Conservation Laws for Continua

1. Conservation Laws for Continua Mass Conservation Linear Momentum Conservation Angular Momentum Conservation

2. Work-Energy Relations Rate of mechanical work done on a material volume Conservation laws in terms of other stresses Mechanical work in terms of other stresses

3. Principle of Virtual Work (alternative statement of BLM) If for all Then on

4. Thermodynamics Temperature Specific Internal Energy Specific Helmholtz free energy Heat flux vector External heat flux Specific entropy First Law of Thermodynamics Second Law of Thermodynamics

5. Transformations under observer changes Transformation of space under a change of observer All physically measurable vectors can be regarded as connecting two points in the inertial frame These must therefore transform like vectors connecting two points under a change of observer Note that time derivatives in the observer’s reference frame have to account for rotation of the reference frame

6. Some Transformations under observer changes

7. Some Transformations under observer changes Objective (frame indifferent) tensors: map a vector from the observed (inertial) frame back onto the inertial frame Invariant tensors: map a vector from the reference configuration back onto the reference configuration Mixed tensors: map a vector from the reference configuration onto the inertial frame

8. Constitutive Laws Equations relating internal force measures to deformation measures are knownas Constitutive Relations • General Assumptions: • Local homogeneity of deformation • (a deformation gradient can always be calculated) • Principle of local action • (stress at a point depends on deformation in a vanishingly small material element surrounding the point) • Restrictions on constitutive relations: • 1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer • 2. Constitutive law must always satisfy the second law of • thermodynamics for any possible deformation/temperature history.

9. Fluids Properties of fluids • No natural reference configuration • Support no shear stress when at rest Kinematics • Only need variables that don’t dependon ref. config Conservation Laws

10. General Constitutive Models for Fluids Objectivity and dissipation inequality show that constitutive relations must have form Internal Energy Entropy Free Energy Stress response function Heat flux response function In addition, the constitutive relations must satisfy where

11. Constitutive Models for Fluids Elastic Fluid Ideal Gas Newtonian Viscous Non-Newtonian

12. Derived Field Equations for Newtonian Fluids Unknowns: Must always satisfy mass conservation Combine BLM With constitutive law. Also recall Compressible Navier-Stokes With density indep viscosity For an incompressible Newtonianviscous fluid Incompressibility reduces mass balance to For an elastic fluid (Euler eq)

13. Derived Field Equations for Fluids Recall vorticity vector Vorticity transport equation (constant temperature, density independent viscosity) For an elastic fluid For an incompressible fluid If flow of an ideal fluid is irrotational at t=0and body forces are curl free, then flow remains irrotational for all time (Potential flow)

14. Derived field equations for fluids For an elastic fluid along streamline • Bernoulli For irrotational flow everywhere For incompressible fluid

15. Normalizing the Navier-Stokes equation Incompressible Navier-Stokes Normalize as Reynolds number Euler number Strouhal number Froude number

16. Limiting cases most frequently used Ideal flow Stokes flow

17. Solving fluids problems: control volume approach Governing equations for a control volume (review)

18. Example Steady 2D flow, ideal fluid Calculate the force acting on the wall Take surrounding pressure to be zero

19. Exact solutions: potential flow If flow irrotational at t=0, remains irrotational; Bernoulli holds everywhere Irrotational: curl(v)=0 so Mass cons Bernoulli

20. Exact solutions: Stokes Flow Steady laminar viscous flow between plates Assume constant pressure gradient in horizontal direction Solve subject to boundary conditions

21. Exact Solutions: Acoustics • Assumptions: • Small amplitude pressure and density fluctuations • Irrotational flow • Negligible heat flow Approximate N-S as: For small perturbations: Mass conservation: Combine: (Wave equation)

22. Wave speed in an ideal gas Assume heat flow can be neglected Entropy equation: so Hence:

23. Application of continuum mechanics to elasticity Material characterized by

24. General structure of constitutive relations Frame indifference, dissipation inequality

25. Forms of constitutive relation used in literature • Strain energy potential

26. Specific forms for free energy function • Neo-Hookean material • Mooney-Rivlin • Generalized polynomial function • Ogden • Arruda-Boyce

27. Solving problems for elastic materials (spherical/axial symmetry) • Kinematics • Assume incompressiblility • Constitutive law (gives ODE for p(r) • Equilibrium (or use PVW) • Boundary conditions

28. Linearized field equations for elastic materials • Approximations: • Linearized kinematics • All stress measures equal • Linearize stress-strain relation Elastic constants related to strain energy/unit vol Isotropic materials:

29. Elastic materials with isotropy

30. Solving linear elasticity problems spherical/axial symmetry • Kinematics • Constitutive law • Equilibrium • Boundary conditions

31. Some simple static linear elasticity solutions Navier equation: Potential Representation (statics): Point force in an infinite solid: Point force normal to a surface:

32. Simple linear elastic solutions Spherical cavity in infinite solid under remote stress:

33. Dynamic elasticity solutions Plane wave solution Navier equation Solutions: