复合材料结构力学试题

Problem Set #1

Handed out: Oct 17th , 2013

Due: Oct 24th , 2013

1. Expand the following tensor equations (Note the Kronecker delta in a and c )

a) 1[(1)]mm T E αβ

αβαβαβευσδσδα=-+-∆ b)

1F F αβσγαβσγαβαβσσσ+= c)

1mn ms n a b δ= d) 1i i B A αα=

（4 points ）

2. Based on the ‘Rigorous handling of 2D model’ in the course note, determine for a

volume fraction ()a /a b +equal to 0.6:

a) The stress in the broken fiber

b) The stress in the unbroken fiber

c) The shear stress in the matrix

Try do this for the length equal to 10, 50, 100 fiber diameters. Base on your graphical results, answer the following questions:

a) What is the effect of changing the overall length?

b) What happens when the overall length is 10 fiber diameters?

c) For the case that the fiber total length is more than 50 times of the fiber

diameter, how far from the break point must one go to achieve 99% of the original applied stress in the broken fiber?

（12 points ）

Note:

1)

0.083= 2) When max 50ζ≥, max tan()1κζ≈, the equations can be simplified.

3) The purpose of this excise is to learn and practice how to write a small

MATLAB program and draw the scientific figures. More attention should be paid to the following MATLAB commands,

a) FOR… END;

b) IF…ELSE…END

c) PLOT;

d) AXIS;

e) XLABEL,YLABEL;

f) TITLE.

MATLAB’s help will provide more details regarding these commands.

3. Consider the following shear lag problem of a piezoceramic crystal (PZT) surface mounted

on a metal plate. Note that the width of the PZT is less than the width of the plate. When the crystal is activated, a strain is induced. The stress in the PZT can be found from the following equation.

()p p p PZT E σεε=-

where

p E is the modulus of the PZT, p σ and p ε are the stress and the strain in the PZT, and PZT ε is the induced strain due to the external voltage applied.

a) For a given induced strain, what does the displacement look like through the thickness?

(Assume that the strain is constant through the width).

b) Determine the coupled differential equations necessary to solve this problem. (The

independent variables can either be the deflections in the PZT and the plate or the stresses in the PZT and the plate. Assume the shear modulus of the adhesive is a G )

c) What are the boundary conditions?

(8 points)

4. Using micromechanics determine the modulus,T E , of the following three cross sections. The

first elemental volume has a square fiber within a rectangular section of matrix. The fiber has sides of a and the unit volume has a height of 3a . Note that in the first one, the fiber volume fraction is 223a a or 1/3. The dimension of the fibers in the second and third elemental volume is again a . First determine the appropriate dimension of the second and third volume so that the fiber volume fractions of both volumes are equal. In order to determine the modulus of both volumes, you can assume constant strain across any plane perpendicular to the applied load in the 2-direction ( Assume that fiber properties are f E and f ν and the matrix properties are m E and m ν).

(6 points)

a a a σσσσ? ? 2 3 σ

σ