Derivation for 3D Using Index Notation

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Transformations

Why needed?

Vectors

Strains

Derivation for 3D using index notation

Derivation for 2D w/o index notation

Transformation using contracted notation

Transformation for engineering strains

Stress

Derivation for 3D using index notation

Derivation for 3D using matrix notation

Derivation for 2D w/o index notation

Transformations

Constitutive matrix (if tensor shear is used… simple derivation)

Thermal and moisture expansion coefficients

alphaBetaTransformation

Maple derivations

Strain

See directory for examples

These will be useful after we cover laminated plate theory

Curvature transformation

ABD transformation

M_N_transformation

Why Transformations?

Isotropic materials

Usually stopped with stresses and strains … both stiffness and strength assumed to be isotropic=>often only need extreme values

Composites (laminates with continuous fiber lamina)

· Fibrous composites have directional properties … (our emphasis is on fibrous composites.)

· Constitutive properties are more easily described in one coordinate system than others but ultimately we need to express properties of all laminae in same coordinate system Þ need transformations

Example (Orthotropic lamina)

· when axes coincide with material axes there are only 9 unique constants (3D)

· when axes do not coincide with material axes, it appears that there are 21 unique constants [C can be fully populated].

Stress and strain

· Maximum s or e may not be of interest, since strength is not isotropic Þ we need to know s and e in a particular coordinate system, eg. assume weak plane at q … we need to know tractions acting on that plane.

Identifying Material Symmetries via Transformations

Malvern, p. 288

Remember that the symmetry under discussion is a directional property and not a positional property. Even when the material has a certain elastic symmetry at each point, the properties may vary from point to point, the properties may vary from point to point in a manner not possessing any symmetry with respect to the shape of the body.

Frederick and Chang.

Equivalent Axes. Two sets of Cartesian axes and are equivalent when the elastic constants and are the same for the two sets of axes.

Covering Operation. A covering operation is a transformation from one set of equivalent axes to another.

Vectors (2D)

(1st Order Tensor)

Difference between 2D and 3D is just range of indices

Matrix Form

The matrix º

Note:

Prove:

Warning: Some folks use opposite definition for aij (eg. Agarwal)

Transformation of Strains

Proof that tensor strains are components of a 2nd order tensor

(If it transforms like one, it is one.)

Therefore, we need to prove that

Matrix Form:

Strain Transformation

Chain Rule

Note

But a and b are dummy indices Þ can switch in second term

must be a second order tensor, since it transforms like one.

Note: Why?

2D Strain Transformation

(without using index notation)

But

Substitute

The transformations for are obtained in a similar manner.

Transformation of e: Contracted Notation

(Voight)

The terms in are still components of a second order tensor…!

What are the ?

The formula gives us the relationship. We just have to organize the terms.

= Homework

Transformation for Engineering Strains

Define: ei º Tensor strains (contracted notation)

ei º Engineering strains

R = Reuter Matrix =

Tensor Strains:

Therefore for engineering strains

· Faster: just divide engineering shear strains by 2, use tensor transformation, and then multiply by 2 to convert back.

Stress Transformation

Derivation #1

Cauchy’s formula:

But

Multiply both sides by (ie. transform to new coordinate system)

But

Þ 2nd order tensor

Derivation #2 Strain Energy Density (SED) = invariant (only valid for elastic material)

But all indices are dummy indices

Þ can switch i and j on left to m and n

Stress Transformation

-Matrix Format-

2D Stress Transformation (without index notation)

Similarly,

Constitutive Property Transformations

There are different ways to derive.

(Voight Notation, Engineering Shear Strains)

1. Invariance of strain energy density

Stiffness

Note:

2. Use of Stress-Strain Relations

3. Use of Strain-Stress Relations

etc.

Why do C and S transform differently?

4. Invariance of Complementary Strain Energy Density

No need for Reuter

5. Use tensor formula and then convert it Voight notation.

Comments:

· Christensen (p. 156-157) has explicit formulas for 3D constitutive matrix. (Tensor strains)

· Frederick & Chang, p. 163-165 has good discussion using contracted notation. (Tensor Strains)

Transformation of Constitutive Matrices

(If tensor shear strains were used)

e = tensor strain

Material Symmetry

· Order of strains

Anisotropic (none)

Monoclinic: One plane of symmetry

Example: Symmetric about x1-x2 plane

Orthotropic: Two perpendicular planes of symmetry (Note: If there are 2, there will be 3.)

Transversely Isotropic: (Isotropic in one plane)

Example: Isotropy in the plane x2-x3

Isotropic

Orthotropic Solid

(Material Coordinate System)

1. No shear-extension coupling

2. No coupling between shears

Compliance Matrix (Christensen, p. 154)

Remove “2’s” for Engineering

shear strains.

· Can you derive this?

· Invert to get stiffness matrix

Orthotropic

Is definition of terms consistent with our previous experience?

Consider:

(a) .

(b) This is not so obvious!

But (recall that we proved !)

strain in direction due to

Now we superimpose effects

*Shear ® trivial

Reduction from 21 Unique Terms… see HW

Test for Equivalence of Coordinate Systems

Expanded example

but

no sum on i

Example: hex_symmetry_2D.mws

Transformation of Thermal and Moisture Expansion Coefficients

Thermal expansion coefficients transform like strains

Consider:

must transform same as

Consider tensor strain definition

define T such that

Curvature Transformation

The curvatures for a plate involve second derivatives of the transverse displacments. The following describes how to transform second derivatives.

(Rotations about x3 axis only Þ

But

Alternate approach to transforming curvature

· zkx must transform just like and

· But “z” does not change with rotation about z axis

Þ must transform just like

Similarly for other curvatures.

Transformation of A, B, D

Assume where the “T” might be defined differently than in previous notes.

But T does not vary with “z” (Q varies with z)

Þ Transformation is the same for A, B, D as it is for Q!

Transformation of Moments

Define:

Transformation of in-plane stress resultants

Define: