Boundary Element Applications to Polymer Fracture

Boundary element applications to polymer fracture

S. Syngellakis & J. Wu

School of Engineering Sciences, University of Southampton, U.K.

Abstract

This paper reports on applications of the boundary element method (BEM) to polymer fracture. Both Laplace transformed and time domain two-dimensional analyses, based on linear viscoelasticity, are developed and applied to centre- cracked plates under tension in order to assess their relative accuracy and efficiency. The time dependence of stress intensity factors is assessed for various viscoelastic models as well as loading conditions. Various approaches for direct assessment of the energy release rate are proposed; its representation through path-independent J-type integrals is also explored. Systematic procedures for J- integral evaluation are developed requiring the derivation of additional boundary integral equations for the displacement gradient distributions in the solid domain. The BEM formulation is extended to the determination of the energy dissipation rate for incremental crack growth. Then, the application of conservation of energy, combined with the knowledge of the critical and current values of strain energy release rate, leads to the assessment of the crack growth rate. Numerical results are compared with other analytical solutions and some experimental measurements. The consistency between BEM predictions and other published results confirms the method as a valid modelling tool for polymer fracture characterisation and investigation under complex conditions.

1 Introduction

The increasing use of polymers has prompted extensive research on their failure mechanisms. Polymer fracture, in particular, has been the subject of many theoretical studies concerned with the identification and determination of parameters governing crack initiation and growth. The finite element method has been mainly applied to simulations of fracture behaviour in polymer matrix

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composites [l]. Relatively few attempts have been made to predct material parameters characterising polymer fracture. The application of the boundary element method (BEM) in this area has been even more limited. An early such attempt was concerned the prediction of stress and displacement fields in the neighbourhood of a crack filled with failed, CO-calledcraze material [2]. More recently 131, a BEM formulation was applied to the evaluation of an expression for the strain energy release rate derived from a functional corresponding to the potential energy in elasticity.

The present BEM analysis attempts to address a wide range of issues relating to the modelling of the problem. Linear viscoelasticity, the most commonly adopted model for polymers, is assumed. The correspondence principle is used to solve the quasi-static problem in the Laplace transform domain with the time- dependent responses determined by numerical inversion. This approach essentially uses the fundamental solutions or Green's functions of the corresponding elastic problems, which can be adapted to viscoelastic models of any complexity and can be chosen to satisfy the traction-free conditions over the crack surface. Solutions are alternatively obtained through time domain BEM formulations derived from viscoelastic reciprocity relations using fundamental solutions specific to the viscoelastic model used. Both formulations require particular attention to the boundary modelling around the crack tip so that the stresses in this region are approximated with reasonable accuracy.

Both transformed and time domain analyses were initially applied to the prediction of near crack-tip stress and crack-opening displacement time histories in order to assess their relative accuracy and efficiency. The time dependence of stress intensity factors was assessed for various viscoelastic models as well as loading and support conditions. Various approaches were proposed and applied for determining the strain energy release rate or crack extension force under isothermal conditions. The representation and of the crack extension force through path-independent J-type integrals was examined. Systematic procedures for their evaluation were developed requiring the derivation of additional boundary integral equations for the displacement gradient distributions in the solid domain. The BEM formulation was extended to the determination of the rate of energy dissipation for incremental crack growth. Then, the application of global conservation of energy, combined with the knowledge of the critical and current values of strain energy release rate, led to an assessment of the crack growth rate. Numerical results were compared with other analytical solutions and experimental measurements.

2 Boundary element modelling

2.1 Laplace transform domain

If the correspondence principle is applied to the linear viscoelastic problem, the relevant boundary integral equation in the Laplace transformed domain is written

= I[B($)U; (S) - (s)~;(s)]~ + f,u;d~a (1) r a

where G, = 0.54, in the case of a smooth boundary, 4 , p,, f, are the Laplace transforms of displacements, tractions and body forces, respectively, and S is the transform space variable; D and rrepresent, respectively, the domain and its boundary; (U;, p,; ) is the elastic fundamental solution for displacements and tractions in which however the elastic constants have been replaced by the corresponding functions in the transformed space according to - 5..= sC.. E V rlk, kl = ~tklEkl (2) where Ekl and Gilklare the Laplace transforms of stresses, strains and relaxation moduli, respectively. The summation convention over repeated indices has been adopted. The boundary element modelling and solution is performed in exactly the same manner as for the respective elasticity problems. Inversions of obtained Laplace transforms to time histories are carried out using established techniques

[4l.

2.2 Time domain

The boundary integral equation can be obtained directly from the reciprocal theorem of linear viscoelasticity [5]leading to ~,,u,(t)= I(U, * dpi- pc *dui )U + J fi * du;d~ (3) r R with the Stieltjes convolution of two functions g(t) and h(t) defined as [5]

The time-dependent fundamental solution U$ (X-4,t) satisfies the field equation in an infinite domain with the body force given by

fi* 46( x-S)H(t) (5) where 6, is the Kronecker delta, 6( X-5) the delta function and H(t) the Heaviside step function. Applying the correspondence principle, ut; (X-5,t) is obtained as the inverse Laplace transform of the corresponding elastic fundamental solution in the transformed space divided by the Laplace domain variable S. A fast general procedure was devised for obtaining time-dependent fundamental solutions standard linear solid (SLS) creep or relaxation models with a high number of springs and dashpots. The modelling in the time domain was based on a linear variation of the boundary variables within a time step. The introduction of a constant spatial model led to a system of equations yielding the solution at any time step in terms of the current boundary conditions and the solutions at all previous time steps.

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3 Fracture models

3.1 Stress intensity factor

The two-dimensional visco-elastic problem considered is schematically drawn in Fig. 1 where, for simplicity, the cracked solid is assumed symmetric relative to the two co-ordinate axes xi (i = 1,2) parallel and perpendicular to the crack with origin at the centre of the crack. Time histories of the stress field D&) in the neighbourhood of the crack tip can be determined by either the Laplace transformed or time domain BEM formulations. The stress intensity factor can then be obtained using

with the stress a22described relative to a polar frame of reference with origin at the crack tip.

J-integral path

/ Crack tip

Figure 1: Centre-crack plate under tension.

It is also possible to obtain the Laplace transform of the stress intensity factor K, (S) from a path-independent integral [6]

7 (S) = 1 - piq,,dr)

r, where I; represents the integration path, a comma followed by an index indicates differentiation with respect to the corresponding co-ordinate and

W =La.($lqj 2 "

The relations between (S) and (S) are similar to those between J and KI in linear elastic fracture mechanics, namely

where p is 1 for plane stress and l-SF for plane strain.

3.2 Strain energy release rate

Crack growth occurs due to the application of either edge displacement Ci(t) or traction jji (t) loading histories. These loads produce stress and strain fields $(t) and ~~(t),respectively, with the total reversibly stored energy E and dissipated energy D, given by [7]

f E t D, = jog(r).Cu (r)d~dr (6) OR where a dot above a symbol indicates differentiation with respect to time.

Assuming small strain rates, that is, linear relations between strain and deformation rates, Eq. (6) can be transformed via the application of Green's theorem to t

E D, = piiidrdr (7) + or The principle of conservation of energy requires that [X] W=E+D,+D,, (8) where W is the total work done by boundary tractions and D, the total surface energy. The crack is assumed to be stationary, that is, after a certain period of loading t = tl, it can be forced to grow instantaneously by a small amount 6a. A crack extension 6a means the removal of the tension pi(x,,O, t) for t > tl over this length along XI,changes in work and energy should therefore be evaluated with this tension reversed, that is, applied as compression. The derivation of an incremental relation similar to Eq. (8) is performed with reference to the geometry and loading of Fig. 1 in order to assess the corresponding rate of surface energy increase. This in turn will lead to an estimate of the corresponding strain energy release rate G(t). The equation of energy conservation is thus written &E D,) 6D.? m= + + (9) where u+Bu m= J P~(x,JP~~(~.% (10)

U+& t &E + Dv) = J j pi (XI > rPui(r)d~&l (1 1) U 1,

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The surface energy stored in the upper crack surface would be 1 6D,s= -G(a,t)& (12) 2 Substituting Eqs. (10)-(12) into Eq. (9) and re-arranging gives U+& U+& r G(a,t)& =2 pi (xl ,t)6ui (t)dq -2 1 1pi (13 ,')S*; (r)d.rkl (13) U tl from which the strain energy release rate can be obtained. In the special case of constant stress field, the incremental solution is essentially elastic, with effective moduli the current values of the creep compliances Dijkjkl(t-tl)defined by E.. = D.. do. IJ ykl * IJ (14) The right-hand-side of Eq. (13) should be equal to the potential energy loss of the external load associated with the incremental boundary displacement: . - 6l7 = I pidui (t)dT , (15)

=P where r, is the loaded part of the boundary. Eq. (15) can provide the strain energy release rate as

at any time t>tl during the viscoelastic response of the polymer.

In the special loading case of a constant boundary displacement, the potential energy does not change and G(t) remains constant for t > tl. Thus only the initial, instantaneous solution is of interest. It is then easily seen from the constitutive relations

q=Gw*d~ii (17) that the effective moduli should be the initial values of the relaxation moduli Gijkl(0).

Since the evaluation of the strain energy release rate G(t) is often based on potential energy changes from an equivalent elastic state, it may be possible to identify a path-independent integral J(t) which would prove to be identical to

G(t) and could be obtained directly from a suitably defined elastic field. For a traction-controlled experiment, the stress field is considered momentarily constant and elastic displacements u,R and strains E{ are determined at time t > tl using appropriate material constants. For tl = 0, E[ can be shown to be identical to ~~~(t)[g];hence, for constant stress fields, the creep moduli DVkl(t)are used as elastic constants. Then the path-independent integral is given by

J = I (Wk2 - piup ,,dT) r ,

where

For an edge-displacement-controlled experiment, stresses o$ and tractions

piR are determined at time tl. For constant strain fields, the relaxation moduli

Gijkl(0)can be used as material constants. Then J = (Wdx2- p!~,,~dT)

'i where

3.4 Crack propagation velocity

The crack will grow when G(a,tl) is greater than a constant critical GC for the material. Then, for a crack extension &, the excess surface energy will be dissipated as viscoelastic deformation taking place within a time increment St. The incremental solution should therefore be viscoelastic accounting for the time-dependence of material properties within this interval. The equation of energy equilibrium is thus written

t,+6t U+& [G(a,t,)- GC]&=2 dr 1 pi (3,t, +r)lidr, (18) 4 a An iterative, trial-and-error process generates the value of 6t for which Eq. (18) is satisfied. Then, the velocity of crack propagation is obtained from

Christensen's predictions of crack propagation velocity [l01 were based on an energy balance relation similar to Eq. (18).

4 Results

The adopted BEM modelling was based on constant elements of variable length. A geometric progression scheme was devised to generate very small elements near the crack tip. The displacement gradients in the domain, required for the evaluation of J integrals, were obtained from Eq. (1) or Eq. (3) with ~i,.= fii,.. The domain stresses were obtained from corresponding integral equations with appropriate kernels. The BEM meshing and modelling was first tested by applying it to the elastic problem shown in Fig. 1. The accuracy of the predictions for crack opening displacement, stress intensity factor and J-integral was considered satisfactory. Computer codes for both transformed and time domain analyses under both plane stress and plane strain conditions were developed and applied initially to

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centre-cracked specimens under constant tension. Their relative accuracy and efficiency was assessed through results for the stress intensity factor for various material models. The agreement with exact solutions was generally good but with the time domain formulation performing slightly better. This formulation was used in all subsequent strain energy release and crack growth rate calculations, which could not be carried out by the Laplace transform approach anyway. The results for the strain energy release rate obtained by the various approaches described earlier with tl=O are shown in Fig. 2. The geometric data were a = 1 mm, W = 12 mm and b = 12 mm. The plate was subjected to a tensile traction of 100 MPa. The material was represented by an SLS model for shear relaxation G(t) = 192+ 4608exp(-0.4t) MPa and a constant Poisson's ratio v =

0.4599. An exact theoretical solution is available in this special case of constant traction and Poisson's ratio. The obvious consistency of the results confirms the validity of the adopted methodology.

+Work at crack tip

II Potential energy loss

0 5 10 15 20

Time (S)

Figure 2: Time history of strain energy release rate for cracked plate under constant tension.

A case of constant applied strain was also analysed with the material and geometric input chosen to simulate closely a reported polymer fracture test [81. The geometric data were a = 30 mm, W = 130 mm and b = 18 mm. The strip was subjected to a strain of 0.1. The material was represented by 5-term SLS model for the shear relaxation modulus, which fitted well to the given material time plot. The Poisson's ratio was taken constant and any other required material parameter was evaluated by the correspondence principle. The results for the strain energy release rate shown in Fig. 3 agree very well with each other. No exact solution is available in this case.

120

100

20 0

0 10 20 30 40 50 60

Time (PS)

Figure 3: Strain energy release rate at various crack growth times for a strip

under constant strain.

Another recent investigation on polymer fracture [l11 was based on correlating tests under constant strain rate with finite element calculations. The present BEM model was applied to the reported test conditions and the time- dependent J-integral calculated. This could not be directly compared to the published result since the latter was only given in dimensionless form, although good qualitative agreement between the two results was noted. Finally, crack growth rates are currently evaluated based on published data and predictions [8,

101 for comparison purposes.

5 Discussion and conclusions

The consistency of the obtained BEM results indicates that the method is a reliable tool for predicting stress and deformation time histories in a viscoelastic solid with a crack. This initial output is subsequently entered in the fracture analysis itself involving the determination of stress intensity factor, strain energy release rate and crack growth rate. These calculations are, to a large extent, based on boundary integral expressions for which BEM is particularly well suited. In the case of J-integral evaluations, BEM has also the advantage for providing reliable domain values at any density along the integration path.

Although the Laplace transform approach yields results almost as accurate as the time domain formulation, the latter provides a wider range of possible solutions to the fracture problem. The strain energy release rate generated by the instantaneous crack growth can only be assessed by this method by applying a sudden change in boundary conditions. ]For this purpose, the original time marching scheme that solves the discretised version of Eq. (3) needs to be enhanced with the incorporation of step changes in loading.

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The demonstrated effectiveness of BEM confirms the method as a valid modelling tool for material characterisation by simulating test procedures and correlating BEM output with experimental data. Provided that reliable values of critical fracture parameters are available, the method can be applied to polymer fracture behaviour investigations under complex geometric and loading conditions. The presented formulations can be fbrther adapted to the analysis of bi- material viscoelastic solids with cracks along their interfaces, a problem particularly relevant to the assessment of local failure mechanisms in polymer matrix composites. Finally, the extension of the method to a non-linear viscoelastic analysis can be achieved by representing the non-linear effect as an additional irreducible domain integral, which is accounted for through an iterative scheme within a time step. The stress singularity in the vicinity of the crack tip is expected to cause considerable strain softening, which can be represented by non-linear material model.

References

Mackerle, J. Finite-element analysis and simulation of polymers: a bibliography. Model. Simul. Mater. Sci Eng., 5, pp. 615-650, 1997. Sun, B. N. & Hsiao, C. C. Viscoelastic boundary element method for

analyzing polymer crazing as quasifracture. Boundary Elements VII, Vol. 1, eds. C, A. Brebbia & G. Maier, Springer-Verlag: Berlin, pp.3169-3/86, 1985. Lee, S. S. & Kim, Y. J. Time-domain boundary element analysis of cracked

linear viscoelastic solids. Eng. Fract. Mech., 51, pp. 585-590, 1995. Cost, T. L. Approximate Laplace transform inversions in viscoelastic stress analysis. AIAA J., 2, pp. 2157-2166, 1964. Gurtin, M. E. & Sternberg, E. On the linear theory of viscoelasticity. Arch. Rat. Mech. Anal., 11, pp. 291-356, 1962.

Nilsson, F. A path-independent integral for transient crack problems. Int. J. Solids Structures, 9, pp. 1107-1 115, 1973. Christensen, R. M. Theory of Viscoelasticity, Academic Press, New York, 1971. Mueller, H. K. & Knauss, W. G. Crack propagation in a linearly

viscoelastic strip. J. Appl. Mech.-Trans. ASME, 38, pp. 483-488, 1971. Schapery, R. A. On some path independent integrals and their use in fiacture of nonlinear viscoelastic media. Int. J. Fracture, 42, pp. 189-207, 1990.

[l01 Christensen, R. M. A rate-dependent criterion for crack growth. Int. J. Fracture, 15, pp. 3-21, 1979. [l l] Gutierrez-Lemini, D. The initiation J-integral for linear viscoelastic solids with constant Poisson's ratio. Int. J. Fracture, 113, pp. 27-37,2002.