Expectations after today’s lecture Know stretch, deformation gradient, and deformation tensor Know the strain descriptions Engineering True Almansi Green Know how to obtain strain from stretch or displacement Be able to transform a state of strain from one system of coordinates to another and find principal strains using: Direct methods Mohr’s circle Eigenvalues and eigenvectors Revisit stress for generalize case. (Previously formulated finite descriptions only for cases without shear stresses.)

Important observation: The part of the brain that’s good for math is different from the part that can communicate math.

“Mathematics has no symbols for confused ideas. ” “Mathematics has no symbols for confused ideas.” George Stigler “Calculus is the language of God.” Richard Feynman

Strain Continuum Mechanics BME 615 Strain Continuum Mechanics

Infinitesimal strain description (with displacements)

Finite Strain Salvador Dali

Patterns of deformation These are cases of uniform strain (or, linearly changing strain vertically but uniform axially in bending) across the specimen. Consider case (a) where L = final length and L0 = initial length. Strain (which is normalized deformation) can be normalized in number of ways. Fung YC, Biomechanics ref. p29   If L = 1.01 and L0 = 1.00, or 1% strain If L = 2 and L0 = 1, Engineering Strain True Strain Almansi Strain Green Strain

1D Strain defined by stretch Define stretch Infinitesimal Strain (engineering strain) Finite Strain (typically used when strain exceeds 10%) In Lagrangian (material) reference system, define Green (St. Venant) strain In Eulerian (spatial) reference system, define Almansi (Hamel) strain

Strain defined by stretch - on a differential element Now define stretch on a diff. element dX2 dx2 dX3 dx3 dX1 dx1 Original shape differential lengths Deformed shape differential lengths In 1D, Lagrangian deformation tensor Note: There can also be shear deformations defined by stretch In 1D, Green strain

Strain defined by stretch – continued 1 In 3D, stretch l becomes Lagrange deformation gradient F In 3D, Lagrange deformation tensor C – also called the right Cauchy-Green deformation tensor is So that Green strain in 3D is where is the identity matrix (ones on diagonal terms and zeros elsewhere)

Strain defined by stretch – continued 2 the Cauchy deformation tensor c in 1D in Eulerian reference system can be used to define Almansi strain e on a differential element Note: Green strain and Almansi strain are consistent measures of normalized finite deformation in their respective reference systems. Engineering strain is not! Engineering strain is a first order approximation that works well when deformations are small (usually < 10%) In 3D, becomes the Eulerian deformation gradient f The 3D Eulerian or Cauchy deformation tensor c is where B is the left Cauchy-Green deformation tensor

Other common measures of strain Spherical strain: Note: Constant relating pressure to spherical strain is called the Bulk Modulus Mean normal strain: Deviatoric strain: where δij is Kronecker delta: = 1 if i = j and = 0 if i ≠ j

Volumetric Change V0 = 1 li = L/L0 = 2 Ai = 4 V = 8

Biaxial Stretch 2 1 1.5 1

Shear Deformation – 45o 2 1 1 1 1 1

Shear Deformation – 45o 2 1 1 v = 0 1 1 1

Shear Deformation – 45o 2 1 1 v = 0 1 1 1 If thickness into plane remains the same, have we lost volume? How do you know?

Shear Deformation – 45o 2 1 Note: Above is affine mapping of where Check corners to prove Then derivatives give the same results!

Definition of “congugates” from Oxford English Dictionary Mathematics: Joined in a reciprocal relation Biology: Fused Chemistry: Related to….. Mechanics: Variables that are defined in such a way they are duals of one another

Energy Conjugates Lagrange Stress T and right Cauchy-Green deformation tensor C Kirchhoff Stress S and Green Strain E (Lagrangian reference system) Cauchy Stress s and Almansi Strain e (Eulerian reference system)

Energy Conjugates & SED (assume incompressibility) V = 1 F = 1 l = L/L0 = 2

Energy Conjugates & SED 0.25 0.5 0.75 Strain Stress 1.0 1.25 1.5 2.0 s = 2 e = 3/8 + W = 3/8 s = 1 e = 1 + W = 1/2 S = 1/2 E = 3/2 + W = 3/8

Finite strain descriptions (with displacements) Y X

Finite strain from deformations Common notation here can be troubling. Do not confuse deformations with displacements for i = 1, 2, 3 where is the original coordinate vector is the deformed coordinate vector is the displacement vector http://en.wikipedia.org/wiki/Finite_strain_theory

3 things that drive me crazy! Stretch ≠ strain Mechanical behavior ≠ material behavior Deformation ≠ displacement

Green strain from deformations for i, j = 1, 2, 3 sum on k Thus, for i = 1, j = 1 Note, for small deformations, higher order terms are not significant. Note, for 1D we can easily go from previous formula for E to current form.

Almansi strain from deformations for i, j = 1, 2, 3 sum on k

Second Order Strain Tensors Any strain formulation in 3D is a 2nd order tensor. Therefore, it has the following properties: Transformation methods that we used for stress hold for principal strain or maximum shear strain or strain on any axis, etc. Mohr’s circle method holds for strain transformations. All the same invariants hold to describe dilatational (or hydrostatic) versus distortional (or deviatoric) strains. The same methods for eigenvalues and eigenvectors hold.

Revisit Cauchy Stress – (s) Eulerian reference of deformed state Unloaded thickness h0 Loaded thickness h Unloaded density ρ0 Unloaded density ρ where

Lagrangian stress (T) revisited (1st Piola-Kirchhoff stress tensor) Lagrangian reference of undeformed state Unloaded thickness h0 Loaded thickness h Unloaded density ρ0 Unloaded density ρ

Kirchhoff Stress (S) revisited (2st Piola-Kirchhoff stress tensor) Kirchhoff stress references the undeformed state Unloaded thickness h0 Loaded thickness h Unloaded density ρ0 Unloaded density ρ

Example Strain Problem BME 615

Consider a deformation that is given by: Xi (i=1~3) represent original coordinates k represents displacement gradient Start with an undeformed unit square and draw the deformation X1, x1 X2, x2 k 1

Evaluate the Lagrangian deformation gradient tensor X1, x1 X2, x2 k 1 Evaluate the Lagrangian deformation gradient tensor

Deformation tensor Green strain tensor

Strain transformations Just like stress, equations from the direct approach can be used for strain Or these equations can be reformulated with a double angle trig identity

Mohr’s circle approach

Strain gage rosettes

Rectangular rosette 3 equations, 3 unknowns relating

Delta rosette Following similar approach, one can obtain principal strains and orientation

Expectations after today’s lecture Know stretch, deformation gradient, and deformation tensor Know the strain descriptions Engineering True Almansi Green Know how to obtain strain from stretch or displacement Be able to transform a state of strain from one system of coordinates to another and find principal strains using: Direct methods Mohr’s circle Eigenvalues and eigenvectors Revisit stress for generalize case. (Previously formulated finite descriptions only for cases without shear stresses.)