On the methodologies of stress analysis of composite structures

Theoretical and Applied Fracture Mechanics 6 (1986) 153-170 North-Holland

153

ON THE METHODOLOGIES OF STRESS ANALYSIS OF COMPOSITE STRUCTURES Part 2: NEW EXPERIMENTAL APPROACHES

J. T. P I N D E R A Department of Ovil Engineering, Faculty of Engineering, University of Waterloo, Waterloo, Ontario, Canada, N2 L 3G1

M.-J. P I N D E R A Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, U.S.A.

1. Introduction New, advanced experimental approaches, based on the principle of physical modelling of real events, have been applied to develop three new groups of experimental methods of stress analysis. They are briefly presented below and their efficacy is illustrated by examples: isodyne methods, strain gradient methods, and thermoelastic effect method. All three are nondestructive methods. The first two methods can be used to determine stress components or their spatial derivatives in materials that are transparent either in the visible band of electromagnetic radiation, or in the infrared, or--in principle--microwave band. The third group of methods is based on application of the thermoelastic effect which occurs in all bodies having non-zero values of the thermal expansion coefficient. These methods appear to be promising. For instance, they allow one to easily distinguish between elastic longitudinal and shear waves.

2. Isodyne methods: Plane and differential elastic isodynes. Photoelastic isodynes. Isodyne coatings 2.1. Plane elastic isodynes

The term 'plane elastic isodyne' denotes a geometric locus of points along which the total normal force intensities A p , , acting on the characteristic sections As between two isodynes, are constant. The characteristic section is any

section in the plane stress field parallel to the characteristic direction which characterizes the related field of isodynes, Fig. 1, Apn = const = Ssm s ,

(1)

where Ss denotes a constant material parameter, r n denotes a dimensionless parameter called 'order of the isodyne', and s+As

Ap. =

~r. d s .

(2)

The plane stress field in a homogeneous, isotropic, and linearly elastic plate is characterized by the two-dimensional equilibrium conditions which are satisfied when the thickness of the plate is infinitesimal but finite, or when the stress state is homogeneous. The hypothetical material of such a plate is characterized by two material parameters only--the modulus of elasticity and the Poisson ratio---and follows Hooke's model of the stress-strain relations. In such a hypothetical plate of an infinitesimal thickness 6, loaded on boundary by force intensities p, all the stress components related to direction z, normal to the plate surface, are equal to zero; the derivatives of all the stress components with respect to z-direction are meaningless. Plane stress fields induced in such hypothetical elastic plates can be conveniently described by the Airy stress function ~(x, y), which yields the known expressions for the components of the stress tensor, with respect to the chosen Cartesian coordinate system (x, y). The isodynes can be expressed in terms of the first derivatives--~x(x, y) and ~y(x, y ) - - o f the Airy stress

0167-8442/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)

J.T. Pindera, M.-J. Pindera / On the methodologies of stress" anal.vsis: Part 2

154

Y

Elastic Isodyne :

Plate of an Infinitesimal Thickness, ~ - ~

I

- r 'x m - 2 ' s x - s x ' ,', sx- ;

a

(Py'B : f o ' y y d x =S s ms, f~Nm] L---j

/

/r.s.o, ) \/

/

where Ss-- s.r N- 1

[~(b)B , \

L.,J

\

"Sx( x'Y = Y°)

Fig. 1. Concept of plane and differential elastic isodynes.

function with respect to coordinates x and y, and in terms of integration functions fr(y) and fy(X) which depend on the boundary conditions, in the following manner: x-isodynes (isodynes related to x-characteristic direction): O

py(X, y) = ~

Cb(x, y) + L ( Y )

= ~ ( x , y) + L ( Y ) = S, msx(X, y) ;

(3)

y-isodynes (isodynes related to y-characteristic direction): 0

sections of isodyne surfaces cut by planes normal to the plate surface and containing the chosen characteristic directions

m,xx = msxx(X, Yo) ,

(7a)

msyy = m,yy(Xo, y) ,

(7b)

where x 0 and Y0 denote chosen values of ordinates x and y. The normal stress components along the chosen cross-sections are proportional to the slopes of corresponding isodyne functions 0

Oyy = S, -~x m'xx(X' Yo) and

px(X, y) = -ff-fy qb(x, y) + fy(x)

0

= cI)v(x, y) + fy(x) = Ssmsy(X, y).

(4)

Thus,

Oryy(X' Y)=

02

0

-Ox - 2 Cb(x, y) = S, -ff-£x m,x(X, y) '

(5)

and 02 O'x~(X, Y)= -Oy2

0 ~(x, y) =

(8a)

S s ~y

msy(X , y )

"

(6)

The function px(x, y) and py(X, y) may be understood as isodyne surfaces spanned over the plate surface. With regard to the technique of evaluating the stress components on the basis of relations (5) and (6), it is convenient to introduce isodyne functions which are the cross-

crxx = S, ~y m,yy(X o, y).

(8b)

It can be shown that the shear stress components O-xyand O-yx are proportional to the slopes of the cross-isodyne functions txy and tyX, which represent shear force intensities acting On chosen cross-sections, and which are the cross-sections of the isodyne surfaces cut by planes normal to the plate surface and normal to the characteristic directions msxy =

m,xy(Xo, y) ,

(9a)

Yo)"

(9b)

msyx = m,yx(X,

Summarizing, two isodyne surfaces related to two mutually perpendicular characteristic direc-

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

tions reliably yield four independent pieces of information on the values of the plane stress tensor: O'x~, O'yy, O-xyand tryx = tr~y. lsodyne functions directly yield the values of the total normal force components acting on characteristic cross-sections which can be used to check the reliability of the procedure by applying the equilibrium conditions, Fig. 1, py(boundary) = py(end),

(10a)

p~(boundary) = px(end),

(10b)

loci of points in the characteristic planes of real plates along which the difference of intensities of normal forces acting on the characteristic section between two differential isodynes in the directions within and normal to the characteristic planes are constant. E.g., Apy,z(X , y, z = zi) = Apy - Ap~

~ss +As and, with regard to the shear forces t,

=

(O'y -- try) ds

= SsAms(y,z)(X , y, z = zi) = const.

txy(boundary ) = t~y(end) ,

(lla)

tyx(boundary ) = txy(end ) .

(lib)

2.2. Differential elastic isodynes The concept of plane elastic isodynes can be reliably and successfully applied as a basis of various procedures of analysis of stress states in plates which are characterized by relatively low values of the gradients of stress components. High gradients of stress components manifest the presence of local or extended three-dimensional stress states, which can not be reliably determined by using procedures based on two-dimensional equilibrium equations. Such three-dimensional stress states in plates, where all the stress components are functions of the distance from the middle plane of the plate, occur in regions of crack tips, notches, local contact and thermal loads, and interference of elastic waves. The knowledge of the actual stress values in such regions is of utmost importance with regard to the load-carrying capacity, damage accumulation susceptibility, and service life of a structure. It is easy to show that the concept of isodynes can be generalized to encompass such cases and that the basic analytical relations are applicable if the characteristic planes (planes containing the characteristic directions) are coplanar with the plate's middle plane. In situations where the values of the stress components in planes coplanar with the middle plane of the plate deviate by not more than 30% from the values at the plate surface it is possible to define differential elastic isodynes in a manner analogous to the definition of plane elastic isodynes. Differential isodynes are the geometric

155

(12)

Equation (12) represents a surface defined as a difference between two related surfaces of the x-isodynes. It should be mentioned that the differential isodynes may not directly yield information on the shear stress components, depending on the magnitude of the out-of-plane stresses. 2.3. Photoelastic isodynes As some in [11,13-15,18,20], it is easy to obtain experimentally fields of photoelastic isodynes, which are identical either to the plane elastic or differential elastic isodynes, using a particular scattered-light technique when a set of specified conditions is satisfactorily approximated [10]. The resulting photoelastic isodynes are given by the lines of constant intensities of particular sheets of light scattered from the primary light beam of intensity I 0 which propagates in one of the characteristic directions and is scanned in the characteristic plane, e.g., IX/Io = (IX), = sin 2 ~rmsx = sin 2 l~bsx = ½(1 - cos ~b,x) = const, (13) where ms~ denotes the normalized linear-relative retardation R/A, and ~b~x denotes the angularrelative retardation, both of which are related to the secondary principal stresses orx1 and orx3 acting in the plane normal to the x-direction of the primary beam. When the conditions specified and discussed in the pertinent references (e.g., in [9]) are satisfied, the following expressions describe satisfactorily the physical event of interest ~b,x = 2~rmsx = 2~rSs I f (tr~ - trX3) dx, etc., where m~, denotes the orders of x-isodynes.

(14)

J.T. Pindera, M.-J. Pindera / On the methodologies o f stress analysis: Part 2

156

E c

.

,,

~

'E o z

.

.

e~

2~

..~ ~

,o

e~

N

-b

b

0

~ L o

Ne-.

o

d

~ N E E E E E E

e;e

xl b

b

I I I

N~N '

=b~ .

E E

g

~

2

0

,"

o

E E E E

m

~

z~

E o N

~._~,

t

ttt

O

L, e-,

I

~o~

d

157

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

three-dimensional stress fields are discussed in the pertinent references. The technique and efficacy of the isodyne method are presented in Figs. 2a,b and 3 which illustrate the determination of the stress components of two- and three-dimensional stress states using plane and differential isodynes. Two basic isodyne measurement techniques have been developed so far: the light-intensity modulation technique illustrated in Fig. 2a, and the modulation technique of the spatial frequency of residual isodynes which is illustrated in Fig. 4a.

Thus, for instance, for the x-characteristic direction one obtains

m,,=s;' I

=Ss- ' ( P y - - P z )

= const.

(15)

One must note that in the regions where high stress gradients occur the generally rectilinear path of the primary beam I 0 can be locally curved, and that both conjugated light beams, ordinary and extraordinary, can separate. The resulting experimental complications and, simultaneously, the additional information on the

Material: Palatal P6 Radiation: h: 6 3 2 . 8 nm (Ss)~ = 56.8 Nmm "t

t,o : 160 h : 5 0 r,i : 140 b : 2 0 ~z:llO o: 6

p : p ( f = const.) =P~ Measurement Planes: 0_< z i < b / 2 Isy = I s y ( X , y , z i) P:O:

z:

0

4

6

9

P=~:

,lO'xx)n

1.4 1.2

"~'e. ,,~

(=zz)n = =zzn ( z )

(O'zz)n'

"4t

~'\

\

(¢zz)n : °'zz (bS~ tl

\

1.0

O.8 0.6 fi4 O.2

(°'xx)n = crxxn(Z)

( O'XX )n =

IC

I1

: Crxx.b(Ss),"

\ \ \

\ '--

I0 0

z[mm]

(CrZZ)n 5

Fig. 3. Exampleof differentialisodynestress analysis(3-D).

~0

158

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2 -% N

~

X

~

=b ;~

+

-7 e.

x

b x

v

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II

l

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g

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+

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0

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

2.4. lsodyne coatings The isodyne coating techniquesmboth the amplitude and the spatial frequency modulation techniques--have been developed to analyze stress states at the surface of nontransparent objects. Reference [16] contains information on the theory and technique of the isodyne coating method. The spatial frequency modulation is particularly useful in the isodyne coating technique because it gives resolution which is about one order of magnitude higher than the resolution of the amplitude modulation technique. An example of application of isodyne coatings in stress analysis of actual objects is given in Fig. 4a,b.

2.5. Instrumentation Detailed information about the principles and techniques of photoelastic isodynes is given in [19]. Measurements can be performed using monochromatic laser radiation within the visible or infrared spectral band, depending on the parameters of the experiment. The isodyne fields can be recorded and evaluated using either standard photographic techniques and manual evaluation, or electronic image-collecting and processing systems.

3. Strain gradient methods

The basic phenomenon taken as the basis of the strain gradient methods, which supplies a quantitative basis for the empirical relations presented in [8], is presented in Fig. 5 in Part 1 of this paper. More extensive data on the strain gradient effects, methods and techniques are given in [5,6,8,13,18,21]. Only a general outline of the method is given below. It has been known for a very long time that the path of light propagating through inhomogeneous bodies is not rectilinear--the light path is curved. A particular example of this phenomenon is the well-known mirage. It has been observed that a noticeable mirage effect may also be produced by stress states in photoelastic objects [1,8,21]. This effect limits the resolution of photoelastic measurements and obscures isochromatic fields in regions of high

159

birefringence gradients [2,8,13]. However, this effect was utilized by Acloque and GuiUemet a quarter of a century ago to measure particular residual stress states [1]. Also, it has been observed that the optical anisotropy caused by an inhomogeneous stress state produces a noticeable separation of the light beam: this effect has been used to develop the theories and techniques of methods that can be used to determine the gradients of some linear functions of stress components and the values of absolute stress-optic coefficients [21]. It is common to present the influence of optical inhomogeneity of a body on the path of light propagation by the simplified relation /~=1~ p

1( n gradn

dng)

-~s

grad Inn

1 dn_ n ds s .

(16)

The most general phenomenological-mathematical model of optical anisotropy produced by stress-strain states proposed by Ramachandran and Ramaseshan in 1961 [22] presents an alteration of an optical parameter as a linear homogeneous function of the components of the stress or strain tensors. The simplified relations derived on the basis of the above-mentioned model, together with the relation (16), lead to the following phenomenological relationships between the curvatures of both ray components propagating with the velocities v 1 and o 2, and the stress tensor components 1 ( dnl~ ) /~1 = ~ ff grad n 1 - - ~ s '

(17)

where, after simplifications involving linearization n x = n o + ClO"1 + C2(o-2 + o-3),

(18)

and 1 ty= 1 ( dn2g) /~2=p-~ n--~ gradn 2 - ds / '

(19)

where n2 = no + C,,~2 + C2(~3 + ,~,).

(20)

Both conjugated rays, E~ and E~ [5,6,13,17,18,21] (Fig. 5 in Part 1 of this paper), are linearly polarized in mutually perpendicular

160

J.T. Pindera, M.-J. Pindera / On the methodologies o f stress analysis: Part 2

planes, regardless of the state of polarization of the original impinging beam. This phenomenon opens various experimental possibilities. Equations (16-20) can be used as a foundation of various mathematical models relating deflections of a light beam traversing a body to gradients of the sum and difference of principal stresses when the limitations which follow from linearization of basic relations are not exceeded significantly. When the stress gradients are not very high, such models can be quite simple and thus they can yield simple, but sufficiently reliable linear relations between the deflections, stress gradients, and values of the absolute and relative stress-optic coefficients C~, C 2, and C~---

y,

C,, C2, C~ = C(a, t ) .

C~ = C l - C 2 = nl -/12 , o'~ -

!

L~

Polarizer5

_1

LW

'~lrnmer sion Screen--~ Tank

Resin P6 (8ASF) I

~(Dx

+ o,,

,

S =

2

-I00

Immersion :

-80 X l { ~ A i r -6o x2 -40

{--

Mineral Oil Dow Corning

704

-2O I

-5 -4 -3 -2 Radiation :

X! = 514.5 nm h2 = 632.8 nrn Polarization : Dx : Ex=~ 0 1 E y = O Dy: E x = O , E y ~ O

-I -20-

(22)

which can be used as an additional piece of information with regard to the determination of C 1, C 2, C~. It must be noted that the curvature and separation of conjugated ordinary and extraordinary rays can be quite noticeable. For instance, as already mentioned, it often occurs that the corn-

L = 16.7m

"

(21)

Relations (16-21) form the basis of gradient photoelasticity. Transmission photoelasticity supplies the following relation:

Y{

!

Material : Polyester

C~ - C2, where

2

3

"" "~ ~ ~ ~ ~

4

d._.SSdy [N / mm3 ]

-40-60-80' - I00-121

Fig. 5. Determination of gradients of the sums of principal stresses using strain gradient method.

J.T. Pindera, M.-J. Pindera / On the methodologies o f stress analysis: Part 2

mon photoelastic materials cause deflections and separations of light beams up to 4 mm over the distance of 100mm. The alters the fields of isochromatics in regions of crack tips, sharp notches, and contact loads. An example of application of the strain gradient method in stress analysis is presented in Fig. 5. More examples are given in [21]. Figure 5 shows that the mean deflections of light beams propagating through a transparent plate are caused by both the curvature of light beams within the plate and the rotation of the faces of the plate. At a shorter wavelength the effect of curvature prevails over the effect of face rotation. So far, these basic relationships are not considered in theories and techniques of experimental methods which use transmitted light to collect information on the stress state, such as the methods of caustics, transmission photoelasticity, etc.

161

where f denotes the specific free energy, s dT denotes the amount of heat supplied to a body, and trij d% denotes the change in strain energy. As mentioned, the basic condition for the validity of the foregoing relationship is small deviation from thermodynamic equilibrium. Assuming that all the pertinent material parameters, E, v, A,/~, a (linear thermal expansion coefficient) and c, (specific heat at constant strain) have constant values, and assuming validity of the thermal equation of state, % = (xe~k)~,j + 2~% -

[(3x + 2 a ) a ( T - T0)]~,~,

(25)

where T O denotes a reference temperature, one arrives at the expression T s - s o = (3A + 2/z)a dekk + c, In ~0 "

For an isothermal deformation process the amount of heat exchanged reversibly with the surroundings can be presented as

4. Thermoelastic methods

(27)

QO = T ( s - S o ) .

The classical phenomenological theory of the thermoelastic effect is based on the model of a continuous, homogeneous, isotropic and elastic body deformed either under isothermal or adiabatic conditions. Being based on the principle of local state, this theory is essentially restricted to small deviations from thermodynamic equilibrium. One of the major assumptions is that all processes of deformation within the elastic limit are reversible. Thus, this theory does not encompass irreversible thermodynamic processes in materials, some of which have been successfully used as a foundation of defect detection techniques [26]. The known Gibbs equation applied to a deformed body can be presented in a familiar form referred to a unit volume: d u = T ds + o-~jd % ,

(23)

where u denotes specific internal energy, s denotes specific entropy, and T denotes absolute temperature. After transformations, introducing the specific Helmholtz function, f, one obtains the known basic relation: d r = - s d T + % deij ,

(26)

(24)

The above leads to the conclusion that the amount of heat exchanged reversibly between a deformed elastic body and surroundings in isothermal conditions can be presented by the relation Q0 = t~ T(3A + 2/x)ekk = a T ~ e

ekk .

(28)

Assuming that the isothermal and adiabatic elastic deformation processes are equivalent regarding the amount of heat exchanged or produced, one obtains Q0 = -c~ A T = a T ~ E

ekk •

(29)

Thus, the temperature alteration produced by an elastic deformation at adiabatic conditions may be expressed as T T-

T O= A T = - a T = -a-

Ce

E

c, 1 - 2 v '~kk

O'kk = Ktrkk ,

(30)

where a may be positive or negative depending on the material. A more rigorous presentation of the ther-

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

162

moelastic effect at an adiabatic (isoentropic) deformation leads to the expression dT T

a d(~rkk) , c

(31)

that could be integrated assuming constant values of material parameters, when the resulting temperature difference is not large. It should be noted that the specific heat at constant strain and the specific heat at constant pressure or constant stress are practically equal for some engineering materials. The common theoretical solutions for thermally induced stress states tend to neglect the major mechanism of heat transfer, namely, radiation. This must lead to an increase in unnecessary discrepancies between the theory and experiment. In fact, radiation represents a major parameter influencing the energy transfer within reversibly deformed body, and between the body and the temperature sensor. It must be noted that some experimental resuits indicate that the common description of energy flow between reversibly deformed body and temperature sensors is not satisfactory. It appears that the presently accepted mathematical models of the involved phenomena must be used very cautiously because they neglect the influence of several major physical parameters related to the spectral distribution of radiant energy. Results of performed experiments, some of which are presented in Part 1 of this paper, Fig. 1, can be summarized as follows [12]: - T h e thermoelastic effect produces thermal strain that can be noticeable in the regions of stress concentrations. It is possible to account for this effect by generalizing the expression for the stress concentration factor an, (Ogn)dynamic = (O~,)g. . . . tr,c[1 -~ (Ogn)th. . . . . ,astic] = (o~.)g[1 + (a.)th ] ,

(32)

where the geometric concentration factor also depends on the curvature radius of the notch tip p, and on the characteristic dimension parameter, for instance, the plate thickness b, (a.)g = a.(p/b

) .

(33)

- The thermoelastic effect influences the shape of the creep curves of viscoelastic materials

during the initial period of loading, in addition to other parameters. The thermoelastic effect can be used to measure the volumetric components of stress or strain tensors in a point-wise manner since the diameter of the temperature gage can be much less than one milimeter. Since elastic shear waves cannot produce any temperature alteration, the thermoelastic effect can be used to easily distinguish between the dilatational and shear waves. The thermocouples used in experiments respond to two mechanisms of the transport of heat energy, conduction and radiation, and to the strain induced in the thermocouples themselves. Consequently, the following forms of the thermoelastic strain gages can be distinguished: • Bonded thermoelastic strain gages responding to the thermoelastic effect produced in gages: strained gages; • Bonded thermoelastic strain gages responding only to the surface temperature alteration in the object: unstrained, contact gages; • Radiation thermoelastic strain gages, responding in a manner depending on the design of experiments: contactless gages.

5. Application of isodynes to problems in composite structures

Two model problems in the mechanics of composite structures have been chosen to illustrate the efficacy of the isodyne technique. The first problem involves a symmetric three-ply construction with the middle ply containing through-thethickness transverse crack normal to the external plies and extending across the width of the structure. The laminated plate is subjected to in-plane tension directed along the longitudinal axis. The second problem deals with a bi-material beam subjected to three-point bending and containing two interlaminar disbonds located symmetrically with respect to the central load. The above two problems have been chosen for several reasons. First of all, they represent models of two different but complementary problems of technological importance encountered in the field of composites. Interlaminar and transverse

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

cracks in composite materials and structures are often difficult to avoid and thus accurate characterization of stress fields in the vicinity of such defects is indispensable from the point of view of defining flaw criticality of advanced composites under various loading conditions. Secondly, the functional form of stress distributions in the vicinity of crack tips for the above configurations as predicted by quasi-static plane elasticity solutions is markedly different, thus presumably leading to different failure modes. As at present there are no generally accepted theories to predict the direction of crack growth and the critical loads of the above cases, the experimental results are of theoretical interest to fracture mechanicians. Thirdly, the possibility of a three-dimensional state of stress in the vicinity of the simulated defects, as has already been demonstrated in the case of homogeneous beams containing surface notches, see Fig. 3, raises the issue of the range of applicability of the plane elasticity solutions in problems of this type. Lastly, experimental models utilized in this study, being inherently threedimensional, can be taken as iconic models of real structures. Consequently, extrapolation of the experimentally obtained stress distributions along various planes within the model to the actual engineering structures is a direct process, a feature of Isodyne Photoelasticity which illustrates both the power and versatility of the developed technique. The experimental results for the first problem are given in Fig. 6a,b, where the maximal shear stress variation o-13in the y = 0 plane at different z i locations above the crack has been determined as a function of the axial distance from the crack tip. The results are presented for two material configurations with different Poisson's ratio for the inner and outer plies: polyester-glass-polyester and polyester-aluminum-polyester systems. The significant influence of the Poisson's ratio on the internal stress distribution in the vicinity of the crack is clearly evident both in the recordings of isodyne fields and the calculated results. It is seen that the loci of the maximum values of the shear stress o'13 occur at different axial distances from the crack tip in the two configurations. For comparison, the predictions of an approximate analytical model [24] are included in Fig. 6b. As the variation of the normal axial stress across the ply thickness has been neglected in the

163

approximate model, the correlation between the analytical solution and isodyne data is mainly of a qualitative nature at points sufficiently remote from the crack. In the region close to the crack tip however, the empirical and analytical results differ radically. Whereas the approximate analytical model yields continuously increasing values for 0-13, the experimental data indicate local maxima as "already mentioned. On the other hand, the local solution for the stress distribution around a crack perpendicular to a bi-material interface developed by Zak and Williams [27] does indeed exhibit such local maxima of 0-~3 near the crack tip [20]. However, the influence of the Poisson's ratio on the localized crack tip stresses predicted by the above plane elasticity solution does not appear to be as significant as indicated by the experimental results. The fracture surface of the polyester-glasspolyester configuration is shown in Fig. 7. As seen in the figure, there is no evidence of debonding along the bi-material interface and the fracture surface has a very smooth appearance away from the crack tip. Evidence of intense stress concentrations in the immediate vicinity of the crack tip is indicated by the local undulations in the fracture plane. The question of whether these undulations are caused by the presence of a highly inhomogeneous three-dimensional state of stress in the vicinity of the crack tip is under investigation. The experimental and analytical results for the second problem are presented in Figs. 8a,b and 9. Figure 8a illustrates the recordings of the x-isodyne field along with the recordings of the dark and light isochromatics fields included for comparison. The disturbances in the stress fields caused by the presence of the two interlaminar disbounds is clearly visible in the two sets of recordings, which also illustrate the near-perfect symmetry of the stress distributions with respect to the centrally applied load. The analytical model chosen for comparison with the experimental results is based on the two-dimensional elasticity formulation which has been proposed by Erdogan and Gupta [4], among others. It employs an infinite Fourier Transform technique and thus is strictly applicable to finite-height, laminated plane geometries extending to -+~ in the x-direction. Consequently, the experimental-analytical correlation is not

164

J.T. Pindera, M.-J. Pindera / On the methodologies of stress atzalysis: Part 2

P

~}/Crock,

i

y

I /-'~l

=I

o- t ~-jo'~z..'

\\

I ~'=z/h , y/b \

/

1 I

~ : x/.,

~- Field of Recordin 9

Fig. 6a. Isodynestress analysis of three-ply structures with internal cracks. Data on experiment, and isodyne fields in outer plies of two specimens.

expected to be valid in regions close to the free end of the structure. Along the formulation yields a physically inadmissible solution in the crack-tip region less than 10-6-10 -4 of the crack length as discussed by Malyshev and Salganik [7], this limitation is frequently deemed of no practical consequence as all infinitesimal elasticity solutions for this type of problem must break

down so close to the crack tip, including those which circumvent the problem of crack-tip material interpenetration (see [25]). The analytical expressions for the tractions in the plane of lamination can be shown to reduce to the following set of singular integral equations coupled in the u and v components of the displacement discontinuities #,(x) and Ov(x ) across

J.T. Pindera, M.-J. Pindera I On the methodologies oj: stress analysis: Part 2

Palatal-Glass-Palatal Specimen With a Crack

Palatal-AI Alloy-Palatal Specimen With a Crock Z/I ~ 2/2 I (o'13)n 4~- /

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= 0.1

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I

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distributions in chosen cross-sections of the specimens presented in Fig. 6a, evaluated analytically and

experimentally.

. . . . . . . . . . . . . . . . . .

Palatal F~-~ (BASF) Glass Original Crack (Defect)

~i!i

;"I-Plane of Sudden Fracture

---

I .--I-z ~D

L.

Fig. 7. Fracture surface of a three-ply composite structure with an internal crack.

166

J.T. Pindera, M.-J. Pindera / On the methodologies o f stress analysis: Part 2

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b

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oJ

,

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c_

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J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

167

Fig. 9. Fracturepatterns in compositebeams with two Localdisbondsunder three-point loadings.Beambefore test (top) and two fractured beams. Primary, or initial fractures at the left. Secondary,or dynamicfractures at the right.

the crack interfaces when the crack faces are free of tractions:

{

-%(x)

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-

0

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B.

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~ l_F~,(s, x )

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' (34)

where

Oo(x), Ou(x) = 0 is the region Ixl > a, a being half the crack length, s = Fourier-transform variable, B n , B12 = constant parameters describing the degree of material property mismatch of the upper and lower beam,

AiAs, x, x'), Fij(s, x; = regular kernels obtained from inversion of the Fourier-transformed structural matrix of the composite beam, + P~, Py = externally applied normal fractions per. unit width. It ought to be mentioned that the applicability of the traction-free condition in the above formulation is verified by experimental data which indicates that the traction components trxy(X) and O'yy(X) vanish on the crack faces. The solution of the above system of equations closely follows the general methodology outlined in [3] where a weighted residual technique is employed to reduce the singular integral equations to a set of algebraic equations in unknown coefficients of Jacobi polynomials used to approximate the displacement discontinuities O,,(x) and O,,(x). Determination of the unknown coefficients, which has been carded out using an 11-term expansion in Jacobi polynomials and 384 Gaussian integration points in the numerical evaluation of the integrals in the above formulation, consequently allows one to obtain the full field interfacial stresses given above. The experimental-analytical correlation of the shear and normal traction components acting in the plane z = 0 along the interface containing the

168

J, T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

disbonds is presented in Fig. 8b for the generalized plane strain case. It ought to be mentioned that the difference between the plane stress and generalized plane strain results for the interlaminar tractions as predicted by the analytical model is not very significant in this case and thus the plane stress results are not shown. Several interesting features of the correlation study are noteworthy. The correlation between the experimentally obtained values of the shear traction component Oxy(X ) and the analytical resuits is very good as the interlaminar crack is approached from the center of the beam, i.e., in the inner region. In the outer region the difference between experimental and analytical results increases somewhat with the distance from the crack tip, although the functional form of the two curves is very similar. This discrepancy may be caused by some function of geometric and material parameters; for instance, it has been observed that the interlaminar stresses in the central region of the bimaterial beam may depend significantly on the viscoelastic behavior of the polyester beam. The comparison between the analytical results for tryy(X) and the experimentally obtained differential stress values (Oryy)diff = Oryy- Orzz appears to indicate the presence of a significant through-the-thickness normal stress component in the polyester beam along the lamination plane in the vicinity of the interlaminar disbonds and the points of external load application. The difference between the externally applied normal tractions and the integral of the difference tryy trzz along the plane of lamination as determined from the isodyne recordings approaches 30%. On the other hand, this difference becomes very small in the plane located half way across the height of the polyester beam where the methods of transmission photoelasticity are applicable. The fracture patterns of two failed bi-material beams are presented in Fig. 9. The primary crack appears to have initiated at the outer tip of the disbond at an angle of approximately 66 ° and then apparently changed direction and propagated dynamically at an angle of 59 ° with respect to the plane of lamination. The subsequent fracture process involved partial delamination of the beam components and clearly dynamic fracture across the height of the polyester beam not far from the inner crack tip of the other symmet-

rically located disbond. The chronological sequence of particular fracture and delamination processes was not determined. The overall fracture process was catastrophic for all the specimens tested with no apparently significant initial quasistatic crack growth along the plane of lamination in the outer region. All fractures occurred several, or more, minutes after application of the critical load. It is interesting to note that the very rapid crack growth initiated at the outer tip where the normal traction component O-yyis compressive as predicted both by the isodyne and analytical calculations. Furthermore, both the experimental data and the analytical model indicate that the shear traction component increases faster near the external crack tip than the internal tip, although the analytical model predicts that this difference is smaller than obtained experimentally. On the other hand, the axial stress component O-xxin the polyester beam along the plane of lamination is significant and has different sign in the vicinity of the outer and inner crack tips: it is tensile as the disbond is approached from the outer region and compressive as the disbond is approached from the inner region. The correlation of predictions of the various theories-primarily of those based on fundamental physical principles [23J--that predict the direction of crack growth and critical loads necessary to initiate crack propagation for the above configuration with the experimental results will be reported at a later time.

6. Conclusions

The presented examples show that the isodyne techniques, strain gradient techniques, and the auxiliary thermoelastic effect techniques can be applied successfully and in a self-contained manner to solve various stress analysis problems of importance in strength and service life optimization of composite structures and materials. These examples illustrate that the isodyne techniques supply, whole field-wise, data on the two dimensional and three dimensional stress states which cannot be obtained by using other, presently available, experimental methods. In particular, the photoelastic isodyne techniques can directly

J.T. Pindera, M.-J. Pindera / On the methodologies of stress analysis: Part 2

supply information on: Total values of the normal and shear forces acting in conveniently chosen cross-sections of a plate; this allows one to check immediately the reliability and accuracy of the performed measurements; - Values of the normal and shear stress components of plane stress fields, which can be checked immediately (four independent pieces of information about three unknown quantities); - L o c a l regions in plates where the stress states are clearly three dimensional; - A l l components of the three dimensional stress states that occur in plates in the regions of notches, cracks, local load application, local heat sources, etc.; - Values of stresses in composite structures, in particular, stresses in the lamination planes, peel stresses, and stress distributions across the thickness of the laminated components; - Values of the stress components in objects of arbitrary shape, by performing measurements in several directions; - Thermal stresses; Residual stresses induced during assembling or caused by local plastic deformations. The isodyne techniques can be applied to determine the internal forces and stress components in objects that are transparent to radiation either within the visible band or the near infrared band or the far infrared band, and that exhibit scattering properties similar to those described by the Rayleigh model. The strain gradient techniques supply data on the gradients of sums and differences of principal stresses, and on the rotation of the lateral faces of plates in regions of high strain gradients related to strong local three-dimensional stress states. The thermoelastic effect techniques are of particular interest in the analysis of vibrations, and in the design of fatigue tests for plastics, ceramics, composite materials and other advanced materials. Of particular interest is the efficacy of this technique in studying the propagation and interference of elastic waves. The ease with which this technique allows one to distinguish between longitudinal and shear strain waves is quite spectacular. There is an urgent need to advance our knowledge and to develop reliable analytical tools in the following areas:

-

169

- Theory of modelling and choice of modelling criteria; Physical basis for reliable analytical presentation of the actual creep and relaxation phenomena; Actual transfer functions of testing and measurement systems and reliability of experimental results (comparison with standard experiments); - Applicability (reliability and ranges of acceptable errors) of linear relations in regions of large deformations; - Transition from stress analysis at the micromechanics (and local) level to stress analysis at the macro (and global) level.

Acknowledgments

The authors gratefully acknowledge the support given by the Natural Sciences and Engineering Research Council of Canada under Grant No. A2939. Also, the authors would like to thank Xin-hua Ji for her assistance in the course of investigating the responses of laminated beams.

References

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J.T. Pindera, M.-J. Pindera / On the methodologies of stress analy,sis: Part 2

[8] J.T. Pindera, "Technique of Photoelastic Studies of Plane Stress States", Rozprawy Inzynierskie .L 109-176 (1955), in Polish. [9] J.T. Pindera, "Rheological Properties of Some Polyester Resins, Part I, II, III", Rozprawy Inzynierskie, Polish Acad. Sci. 3, 361-411 and 481-540 (1959), in Polish. [10] J.T. Pindera and E Straka, "Response of the Integrated Polariscope", J. Strain Analysis 8, 65-76 (1973). [ll] J.T. Pindera and S.B. Mazurkiewicz, "Photoelastic Isodynes: A New Type of Stress Modulated Light Intensity Distribution", Mech. Res. Communications, 247-252 (1977). [12] J.T. Pindera, P. Straka and M.F. Tschinke, "Actual Thermoelastic Response of Some Engineering Materials and its Applicability in Investigations of Dynamic Response of Structures", VDI-Berichte No. 313, VDI Verlag, Dfisseldoff, 579-584 (1978). [13] J.T. Pindera, "Foundations of Experimental Mechanics: Principles of Modelling, Observation and Experimentation", in: J.T. Pindera, ed., New Physical Trends in Experimental Mechanics, (CISM Courses and Lectures No. 264), Springer-Verlag, Wien, 188-326 (1981). [14] J.T. Pindera and B.R. Krasnowski, "Determination of Stress Intensity Factors in Thin and Thick Plates Using Isodyne Photoelasticity", in Leonard A. Simpson, ed., Fracture Problems and Solutions in the Energy Industry, Pergamon Press, New York, 147-156 (1981). [15] J.T. Pindera and S.B. Mazurkiewicz, "Studies of Contact Problems Using Photoelastic Isodynes", Experimental Mech. 21, 448-455 (1981). [16] J.T. Pindera, S.S. Issa and B.R. Krasnowski, "lsodyne Coatings in Strain Analysis", Proc. 1981 Springer Meeting, SESA (Dearborn, Michigan, May 31-June 4, 1981), SEM (Formerly SESA), 111-117 (1981). [17] J.T. Pindera, F.W. Hecker and B.R. Krasnowski, "Gradient Photoelasticity", Mech. Res. Communications 9, 197-204 (1982).

[18] J.T. Pindera, "New Development in Photoelastic Studies: Isodyne and Gradient Photoelasticity", in: F.P. Chiang, ed., Incoherent Optical Techniques in Experimental Mechanics, Optical Engineering 21, 672-678 (1982). [19] J.T. Pindera, "Apparatus for Determination of Elastic Isodynes and of the General State of Birefringence Whole Field-Wise Using the Device for Birefringence Measurements in the Scanning Mode", Canadian Patent 1156488 (1983). [20] J.T. Pindera, B.R. Krasnowski and M.-J. Pindera, "Theory of Elastic and Photoelastic Isodynes. Samples of Applications in Composite Structures", Experimental Mech. 25, 272-281 (1985). [21] J.T. Pindera and F.W. Hecker, "Basic Theories and Experimental Methods of Gradient Photoelasticity", Experimental Mech., to be published. [22] G.N. Ramachandran and S. Ramaseshan, "Crystal Optics", in: S. Flfigge, ed., Handbook of Physics, Vol. 25, Springer-Verlag, Berlin, 1-217 (1961). [23] G.C. Sih, "Mechanics and Physics of Energy Density Theory", Theoret. Appl. Fracture Mech. 4, 157-173 (1985). [24] V.V. Vasil'ev, A.A. Dudchenko and A.N. Elpatevskii, "Analysis of the Tensile Deformation of Glass-Reinforced Plastics", Mekhanika Polimerov 1, 144-147 (1970k [25] S.S. Wang and I. Choi, "Boundary-Layer Effects in Composite Laminates: Part lI--Free-Edge Stress Solutions and Basic Characteristics", J. Appl. Mech. 49, 549-560 (1982). [26] D.W. Wilson and J.A. Charles, "Thermographic Detection of Adhesive-bond and Interlaminar Flaws in Composites", Experimental Mech, 21, 276-280 (1981). [27] A.R. Zak and M.L. Williams, "Crack Point Stress Singularities at a Bi-Material Interface", J. Appl. Mech. 30, 142-143 (1963).