1. Continuum Mechanics for Hillslopes: Part V
Focus on constitutive relationships
Reading for next week: (Korup et al., 2012)
Homework:
Translate and improve one of the lectures that has already been given based on the reading by Major, Add good, physical examples of how these concepts are applied. The best of these will be used in future course offerings. Due 26 Sept.
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2. Constitutive Relations
Definition:
Constitutive Equations: expressions that describe the relationships between stress and strain, or stress and rates of distortion.
Goal: to relate stress tensor to strain tensor
Not derived from general laws of mechanics but rather from empirical (laboratory or field) observations!
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3. Constitutive Relations
Linearly Viscous Fluid
Rates of strain are related linearly to stresses
Strains immediately and indefinitely upon application of shear stress
Non-newtonian fluids: non-linear relationship or threshold behavior
Linearly Elastic Material
Strains immediately upon application of a stress
Once the stress is removed, all the strain is recovered (Think “spring”)
Applicable to small elastic deformation of materials
Travel of seismic waves through rock
Flexure of lithospheric plates under large loads
Strains are linearly related to stresses
Relationships between stress and normal strain
Relationships between shear stress and shear strain
Relationship between pressure and dilatation
“Hookian” elastic solid
Plasticity – the Coulomb Failure Rule
No strain occurs until reaching a threshold (yield stress).
Once crossed, the material strains indefinitely. (Think, “block on an inclined plane”)
Not proportional to stress and not recoverable after stress is removed.
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4. Linearly Elastic Material: Relationships between stress and normal strain
An ideal linearly elastic material:
stress is linearly proportional to strain
(Hooke’s Law)
for uniaxial normal stress:
σ represents a uniaxial normal stress (positive in tension)
ε represents strain (positive in elongation, - for comp.)
E is the constant of proportionality: Young’s modulus
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5. Linearly Elastic Material: Relationships between stress and normal strain
If a tensional normal stress is exerted in the y-coordinate direction, Hooke’s law can be written as:
The ratio of contraction to extension is known as Poisson’s ratio and is expressed for uniaxial stress as:
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6. Linearly Elastic Material: Relationships between stress and normal strain
Elongation in the y-direction also causes contraction in the z-direction.
For an isotropic material, under uniaxial stress, the amount of contraction in the z-direction is identical to the amount of contraction in the x-direction.
If the material is incompressible, thus conserving volume, dilation is zero: z
θ = εxx+ εyy+ εzz = 0
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7. Linearly Elastic Material: Relationships between stress and normal strain
For an isotropic material subject to triaxial rather than uniaxial stress, we can write the normal strains along the coordinate directions in terms of Young’s modulus and Poisson’s ratio as:
Using:
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8. Linearly Elastic Material: Relationships between stress and normal strain
We see that normal strains in any single coordinate direction are related to normal stresses applied in all coordinate directions,
not just to the normal stress applied in the
direction coincident with the normal strain.
These relations can also be written in terms of principal stresses and principal strains (1,2,3 instead of xx, yy, zz):
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9. Linearly Elastic Material: Relationships between shear stress and shear strain
Relationships among shear stress, shear strain, and elastic moduli are derived by considering a special case of plane stress that is known as pure shear
For this case, , and we shall assume
Furthermore, we shall consider the strain that occurs within a plane in a coordinate system that is rotated with respect to the principal stress axes.
Under these constraints :
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10. Linearly Elastic Material: Relationships between shear stress and shear strain
-considering this special case of
plane stress called pure shear.
applying:
to:
we get:
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11. Linearly Elastic Material: Relationships between shear stress and shear strain
Given:
Maximum shear stress occurs on a plane that makes a 45° angle to the principal stress axes If
Then
Thus from we get:
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13. Linearly Viscous Fluid:
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14. Linearly Viscous Fluid:
For an isotropic, incompressible fluid eqn [99] can be generalized to three dimensions as the the viscous stress Tensor:
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15. Linearly Viscous Fluid:
For an isotropic, incompressible, linearly viscous fluid, we can write the above as:
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16. Linearly Viscous Fluid:
If the viscosity of the fluid is constant:
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17. Linearly Viscous Fluid:
For an incompressible fluid, conservation of mass dictates that the bracketed third term on the right hand side of the equation, known as the divergence of the velocity field, is zero.
This expression is the x-direction component of an equation known as the ‘Navier–Stokes equation’ for an incompressible, linearly viscous fluid of constant viscosity.
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18. Linearly Viscous Fluid:
The viscous stress term on the right-hand side of eqn [105] can be written more compactly using the Laplacian operator as:
The Navier–Stokes equation, derived using the conservation of momentum equation, is a control volume representation of Newton’s second law for an incompressible, linearly viscous
fluid. Therefore, eqn [107] states that the inertial acceleration (left-hand term) equals the sum of forces (right-hand terms) acting on the system. Thus, the left-hand term can be viewed as an inertial force, and the right-hand terms as pressure, viscous, and body forces, respectively.
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19. Linearly Viscous Fluid:
If we now introduce the concept of the Reynolds number (Re), which represents the ratio of inertial forces to viscous forces in a fluid, we can highlight some limiting applications of the Navier–Stokes equation. For low- Re flows, viscous forces greatly exceed inertial forces; thus, the left-hand term is negligibly small and eqn [107] can be simplified as:
For high-Re flows viscous forces are assumed to be relatively unimportant, and eqn [107] can be simplified to: which is known as Euler’s equation. This equation, which chiefly balances inertial and pressure forces, has applications, for example, in river hydraulics.
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21. (And owing to symmetry of shear stresses and shear strains, εxy = εyx)
Linearly Elastic Material: Relationships between shear stress and shear strain
Because we find that:
We learn that for
Thus for rotated coodinates:
Giving:
(And owing to symmetry of shear stresses and shear strains, εxy = εyx)
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22. Linearly Elastic Material: Relationships between shear stress and shear strain
Noting that εxy is the average shear strain in the xy plane
and is equal to ½γxy
(where γxy is the total, or engineering, shear strain in the xy plane)
Or:
Where:
(the rigidity modulus, or shear modulus)
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23. Plasticity – The Coulomb Failure Rule
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24. Example Application No time for this in this lecture.
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25. Conservation of Momentum and Stress Equilibrium
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26. Effective Stress and Effective Stress Equilibrium
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27. Effective Stress and Elastic Strain
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28. Displacement Formulation of Constitutive Relations and Groundwater Flow
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