Kinematics Deformation Flow Kinematics Finite Strains

Deformation and Flow Solid continua deform and fluid continua flow Deformation is measured by the strain tensor, which describes change of shape Flow is measured by the rate-of-deformation tensor which is related to velocity gradients. This lecture focuses on deformation Feb 2001 Added some extra slides and animated Jan2000 Fixed typos on slides 4, 11, 12, 15 Feb2000 This has been shortened and slightly edited from BE250A topic 4

Kinematics u = x ­ X X x Reference State Current State (“undeformed”) (“deformed”) Time: t=0 t=t Region: R0 R P x u = x ­ X O P X undeformed (or “material”) coordinates current (or “spatial”) coordinates

Lagrangian & Eulerian Descriptions Lagrangian or “material” description of motion Motion as seen by the convecting material particle Undeformed coordinates label material points Most useful for solid mechanics problems Eulerian or “spatial” description of motion X x = ( , ) t Motion as seen by a fixed spatial observer Current spatial coordinates of material points change Most useful for fluid flow problems

Displacement Vector The displacement vector, u

Deformation Gradient Tensor

Displacement Gradient Tensor The displacement vector, u

Polar Decomposition Theorem

(Finite) Strain Tensors Strain is a measure of change of shape independent of rotation. Change of shape corresponds to change of length (i.e. stretch) Lagrangian Green’s Strain Tensor Eulerian Almansi’s Strain Tensor

Infinitesimal (Cauchy) Strain The Green’s and Alamansi strain tensors are exact measures of shape change for any finite deformation, but they are nonlinear. In terms of the displacement gradients, the finite strains are quadratic, e.g. When the displacement gradients are small enough (<1%), we may linearize the finite strains to obtain the infinitesimal Cauchy strain tensor:

Strain is Change in Length Consider the squared length elements: Hence, E RS is a measure of squared length-change ds dS dX 2 - = R S

Simple extension Uniform extension

Simple Shear X2 Simple Shear x X = + tan γ 𝛾𝛾 X1 tan γ 3 = + tan γ 𝛾𝛾 X1 tan 2 γ vanishingly small for small strain

Pure Torsion of a Cylinder = Θ + θ α Z z α is the twist per unit length of the tube

Length, Area and Volume Change The stretch and strain tensors U, V, C, B, E and e all describe how material elements of length change from the undeformed dL to deformed dl states, e.g. Nanson’s formula relates elements of area in the deformed da and undeformed dA states: Elements of volume in the deformed dv and undeformed dV states are related by detF: dv = detF dV

Measurement Considerations To overcome problem of defining natural state in an experimental setting, reference dimensions in uniaxial tension tests are often recorded at a finite extension, λ* Measured or Nominal stress where A = reference (undeformed) area Cauchy or true stress a = deformed area, λ = extension (stretch) ratio