1. Strain at a point by using image analysis Prepared by Prof. John Brunski, –Biomedical Engineering Department, Rensselaer Polytechnic Institute Copyright  2000 –Permission is granted for use in lecture courses. –Rensselaer Polytechnic Institute

2. Strain at a point by using image analysis Given: deformation of a material Find: a mathematical description of the deformation. Answer: strain at a point

3. Activities: determine state of strain in these two cases Stretched rubber band Compressed articular cartilage From Ann Biomed Eng. 1996, Schinagl et al.

4. Example: detailed strain analysis by digital image correlation… of the outlined area, inside of one thread of an implant in bone

5. Deformation Note ink lines, rectangles and triangles drawn on a rubber band. Stretch the rubber band in uniaxial tension by hand. Note that the figures drawn on the rubber band seem to deform differently. However, intuition tells us that the whole piece of rubber band deforms the same – at least in our case of “uniaxial” tension.

6. Strain Strain is a (mathematical) way of describing what happens during deformation of a material. We normally define strain as something that happens “at a point” –the size of the point can vary depending on the goals of the analysis. –In the general case, strain will not be the same from point to point in a material – see example of a soap eraser being deformed.

7. Strain is not a scalar It’s clear also that strain “happens” in different directions. For example: –On a stretched rubber band, there is extension in one direction but contraction in another. –Also, angles between lines may change. It becomes clear that we need more than one or two numbers to “capture” what’s happening during strain.

8. Beyond scalars Strain is a 2 nd -rank tensor (unfortunately!?) –A scalar is a tensor of rank 0; it has 3 0 = 1 component and is invariant with respect to coordinate transformations. –A vector is a tensor of rank 1; it has 3 1 = 3 components and these components vary with the coordinate system while the vector magnitude is invariant. –Strain is a tensor of rank 2; it has 3 2 = 9 components which depend on the coordinate system. (Strain has invariants, but we will ignore them for the present.)

9. Strain in 3D Strain, E*, is usually written as a 3 x 3 matrix for a 3D situation: The matrix is symmetric. –E* 12 = E* 21, E* 13 =E* 31 etc. –There are really only 6 numbers to determine. –The values in E* depend on the coordinate system that’s chosen.

10. Meaning of strain components The diagonal terms are simple stretches or compressions along the 3 coordinate axes. The off-diagonal terms relate to changes in angles between lines.

11. How do we compute E*? Given: deformation of a material. Find: the (finite) strain tensor.

12. 2D analysis The strain tensor really only involves 3 relevant entries in the 2D case: –We need to determine E* 11, E* 12 = E* 21, and E* 22 when the x1 and x2 coordinate system is in the plane of the rubber band and cartilage.

13. Key results from strain theory Measure (L f 2 – L o 2 )/2L o 2 in the direction of a unit vector n. There are 3 unit vectors, one for each side of a chosen triangle. Compute (L f 2 – L o 2 )/2L o 2 for each side; set equal to n E*n You’ll get 3 eqns. in 3 unknowns, E* 11, E* 12 and E* 22 ; solve

14. The entries in the strain tensor depend on the coordinate system that’s used. In our examples we are using an x1-x2 system in which the origin is located in the upper left of the image. The positive x1 points to the right and the positive x2 points down. From observing our rubber band, we expect ahead of time that strain E 11 is most likely going to be largest, with E 22 smaller, and with the shear strains also being small – given that the rectangles on the rubber band remain more or less as rectangles with little angular change.

15. Measurements Use Image-Tool to measure L o and L f for 3 sides of a chosen triangle on rubber band and cartilage. –Do this by using position vectors and vector concepts. Give results to TAs for input into Maple program, which solves 3 eqns. for the 3 unknowns. We compute the strain tensor referred to the x 1 – x 2 axes.