A review of the biaxial strength of fibre-reinforced plastics

PII: S1359-835X(97)00090-0

Composites Part A 29A (1998) 869–886 1359-835X/98/$ - see front matter q 1998 Published by Elsevier Science Ltd. All rights reserved.

A review of the biaxial strength of fibrereinforced plastics

Holger Thom Bundesanstalt fu¨r Materialforschung und -pru¨fung, 12200 Berlin, Germany (Received 10 February 1997; revised 5 August 1997; accepted 26 August 1997)

Several strength criteria for laminae and laminates are presented and compared numerically. It is concluded that the von Mises hypothesis is not suitable for composites, instead strength criteria should be derived from Mohr’s hypothesis. In industry mostly laminates are used. It would be desirable if the strength properties could be determined for any arbitrary laminate from those of the laminae included. Since this is spoiled by interaction between the layers, some of these interaction effects are described. In order to become established, a strength criterion has to be verified experimentally. Thus, different techniques for biaxial testing with their characteristic problems are shown. q 1998 Published by Elsevier Science Ltd. All rights reserved. (Keywords: biaxial strength; D. mechanical testing; strength criteria)

Nomenclature a,A a,A d E « e F G n P J S j t t

scalar vector, tensor diameter elastic or Young’s modulus strain ultimate strain factor of a strength tensor shear modulus Poisson’s ratio pressure fibre volume fraction ultimate stress (absolute value) normal stress wall thickness shear stress Subscripts

1 2,3 n,t x,y,z

fibre longitudinal direction (also principal axis of orthotropy) fibre transverse directions normal, transversal to plane arbitrary directions Superscripts

þ ¹

property for tension property for compression

Calculations are done with the following material data for carbon/ epoxy-HT: S1þ ¼ 2150 MPa, S1¹ ¼ 1500 MPa, S2þ ¼ 35 MPa, S2¹ ¼ 200 MPa, S12 ¼ 80 MPa.

INTRODUCTION Initially, research in the field of anisotropic materials was carried out on materials such as timber or crystals. In the mid 1960s, research started to focus on fibre-reinforced plastics (FRPs). In contrast to isotropic materials, no failure criterion has been established so far for these materials. This lack of a criterion is due to the following additional problems. • The criterion cannot be based on the principal stresses due to anisotropy (like the von Mises hypothesis), but has to be formulated with respect to the axes of anisotropy. • Fibres ending at free edges of laminates induce interlaminar stresses 1. • FRPs are more sensitive to local stress concentrations, therefore specimens have to be designed more accurately 2. • Owing to the high modulus ratio in the two axes of orthotropy, the mode of failure has to be checked (buckling) 2. • The transformation of the strength properties of different plies to the properties of a laminate is not at all obvious; Gerharz and Schu¨tz3 even said that this was impossible. • The term ‘failure’ is a question of definition, as with ductile materials. The first change in behaviour is due commonly not to yielding but to initial fibre fractures or microcracks in the resin of the off-axis plies. The lack of a reliable failure criterion is disadvantageous, especially for composite materials. FRPs are mostly used in lightweight constructions. If the point of failure is only vaguely known, high safety factors may cause unnecessary weight. Nevertheless industry uses very simple and imprecise failure criteria like the maximum stress criterion.

869

A review of the biaxial strength of fibre-reinforced plastics: H. Thom This is due to today’s methods of production which still cause high scattering, and because fatigue or impact are usually the design restrictions of the component. However, with the goal of composite design being theoretically substantiated and not only experimentally proven, a reliable criterion describing static strength will be one of the basic constituents. Reading the literature, one observes that initially criteria were formulated theoretically but not verified experimentally. Only later, when testing methods were developed and materials were cheaper, can trustworthy results be found in the literature. Still, there are a lot of unanswered questions, especially since most researchers concentrated on laminates without searching for the effects leading to the difference between laminae and laminates. A good failure criterion should take the following rules into consideration; • it should describe as many different types of materials and loadings as possible, • a minimum of experimental work should be necessary to determine the criterion’s coefficients. The establishment of a failure criterion involves three steps: the formulation of the criterion, the experimental verification, and the convincing of future users. This survey presents and compares some of the common criteria. Afterwards, different geometries of specimen and their problems are shown. The successful path from the scientist’s ivory tower to the design department cannot be found in this study. First we present a survey, although very brief, concerning the different modes of fracture and effects not known from isotropic monoliths.

FAILURE MECHANISMS Tensile fibre failure Typical for tensile fibre failure is pull-out on the failure surface. This also happens under very cold and dry, i.e. brittle, conditions 4. A rise in temperature increases the pullout length. These observations show that local fibre fractures occur earlier than the final fracture, owing to misalignment and scatter during the manufacture of the fibres. The stresses of broken fibres are transferred to intact fibres by the resin. A further increase in the stress results in debonding of the fibre due to stress peaks at its ends. While Miller and Wingert4 state that ‘‘specimen fracture is the cumulative total of a series of nearly independent microstructural fracture events’’, Purslow 5 reported that he was able to determine the direction of crack propagation on small surfaces. Since the crack process on a large area is chaotic and propagates on several surfaces, the origin of the crack cannot be ascertained. Compressive fibre failure This mode of failure is to a lesser degree determined not by the strength of the fibres but rather by their stability. The

870

Figure 1 Compressive strength of GFRP

failure results from microbuckling or kinking of the fibres within the resin. On SEM pictures, tensile, compressive and neutral zones on the fibres can often be discovered. Microbuckling is more likely to initiate at free edges or close to inner voids, where lateral support is reduced. It can be distinguished from kinking bands by the multiple fractures of the fibres. Dow and Rosen6 tried to find the actual mode of compressive failure. They distinguished an extension and a shear mode (Figure 1). In the extension mode, the adjacent fibres will buckle with 1808 phase shift, thus extending the matrix mainly in a direction perpendicular to the fibres. Shear strains are the main deformation in the shear mode, where the fibres buckle in phase. The result of these analytical models is that a lamina of commonly used fibre volume fraction will fail in the shear mode. The affiliated compressive strength is given by S1¹ ¼

Gm 1¹J

(1)

Since the linear model caused overestimation of the compressive strength at high fibre volume fractions, they also considered non-linear elastic behaviour. The compressive strength of the non-linear modeal was still higher than the experimentally determined value. Wisnom7 looked at the influence of fibre misalignment on the compressive strength. Only 0.258 misalignment reduced the predicted strength from 2720 MPa to 1850 MPa for a carbon/epoxy lamina. At 38 misalignment, a strength of only 700 MPa was left. A periodic deflection of the fibres was examined by Hahn and Williams8. They found the compressive strength to be g12 (2) S1¹ ¼ JG12 g12 þ pfh =l G12 is the secant composite shear modulus, f0 the amplitude of the fibre deflection, and l is half of its wavelength. This is in contrast to Rosen’s theory, since the equation

A review of the biaxial strength of fibre-reinforced plastics: H. Thom above becomes S1¹ ¼ JG12

(3)

if no initial deflection is assumed. The difference is due to the selection of the free body. In the model of Hahn and Williams the forces are applied only to the fibre. No matter which theory is used, the important fact is that the compressive strength of the composite is somehow proportional to the shear modulus of the matrix. This has also been verified experimentally. In practice, the theoretical compressive strength can never be reached owing to fibre misalignment and fibre curvature. Figure 2 Delamination due to microcracks in off-axis plies

Matrix failure The mechanism of matrix failure is the most complicated. It is presented here only very briefly. Matrix failure in a ply is called ‘inter-fibre’ or ‘intralaminar’ fracture, the separation of two plies ‘delamination’ or ‘interlaminar’ fracture. The fracture mechanics differs according to the fracture mode of normal or shear stress (mode 1, 2). The morphologies of both modes can be distinguished clearly 9. It is believed that inter-fibre fracture starts at the interface of fibre and resin10. The crack then propagates through the interface into the resin. This process produces river markings and textured microflow in the resin when loaded in mode 1, and stacked lamellae (also called hackle marks) when loaded in mode 2. Delamination is caused by interlaminar stresses (see freeedge effect). Interlaminar stresses often result out of microcracks in off-axis plies. Thus, delamination is not dependent primarily on the homogenous stress state11. A unidirectional lamina failure can be attributed to any of the reasons mentioned above, whereas a laminate fails ultimately usually owing to fibre failure12 (an exception is delamination or oblique, compressive inter-fibre cracking13). Free-edge effect A speciality of compounds is the free-edge effect. Because of this effect, geometry can influence the macroscopic strength. Laminate theory does not include the constraints of a compound. According to this theory it follows that multidirectional compounds would have normal and shear stresses on edges with ending fibres. Pipes and Pagano14 calculated the stresses near a free edge with finite differences. They found a strong increase in the shear stress txz and to a lesser degree in jz near the free edge. This three-dimensional stress state reaches approximately as far into the compound as it is thick. Pipes et al. showed15 that failure due to the free-edge effect also depends on the angle of two adjacent plies. The strength of a 6308 specimen is greatly reduced by the free-edge effect. The ultimate strain is equivalent to that of the 308 off-axis specimen. However, the ultimate strain of a 6 458 specimen is higher than that of the 458 off-axis specimen. The free-edge effect does not seem to influence the 6 458 specimen. An explanation for this difference is the value of txz . Near the free edge and the

interface of two plies this stress is much higher for the 6308 specimen than for the 6 458 specimen. The free-edge effect is also responsible for the difference in strength of differently stacked, but otherwise similar, laminates. Schulte and Stinchcomb16 reported that the strength of a (08/908) laminate depended on the orientation of the outer ply. Also Whitney and Browning17 observed on a ( 6 458,908) s laminate that only one stacking sequence led to delamination. In the case of delamination jz was greater than zero, in the other stacking sequence smaller than zero. Pagano and Pipes18 examined this correlation in detail and concluded that the difference in strength can be put down to the interlaminar normal stress. A correlation with interlaminar shear stresses is less significant. Cook–Gordon effect19 This effect also leads to differences in strength between unidirectional laminae and multidirectional laminates. Swanson20 observed that a lamina had a higher strength than a laminate, independent of the free-edge effect. This became obvious when a tough resin was applied. Whereas a normal, brittle resin decreases the ultimate strain by 5%, the tough resin reduced it by 30%. An explanation is delivered by the Cook–Gordon effect. In off-axis plies of laminates, microcracks occur long before failure of the longitudinal fibres (Figure 2). These microcracks lead to stress peaks in the adjacent longitudinal fibres. Those stress peaks are reduced by a debonding of the fibres. In case of a strong interface, like the interface of the tough resin, this mechanism no longer works as well. Geometric influences on strength The strength of compounds also depends on further geometric parameters. There is a definite correlation between the thickness and the strength of a compound and of a laminate’s ply16,21. Increasing thickness decreases the strength. O’Brian and Salpekar21 found with laminates consisting of 6, 8, 16, 32, 64 plies, a maximum strength at a thickness of 8 plies. For thin 908 off-axis plies the strain for the first microcrack increases progressively with decreasing thickness (see Cook–Gordon effect)22. The strain, before the first microcracking occurs, does not depend on S 2 but on

871

A review of the biaxial strength of fibre-reinforced plastics: H. Thom Table 1 Survey of the presented strength criteria Criterion

Distinction between failure modes

Interaction term

Application to UD-lamina, MD-laminate, not specified

Analytic Maximum stress/strain Tsai–Hill Hoffman Franklin–Marin Tsai–Wu Puppo–Evensen Hashin–Rotem Wu–Scheublein Tennyson–MacDonald–Nanyaro Chang Chang Truncated maximum strain Cuntze Puck

þ ¹ ¹ ¹ ¹ ¹ þ ¹ ¹ þ þ þ þ

Experimental

¹ þ þ ¹ ¹ þ ¹ ¹ ¹ ¹ ¹ ¹ ¹

the energy release rate. Under certain thicknesses, microcracks are even suppressed completely. The strain necessary for microcracking also depends on the location of the ply in a laminate. Microcracks in an outer ply occur at lower stress levels than in inner plies. Some of these measured differences can also occur as a result of the manufacturing process of the specimens. Clements and Lee23 increased the strength of 08 specimens by 15%–25% after polishing the edges. They concluded that the strength was decreased by micro damage to fibres and by the different ratio of surface to volume of previously cut fibres. When testing two 6 458 off-axis specimens, one narrow and the other wide, one will observe that the narrow specimen will fail owing to matrix failure, whereas the wide specimen fails owing to fibre failure. The wide specimen has long continuous fibres, the narrow one has only short fibres with much less overlap24. This can also change Young’s modulus25.

STRENGTH CRITERIA Over the years, many strength criteria have been developed. This section presents some of the most important ones and explains their formulations. The criteria presented are all macroscopic. Input data consist of the strength of a single ply or a whole laminate as well as the stresses. From the macroscopic criteria, microscopic criteria have to be delimited, which are mostly used in combination with finite elements. They are based more on the physical processes of crack propagation, mainly on the energy release rate. Because of their need for immense CPU time and lavish modelling, their application in industry for large models is doubtful in the future. Further surveys of strength criteria can be found in references26–36. To start with, Table 1 shows the presented criteria, their structure, and their application.

UD n.s. n.s. n.s. n.s. n.s. UD MD n.s. UD MD UD UD

¹ ¹ ¹ þ þ ¹ ¹ þ þ ¹ ¹ þ þ

Since the strength of anisotropic materials cannot be described in a principal stress state, he added shear stresses. It is formulated for stresses parallel to the axes of orthotropy. Interaction terms are not taken into consideration. The different modes of failure are distinguished. For the plane stress state it is of the following form: j1 ¼ S1þ (j1 . 0) j1 ¼ ¹ S1¹ (1 , 0)

(4)

j2 ¼ S2þ (j2 . 0) j2 ¼ ¹ S2¹ (j2 , 0) jt12 j ¼ S12 38

Pettit and Waddoups supplemented the maximum strain criterion in order to consider non-linear elastic responses. Tsai–Hill (or modified Hill) (1965) criterion39 Hill derived the criterion from the von Mises hypothesis for anisotropic materials. Later this was altered by Tsai for composites. The criterion has interaction terms, but it does not distinguish between tensile and compressive properties. Fan 40 suggested the use of the tensile, or compressive, strength as a function of the actual stress sign. j j j j t (5) ( 1 )2 ¹ 1 2 2 þ ( 2 )2 þ ( 12 )2 ¼ 1 S1 S2 S12 S1 Hoffman (1967) criterion41 The Hoffman criterion is a supplement to the Tsai–Hill criterion. Linear terms were added to consider the difference of tensile and compressive strength: (

1 1 1 1 j21 j j ¹ )j þ ( ¹ )j þ ¹ 1 2 S1þ S1¹ 1 S2þ S2¹ 2 S1þ S1¹ S1þ S1¹ þ

j22 þ ¹ S2 S2

þ(

t12 2 ) ¼1 S12

ð6Þ

Franklin–Marin (or modified Marin) (1968) criterion27 Maximum stress/strain criterion (1920)

37

This is the most simple criterion. Jenkins derived it from the principal stress criterion for brittle, isotropic materials.

872

Franklin modified the criterion of Marin proposed in 1956. Marin had added terms to the von Mises hypothesis to consider the different properties for tension and

A review of the biaxial strength of fibre-reinforced plastics: H. Thom compression. He did not add a term for the shear stress (although it is formulated in the system of orthotropy), and also did not consider the difference between compression and tension for the transverse strength. Franklin removed these weak points and added a further term for biaxial strength, which has to be determined experimentally. This term is supposed to increase the accuracy of the criterion. The resulting equation is S1¹ ¹ S1þ S2¹ ¹ S2þ j21 K12 j1 j2 þ ¹ j1 þ þ ¹ j2 þ þ ¹ ¹ S1 S1 S2 S2 S1 S1 S1þ S1¹ þ

j22 þ ¹ S 2 S2

þ

t212 ¼1 S212

ð7Þ

K12 is determined by an experiment with biaxial tension. With the stresses j1 ¼ j2 ¼ j, K12 is determined by  þ ¹ S1þ S1¹ 1 ¹ þ ¹ þ S1 S1 K12 ¼ 1 þ þ ¹ þ (S1 ¹ S1 ) þ (S2 ¹ S2 ) þ ¹ j S 2 S2 S2 S2 ¹

S1þ S1¹ j2

ð8Þ

Tsai–Wu (1971) criterion42

Tsai and Wu neglected all summation terms beginning with the cubic. The cubic term not only increases the number of unknowns significantly, but would also be responsible for an open-ended failure envelope. The same formulation was already used by Zacharov (1963), but is not found in Western literature29. The formulation by Gol’denblat and Kopnov (like the polynomial of Tsai–Wu, but square root of the second summation term) is not accorded more generality by Tsai and Wu, since the equation can be transformed to 2Fi ji þ Fij ji jj ¹ (Fi ji ) ¼ 1 2

(10)

With regard to curve fitting, the formulation is no better than that of Tsai and Wu, though more complicated. For transversly isotropic materials, the tensor of the second order consists only of coefficients on the main diagonal, owing to symmetry. Since strength cannot be dependent on the sign of shear stresses, coefficients with a single 4, 5 or 6 in the index must vanish. For transverse isotropy the indices are equivalent. Under these assumptions, the equations for the coefficients are the following:

F11 ¼

1 S1þ S1¹

F44 ¼

1 S223

F2 ¼

F22 ¼

F55 ¼

1 S231

1 1 þ ¹ ¹ S2 S2

1 S2þ S2¹ F66 ¼

1 1 1 1 j21 þ ¹ ¹ )j1 þ ( þ ¹ ¹ )j 2 þ þ ¹ S1 S2 S1 S1 S1 S2 þ F12 j1 j2 þ

The Tsai–Wu criterion is a tensor polynomial. The formulation has the following form in Voigt’s tensor notation: (9) Fi ji þ Fij ji jj þ … ¼ 1, i ¼ 1…6

1 1 ¹ ¹ ¹ S1 S1

The remaining coefficients F12 , F13 and F23 have to be determined experimentally. For plane stress and transverse isotropy, the criterion can be written as (

The Hoffman criterian coincides when K12 ¼ 1.

F1 ¼

Figure 3 Sensitivity of jbiax on F12

F3 ¼

F33 ¼ 1 S212

1 1 þ ¹ ¹ S3 S3

1 S3þ S3¹ ð11Þ

j22 þ ¹ S 2 S2

þ

t12 ¼1 S12

ð12Þ

The term F12 has to be determined by a biaxial experiment. This determination is very complicated though, since F12 has very little sensitivity to the biaxial stresses. Tsai and Wu mentioned the following experiments for the determination: biaxial tension, a 458 off-axis specimen (recommended by many Russian scientists), and a 458 shear test (recommended by Tsai and Wu). Since the best experiment is also dependent on the material, a sensitivity study, as p on shown in Figure 3, should be done. The parameter F12 the abscissa is the standardized value of F12 : F12 p  ¼ p F12 F11 F22

(13)

Since Tsai and Wu also introduced a stability criterion to p has to be in the guarantee a closed failure envelope, F12 p range of ¹ 1 # F12 # 1. The stability criterion has the following definition: Fii Fjj ¹ Fij2 $ 0 no sum over i and j

(14)

Owing to problems with the experimental determination of F12 , several estimates can be found in the literature. Tsai and p ¼ ¹ 0:5, in order to fulfil the Hahn43 proposed the value F12 von Mises hypothesis in case of the isotropy. Narayanaswani and Adelman44 even proposed to remove the interacp tion term, since Pipes and Cole45 measured values of F12 contradicting the stability criterion. Pipes and Cole concluded, though, that their experiment was unsuitable for the determination of the interaction factor. In Figure 4 it can be p is not negligible for jx ¼ 6 jy ¼ 6 j, txy ¼ 0. seen that F12 The stability criterion mentioned above was criticized by Sendeckyj28, since strength under hydrostatic pressure is unlimited for most materials. This criticism is unimportant for the plane stress state, though.

873

A review of the biaxial strength of fibre-reinforced plastics: H. Thom 2 6 6 A3 ¼ 6 4 2 6 6 B¼6 4

b=S2x

¹ 1=2S2z

¹ b=2S2x

7 7 ¹ a=2S2y 7 5

a=S2y

1=S2yz

0 1=S2zx

3

1=S2z 3

0

7 7 0 7 5

ð18Þ

1=S2xy

Figure 4 Influence of F12 on the strength

Puppo–Evensen (1972) criterion46 Puppo and Evensen criticized the fact that some strength criteria were not invariant against coordinate transformation. They described the ‘paradox of the Hill-type criterion’. A balanced (equal number of identical plies in each direction) (08,908) s laminate has two sets of axes of orthotropy (in 98/908 and 6458). The failure envelope calculated with a non-invariant criterion is different for each set. Consequently the question arises, which set of axes has to be chosen, when the calculation is not done plywise. Therefore, an interaction factor g is defined: S2xy g¼3 Sx Sy

(15)

For isotropic materials this factor is equal to unity, for noninteracting materials zero, and for only shear reinforced, otherwise isotropic materials, larger than unity. With this factor the axes of principal strength can now be defined as those axes for which g is minimal. In the three-dimensional case there are three interaction factors: a¼3

S2yz Sy Sz

b¼3

S2zx Sz Sx

g¼3

S2xy Sx Sy

(16)

The strength criterion is like that by Hill of a quadratic nature. In Voigt’s tensor notation it has the form " # Ai 0 T i ¼ 1, 2, 3 (17) j Ri j ¼ 1 Ri ¼ 0 B Thus it consists of three intersecting failure envelopes. The submatrices Ai and B are defined by the following equations: 2 2 3 1=Sx ¹ g=2S2y ¹ b=2S2z 6 7 6 7 A1 ¼ 6 g=S2y ¹ 1=2S2x 7 4 5 2 6 6 A2 ¼ 6 4

b=S2z g=S2x

g=2S2x 1=S2y

¹ 1=2S2y

7 7 ¹ a=2S2z 7 5 a=S2z

874

3

Depending on the actual stress sign, the tensile or the compressive strength is used for Sx , Sy , and Sz . Further properties of this criterion are unlimited strength for hydrostatic pressure, and an equivalence to the von Mises hypothesis for isotropic materials. The factor g can be further adjusted to certain materials by introducing an exponent. So far, nobody seems to have made use of this potential. For plane stress the criterion can be rewritten as* ( g(

j1 2 S j j j t ) ¹ g( 1 )( 1 )( 2 ) þ g( 2 )2 þ ( 12 )2 ¼ 1 S1 S2 S1 S2 S2 S2 j1 2 S j j j t ) ¹ g( 2 )( 1 )( 2 ) þ ( 2 )2 þ ( 12 )2 ¼ 1 (g , 1) S1 S1 S1 S2 S2 S12 ð19Þ

The axes of principal strength can differ from those of orthotropy, when g is larger than unity. In this case they do not need to be orthogonal too. When g . 1 the following equation has to be used, which in turn is derived from the system rotated by 458: j j j j j j t ( 1 )2 ¹ 2h(g)( 1 )( 2 ) þ ( 2 )2 ¹ 2g(g)[( 1 ) 6 ( 2 )]( 12 ) S1 S1 S2 S2 S1 S2 S12 t þ ( 12 )2 ¼ 1 (g . 1) S12 1 h(g) ¼ 1 ¹ f (g) 2 p g(g) ¼ g=12[3f (g)=(f (g) ¹ 4) þ 1]f (g) q f (g) ¼ 0, 1[ (3=g þ 4)2 þ 240=g ¹ (3=g þ 4)]

(20)

In the case that this criterion is used in the system of orthotropy, the axes of principal strength do not have to be calculated. A check whether g is larger or smaller than unity is sufficient. In the case g ¼ 1 the difference between eqn (19) and eqn (20) vanishes. Hashin–Rotem (1973/1980) criterion

47,48

Apart from the maximum stress/strain criterion, the criteria presented so far do not distinguish between the different failure modes, as they are based solely on curve fitting. It is not possible to determine whether it is the fibre or the resin that fails. The calculated strength in the case of * The reference has a mistake in this equation.

A review of the biaxial strength of fibre-reinforced plastics: H. Thom biaxial tension also depends on the compressive strength in the Tsai–Wu and the Hoffman criteria. To take remedial action, Hashin supplemented a criterion, which was developed by Hashin and Rotem, to describe fatigue of composites. It takes distinct fibre and resin failure modes into consideration. The criterion starts out with transverse isotropy (S2 ¼ S3 , S23 ¼ S31 ). In order to guarantee invariance against rotation around the fibre axis, the invariants of the stress tensor are used to derive the criterion. These invariants are I1 ¼ j11

considered 1 S2þ 2

1 S2¹ 2 1 [( ) ¹ 1](j2 þ j3 ) þ 2 (j2 þ j3 )2 S2¹ 2S23 4S23

I4 ¼ j212 þ j213

þ

A1 I1 þ B1 I12 þ A2 I2 þ B2 I22 þ C12 I1 I2 þ A3 I3 þ A4 I4 ¼ 1 (22) From pure transverse and respectively longitudinal shear stress A3 and A4 are A4 ¼

1 S212

(23)

(28) The distinction, if j2 þ j3 is larger or smaller than zero, stands for the sign of the normal stress on the fracture plane. For the derivation, please refer to the reference. However, from this derivation it can be concluded that matrix failure occurs in quadratic approximation always in a plane of maximal, transversal shear stress. For the plane stress state, the equations can be rewritten as j t ( þ1 )2 þ ( 12 )2 ¼ 1 (j1 . 0) S12 S1 j1 ¼ ¹ S1¹ (j1 , 0)

Only stresses acting on the fracture plane are taken into account for the different failure modes. It is assumed that tensile and shear stresses are mutually weakening, therefore causing fibre failure j 1 ( þ1 )2 þ 2 (j212 þ j213 ) ¼ 1 (j1 . 0) S1 S12

(24)

The linear term of j1 is neglected, since a uniaxial tensile test delivers only one value. An even more simple approximation is achieved in the use of the maximum stress criterion: j1 ¼ S1þ

(j1 . 0)

(25) 49

It is also used for the compressive strength. Rosen reported that microbuckling takes place in a shear mode, but it was not known whether additional shear stress is weakening or strengthening. Therefore j1 ¼ ¹ S1¹ (j1 , 0)

1 2 1 (j23 ¹ j2 j3 ) þ 2 (j212 þ j213 ) ¼ 1 (j2 þ j3 , 0) S223 S12

(21)

I5 cannot be used, owing to the linear shear stresses. Using a quadratic formulation, it follows that

1 S223

ð27Þ

In the case of matrix failure due to compression, an additional condition can be taken into account. Analogous to hydrostatic pressure, it is assumed that the strength is significantly higher in the case of equal stresses in both transverse directions than in the case of uniaxial stress. By adding the linear terms to the previous equation, one arrives at

I3 ¼ j223 ¹ j22 j33

A3 ¼

1 2 1 (j23 ¹ j2 j3 ) þ 2 (j212 þ j213 ) 2 S23 S12

¼ 1 (j1 þ j2 . 0)

I2 ¼ j22 þ j33

I5 ¼ 2j12 j23 j13 ¹ j22 j213 ¹ j33 j212

(j2 þ j3 )2 þ

(26)

The description of the matrix strength is more complicated, since the plane of fracture is not known, a priori. Therefore the plane would have to be identified by a rotation, similar to Mohr’s circle, for which the strength criterion has a maximum. This is not only complicated, but also the criterion would no longer be quadratic. The normal stress in the fibre direction is neglected. Again, the linear term of the transversal normal stress cannot be

( (

j2 2 t12 2 ) ¼ 1 (j2 . 0) þ) þ( S12 S2

j2 2 S¹ j t ) þ [( 2 )2 ¹ 1] ¹2 þ ( 12 )2 ¼ 1 (j2 , 0) 2S23 2S23 S2 S12

(29)

It should be emphasized that it is very hard to measure S23 . However, Puck11 concluded that S23 ¼ S2þ since the plane of fracture is always rotated 458 to the pure shear stress state. Wu–Scheublein (1974) criterion50 The authors of this criterion saw the problem of quadratic approximation was that only orthotropic laminates can be handled without plywise analysis. Since a multidirectional laminate is often anisotropic, they added the cubic tensor in the tensor polynomial used by Tsai and Wu. This criterion is not to be applied to unidirectional plies. Because of the number of coefficients, Wu and Scheublein restricted the derivation to the plane stress state. Analogous to Tsai and Wu, they figured F1 j1 þ F2 j2 þ F11 j21 þ F22 j22 þ F66 j26 þ 2F12 j1 j2 þ 3F112 j21 j2 þ 3F221 j22 j1 þ 3F166 j1 j26 þ 3F266 j2 j26 ¼ 1

ð30Þ

The diagonal terms for the linear and quadratic tensor

875

A review of the biaxial strength of fibre-reinforced plastics: H. Thom Chang Chang (1984) criterion54–56

remain the same: F1 ¼ F22 ¼

1 1 ¹ S1þ S1¹ 1 S2þ S2¹

F2 ¼

F66 ¼

1 1 ¹ S2þ S2¹

F11 ¼

1 S1þ S1¹

1 S212

ð31Þ

An experimental determination of the interaction terms would end in an enormous number of tests. The best suited ratio of stresses for the determination depends also on the interaction term, to be measured. Thus, a number of iterations would be necessary to give reliable results. Needing n iterations, 5 þ 5n experiments would be necessary to determine the ten coefficients. Since one normally averages each value out of several tests, the number of tests would further increase. Consequently, Wu and Scheublein used a hybrid method to reduce the number of experiments. In the first iteration, the strengths are calculated by using a plywise analysis. The calculated points of the failure envelope can be expressed piece by piece by a polynomial of the third order. The calculated failure envelope delivers the biaxial strength of the laminate for further iterations. Finally the optimal ratio of stresses will be known. The number of experiments has been reduced from 5 þ 5n to ten tests. The experimental interaction term has to be compared with the calculated value, and in the case of significant deviation, an experimental iteration has to follow.

The Chang Chang criterion considers also non-linear elastic behaviour of the matrix. It distinguishes between matrix cracking and fibre breakage or fibre–matrix shearing. Since non-linearities between shear stresses and shear strains are more important than those between normal stresses and strains for composites, only the shear stress– strain relation is considered to be non-linear. To incorporate the strain history in the stresses Chang, Scott and Springer combined the Yamada–Sun criterion57 with Sandhu’s strain energy criterion58,59. The resulting criterion is for both failure modes of the form Zg12 j12 dg12 j ¼1 (35) ( 1=2 )2 þ Z0e12 S1=2 j12 dg12 0

The non-linear shear stress–strain relationship is expressed according to Hahn and Tsai60 as j (36) g12 ¼ 12 þ aj312 G12 G12 is the initial ply shear modulus. The constant a can be calculated from the stress–strain curve of an off-axis test under an arbitrary angle with «y ¹ «yelast (37) a¼ j3x Using this non-linear relationship, the integration above can be performed, thus:

Tennyson–MacDonald–Nanyaro (1980) criterion51–53 Following the same reasoning already mentioned by Wu and Scheublein, Tennyson et al. also thought of using a cubic tensor polynomial. The derivation is similar up to the point of the determination of the cubic interaction terms. They too used the hybrid method, but needed only one experiment to determine F12 (plus five tests for uniaxial strengths). The cubic interaction terms are estimated with this quadratic interaction term by setting the discriminant of the cubic tensor to zero. The stresses can be replaced by a loading factor l:

j212 3 þ aj4 j1 2 2G12 4 12 ¼1 ( ) þ 2 S1 S12cr 3 4 þ aS 2G12 4 12cr

j 1 ¼ k1 l j 2 ¼ k2 l j 6 ¼ k6 l

According to Yamada and Sun, the shear strength of a crossply laminate should be used instead of that of a lamina, since the shear strengths of laminates are usually higher than the strenghths of laminae.

(32)

eqn (30) can be rewritten as al3 þ bl2 þ cl þ d ¼ 0

j212 3 þ aj4 j2 2 2G12 4 12 ¼1 ( ) þ 2 S2 S12cr 3 þ aS412cr 2G12 4

(38)

a ¼ 3(F112 k12 k2 þ F221 k22 k1 þ F166 k1 k62 þ F266 k2 k62 ) Truncated maximum strain (1990) criterion61,62

b ¼ F11 k12 þ F22 k22 þ F66 k62 þ 2F12 k1 k2 c ¼ F1 k 1 þ F2 k 2 d¼ ¹1

(33)

Setting the discriminant to zero, it follows that 27a2 þ a(4c3 þ 18bc) ¹ 4b3 ¹ b2 c2 ¼ 0

(34)

The cubic interaction terms can now be calculated by inserting the stresses responding to the three uniaxial and the one biaxial strength.

876

Hart-Smith created a very simple criterion based on the Tresca yielding condition. Its application is limited to thin laminates (plane stress state) where ultimate failure is fibre dominated. These laminates have to be made of strong stiff fibres in a soft matrix, i.e. composites with n21 < 0. For such a laminate, only e1þ , e1¹ and n12 are needed. For better visualization, the construction of the failure envelope is shown instead of the descriptive formulae (Figure 5). The parameter gcrit is calculated by gcrit ¼ (1 þ n12 )max(e1þ ; je1¹ j)

(39)

A review of the biaxial strength of fibre-reinforced plastics: H. Thom þ The factor p12 indicates the absolute value of the failure envelope’s slope when intersecting the t12 axis. It has to be determined experimentally and is supposed to increase the accuracy. The section of the failure envelope characterized by mode B is expressed by a parabolic equation, thus: q 1 ¹ j )2 þ p ¹ j ) ¼ 1 ( S212 þ (p12 12 2 2 S12

(j2 , 0, 0 # j

Figure 5 Tresca’s and truncated maximum strength criterion’s failure envelopes

Puck (1996) criterion11 Like Hashin and Rotem, Puck also starts from Mohr’s hypothesis, which states that in the case of brittle materials only stresses acting on the crack plane are responsible for fracture. Because of the increased potential of computers, he no longer neglected the determination of the fracture plane for matrix failure. The derivation of the criterion, presented in this survey, is restricted to the plain stress state, since interesting simplifications occur in this stress state. The three-dimensional derivation is described in the reference. Fibre failure is assumed to be independent of the stresses other than that in the direction of the fibres. However, owing to the distinct Poisson ratios of fibre and resin, normal fibre stresses are induced by transverse stresses. The ultimate stress is somewhere between the uniaxial ultimate stress and the stress calculated from ultimate strain. Since the difference is only very small, the strength is approximated by 1 j 1 «1 j þ 6j ¼ 1 e1 2 S6 1

(40)

Puck mentions, though, that the hypothesis of maximum fibre stress is not acceptable to others, and therefore needs further investigation. The inter-fibre fracture is described by three equations referring to different failure modes. Failure due to tension and shear is called mode A, failure due to shear and lesser compression mode B, and shear with stronger compression mode C. While the fracture plane is perpendicular to the transverse stress for modes A and B, it is oblique for mode C. In mode C the fracture occurs in a plane of maximal shear stress. The failure envelope is described for mode A by an elliptical equation: s þ t þ j2 þ S2 2 j 2 2 ) ( ) þ p12 ¼ 1 (j2 . 0) ( 12 )2 þ (1 ¹ p12 S12 S12 S2þ S12 (41)

j2 R23 ) j # þ p ¹ j12 S12 1 þ 2p23

(42)

The inclination parameter again indicates the slope at the t12 axis. In this case it can be interpreted as a measure for Coulomb’s friction, which increases the shear strength in the case of pressure on the fracture plane. The factor R23 will soon be explained. The fracture angle is calculated by searching for the global maximum of the effort (ratio of current loading to possible strength): r j j ¹ jn )2 þ p ¹ j n (43) E(v) ¼ ( nt )2 þ ( nl )2 þ (p12 12 R23 S12 S12 S12 One will observe that jn will remain constant at the value ¹ R23 when calculating the fracture plane for mode C. Thus, the fracture plane can simply be calculated by s R23 (44) v ¼ arccos ¹ j2 The resistance of the material to transverse shear stress on the fracture plane is called R23 by Puck. Since a specimen loaded with pure transverse shear stress fails at an angle of 458, the reason for failure is tensile and not shear stress on the fracture plane. Therefore R23 is not equivalent to the strength S23 . Instead, it can be measured by uniaxial compression, where R23 is distorted by Coulomb’s friction, though. However, Puck derived R23 from Mohr’s circle based on transverse compression: q S¹ S¹ ¹v (45) ¹ p23 ) R23 ¼ 2 cotanv ¼ 2 ( 1 þ p23 2 2 ¹ ¹ is assumed to be tied to p12 by The inclination parameter p23 ¹ p23 p¹ ¼ 12 S23 S12

(46)

Now, the failure envelope for mode C can be expressed by the following elliptical equation: [(

t12 j S¹ )2 þ ( ¹2 )2 ] 2 ¼ 1 ¹ 2(1 þ p23 )S12 S2 ¹ j2

p ¹ S12 1 þ 2p23 j12 ) (j2 , 0, 0 # j j # j2 R23

(47)

The equations used for the derivation of the elliptical equation for mode C started out by neglecting the longitudinal stress. Instead, Puck assumes that high longitudinal stresses (0.7S1 ) reduce the transverse strength due to local fibre failure and fibre debonding. This reduction is expressed by an ellipse, which reduces the strength to

877

A review of the biaxial strength of fibre-reinforced plastics: H. Thom 50% progressively, beginning at 0.7S1 : fw2 þ

I3 ¼ t231 þ t212

1 j (j 61 j ¹ 0:7)2 ¼ 1 0:12 S1

(48)

Unity on the right side of the equation sign in eqns (41) and (42) and eqn (47) can then be replaced by fw . The determination of the fracture angle offers the additional benefit that the danger of delamination due to an oblique crack (mode C) can be noticed. In the case of mixed modes, the three-dimensional determination of fracture angles becomes numerically unstable. This can also be deduced from the physical view, since the orientation of the voids also controls the fracture angle when the effort is comparatively indifferent to the angle. Therefore, corners can be rounded with a probabilistic formulation. The area of significant effort (more than 50%) over the variable fracture angle is an appropriate measure for the reduction of strength. Cuntze (1995) criterion

F1j :

j : F12

at1 I1 S1¹

þ

aj1 I1 ¼1 S1þ bt1 I2 (þ S1¹

ct1 I3 S2

)¼1

at2 I2 bt2 I4 ct2 I3 d2t I12 þ ( þ þ ¹2) ¼ 1 2 S2¹ S212 S2¹ S2

I2 ¼ j2 þ j3 † Formula of ref.63 changed according to64.

878

j : aj12 F12

2 j2 j2 2 Jj j j j 2 t12 þ ej12 ( þ1 )2 ) ¼ 1 þ þ b12 ( þ ) ( þ c12 3 S2 S2 S1 S12

F12 : a12 ( F2t : at2

Jj1 j t þ bt1 ¹2 ( þ ct1 ( 12 )2 ) ¼ 1 ¹ S1 S1 R12

t12 2 j t2 Jj ) þ b12 2 3 12 ( þ c12 ( þ1 )2 ) ¼ 1 S12 S1 S12

j2 Jj t j2 2 t t12 2 ) þ d2t ( ¹1 )2 ) ¼ 1 (51) ¹ þ b2 ( ¹ ) ( þ c 2 ( S2 S2 S12 S1

There are different ways to apply criteria without distinct failure modes to multidirectional laminates:

(49)

The summation terms in brackets describe mixed mode failure. A probabilistic calculation is replaced by these correction terms. In the case of a porous matrix, also a ‘crumbly’ fracture mode can be considered, which leads to failure under hydrostatic pressure. The invariants are given by I1 ¼ Jj1q

F1t : at1

Jj1 ¼1 S1þ

Application of the strength criteria without distinct failure modes

a12 I3 b12 I2 I3 c12 I5 d12 I12 þ þ ( þ 2 )¼1 S212 S312 S312 S1þ

F2t :

(50)

The first invariant is not based on the ‘smudged’ longitudinal laminate stress but on the fibre stress times the fibre volume ratio. This is done because longitudinal strength is assumed to depend only on fibre strength. It first appears to be formalism but in case of thermal stresses it might just lead to a difference. One will observe that the strength depends on the sign of the shear stresses due to I5 . This mistake was made on purpose, in order to take into consideration the distinct interactions of j2 (j3 respectively) with t12 and t31 . In the plane stress state the linear shear stress terms vanish. For the plane stress state the equations are F1j : aj1

j aj12 I2 bj12 I4 cj12 I22 ej12 I2 I3 f12 I5 gj12 I12 þ þ ( þ þ þ þ2 ) ¼ 1 2 2 S2þ S312 S312 S2þ S2þ S1

F12 :

I5 ¼ (j2 ¹ j3 )(t231 ¹ t212 ) þ 4t12 t23 t31

63

The strength criterion of Cuntze is chronologically misplaced. However, it was influenced by the development of the Puck criterion, which goes back to the year 1970. In contrast to Puck, Cuntze did not determine the fracture angle. Instead, he used those invariants of the elasticity tensor that included the stresses responsible for the distinct failure modes corresponding to Puck’s analysis. Also isotropic materials or fabrics can be dealt with by using a different set of invariants. For a transverse isotropic material the equations for the different failure modes are the following†:

F1t :

I4 ¼ (j2 ¹ j3 )2 þ 4t223

• a plywise analysis, • application to the laminate as a whole, • determination of the strengths in the direction of the axes of orthotropy with a plywise analysis, which are then inserted in the criterion for the laminate as a whole. The first method allows determination of the laminate’s strength with the data of the unidirectional plies. These data can be found in tables. Experiments could sometimes be spared. However, the stresses in the plies would have to be calculated anew for every stress state. These calculations can be avoided by using the second method. Instead, the strengths have to be determined experimentally for every distinct laminate structure. The hybrid method tries to combine the advantages of the previous two methods. The strengths, which were determined experimentally in the second method, are now calculated with the first method. Further plywise analysis is not necessary. However, it is not possible to determine the failing ply. This can only be done by using the first method.

A review of the biaxial strength of fibre-reinforced plastics: H. Thom

Figure 6

Comparison of the strength criteria in the j1 ¹ j2 graph

Figure 7

Comparison of the strength criterion in the j1 ¹ j12 and j2 ¹ j12 graph

Comparison of the strength criteria The following shows graphically some of the presented strength criteria. The calculations are based on a carbon/ epoxy-HT lamina. With the exception of the Tsai–Wu and the Puck criteria, the criteria needing experimental data are left out. For the Tsai–Wu criterion the value suggested by Tsai and Hahn was used for the interaction term p ¼ ¹ 0:5). According to Puck the strength S23 was (F12 replaced by S2þ for the Hashin–Rotem criterion. The inclination parameters in the Puck criterion are taken from ¹ þ ¼ 0:2=p12 ¼ 0:3. the reference and are p12 The most significant difference can be seen in the j1 ¹ j2 graph (Figure 6). The Tsai–Hill criterion delivers the most conservative values (the suitable strength is used for each quadrant). The increase in biaxial strength is the highest for the Tsai–Wu criterion. The maximum stress and the Hashin–Rotem criterion have exactly the same failure envelope and almost the same envelope as the Puck criterion in this graph. The Puppo–Evensen criterion seems to work only with laminates with less orthotropy, since its failure envelope is not closed.

The difference in the diagrams in which j12 over j1 or j2 is drawn is no longer clear. The criteria similar to the Tsai– Wu criterion, only distinct in the interaction term, form the same failure envelope (Figure 7). From these graphs one would conclude that a biaxial test with the two normal stresses would be best suited to determine the most accurate criterion. However, the Puck criterion seems to be very plausible and therefore a test in the j2 ¹ t12 stress state seems to be more useful. However, it has not yet been ‘proven’ that the longitudinal strength is independent of the transverse stresses. This still has to be clarified. Hashin and Rotem, Puck, as well as Cuntze derived their criteria from this assumption. Later this year the results of a project funded by the German Ministry for Education and Science will be published in which the three-dimensional criterion of Puck was experimentally examined 65. While tensor polynomials seem to be the most accurate for laminae (Puck and Cuntze criterion are excluded), one observes with laminates that the maximum stress/strain criterion describes failure better than the others. The reason

879

A review of the biaxial strength of fibre-reinforced plastics: H. Thom

Figure 8

Determination of strength with maximum stress and Tsai-Wu criterion for a [08, 6 608] s laminate

is the definition of the term failure. Experiments normally record ultimate fracture as failure. Ultimate fracture is in most cases caused by fibre failure. Consequently, the test data are close to the lines of fibre failure in the maximum stress/strain criterion (see Figure 8‡). Tensor polynomials and other criteria, that do not distinguish between failure modes, indicate failure at plainly lower stress levels, when used in plywise analysis. This failure is mainly characterized by matrix fracture. Since matrix failure causes nonlinear behaviour and reduces Young’s modulus, an accurate determination of the beginning point of this fracture mode is also of interest. A compromise solution could be the use of the tensor polynomial for the determination of matrix failure and the maximum stress criterion or eqn (24) for fibre failure. This aspect might just be of historic value owing to the expected superiority of the Puck and Cuntze criteria.

can be seen in Figure 9, the different specimens also have different properties. While the differences in strength and modulus are small for tensile specimens, they become more significant for compressive or in-plane shear tests. Much higher compressive strengths can be achieved with the specimen developed by Matthews and Haeberle67 as well as with that developed by Curtis et al.68. The latter can be used with the Celanese rig. Scatter in tensile tests can be reduced by cross-plies in the 908 direction69.

SPECIMEN TYPES Just as there are macroscopic and microscopic failure criteria, there are also two principal types of specimen. Specimens of the first type are designed for a homogeneous stress state in the test section, whereas those of the second type have local stress peaks due to holes or notches. To verify macroscopic failure criteria, specimens with a homogenous stress state are much better suited since they satisfy the demand of a homogeneous stress state in a statistically significant test volume. Further requirements on such a specimen are that failure has to start in the test section and that it must in no case be caused by buckling. Unidirectional (UD) laminae can be tested as well as multidirectional laminates. Testing of UD laminae allows the construction of a database for the different materials. Interaction of stacked laminae can be examined by testing laminates. The uniaxial strengths needed for the failure criteria are normally determined with special specimens which are described by special standards, such as ASTM or CRAG. As ‡ Data points are not measured but only used for illustration. Reduction of strength is not considered.

880

Figure 9 Comparison of different compression tests (average and standard deviation)66

A review of the biaxial strength of fibre-reinforced

Figure 12 Crossbeam by Boehler and Demmerle

Figure 10 Comparison of the failure criteria with the off-axis test

Figure 13 Flat plate

Contraction due to Poisson’s ratio is not supported by the rigid tabs. Special attention has to be paid to small angles ensuring that no single fibre is clamped by both grips.

Crossbeam Figure 11 Crossbeam

Off-axis specimen The off-axis specimen has the most simple geometry. It is a flat plate with a distinct angle between uniaxial loading and fibre orientation. The stress state is biaxial when transformed parallel to the axes of orthotropy. The three plane stresses cannot be varied independently since they are tied to each other by the transformation tensor. The difference between the distinct criteria is not significant as can be seen in Figure 10. Pipes and Cole reported in ref.45 that the off-axis sample is not suitable for determination of the interaction term F12 . Matrix failure may be caused by the free-edge effect which can result in ultimate failure before the actual strength is reached. The specimen has either to be designed with a high aspect ratio or clamped with a pivoted grip to prevent moments induced by orthotropy. Another method is the use of oblique rigid end-tabs70. The angle to the longitudinal direction of the specimen is given by exy ¯ gxy C¯ C16 ¼ cotv ¼ ¹ ¯ 16 C¯ 11 ¼ jxx jxx C 11

(52)

The crossbeam is a sandwich construction in which the laminate to be tested is on one of the outer coats (Figure 11). The stress state is induced by bending of the beams. Shear can be engendered by changing the orientation of the fibres. Bert et al.71 proposed an elliptical reduction of thickness for better homogeneity of stresses. The ellipse is described by r b j2 ¼ (53) j1 a It also guarantees failure in the gauge section. The stress state, however, in the test section is not determined by loads applied, and not by the geometry72. The only advantage to be found is in the quadrant compression/compression, whereby the sandwich construction increases stability. Nowadays, this specimen is becoming popular again. Boehler and Demmerle73 optimized the test section with regard to the stress state and also with regard to the certainty of the stresses with finite element calculations. As parameters they used the sizes shown in Figure 12 (1/8 of the specimen) and in addition the ratio of the cross-sectional area of limbs and slots, the distribution of the slot width and the limb width and the number of slots. The halfwidth t and the number of slots were fixed for the numerical optimization. The resulting geometry depends on the material. Loads are applied by in-plane tension.

881

A review of the biaxial strength of fibre-reinforced plastics: H. Thom elements for glass/epoxy in the range of ¹ 158 # a # 208. The analysis leads to the following conclusions: at 08 load angle the stress state is almost homogenous; the higher the absolute value of the load angle, the less homogenous the stress state; determination of the shear modulus is accurate while the measured shear strength is less, compared to other types of specimen. Thin-walled tube

Figure 14 Circular specimen

Flat plate The flat plate is pulled or pushed at all four edges (Figure 13). Shear stresses are controlled by the fibre orientation. Whiffle-tree linkage grips are used to allow contraction. Nevertheless the stress state is not homogenous74, since the grips are not infinitely small. An improvement can be achieved with an elliptical reduction of the thickness. This geometry is mainly accepted for notched specimens. Circular specimen Arcan, Hashin and Voloshin75 introduced a new geometry for biaxial tests in 1978 (Figure 14). The biaxial stress state is controlled by the load angle a (a ¼ 08 corresponding to pure shear): jx ¼ ja sina jy ¼ ja sina

jr ¼

txy ¼ ja cosa ja ¼

Pa Atest section

(54)

The cross-section is measured at the narrowest spot of the gauge section. Marloff76 examined this specimen geometry with finite

Figure 15 Tube design by Swanson

882

The tube is the most versatile specimen. All three plane stresses can be applied independently from each other. The free-edge effect is not present. Axial stresses are applied by axial tension or compression, hoop stresses by inner or outer pressure, and shear stress by torsion. Three different kinds of fibre orientation can be distinguished: unidirectional laminae in axial or hoop orientation, helically wound laminae, and multidirectional laminates. It is argued that helically wound tubes are more general since the transformation of elasticity is also checked77. The main concern might be though to find a compromise between high axial loads for 08 and high pressures needed for 908 wound tubes. However, owing to orthotropy, additional stresses are induced which may lead to other difficulties. The term ‘thin’ has to be defined in a more conservative manner than for isotropic materials, owing to orthotropy. Rizzo and Vicario77 defined it as t=d # 0:02. It has to be mentioned that their analysis was for a helical angle of 308 which is very unfavourable for a homogenous stress state. The most disadvantageous angles of helically wound laminae are 308 for axial loads and 608 for torsion and pressure78. There are several analytic solutions in the literature for the stress state in a tubular specimen, some of them taking into consideration the clamping constraints78–82. In the age of less and less expensive CPU time, finite element methods should have preference, since they require less simplification of the problem. Choo and Hull83 remark that inner pressure induces a three-dimensional stress state on the inside of a tube. The radial stress is calculated according to Timoshenko84 with Pi ri r [1 ¹ ( o )2 ] 2 r ¹ ri

ro2

(55)

For a thin tube this can be approximated to a linear decrease in radial stresses. The radial stress decreases for example the shear stress in the 458 plane, when axially compressed. This shear stress has to be zero on the outside owing to equilibrium. Consequently this influence cannot be neglected in general, as has been the case. Instead the radial stress should be inserted in a three-dimensional criterion with 0–0.5Pi . The major source of stress inhomogeneities are the collets, especially when loaded with pressure. Also in the case of torsion or when helically wound tubes are tested, attention has to be paid to the length of the specimen. When tubes without reinforced end sections are used, failure normally occurs near the grips. The most common design for the reduction of stress concentrations is thickened end sections. A very successful design has been made by

A review of the biaxial strength of fibre-reinforced plastics: H. Thom

Figure 16 Test facility by Cole and Pipes

Figure 18 Typical fabrication set-up

Figure 17 Collet by Lindholm et al.

Swanson (Figure 15)85,86,2. Much more complicated is the attempt to match the radial displacement of the end sections with that of the gauge section. Such radially adjustable collets have been developed by Cole and Pipes72 as well as Lindholm et al.87,88. The more expensive test facility is justified by cheaper, because shorter, specimens. Cole and Pipes used an arrangement as shown in Figure 16. There are no conventional collets but tabs sealed with O-rings. Axial and hoop stresses are loaded only by hydraulic pressures. The pressures Pi and Po control the hoop stresses. Both pressures together respectively Pax are used to apply axial loads. Radial displacement of the stiffer tab section is forced by P1 and P2 . Torsion is loaded by a mechanical link (not shown in the figure). A total elimination of the constraints, especially when the tube is loaded simultaneously axially and torsionally, is not achieved. Stress concentrations are induced mainly, although less significant when compared to conventional collets, by local bending of the tabs. Owing to this concept of applying loads, a new problem appears: the threedimensional stress state, when loaded in axial tension, can no longer be neglected. This is especially true for unidirectional lamina owing to the high Poisson ratio n32 . Lindholm et al. designed a radially adjustable collet without changing the loading procedure. As can be seen in Figure 17, the collet consists of a grip separated in 2 3 12 segments. The clamping pressure is summed up by P1 and P2 . For radial displacement one of the pressures is increased while the other is decreased. To reduce the influences of the clamping constraints, one would be tempted intuitively to increase the length of the specimen. According to a finite element analysis by Rizzo and Vicario77, a sample with high aspect ratio (l/d) leads to higher stress gradients in the crosssection. Since strength is controlled by the failure mode, the mode

has to be examined. Wang and Socie1 reported that a tube failed under outer pressure by delamination while a flat specimen failed at higher stresses due to kinking. Krempl and Niu89 observed that a 6 458 wound tube had a different torsional strength for both turning directions. They concluded that strength depended on the orientation of the outer ply, and thus buckling was the actual failure mode. The explosion-like fracture impedes the determination of the failure mode, hence accompanying calculations and SEM investigations sometimes have to be used. For the calculation of the critical buckling stresses, solutions can be found in the references90–94. Since Ley et al.94 described buckling of ring-stiffened tubes, clamping constraints also can be taken into consideration. The manufacturing process has to generate a laminate structure comparable with industrially produced laminates on the one hand, and to reduce scatter due to production tolerances on the other. Consequently, the fibre volume ratio J has to be at least 0.6. Tubes can be produced by filament winding (winding with very thin rovings), and winding with larger rovings, wide tapes, or fabrics. The use of tapes (at least for 908 winding angles) and fabrics for tubes is complicated, since the generation of continuous plies is impossible. Either stiffness or strength is discontinuous, depending on the chosen joint33. Cole and Pipes72 though, could not find a correlation between the location of discontinuity and of transverse failure. They paid attention to the angle of the lap joint, so that it did not coincide with any direction of minimal strength. Thick rovings often lead to an inhomogenous fibre density95. At the top edges of these rovings, regions with low fibre content are formed. The highest quality can be achieved by filament winding. To achieve high fibre volume contents, surplus resin has to be pressed out of the laminate. This is done by winding the roving on a hollow mandrel, which can be internally pressurized. Typical cavity tools can be seen in Figures 18 and 19. Before the winding of rovings, a vent cloth to draw off volatile gases and a bleeder for surplus resin are wound around the mandrel. Separation between them and the laminate is achieved by perforated Teflon films for example81. Byung and Lehnhoff 96 even vacuumed surplus resin and gases through porous spacers at the ends of the tube. Vent cloth and bleeder are therefore not necessary to

883

A review of the biaxial strength of fibre-reinforced CONCLUSIONS

Figure 19 Cavity tool by Whitney

receive a laminate with high fibre volume ratio and little voids. Filamentary wound tubes do not necessarily need removal of surplus resin since the winding process already presses the resin out of the laminate. Determination of the fibre volume ratio is difficult then, because some surplus resin remains on the outer surface. Inner layers can be weakened by winding tubes several layers thick. The final strength of a layer is almost independent of the applied tension of the winding process as long as the layer tension is greater than zero. For no tension, or worse, compression, strength is reduced by inaccurate fibre orientation. An applied tension of 5–10 N ensures parallel, well orientated fibres. However, when several layers are wound, each new layer compresses the mandrel and the previously wound layers. Knight97 evaluated with calculations and experiments that a kevlar/ epoxy tube with 64 plies and 5.9 N applied tension on a polymethylenethacrylate mandrel (d ¼ 150 mm, t ¼ 3:3 mm) with more than half of the plies cured in a state of compression. With a tube consisting of 36 plies, 8.9 N applied tension, and an aluminium mandrel (d ¼ 150 mm, t ¼ 6:2 mm) the layer tension fluctuated between 125 MPa and 180 MPa. Therefore the mandrel could already be inflated during the winding process. When choosing the applied tension, it also should be considered that lower applied tensions do not lead to coincident tension in the laminate, owing to rapid resin flow and cross-section flattening97. Knight states this correlation between layer and applied tension for kevlar/epoxy laminates q 2 ¹ 4, 52 (56) Flayer ¼ Fapplied Mistakes can also be made when measuring the loads. Friction in the test set-up can reduce the applied loads or prevent contraction. Internal combined with external pressures induce axial stresses in the test set-up but not in the sample. When high hoop strains occur, the actual diameter instead of the nominal diameter should be used to calculate the hoop stress.

884

During the last 30 years many strength criteria have been formulated. Most of them are based on the von Mises hypothesis. This hypothesis though was formulated to describe yielding of ductile, isotropic materials. Its application to brittle, anisotropic, fibre-reinforced plastics is doubtful. For example lower transverse tensile strength would lead to higher biaxial compressive strength. Instead, it is Mohr’s hypothesis that seems to describe the micromechanical relationships more realistically. This hypothesis takes only stresses acting on the failure plane into account. Mohr’s hypothesis was further derived for composite materials by Hashin and Rotem, Puck and Cuntze. A good failure criterion should distinguish the different failure modes. This is not only a step towards failure mechanics but also facilitates a much better description of a laminate’s strength. Since distinct failure modes produce cornered failure envelopes, probabilistic aspects should be considered to round off these corners. This is physically plausible and will increase further the accuracy of the predicted strength. Strength criteria from the 1960s described unidirectional laminae. Later it was thought that the strength of laminates could only be derived inaccurately from the properties of laminae. Thus, criteria for whole laminates were formulated. Nowadays the initial concept is becoming more popular again. A distinction between failure modes is therefore necessary. Should the strength be described accurately by the new criteria, the next step would be to understand the interactions of the laminae in a laminate. Also the application to woven fabrics is of interest, since strength is reduced by the curvature of the fibres and changing thickness of the fabric. For biaxial experiments the tubular specimen is widely accepted. Besides the complicated fabrication processes, the introduction of loads causes problems. The influence of radial stresses due to internal or external pressure has not yet been evaluated. Also testing of fabrics and prepeg tapes is not easy with this design. However, no other sample geometry causes less problems. When one is familiar with the problems, designs, manufactures and when one measures accurately, the tubular specimen is still best suited for the determination of biaxial strength. When the interaction of laminae in laminates is being investigated, the absence of the free-edge effect might be undesirable. Whether the off-axis test will provide the alternative is doubtful. Probably a combination of different specimens will be needed, particularly so that differences in strength due to geometry can be detected and measured. As fabrics gain more and more popularity a new testing machine for flat specimens would be desirable. The crossbeam might just be the starting point.

ACKNOWLEDGEMENTS This work was supported by DFG through GraduiertenKollegg Polymerwerkstoffe at Technische Universita¨t Berlin.

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