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Transformations

1. Show that the principal directions for stress and strain need not coincide for an anisotropic material.

2. Show that

3. Tensor strains (and stresses) can be transformed using the formula .

Using contracted notation we can express this as i' = Tijj.

Derive the matrix Tij for 3D analysis. (Order of strains for this HW = 11 22 33 12 23 13)

Maple solution: coordinateTransformations.mw

4. Write computer subroutines to calculate i', i', Cij' given i, i, cij and the angle between the xi and xi' axes. (2D only) Use engineering shear strain.

Answer

Solution: planeStressMatrix.mw transformations_2D.mw

5. Use the 2D subroutines from (4) to obtain the following plots: *Also, tabulate for theta=0 to 90 for each 10 deg increment

(a) Given: 1, 2, 3 = 5, 7, -2

Plot: 3' versus 1' for theta = 1-> 180

(b) Given 1, 2, 3 = .01, -.06, .02

Plot: 3' versus 1' for theta = 1-> 180

(c) Given the following properties for graphite/epoxy:

  • E11 = 30e6,E22 = .75e6,12 = .25, G12 = .375e6
  • Plot:Q'11, Q'22, Q'33 , Q'23 versus theta for theta = 1, 180(this is plane stress). In this case, theta is the rotation angle for the lamina and is positive counter-clockwise. The primed quantities are the properties in the original coordinate system.

Solution: variationWithTheta.mw <=check this solution!

6. Plot the effective engineering modulus Ex for a lamina orientation of 0 - 90 deg.

The material properties of the lamina in the material coordinate system are

E11 := 100: E22 := 10: E33:=10:

G12 :=5: G23 :=3.845: G13:=5:

v12:=.35: v23:=.3: v13:=.35:

Positive angle is taken to be counter-clockwise from the x-axis to the x1 axis (x1 = fiber direction). Calculate two ways:

(a) Constrained conditions (ie. zero shear strain during extension)

(b)Unconstrained conditions (ie. zero shear stress during extension)

Solution: constrainedUnconstrained_Exx.mw

7. Analysis of a flat tape laminate will involve a simple material transformation… a single rotation about one axis. If the laminate is not flat or if you are analyzing a textile composite, the orientation of the material might require a sequence of rotations. Assume that the material is subjected to a sequence of three rotations: first about the x1-axis (alpha), then about the x2-axis (beta), and then about the x3-axis (theta). Each rotation follows the right hand rule for the sign convention. To perform the analysis, the constitutive matrix must be expressed in the original (unrotated) coordinate system. Derive the formula for the transformation. (Use Euler angles)

Material Symmetries

8. Show that if (where are thermal expansion coefficients) for one coordinate system, then in all coordinate systems.

9.Show that for a hexagonal array, for all coordinate systems.(i.e. for any rotation about the x3 axis)

Solution: diffusionSymmetry.mw

10. Prove that for an orthotropic material there can be at most 9 independent stiffness coefficients. Start with the 4th order stiffness tensor (i.e. do not use contracted notation).

orthotropicMaterial_symmetries.doc Solutions\orthoMaterialSymmetry_contracted.mw