special case of Boltzmann model • 2) Relaxation function can be approximated by Fourier series

49. a Spectrum of relaxation func. n n1 n5 n2 n4 n3 an: coefficient vn: characteristic frequency an (vn ) : discrete spectrum Note: in living tissue such as mesentery continuous spectrum is required

50. Generalization to viscoelastic materials • Assumptions: small deformation, infinitesimal displacements, strains and velocities F  s, u  e, c, k  tensor