Local Stress Concentrations in Imperfect Filamentary Composite Materials

Journal of Composite Materials http://jcm.sagepub.com/

Local Stress Concentrations in Imperfect Filamentary Composite Materials John M. John M. Hedgepeth and Peter Van Dyke Journal of Composite Materials 1967 1: 294 DOI: 10.1177/002199836700100305

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Local Stress Concentrations in

Imperfect

Filamentary Composite Materials JOHN

M. HEDGEPETH AND PETER VAN DYKE

Martin Marietta

, Maryland Corporation Baltimore

,

Solutions are presented for two stress distribution problems which result from breaking of the filaments in a composite material composed of high modulus elements embedded in a low modulus matrix. Both problems represent extensions of the two-dimensional filamentary structure stress concentration problem: the first concerns the determination of static stress concentration factors in the unbroken elements of a three-dimensional square or hexagonal array where specified filaments are broken; the second involves the stress concentration factor in the element adjacent to a broken filament in a two-dimensional array where the shear stress in the matrix adjacent to the broken filament is restricted by a limit stress in

an

ideally plastic sense.

INTRODUCTION

fabricated from composite materials made from a matrix S in which filaments are wound, woven, or plied are becoming prevalent in aerospace applications. Solid-propellant rocket-motor cases, for instance, are being constructed by winding resin-coated glass filaments on The high-strength glass filaments carry the pressurization a mandrel. loads and the resin forms the matrix which produces unified, efficient material. Other applications make use of the good foldability of coated fabrics to package large, low-density structures into small volumes until their erection by mechanical means or inflation is desired. Whenever one or more fibers is broken in filamentary structure under stress, the load in the broken fiber or fibers must be transferred through the matrix to the adjacent elements in order to restore equilibrational design using composite materials, the local rium ; to ensure stress concentrations created by this redistribution must be understood. STRUCTURES

a

a

a

filamentary structure stress concentration problem in two dimensions (a plane of filaments) has been formulated and solved by Hedgepethlll. This original problem considered the effects of number of the stress consecutively broken filaments in the planar array trations in the elements adjacent to the last of the broken filaments. A

a

concen-

on

294 Downloaded from jcm.sagepub.com at Bibliotheek TU Delft on April 14, 2014

Results for both static and dynamic stress concentrations under a uniform tensile loading were obtained. The investigation was limited, however, to the two-dimensional problem, and by the assumption that both the filaments and matrix behave elastically. The present paper describes extensions of these first results which include the effects of three dimensions and plasticity. The influence-function technique introduced in Reference 1 is used to obtain results for the following two additional problems concerning filamentary material under uniform tensile loading: first, static stress concentration factors are found in unbroken elements in three-dimensional square or hexagonal array of elements, where specified filaments are broken; secondly, the static stress concentration factor is found in the element adjacent to a broken filament in a two-dimensional array, where the shear stress in the matrix adjacent to the broken filament is restricted by a limit stress for an ideally plastic material. a

FORMULATION

OF

THE

PROBLEMS

THREE-DIMENSIONAL

The model which is considered is

shear-lag analyses; it matrix which connected by

common

to

composed of tension-carrying elements carries only shear. The first problems to be

is

a

considered involve a threedimensional model where the elements are all oriented in a uni-directional manner and are distributed evenly throughout the material. Both square and hexagonal filament distributions will be considered.

Square Array: The square array of filaments is shown in Figure la; the elements are spaced an equal distance d apart, and are aligned in the x direction. in the plane The displacement of the ( n, m ) th element, with n and normal to the filaments, is given by Un, m(x); the tensile force in this element is Pn, m ( x ), and is related to Un, ,n by m

where

EA

is the extensional stiffness of the filaments.

Figure 1(a). Three dimensional element, square array.


The interactions among the elements in the array are complex in nature, and as a first approximation it is assumed that the behavior of a given element is influenced only by its nearest neighbors. The shear force per unit length between the ( n, rn ) th and (n + 1, m)th filament is thus assumed to be where Gh is the effective matrix shear stiffness. The equilibrium equation treating the ( n, m ) th element and its neighbors is then

The

boundary

element

conditions

are

given by observing that for

a

broken

while for all other elements

For

they

large are

values of

loads in all elements, whether or not the uniform tensile load P, and so the re-

x, the tensile

broken, approaches

maining boundary conditions

are

Non-dimensionalization of the load, displacement, and coordinates is carried out by defining the dimensionless quantities

The

equilibrium equation (3)

then becomes

with

and the

boundary conditions

boundary conditions at ~ 0, it is convenient to the influence-function technique introduced in Reference 1. After imposing unit displacement the filament = m - 0, and maintaining displacement at 1 0 in all other elements, the element forces and displacements are represented by qn, ~, ( ~ ) and Because of the existence of the mixed

-

use

a

a

zero

on

=


n

vn, &dquo;t ( ~ ) . These influence functions

problem by

where the summations

are

superimposed

to obtain the

actual

extended only over the broken elements since the displacements vanish at ~ = 0 in unbroken elements; the subscripts i directions. The force and j represent broken elements in the n and boundary condition then gives the required equations for the unknowns are

m

Ui, J( 0)

with the number of equations and unknowns being equal to the number of broken elements. The determination of the influence functions qn, In is carried out by solution of Eq. (8) with Vn, In replacing ic,z, 7rL under the boundary conditions

Applying the transformation

whose inverse is

transforms the equation for the influence-function

with

displacements

to

boundary conditions


The solution (16) is

with the

Eq. (15) satisfying

to

boundary

conditions of

Eq.

angles /3 and y given by

The inversion may be written

Once

the

the

required

and the qn,rn in turn used

v,~, &dquo;,

are

as

determined, the qn, In

are

determined from

Eq. ( 12 ) to find the required uL, ~ ( 0 ) . These are only broken through Eq. (11), with the summations elements, to obtain the complete solution. When only the stress to be obtained, the values of q,L, &dquo;z ( ~ ) must be detrations at 1 = 0 termined. These given by are

used in

over

concen-

are

are

carried out integration required by the inversion small, high-speed using simple numerical integration procedure digital computer. Although the first integration necessary for the solu= 0 may be put in the form of tion of Eq. (21) for elliptic integral numerical integration provided the generality reof the second kind, The double

was

a

on

a

n

an

a

quired for all n values, integration

to

form

a

and

was

easily combined

with the

required second

single integration procedure. Separate computer

used to obtain the required influence-function solutions and then to use these solutions to calculate stress concentrations for particular broken filament configurations.

programs

were

Hexagonal Array: of elements is shown in Figure Ib, but the coordinates along which the subscripts n and m denote elements are now at a 60 degree angle, thus reflecting the new symmetry of the problem. The only variation from the square array problem is in the equilibrium equation of the ( n, m ) tla element, which becomes The

hexagonal array


Figure 1(b). Three dimensional element, hexagonal array.

Applying for the

the influence-function

qn, m ( 0 )

Evaluation of the

approach

leads to the inversion

integral

again made employing numerirequired integrals cal integration procedures. The symmetry of the element array then used in the solution of particular problems using the influence-function was

was

formulation.

RESULTS

FOR

THE

THREE-DIMENSIONAL

PROBLEMS

Stress concentrations have been calculated in unbroken elements when various numbers of filaments are broken in square and hexagonal filament arrays. An automated procedure was developed to combine the influence coefficients, solve the simultaneous equations for the quantities un,~,(0), and then obtain the required stress concentrations. As a check

the three-dimensional calculations, straight rows of filaments were broken in both arrays and the stress concentrations at the row adjacent to the last broken row were obtained. In both the square and hexagonal array cases the stress concentrations at the centers of the adjacent rows approached the known two-dimensional results for the index representing the number of broken rows. Solutions for various configurations of broken filaments were then obtained. In particular, for the square element array, the filaments were broken in such a way as to form circular regions, and the stress concentrations in the element on the major diameter adjacent to the broken filaments was calculated. The way in which the broken filaments were chosen is shown in Figure 2; the number N represents the number of broken filaments on the major diameter, and all of the filaments M on


Figure 2. Broken

element configura-

tions, square array.

within a circle of diameter Nd were broken. The results for values of N from 1 to 7 are given in Table I. The ratio of the three-dimensional concentration to the

corresponding two-dimensional result, X, shown in Figure the values of x for increasing N

and The

are

pattern of broken elements

required

to fill

was

calculated,

3.

the circular

area

of

with each increase in the parameter N. For the case where N is 2, the number of broken filaments is 2, and the stress concentration factor is comparatively low as compared with the case where N is three and there are 9 broken filaments. This variation in the number of broken filaments accounts for the scatter in the results for A at small values of N. The solution to the continuous problem of the stress concentration diameter

Nd

changes

Table I. Square Array Stress Concentration Factors Circular Broken Filament Configuration.


Figure 3. Ratio of concentration factors,

square array.

the edge of a disk-shaped void in a three-dimensional body composed of elements and joining matrix under a uniform tensile loading may be obtained by considering the analogous problem of the flow of a perfect fluid around a disk~2~. The solution for the stress concentration as the edge of the void is approached is (2/~(f―~))~ with r the dimensionless radial coordinate from the center of the unit radius void in its plane. For the two-dimensional problem of a crack in a sheet, the stress concentration as the edge of the crack is approached is ( 1 /2 ( y -1 ) ) 1i~ with y the coordinate from the center along the crack of length 2. The ratio of the three-dimensional to two-dimensional continuous solutions is 2/7r, and is shown in Figure 3 as the apparent limit of x as N becomes large. For the square array case, a square broken filament configuration was also investigated. The results, for N from 1 to 6, are presented in Table

near

II.

Table II. Square Array Stress Concentration Factors Square Broken Filament Configuration.


Figure 4. Broken element configuration, hexagonal array.

The broken filaments in the hexagonal array are shown in Figure 4; N represents the number of filaments on the major diagonal of a hexagonshaped array of broken elements. Stress concentrations were obtained for both the element adjacent to the broken group on the major diagonal, and also for the element in the hexagonal ring of elements surrounding the broken filaments whose stress concentration is greatest. These results are summarized in Table III.

Table III. Hexagonal Array Stress Concentration Factors Hexagonal Broken Filament Configuration.


Figure 5. Ratio of concentration factors, hexagonal array.

of the ratio x for these two elements as N increases are shown in Figure 5. The curve of maximum stress concentration is well above the apparent continuous x limit, while the diagonal element ratio is below this limit. Had the hexagonal array elements been broken in such a way as to approximate a circular area, as was done with the square array, the concentration ratio x would approach the 2/7r conTwo

curves

tinuous limit.

FORMULATION OF THE PLASTICITY PROBLEM

The two-dimensional filament model considers a single layer of evenly spaced elements. In the problem treated by Reference 1, the

elements and the matrix

elastically.

The

connecting the elements

possibility

matrix material will exceed

that the a

shearing

maximum

are

assumed to behave

stress in

the low modulus elastic stress can be accounted

by formulating a model in which the matrix material acts The shear stress may increase elastically to plastic

for

in an ideally a limit plastic

manner.

value T, after which the stress remains constant with increased displacement of the bounding filaments. The configuration for this twodimensional plasticity problem is shown in Figure 6. The limit shear force per unit width Th is present in the region between the broken element n 0 and its neighbors n = ± I. The region in which this stress exists, a, extends in both the positive and negative x directions. The

stress

=


Figure

6.

ration.

stress concentration

which is

to

Plasticity problem configu-

be found is in the elements n

=

± 1 at

x==0.

The nondimensionalizations represented by Eq. (7) are used to define dimensionless forms of the limit shear force per unit width Th and

plastic region

a

governing elements beyond those which

The equilibrium equation the limit shear region is

For the element n

The

=

1, the

symmetry about n =

holds for element n

=

-

ment is

Defining the function f ( ~ )

equilibrium condition

bound

is

0 implies that un = u_ n, and the same equation l. The equilibrium equation for the n = 0 ele-

as


and employing the single-dimensional fined by Eq. (14) yields the equation

for the influence function

_

±

of the transform de-

displacement quantity, with

The boundary conditions are, at ~

and, at ~

equivalent

=0,

oc

equation ( 29 ) holds only in the region where ~ < a, where f ( ~ ) is positive quantity. At the boundary 6 = a, the function the right hand side of Eq. (29) vanishes. The f ( ~ ) is zero, and for ~ > particular solution to Eq. (29) is The differential a

a

The

complementary function solutions

are

in the form

Applying the boundary conditions, Eq. (31), at the boundary 4 yields =

and

matching the

the

single-dimensional

functions

a

with

Employing the inverse transform, given by of Eq. (14), yields

form

Defining the integral functions


allows un to be written

Solving for f ( ~ ), using Eq. (28), yields

where the

angle qf is 0/2, 8 is 2 sin

After the

~, and

integral equation (38)

is

solved for f ( ~ ), the stress

centration in the element n = 1 is found tion forces qn from

by finding

con-

the influence-func-

with vn found from Eq. (37). For the single broken filament application of the influence-function technique, the single unknown displacement is

( see Eq. ( 12 ) )

and

so

the force in the first element is (see

The solution to the

Eq. ( 11 ) )

integral equation is

carried out using a numerical procedure (see Reference 3); the extent of the plastic region a is assumed, and the plastic shear T is found after employing the restriction that t ( ~ ) vanish at ~ = a. During the course of the calculations it was found that a change of variable of the form

with k equal

to 10 increased

the range of the variable

could be obtained using the

single,

for which results uniform-mesh, numerical solution a

technique. RESULTS FOR THE

PLASTICITY

PROBLEM

The numerical solution to the integral equation, carried out by assuming the extent of the plastic region and then obtaining the limit shear T and the function f( g), allows the determination of the stress concentration in the element adjacent to the broken filament. Figures 7 shows the results of the numerical evaluation for the constant limit shear stress


Figure 7. Limit shear

stress for

in-

creased load factor.

Figure 8. Extent of plastic region for increased load factor.

Figure 9. Stress concentrations for

in-

creased load factor.


in the region between the broken filament and its immediate neighbors as a function of the limit load factor. This load factor is defined as the ratio of the load at infinity to the load at infinity required to raise the shear stress to only the limit value. The shear stress T decreases rapidly from its initial value of one as the load factor is increased. The extent of the plastic region as the load factor is increased is shown in Figure 8;

the curve is almost linear, with a slightly increasing slope with increased load factor. The stress concentration factor as a function of load factor is shown in Figure 9. The stress concentration initially drops quickly with increased load factor, but then begins to level off at higher load values.

CONCLUSIONS

that have been treated here are small segment of the many stress concentration problems whose formulation could be based on the two-dimensional plastic and three-dimensional elastic models. For example, the two-dimensional plastic model can be used as basis for the solution of problems where more than one filament is broken. It also provides a framework for the formulation and solution of problems involving different physical situations; for example, the case where the bond between filament and matrix is the factor which limits the allowable matrix shear stress can be treated by a small modification to the The

problems

a

a

governing integral equation.

The three-dimensional model used to obtain the elastic stress concentrations was based on a &dquo;nearest neighbor&dquo; approximation. The accuracy of this approximation can be determined by also considering the &dquo;next nearest&dquo; neighbors surrounding the broken filament when determining the governing equilibrium equation. Plastic matrix influence coefficients can also be determined for the three-dimensional model, thus leading to solutions of three-dimensional single and multiple broken

filament plasticity problems. Definition of macroscopic failure criteria in a statistical sense should be possible with the stress field definitions provided by the two-dimensional results of Reference 1 and the additional results presented in this paper. In addition to the extensions, mentioned above, to the existing results, combination of these results through statistical analysis to obtain overall structural failure estimates would seem to be an area deserving of continued efforts. 308 Downloaded from jcm.sagepub.com at Bibliotheek TU Delft on April 14, 2014

NOMENCLATURE a

Length of plastic region in matrix Integral functions, Eqs. (36), (39)

=

Bn, Cn, B,

C

=

d

=

EA

=

M

Filament spacing Extensional stiffness of a filament Plasticity solution function, Eq. (28) = Effective matrix shear stiffness = Kernel function, Eq. (34) = Total number of broken filaments in

N

= Number of filaments

f

=

Gh K

figuration

=

P n In

major diameter, diagonal,

or

con-

side of

a

broken filament configuration Applied force on each filament at infinity Force in (n, m) th filament Dimensionless load in (n, m ) th filament, Eq. (7) Dimensionless load in (n, m ) th filament for influence-function solution Alternate dummy variable, Eq. (43) Limit shear force per unit width for ideally plastic matrix

=

p

on

broken filament

a

pn, &dquo;,, g,t, &dquo;L

= =

s

=

Th

=

t

= Dummy variable, Eq. (34) Displacement in (n, m ) th filament Dimensionless displacement in (n, m ) th filament, Eq. (7) Dimensionless displacement in (n, m ) th filament for influencefunction solution, Eq. (7) Transformed dimensionless displacement, Eq. (14)

’

Un,

’

=

,n

un, In

=

vn, &dquo;~

=

~

v

=

Coordinate parallel to filaments Dimensionless length of plastic region, Eq. (24) Angle defined in Eq. (18) Angle defined in Eq. (18) Trigonometric function defined in Eq. (30) Alternate coordinate variable, Eq. (43) Transform variable Ratio of three-dimensional to two-dimensional stress concentration Dimensionless coordinate parallel to filaments, Eq. (7) Dimensionless limit shear force per unit length, Eq. (24) Transform variable

=

x

=

a

~

=

=

y

8

=

1

=

0 x

=

=

i

=

T

=

(~ If

=

Angle equal to 9/2

=

i, j, n,

In

=

Indices

REFERENCES

1.

Hedgepeth, "Stress Concentrations D-882, Langley Research Center (1961).

J.

M.

2. P. Morse and H.

(1953), p. 1293.

Feshbach,

Collatz, The Numerical ( 1960), Chap. VI.

3. L.

Methods

Treatment

in

Filamentary Structures,"

, of Theoretical Physics II

NASA TN

McGraw-Hill

of Differential Equations , Springer-Verlag

( received June 7, 1967 )

309

Local Stress Concentrations in Imperfect Filamentary Composite Materials

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