Fatigue crack propagation in polyvinylchloride and polyethersulfone polymer foams

Abstract

Crack growth in cross-linked polyvinylchloride and thermoplastic polyethersulfone foams under mode $I$ fatigue cycling has been experimentally examined. Polyvinylchloride foams of densities ranging from 45 to 100 kg/m³ and polyethersulfone foams of densities ranging from 60 to 130 kg/m³ were considered. All cyclic tests employed the sandwich double cantilever beam test specimen. The test results revealed that crack growth rate (da/dN) compared at a constant cyclic energy release rate (ΔG) decreased with increasing density of the foam. The crack propagation rates in low-density polyethersulfone foams were much less than in polyvinylchloride foams of similar density. The crack growth rate in the high-density polyethersulfone foam was highly dependent on the loading R-ratio. The crack front in the low-density polyethersulfone foam displayed marked deviations from a straight shape, where the edge regions propagated faster than the center of the specimen. All polyvinylchloride foams and high-density polyethersulfone foam displayed straight crack fronts.

Keywords

Fatigue of foam fatigue crack growth polyvinylchloride polyethersulfone double cantilever beam fracture specimen

Introduction

Sandwich composites with low-density polymer foam cores are increasingly being utilized in lightweight structures. Polymer foams are typically weak and brittle, and may govern failure of fatigue-loaded sandwich panels. It is widely recognized that defects may be present in the foam as a result of lack of control of the foam manufacturing process. A small defect in the foam structure is very difficult to detect, but has the potential to grow during its service life. Such a defect can eventually grow to some critical size causing rapid propagation and failure of the structure. Studies to understand the crack growth behavior in foams are thus an important part of the reliability assessment process.

When structures are flawed from the beginning of their service life, i.e. they contain a small crack, cyclic loading produces a stress intensity range, ΔK, for each cycle, causing the crack to grow, which is quantified by the crack growth rate, da/dN, where N is the number of fatigue cycles. The most common approach of characterizing fatigue crack growth is to apply linear elastic fracture mechanics (LEFM). According to this approach, the rate of crack growth is related to ΔK in the form of Paris law [1]; da/dN = AΔK^m, where A is a constant and m is the growth exponent. Zenkert et al. [2] examined cyclic mode I crack growth in polyvinylchloride (PVC) foams and found an exponent, m ≈ 6. Olurin et al. [3] found that fatigue cracks in aluminum foams grow very fast, with exponents (m) of the order of 20. Similar to the results for PVC by Zenkert et al. [2], Olurin et al. [3] found that increased foam density leads to reduced crack growth rate at a given ΔK. Face/core interface cracks in sandwich structures with a foam core often grow in the foam core, near the actual face/core interface. The growth rate $(da / dN)$ has been related to $Δ K$ [4]. More common for such tests, however, is to express the rate of interface crack growth in terms of the cyclic energy release rate, ΔG, i.e.; da/dN = BΔGⁿ, where B is a constant and n is the exponent. Such an approach is convenient because G is a well-characterized (energy based) quantity, see e.g. [5,6].

Fatigue of foams has been modeled on a micro-scale by extending a static fracture model by Maiti et al. [7] to cyclic loading. The approach, suggested by Gibson and Ashby [8] considers a crack embedded in a hexagonal cell structure where the cell edge just in front of the crack is loaded in cyclic bending until it fails. When the cell wall breaks after a certain number of loading cycles the crack advances a finite length equal to the cell size. For closed cell foams, however, the cell edges are connected by plate-like membranes, and if the walls are thick enough, cell wall stretching and cracking become additional mechanisms contributing to the fatigue life, see Zenkert et al. [2]. An in-situ SEM study of the fatigue crack growth in closed cell PVC and polyethersulfone (PES) foams has recently been conducted by Saenz et al. [9]. This study indicates that extensional deformation of entire cells ahead of the crack front dominates the low-cycle fatigue behavior, especially for the ductile PES foam.

Fatigue crack growth characterization of recently introduced thermoplastic PES foams has not been reported in the open literature. Our previous study [9] was limited to low-cycle in-situ SEM fatigue cycling of PVC and PES foam specimens. This paper presents an experimental study on cyclic crack propagation rates in macroscopic size PVC and PES foams over a range of load amplitudes and load ratios. All tests utilized a sandwich double cantilever beam (DCB) specimen loaded cyclically under displacement control.

Experimental

Materials and test specimens

The major objective of this study is to experimentally characterize the mode I fatigue crack growth rate in PVC and PES foams. Both types of foams were obtained from DIAB [10]. The base PVC and PES polymers are both amorphous and ductile plastics in their solid dense forms. During foaming of the PVC, ico-cynantes mixed with the foaming agent slightly cross-link the PVC polymer, which increases the modulus and strength, but reduces the ductility of the material. For PES, the base polymer remains unmodified during the foaming process. Hence, the mechanical properties of solid PES should be representative for the solid constituent in the foams although the very small dimensions of the cell edges and cell walls may alter the response.

Three foam densities of each polymer were examined, i.e. H45, H60 and H100 for the PVC foams and F50, F90 and F130 for the PES foams. The numbers following the foam type letters H and F indicate the nominal foam density in kg/m³. The mechanical properties of the foams are listed in Table 1. Most properties listed are obtained from the foam manufacturer’s data sheet [10], although some properties (indicated by an asterisk) were determined by Saenz et al. [11,12]. The large difference between the tensile and compressive moduli for the F series foams listed in Table 1 could be attributed to the fact that the compressive modulus is measured through the thickness of a foam panel while the tensile modulus is measured in-plane. Any anisotropy of the foam would cause a difference. Taher et al. [13] found that the out-of-plane modulus of polymer foams may exceed the in-plane modulus by more than a factor or two. Furthermore, there are likely batch-to-batch variations in foam structure and properties. The static fracture toughnesses listed in Table 1 were determined by the sandwich DCB test [12], Figure 1.

Table 1.

Mechanical properties H and F series foams listed without an asterisk are according to the manufacturer [10].

Property	Units	H45	H60	H100	F50	F90	F130
Density	kg/m³	48.0	60.0	100	50.0	90.0	130
Tensile Modulus	MPa	55.0	75.0	130	17.6*	22.7*	66.1*
Tensile Strength	MPa	1.40	1.80	3.50	1.60	2.20	2.70
Compressive Modulus	MPa	50.0	70.0	135	30.0	40.0	50.0
Shear Modulus	MPa	15.0	20.0	35.0	7.50	9.50	11.5
Fracture Toughness (G_Ic)	kJ/m²	0.24*	0.38*	0.89*	0.58*	0.72*	1.53*

Data from Saenz [11,12].

Similar to the previous static fracture study [12], fatigue testing employed the sandwich DCB specimen shown in Figure 1. The sandwich DCB specimen consists of a 12.7 mm thick layer of foam adhesively bonded to 6.25 mm thick aluminum adherends. The layer of foam is precracked in the center at the loaded end. The lower aluminum adherend is pin-supported near the specimen end while the upper adherend is pin connected to the moving piston of the servo hydraulic test machine, and loaded by a vertical force, P, or displacement, δ, see Figure 1. Figure 2 shows details on the load introduction and support conditions. Loading tabs with a hole for the loading pins were screwed into the upper and lower aluminum adherends. The specimens were connected to the test machine using a clevis arrangement. The overhanging length at the left end of the specimen is 15.6 mm. The PVC and PES foams were delivered as 12.7 mm thick panels from DIAB. The foam was cut into 25.4B × 12.7 T × 200 L (mm) blocks, where B, T and L denote the width, thickness and length of the blocks. The foam blocks were adhesively bonded to the aluminum adherends to achieve the sandwich DCB test configuration shown in Figure 1 using a Hysol EA 9309.3 NA epoxy adhesive. A 45-mm long prenotch was machined at the foam midplane at the front end of each specimen, using a 0.45-mm thick “slitting saw blade”. A fresh razor was then used to sharpen the crack front.

The calculation of the energy release rate, G, for the sandwich DCB specimen is based on an analytic foundation model expression for the specimen compliance, C, as a function of crack length [14],

C = \frac{4}{E_{x} {Bh}^{3} λ^{3}} (2 λ^{3} a^{3} + 6 λ^{2} a^{2} + 6 λ a + 3)

(1)

where

λ = (\frac{6 E c}{h c h 3 E x}) 1 / 4

(2)

where E_c is the core modulus (in the thickness direction) and E_x is the effective bending modulus of the upper and lower legs of the DCB specimen [14], h_c and 2 h are the thicknesses of the foam layer and DCB specimen, respectively, see Figure 1.

The general compliance expression [14] identifies the uncracked length, c, Figure 1, as an important parameter, for limiting the crack extension in static and cyclic fracture tests. Equation (1) is an asymptotic solution of the general compliance expression, valid when the uncracked length, c, of the specimen exceeds a limiting value given by [14],

c > \frac{2.645}{λ}

(3)

Table 2 provides the limiting values of the uncracked length for the DCB specimens examined here calculated based on the compression modulus of the foams listed in Table 1. As indicated in Figure 1, the total length of the specimen (200 mm) is defined by the overhanging portion left of the load introduction and the sum of the crack length (a) and uncracked length (c). To avoid the non-linearities associated with the hyperbolic functions encountered at short uncracked lengths, (c), fatigue testing should be stopped before the limiting value,

c_{min}

, is reached. For the foams examined here, the results in Table 2 indicate a minimum length, c, in the range from 61 to 88 mm, which corresponds to maximum crack length in the range from about 96 to 123 mm.

Table 2.

Tensile modulus and limiting uncracked length (c = c_min), for DCB specimens with a 12.7 mm thick PVC and PES foam cores calculated from equation (3).

Foam	E_c (MPa)	λ (m^-1)	c_min (mm)
PVC H45	50	33.9	78
PVC H60	70	36.9	72
PVC H100	135	43.3	61
PES F50	30	29.9	88
PES F90	40	32.1	82
PES F130	50	33.9	78

PES: polyethersulfone; PVC: polyvinylchloride.

Fatigue testing

The cyclic range of energy release rate may be expressed in terms of the applied cyclic loads and the compliance derivative,

Δ G = \frac{P_{max}^{2} - P_{min}^{2}}{2 B} \frac{dC}{da}

(4)

where P_max and P_min are the maximum and minimum loads applied to the specimen in any given cycle, and B is the specimen width. The compliance derivative, dC/da is obtained from equation (1),

\frac{dC}{da} = \frac{4}{E_{x} {Bh}^{3} λ^{3}} [6 λ^{3} a^{2} + 12 λ^{2} a + 6 λ]

(5)

The fatigue tests presented here were all conducted in displacement control. According to this procedure, a cyclic (sinusoidal) displacement is imposed on the specimen where the maximum and minimum displacements imposed, δ_max and δ_min, are held constant, see Figure 3. It should be pointed out that in displacement control, the maximum and minimum loads will decrease as the crack extends in the DCB specimen leading to decreasing ΔG. This would eventually lead to crack arrest after a number of cycles if ΔG becomes insufficient to further extend the crack. For displacement controlled fatigue testing, the loading ratio, R, is given by,

R = \frac{δ_{min}}{δ_{max}}

(6)

where δ_min and δ_max are illustrated in Figure 3. For a linear elastic specimen, this definition coincides with the load-based definition of R.

δ_{max}

is selected to keep the maximum dynamic load, P_max, at least 25% below the static fracture load,

P_{c}

P_{c}

may be calculated from the fracture toughness data,

G_{Ic}

, listed in Table 1,

G_{Ic} = \frac{P_{c}^{2}}{2 B} \frac{dC}{da}

(7)

where

dC / da

is given by equation (5). The minimum displacement was chosen to prevent closure of the fracture surfaces, which was observed to occur at displacements between 0.25 and 0.5 mm.

Each DCB specimen was tested in a 100-kN servo-hydraulic MTS test machine with a 2.22-kN fatigue-rated load cell. Due to deformations of the load cell and fixture, a compliance correction was done by replacing the sandwich DCB specimen with a very stiff aluminum block. The aluminum block was loaded to about 1 kN and the displacements recorded. Figure 4 shows the displacement vs. load data and a second order polynomial curve fit to the data. The actual displacement of the sandwich DCB specimens is obtained from,

δ = δ_{q} - δ_{i}

(8)

where δ_q is the total measured displacement and δ_i is the displacement measured for the rigid aluminum block (Figure 4). This correction applies to any load, P, encountered.

Crack length monitoring

Two methods were used to monitor the crack length. The first method used a traveling microscope where the crack tip is monitored with a crosshair, see Figure 5(a). The lead screw-driven microscope travels parallel to the crack using a hand crank. The fine scale on the traveling microscope allows positioning of the crosshair within ±0.01 mm, Figure 5(b). However, it is a difficult task to accurately determine the crack length in the sandwich DCB test specimen since the exact crack location is obstructed by the irregular coarse cellular foam structure. Several trials for a single specimen revealed an accuracy of about ±1.5 mm.

A second technique for determining crack growth in a sandwich DCB specimen without the need of visual inspection is based on the specimen compliance. This method utilizes the analytic expression for the compliance, C, of the sandwich DCB specimen, equation (1). With knowledge of C, and the mechanical properties of the constituent materials, this procedure thus allows backing out of the crack length, a.

Discrete load and displacement data, P_i and δ_i, were sampled as indicated by the dots on the curves in Figure 6. A plot of displacement vs. load data for the selected loading cycle (N_i) provides the compliance, C_i. This procedure was implemented in a MATLAB computer program for processing of the recorded data which was sampled at 10 Hz using the MTS servo hydraulic controller. Once the compliance at any given cycle was determined, it is then possible to determine the crack length, a, by using a built-in MATLAB algorithm that solves equation (1) for the crack length. As discussed earlier, the F-series foams exhibit substantial orthotropy as manifested by the large difference between the in-plane (tensile) and out-of-plane (compressive) moduli, Table 1. Since the foundation model is based on the out-of-plane modulus of the foam (Figure 1), we consistently used the out-of-plane modulus values listed in Table 1 in the compliance calculations. Notice that the in-plane and out-of-plane moduli of the H-series foams are similar, indicating isotropy. Note also that the program first corrects the displacements by removing the system compliance according to equation (8). It should be pointed out that the compliance method provides an average crack length whereas the microscopic measurement provides a crack length based on observations of one edge only.

Several techniques exist for determining the cyclic crack growth rate (da/dN), e.g. the point-to-point method and a curve fit method [15]. The crack growth rate according to the point-to-point method is,

\frac{d a}{dN} = \frac{a_{i + 1} - a_{i}}{N_{i + 1} - N_{i}}

(9)

where i and i + 1, represent two consecutive loading cycles. This method is straight-forward, but in our displacement controlled fatigue tests, however, crack growth rates are small and this method is associated with large scatter and uncertainty in the growth data. An alternative method for determining the crack growth rate is based on a fitted curve to the “a vs. N” curve. A typical curve fit employs a power function,

a = a_{o} + c_{1} N^{β}

(10)

The crack growth rate is given by,

\frac{d a}{dN} = β c_{1} N^{(β - 1)}

(11)

This method produces a more consistent evaluation of the crack growth rate and was used for all specimens considered in this study.

Sandwich DCB specimens were tested in fatigue to determine cyclic fatigue crack growth rates in the foams considered, i.e. H45, H60 and H100 (PVC) foams and F50, F90 and F130 (PES) foams. Initial fatigue testing of the DCB specimens with low density PES foams, however, showed a tendency for crack kinking after a relatively small number of cycles. A study of crack kinking in sandwich DCB specimens was presented in [14] where it was determined that specimens with thinner cores should be more stable. This was verified by static fracture testing of specimens with “thin” and “thick” cores (12.7 vs. 25.4 mm). Cyclic loading, however, is obviously affecting the crack stability and stress distribution ahead of the crack tip differently than the static loading. To suppress or prevent crack kinking in the fatigue loaded DCB test specimens with PES foam cores, 1.5-mm deep semi-circular grooves of 6.35 mm radius were machined in the specimen edges using a 12.7-mm diameter ball end mill as shown in Figure 7. Edge grooving was suggested by Anderson [16] to promote self-similar crack growth in fracture specimens. To aid in locating the position of the crack front, black paint was sprayed on one edge of the DCB specimens.

The fatigue test program is summarized in Table 3. The sandwich specimens were tested for a minimum of 43,000 cycles and some up to 100,000 cycles. Because of inaccuracies in the small displacements imposed and possible crack closure effects, the R ratios based on the measured loads

(R = P_{min} / P_{max})

may differ slightly from the ones based on the displacements listed in Table 3. Results will be presented in the next section. The fatigue tests were conducted at a low frequency, 0.5 Hz, to avoid hysteretic heating during cyclic loading.

Table 3.

Fatigue test matrix for sandwich DCB specimens.

Test 1	Test 2	Test 3	Test 4
H45	R	0.2	0.125
δ_max (mm)	1.91	2.03
δ_min (mm)	0.38	0.25
a_o → a_f (mm)	51.9→86.4	61.8→91.4
Total cycles	96,000	51,000
H60	R	0.342	0.342	0.217	0.217
δ_max (mm)	0.97	0.97	1.52	1.52
δ_min (mm)	0.33	0.33	0.33	0.33
a_o → a_f (mm)	26.8→43.8	25.7→42.2	44.6→65.9	42.2→65.7
Total cycles	45,000	45,000	47,000	43,000
H100	R	0.1	0.1	0.25
δ_max (mm)	2.79	3.05	1.14
δ_min (mm)	0.25	0.30	0.29
a_o → a_f (mm)	64.1→83.6	58.4→92.7	23.3→37.5
Total cycles	50,000	100,000	50,000
F50	R	0.5
δ_max (mm)	1.02
δ_min (mm)	0.51
a_o → a_f (mm)	24.8→35.8
Total cycles	50,000
F90	R	0.429
δ_max (mm)	0.89
δ_min (mm)	0.38
a_o → a_f (mm)	22→44.3
Total cycles	117,000
F130	R	0.33	0.5
δ_max (mm)	1.52	1.02
δ_min (mm)	0.51	0.51
a_o → a_f (mm)	22.9→60.6	21.8→37.4
Total cycles	45,000	75,000

DCB: double cantilever beam.

Results and discussion

Verification of crack length measurements

To examine the accuracies of the two methods of crack length measurement, fatigue testing of a DCB specimen with a H100 core was first conducted. The crack length was determined from the measured compliance and measurements with a traveling microscope on one side of the specimen, Figure 5. A total of 50,000 loading cycles, having maximum and minimum opening displacements of 2.79 and 0.25 mm (R = 0.09), were imposed on the H100 specimen. Crack length was measured at cycle number 0, 2.5 k, 10 k, 18 k and 45 k and plotted along with the compliance-based crack length vs. the number of loading cycles in Figure 8. It is noted that both crack lengths are in close agreement. Figure 9 shows examples of crack length results for the H45 and H60 PVC foams. Crack lengths determined from the compliance and traveling microscope are within ±1.5 mm.

Crack growth results for the PES foams are shown in Figure 10. For all PES specimens, the compliance method underestimates the crack length. The crack lengths determined from the compliance and traveling microscope are far apart for the F50 specimen. For the F50 specimen, Figure 10(a), the compliance method underestimates the initial crack length by about 10 mm. For the F90 and F130 specimens, Figure 10(b) and (c), the difference between the two methods was less than about 5 mm during the range of crack lengths examined.

The difference in crack length between the edge measurement and the compliance method at the beginning of the fatigue test must be attributed to the compliance method. The machined initial crack front is straight and well-defined and is quite easily measured using the traveling microscope. Examination of equation (1) shows that the major contribution to the compliance comes from the modulus of the foam. Variations in foam density and mechanical properties within and between foam panels are common. The foam manufacturing process where a solid polymer is expanded into a foam is not very well controlled, leading to variations in the foam density and modulus.

Another major difficulty in measuring crack length especially for the F50 specimens is that the crack front becomes highly non-uniform after crack extension. Black paint was applied to the crack surfaces for some F50 and F90 specimens cycled to about 100,000 cycles to identify the shape of the crack front. After the paint dried, the specimens were fractured statically by application of a continuously increasing load to the DCB specimen. The newly created fracture surfaces and crack front were clearly distinguishable. This procedure revealed that the crack was longer at the specimen edges than at the center.

Figure 11 shows a schematic of the crack front in a F50 foam specimen after about 100,000 load cycles. This phenomenon is opposite to the tunneling commonly encountered in mode I fatigue testing of solid metal alloys where the crack tends to grow faster in the center than at the edges [17,18]. For the F50 specimen, the crack length was 10–12 mm longer at the edges than in the center. Cyclic loading of the specimens with a low-density PES foam evidently results in faster growth of the crack at the specimen edges than at the center. It is not clear what mechanisms are responsible for this effect. It is possible that the edge grooving, Figure 7, may be a factor that promotes non-uniform growth.

One possible explanation of reverse tunneling may be found in the multi-axial state of stress that varies along the crack front for a 3D crack. In solid materials, a multi-axial state of stress significantly alters ductility, fracture and fatigue properties [19]. For example, brittle materials become ductile when subjected to very high pure hydrostatic pressure (and vice versa). Recent numerical results [20] investigating 3D crack propagation during cyclic loading suggested that when the internal constraints and the associated change in fatigue properties are ignored, reverse tunneling is observed. The “classic” forward tunneling could only be obtained when the constraint effect of the propagation criterion was considered. Since a foam is not homogeneous, the constraint effect may be small or non-existing. Thus, the observation of a “reverse” tunneling is consistent with this numerical observation. However, more work needs to be done to make a true assessment to this regard.

Undoubtedly, such a large difference in crack length would invalidate the data reduction assuming uniform crack extension. ASTM E399 [21] standard for mode I fracture testing of metals specifies that the crack length difference shall not exceed 15%. Furthermore the ASTM D5528 [22] standard for mode I DCB fracture testing of composite laminates requires that the crack length measured along the crack front should not deviate more than 2 mm.

To enforce agreement between measured and calculated initial crack length for the DCB specimens with PES foams (Figure 10), the out-of-plane modulus of the foam $(E_{c})$ was adjusted (increased to higher values). Corrected crack growth results for the PES foams are shown in Figure 12. The out-of-plane moduls are listed in the figure caption. Inspection of the results shown in Figure 12 reveals that the adjustment of the foam modulus to achieve agreement between the calculated and measured initial cracks length also improved the overall agreement between the two methods. For the F50 foam, however, the inverse tunneling effect observed after about 100 k cycles cannot readily be compensated for. The test results for this foam were therefore discarded after 50 k cycles.

Crack growth rates

Empirically fitting a power function, equation (10), to the crack length vs. number of cycles, Figures 8, 9 and 12, provides good fits. An example of such a fit is shown for a H45 specimen in Figure 9(a). Differentiation of the empirical function according to equation (11) provides the cyclic growth rate, $da / dN$ The cyclic energy release rate, ΔG, is determined for each specimen from equation (4) using the recorded load levels, P_max and P_min, crack length and the foam modulus, $E_{c}$ (adjusted for the PES foams as discussed above). With the growth data (da/dN) and cyclic energy release rate (ΔG) established, it is possible to construct cyclic growth curves and fit the linear region to Paris law,

\frac{da}{dN} = B Δ G^{n}

(12)

Figure 13 shows fatigue crack growth curves for sandwich DCB specimens with H45 foam tested at R-ratios of 0.2 and 0.125. It is noted that the growth data follow Paris law at lower ΔG values, but exceed the linear relation (Paris law) at larger cyclic energy release rates. This behavior is expected because the maximum fatigue loads encountered start to approach those required for static fracture. Figure 14 shows fatigue crack growth curves for four sandwich DCB specimens with H60 foam tested at R-ratios of 0.342 and 0.217. Comparison of the results for two replicate test specimens shows small specimen-specimen variation. The fatigue crack growth curves for H100 foam tested at R-ratios of 0.25 and 0.1 shown in Figure 15 are qualitatively similar to those obtained for the H45 and H60 foams, Figures 13 and 14.

As discussed earlier, fatigue testing of the low-density PES foams (especially F50 foam) is associated with substantial difficulties. Non-uniform crack growth and kinking invalidated several of the fatigue tests. Only a single DCB specimen with F50 foam was successfully tested. This specimen was tested at an R-ratio of 0.5. The growth results are shown in Figure 16. For large values of the cyclic energy release rate, the F50 specimen displayed fast crack growth. This range corresponds to the initial part of the test, Figure 12(a). As mentioned earlier, this specimen was cycled only for 50 k cycles (Figure 10(a)) to suppress the inverse tunneling effect.

The crack growth results for the one successfully tested F90 specimen are shown in Figure 17. The Paris regime is clearly identified. Fatigue crack growth curves for the F130 foam tested at two R-ratios are shown in Figure 18 with the Paris regions indicated by dashed lines.

Table 4 provides a summary of the crack growth rates (da/dN) for all the foams at a constant cyclic energy release rate of ΔG = 200 J/m². The R-ratios based on cyclic displacements and cyclic loads are listed. The effect of foam density on the crack growth rates is first discussed. For the PVC foams, the density has a great effect on the crack rates. As the density increases, the crack growth rate substantially decreases. It is noted, however, that foam density appears not to influence the crack growth exponent, n, which is about three for all foams tested. Zenkert et al. [2] found that the growth exponent (m) (da/dN = AΔK^m), for a large range of PVC foams was

m \approx 6

. This result seems to be consistent with our PVC data, da/dN = BΔGⁿ, where

n \approx 3

considering the square root relation between K and G (K ∼ G^1/2).

Table 4.

Fatigue crack growth rate at ΔG = 200 J/m², and exponent (m) for Paris’ law.

R		da/dN (µm/cycle)	n
δ_min/δ_max	P_min/P_max	da/dN (µm/cycle)	n
H45	0.125	0.06	14.4	3.1
0.20	0.05	20.4	3.6
H60	0.217	0.08	5.32	3.0
0.217	0.14	3.86	3.0
0.342	0.26	3.91	3.3
0.342	0.24	3.91	3.3
H100	0.10	0.07	0.29	2.8
0.10	0.01	0.29	2.8
0.25	0.18	0.12	4.1
F50	0.50	0.3	0.90	2.0
F90	0.43	0.38	0.90	3.1
F130	0.33	0.21	1.20	2.1
0.50	0.37	0.30	1.2

For the PES foams, a range of crack growth exponents (n) was noted $(1.2 < n < 2.3)$ . Hence, the growth exponent is substantially less than for the PVC foams (n ≈ 3.0). This could possibly be attributed to the high toughness of the PES foams, although, as discussed earlier, Olurin et al. [3] found very large growth exponents (m) for crack growth in aluminum foams. For ΔG = 200 J/m² da/dN for the PES F130 foam ranged between 0.3 and 1.2 µm/cycle depending on the R ratio. The results provide no clear indication about the influence of foam density on the crack growth rate for the PES foams examined.

Comparison of the crack growth rates for the PVC and PES foams at similar densities shows that the F50 and F90 PES foams display slower crack growth rates and smaller crack growth exponents than the similar density PVC foams. The growth rate for the F130 foam is higher than for the H100 foam, but there is a very large influence of the R ratio.

Conclusions

An experimental study on fatigue crack growth in cross-linked PVC and thermoplastic PES foams has been presented. A range of PVC and PES foam densities was considered. The fatigue tests employed the sandwich DCB test specimen. The test results revealed that crack growth rates compared at a constant cyclic energy release rate decreased with increasing density of the foam. At a constant cyclic energy release rate, the crack propagation rates in low-density PES foams were much less than in PVC foams of similar density. The growth exponent, n, in Paris law was approximately constant ( $n \approx 3$ ) for the PVC foams, but varied from 1.2 to 3.1 for the PES foams depending on the R ratio and foam density. For the high-density PES foam (F130) the growth rate was highly sensitive to the R ratio. The crack front in the low-density PES foams displayed substantial deviations from the ideal straight shape. The edges propagated faster than the center of the specimen.

Footnotes

Acknowledgements

Support for this research was provided by the National Science Foundation (CMMI-0824827) under a sub-contract from University of Delaware. Also, special thanks go to Chris Kilbourn and James Jones of DIAB Desoto, Texas, who provided foam materials free of charge. Thanks go also to Mark S Hoerber Jr for his assistance with typing of this manuscript.

References

Paris

Gomez

Anderson

. A rational analytic theory of fatigue. Foundations of Linear Elastic Fracture Mechanics 1961; 13: 9–14.

Zenkert

Shipsha

Burman

. Fatigue of closed cell foams. J Sandw Struct Mater 2006; 8: 517–538.

Olurin

McCullough

KYG

Fleck

. Ashby, Fatigue Crack Propagation in Aluminum Alloy Foams. Int J Fatigue 2001; 23: 375–382.

Shipsha

Burman

Zenkert

. Interfacial fatigue growth in foam core sandwich structures. Fatigue Fract Eng Mater Struct 1999; 22: 123–131.

Gillespie

Carlsson

Gawandi

. Fatigue crack growth at the face sheet-core interface in a discontinuous ceramic-tile core sandwich structure. Compos Struct 2012; 94: 3186–3193.

Manca M, Quispitupa A, Berggreen C, et al. Face/core debond fatigue crack growth characterization using the sandwich mixed mode bending specimen. Compos: Part A 2012; 43: 2120–2127.

Maiti

Ashby

Gibson

. Fracture toughness of brittle cellular solids. Scripta Metall 1984; 18: 213–217.

Gibson

Ashby

. Cellular solids structure and properties, 1st ed. Cambridge: Pergamon Press, 1988.

Saenz

Carlsson

Karlsson

. In situ analysis of fatigue crack propagation in polymer foams. Eng Fract Mech 2013; 101: 23–32.

10.

DIAB Group, DIAB H Series Data, DIAB F Series Data, DIAB Products - Divinycell. Online. http://www.diabgroup.com/americas/u_products/u_prods_2.html.

11.

Saenz E. Fatigue and fracture of foam cores used in sandwich composites. Master Thesis. Florida Atlantic University, 2012.

12.

Saenz

Carlsson

Karlsson

. Characterization of fracture toughness G_C of PVC and PES foams. J Mater Sci 2011; 46: 3207–3215.

13.

Taher

Thomsen

Dulieu-Barton

. Determination of mechanical properties of PVC foam using a modified arcan fixture. Compos Part A 2012; 43: 1698–1708.

14.

Saenz

Hernandez-Perez

Carlsson

. A DCB specimen for foam core fracture characterization. J Sandw Struct Mater 2012; 14: 281–295.

15.

ASTM E647-11e1, Test Method for Measurement of Fatigue Crack Growth Rates, ASTM, 2001.

16.

Anderson

. Fracture mechanics: fundamentals and applications, Boca Raton: CRC Press, 2005.

17.

James

Newman

. The effect of crack tunneling on crack growth: experiments and CTOA analyses. Eng Fract Mech 2003; 70: 457–468.

18.

Lan

Deng

Sutton

. Investigation of crack tunneling in ductile materials. Eng Fract Mech 2010; 77: 2800–2812.

19.

Zamrik SY, Mirdamamdi M and Davis DC. A proposed model for biaxial fatigue analysis using the triaxiality factor concept. Advances in Multiaxial Fatigue. ASTM STP 1191 (1993) 85.

20.

Nittur

Karlsson

Carlsson

. Implementation of a plastically dissipated energy criterion for three-dimensional modeling of fatigue crack growth. Int J Fatigue 2013; 54: 47–55.

21.

ASTM E399-09e2, Standard test method for linear-elastic plane-strain fracture toughness of metallic materials, ASTM, 2001.

22.

ASTM D5528-01, Standard test method for mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites, ASTM, 2001.