Stress, Strain, and Viscosity

Stress, Strain, and Viscosity San Andreas Fault Palmdale Solids and Liquids Solid Behavior: Liquid Behavior: - elastic - fluid - rebound - no rebound - retain original shape - shape changes - small deformations are temporary - permanent deformation - (e.g. Steel, concrete, wood, - (e.g. Water, oil, rock, lithosphere) melted chocolate, lava) Solids and Liquids Linear Viscous Fluid: - Rate of deformation is proportional to the applied stress - A “linear” viscous fluid (w.r.t. stress and strain rate) - Also known as a “Newtonian Fluid” Solids > ? > Liquids Is there anything that behaves in a way between solid and liquid ? Plastic Material - solid - but deforms permanently - malleable - ductile Ductile or malleable materials are “non-linear” Solids and Liquids Are all materials either a solid or a fluid, all the time ? Applied heat can cause solid materials to behave like a fluid Some material may be elastic when small forces are applied but deform permanently with larger applied forces Elastic: Deformed material returns to original shape  Ductile: Stress exceeds the elastic limit and deforms material permanently  This rock responded to stress by folding and flowing by ductile deformation. Occurs under high heat and high pressure This rock fractured under stress by brittle deformation. Occurs under low heat and at shallow depths in the Earth's crust. Three types of stress Stress: is the force acting on a surface, per unit area – may be greater in certain directions. TENSIONAL STRESS Figure 9.2 Unstressed cube of rock COMPRESSIONAL STRESS SHEAR STRESS Strain V B B' D D' h A C Consider a layer of fluid between 2 plates The top plate moves with velocity, V The shape change can be written as V t / H The rate of shape change is d/dt(shape change) . rate of deformation =  = V/H (units of 1/time) Stress V B B' D D' h A C L Pressure is applied to move the fluid Stress is described as force per unit area (units of pressure, Pa)  = force/area = F/LW Viscous Fluid V B B' D D' h A C L A viscous fluid is defined by the relationship of stress to strain rate .  2 Viscosity () is the constant of proportionality (units of Pas) This “constitutive” equation describes the mechanical properties of A material Viscous Fluid .  2 If material has a high viscosity (), it will strain less for a given applied stress ()  = / 2 Physical Laws of Motion Newton's 1st Law: Object in motion stays in motion; Action and reaction, Velocity motion imparted by the top plate induces a reaction of the fluid below Physical Laws of Motion Newton's 2nd Law: Acceleration of fluid is proportional to the net force What does this mean? - If there is no acceleration, then forces balance, that is things move but don't accelerate. In this case, forces balance and there is “no net force” For all viscous fluids, the net force = 0 Velocities in the Earth's mantle are small and accelerations are negligible Momentum is also negligible in slow viscous fluids Conservation of Mass Mass is conserved in fluid flow (density changes are negligible) Fluid is “incompressible” . Rate of fluid flow into box = flow out of box No net accumulation of material See Class notes – Formal treatment of stress and strain Stress > Strain > Viscosity > Hydrostatic Stress  Hydrostatic stress is defined as confining pressure  Normal stresses acting on a particle are equal on all sides  Known as “isotropic” 11 = 22 = 33 = -P  With no tangential components 12 =  = 3 = 0 Deviatoric Stress  Mantle flow is not driven by hydrostatic pressure  But is driven by deviations from it, known as deviatoric stress  Let's consider an average normal stress ave11+ 22+ 33 = ii/ 3 3  Then deviatoric stress is given by devijave the applied stress - average normal stress = difference (deviatoric stress) Deviatoric Stress Minimum stress devijave  Maximum stress If deviatoric stress is non-zero, than fluid flow proceeds 11 – ave 12 21 22 -ave 31 32 Deviatoric stress 13 23 33 - ave If the diagonals are all equal, then there is no deviatoric stress  And there is no fluid flow  Lithostatic Stress  A special case of hydrostatic stress  Hydrostatic stress increases with depth in the Earth  But at each depth, normal stresses are balanced As a slab sinks into the Earth's interior, it experiences progressively increasing hydrostatic stress with depth known as lithostatic stress.  Measuring Lithostatic Stress Near the surface, stress measurements are strongly effected by faults, joints, etc.  At deeper depths, pressure closes faults and fractures and stresses are transmitted across faults  Activity: calculate deviatoric stress in a continental block  Normal faults in El Salvador Jointing in a rock Depth Measuring Lithostatic Stress Lin et al., 2006 Fracture density measured in the field in south Africa  Boreholes show reduced hydraulic conductivity at deeper depths  Indicating that fractures close with increasing lithostatic stress  Strain Strain is a measure of deformation When deformation occurs, different parts of a body are displaced by different amounts x2 p3 p1 p2 x1 p4 u1 = b x2 Displacements (u) in x1 direction increase with increasing x2 . No displacement in x2 direction: u2 = 0 Strain x2  p1 p2 x1 p3 p4 Displacement of p1 and p2 are not equal This implies a gradient of displacement d u1 = b = - tan  d x2 Where  is the angle through which the body deforms Shearing deformation changes along the x2 direction. x2 Strain: Solid Body Rotation  p1  p2 x1 If 1 = 2 , then we have rotation only and no deformation In this case, the gradient in deformation can be written as: tan  tan  d u2 d x1 - d u1 d x2 Solid body rotation suggests the gradient of deformations are equal Strain: How do we distinguish deformation and rotation ? Think about how 1 and 2 relate to each other   ( Rotation only (nonzero Deformation (0 Pure shear ( Simple shear 0 Strain: How do we distinguish deformation and rotation ? We generally write this in terms of the tangent of the rotation angle  d u2 12 = ½ ( tan tan ½ ( d x1 Deformation: d u1 12 = ½ ( tan tan ½ ( d x2  Rotation: d u1 ) d x2 d u2 + ) d x1 The Strain Tensor The strain tensor can them be written And the rotation tensor can them be written d uj ij =½ ( d xi d ui ) d xj  d uj d ui ij =½ ( + ) d xi d xj  The Strain Tensor : Test The strain tensor can them be written d uj d ui ij =½ ( + ) d xi d xj Let's try a test: In the case of a box that is “stretched” What is i and j in ij ? Then find the solution The Strain Tensor : Case of Volume Change If a cube expands (or shrinks) V – Vo Vo ij = u = 11 + 22 + 33 d ui  d xi u Indicates volume change, dilitation, divergence The Strain Tensor : Case of Volume Change u =0 Means there is no volume change and fluid is incompressible Strain Rate Strain rate describes deformation change over time How fast can a material deform ? How fast can fluid flow ? Strain Rate Strain rate describes deformation change over time How fast can a material deform ? How fast can fluid flow ? Rates are concerned with velocity d Vj d Vi ij =½ ( + ) d xi d xj We can use velocity gradient to measure the rate of fluid flow or shear