WO2007071242A1 - Method for constructing a product exposed to load, especially a biomedical joint implant comprising nanocomposites - Google Patents

Abstract

Method for constructing a product with a high degree of long term deformation resistance against cyclic load, preferably a biomedical joint implant, where the product comprises nanocomposite material containing polymer and nanoclay.

Description

Method for constructing a product exposed to load, especially a biomedical joint implant comprising nanocomposites

FIELD OF THE INVENTION The present invention relates to nanocomposites, especially the use of nanocomposites in biomedical joint implants.

BACKGROUND OF THE INVENTION Ultra High Molecular Weight PolyEthylene (UHMWPE) has been utilized as the syn- thetic replacement for articular cartilage in total joint replacements for nearly four decades. This is described in more detail by Krzypow D. J. and Rimnac CM in "Cyclic steady state stress-strain behavior of UHMWPE" published in Biomaterials 21 (2000) 2081-2087; and by Bergstrom J.S., Rimnac CM. and Kurtz S.M. in "An augmented hybrid constitutive model for simulation of unloading and cyclic loading be- haviour of conventional and highly cross linked UHMWPE" published in Biomaterials 25 (2004) 2171-2178; and by Meyer R. W. and Pruitt, L.A. in "The effect of cyclic true strain on the morphology, structure and relaxation behaviour of ultra high molecular weight polyethylene." published in Polymer 43 (2001) 5293-5306.

UHMWPE has superior biomechanical properties including high toughness, low friction, and good biocompatibility. The reasons for the use of UHMWPE as compared to other polyethylene groups are probably higher Young's modulus, tensile strength, impact toughness, mechanical stability at higher temperatures and abrasion resistance. Chemically, the only difference between UHMWPE and High Density Polyethylene (HDPE) is the size of the molecules. Thus, UHMWPE is really a subgroup of HDPE (Plastics; Materials and processing, A. Brent Strong, Pearson, Prentice Hall 3^rd ed. Pp 231, 2006). Low Density Polyethylene and Linear Low Density Polyethylene (LDPE and LLDPE) are chemically similar to HDPE but contain more branches per chain length. This affects crystallization and contributes to a steric difference between LDPE and HDPE. The use of UHMWPE in the medical industry is a trade off between properties of the polymer and price and processing difficulties. UHMWPE is much more difficult to process than, for instance, HDPE.

Recent efforts in the Polyethylene area to improve materials have focused on resin type, sterilization method, radiation cross linking and thermal treatments as it is described by Bergstrδm J.S., Rimnac CM. and Kurtz S.M. in the aforementioned article with title "An augmented hybrid constitutive model for simulation of unloading and cyclic loading behaviour of conventional and highly cross linked UHMWPE" pub- lished in Biomaterials 25 (2004) 2171-2178.

The lifetime of total joint components made of UHMWPE is a problem. Problems are described in literature, and as discussions with different surgeons has revealed, there are unresolved problems in this area. Loosening and infections have been correlated to the foreign body reaction from UHMWPE debris found in the surrounding tissues due to wear problem. Oxidation and sterilization have been mentioned as being one of the causes for the debris, however, as disclosed in the aforementioned references, the nature of the deformation in cyclic loading is identified as a cause also. The total plastic deformation plays an important role.

Another problem is the Cyclic softening of UHMWPE. When loading and unloading the material, it gradually softens as the modulus of elasticity (Youngs modulus) decreases.

There is a need in the art of medical joint and bone replacement to steadily improve the material.

DESCRIPTION / SUMMARY OF THE INVENTION

Therefore, it is the purpose of the invention to provide an improved material medical implants. Especially, it is the purpose to provide a biomedical joint implant with a high resistance against long term deformation due to cyclic load. It is a further object of the invention to provide a design method for improved products that are exposed to cyclic load. This purpose is achieved by a product for a biomedical joint implant or bone implant, the product comprising nanocomposite material containing polymer and nano-material for resistance against stress softening, creep or for resistance against total plastic deformation due to cyclic loads, wherein the nano-material is in the form of nano- platelets with a ratio between an average height of the platelets and an average width of the platelets of at least 1 :20.

As will be more apparent from the following, there has been a surprising finding that nanocomposite material containing polymer and nano-platelets, such as in nanoclay, has a high resistance against deformation due to cyclic load. It shows a significant reduction in the total plastic deformation after several load cycles compared to the unfilled polymer. Creep properties are improved, especially at high loads and the total plastic deformation is significantly reduced. To our knowledge, the deformation resistance has not yet been investigated in detail for such kind of nanocomposites, and polymer with nanoclay has not yet been proposed for the reason of withstanding cyclic stress on the long term basis.

Though a ratio between the average thickness and width of the platelets has been defined above to be at least 1 :20 to be useful for the invention, preferably, the ratio is at least 1:200, or even better at least 1:1000. In the experiments, good results have been achieved with ratios in the order of 1 :2000, as it may be the case for nanoclay, for example Montmorillonite. Thus, an example of the dimensions of large nanoclay platelets is one nanometer in thickness and of the order of one or two micrometer in diameter.

A different approach for quantifying the nano-plate structure useful for the invention is by the radius of gyration Rg. Rg is the average of the possible end-to-end distances for the polymer chains in the polymer product. For conventional polyolefin, Rg is typically between 2 ran and 10 nm in average, whereas the polymer chain length itself is in the order of micrometer. The platelet width of the nano-platelets should be much larger than Rg in order to have a substantial effect. In experiments, an effect is observed for nano-platelets with a diameter of 20 nm, the effect increasing with increasing diameter. The platelets need not to be single platelets, but may as well be aggregations of platelets as it typically is seen in not completely exfoliated clay.

Different deformation effects are found in polymers exposed to load. Terms used in the patent will be defined as creep, total plastic deformation, and cyclic stress softening as follows.

Creep: Gradual alteration (deformation) in length of a part subjected to a load causing stresses to arise in the part. The total deformation and speed of deformation is a func- tion of variables like time, stress and temperature. The total deformation consist of an elastic (recoverable) part, and a viscoelastic part, where some part of this deformation might be recovered over time and another part be permanent (irrecoverable).

Total plastic deformation: Subjecting a part to a high load can cause it to respond not only elastic (recoverable) but also give a permanent plastic deformation (irrecoverable). If the timescales are short, one will usually not use the word creep about the processes involved in a loading-unloading experiment. The total plastic deformation is used to describe the alteration in length occurring after loading-unloading experiments. In a uniaxial tensile test experiment, stress can be plotted on one axis and de- formation on the other axis. After loading-unloading into the plastic region of the material, it can be observed that at unloading the curve does not return to zero deformation, but a certain degree of plastic deformation has taken place. If the test is repeated, the degree of plastic deformation can again change and the sum of these plastic deformations is called the total plastic deformation. In a medical implant, the patient will be active, and this way subjects the implant to various stresses where plastic deformation might take place. The sum of these plastic deformations is in scientific literature believed to play a major role for the life time of an implant.

Cyclic stress softening: In a uniaxial tensile test experiment, stress can be plotted on one axis and deformation on the other axis. The slope of the curve in the beginning of the loading (small stresses) is usually a straight line and at unloading the material will return to its original shape. This is a pure elastic behaviour, and the coefficient of proportionality between stress and deformation called Youngs modulus (stress = Youngs modulus multiplied by the deformation). The higher Youngs modulus, the stiffer the material. After a series of loading, unloading experiments where loads are high (as described above under total plastic deformation) it can be observed, that Youngs modulus changes, it decreases. This is termed "cyclic stress softening".

Though nanocomposites have been reported for certain biomedical applications, the cyclic stress behaviour has not been discussed in this connection. The Manias group of the Perm. State Univ. reported some preliminary results on barrier properties of nanocomposites for breast implants. These composites were based on Montmorillonite (MMT) and polydimethylsiloxane (PDMS). The same group reported also on barrier properties of another system based on MMT and polyurethane (PU) for biomedical applications (J. Biomed. Mater. Res. A 64(1): 114-119, 2003). The Martin group from the University of Queensland reported on improvement of mechanical properties of nanocomposites (MMT/PU) with applications to polyurethane breast implant shells and membranes for artificial blood pumps. Structure-property relations for MMT/PU nanocomposites without references to biological applications have been reported by Finnigan et al. (Polymer 45: 2249, 2004).

However, applications of polymers with nanoclay in order to reduce the total plastic deformation reduce cyclic softening and improve creep properties, have to our knowl- edge not been reported, especially not in biomedical applications. Furthermore, biomedical application with polyolefins, especially HDPE or UHMWPE, containing nanoclay, has to our knowledge not yet been reported. The use of polyolefins with clay has in several articles been reported to be difficult due to the high polarity of the clay compared to the very low polarity of polypropylene of polyethylene.

According to the invention, when used as an implant, it has to be determined whether the implant comprises the nanocomposite material as part or whole of its surface and whether it comprises the nanocomposite material as part or whole of its bulk material. The practical embodiment of the nanocomposite may depend on the specific applica- tion. As mentioned above, nanocomposites reveal a high degree of resistance against long term deformation due to cyclic load, such that the nanocomposite material may, for example, be employed in those parts of the product, where the cyclic stress is high. In some instances, a surface coating may suffice, in other instances the bulk material or part of it is advantageously made of the nanocomposite. In experiments, polypropylene and High Density Polyethylene (HDPE) has been used as the polymer in the nanocomposite. However, the experimental data are considered to be generally representative for polyolefins with nanoclay. Low Density Polyethylene or Linear Low Density Polyethylene may be used in connection with the inven- tion. However, preferred polyolefins for joint or bone implants may be High Density Polyethylene (HDPE) or even Ultra High Molecular Weight Polyethylene (UHMWPE) due to their higher strength.

The clay may be smectite type clay, for example Montmorillonite. However, other clays may be used.

In addition, the product may comprise a ceramic filler material in order to adjust the physical properties of the product.

Useful applications of the invention may be any place where polyethylene is used for implants, and the implant has to resist a mechanical deformation over time. Examples are for instance hip joints, knee joints, acetabular sockets, finger joints, facial implants and any other kind of bone joint or part of bones. The implant can be a total implant or a partial implant used in combination with e.g. metals or ceramics.

Surprisingly, the above purpose of the invention is also achieved by a polyolefin material, for example HDPE, which has been subjected to annealing at a temperature higher than HO⁰C, for example between 110°C and 130°C. Experiments have shown that an annealing time of 2 hours is appropriate, however, the time may vary, for ex- ample between 30 minutes and 4 hours, preferably between 1 and 3 hours.

The invention also comprises a method for minimising stress softening, creep or plastic deformation due to cyclic loads in a product for a biomedical joint implant or bone implant, the method comprising the steps of - providing a thermoplastic polymer

- mixing nano-material in the form of nano-platelets into the polymer, the nano platelets having a ratio between the average height of the platelets and the average width of the platelets of at least 1:20.

- heat moulding the mixture. Preferably the ratio is at least 1:200, for example at least 1:1000 or even 1:2000, as it may be the case for a nanoclay, for example Montmorillonite.

Thus, the method also implies constructing a product for a biomedical joint implant or bone implant with a high degree of resistance against stress softening, creep or of resistance against plastic deformation due to cyclic loads.

In order to achieve an even better resistance, the method may comprise annealing the moulded product. The reason is to be seen in the fact that a typical moulding of a thermoplastic polymer product leads to a large degree of crystallisation in the middle of the product due to the slower cooling than at the outer parts of the product. If the crystallisation is increased by a factor of 30%-50%, it has been found that the Young's modulus increases by a factor of between 2 and 3.

The annealing may be performed at a temperature of more than 110°C for a predetermined time, for example more than one hour. The annealing temperature is dependent on the product. The temperature should be high enough to promote crystallisation but not so high that severe thermal disintegration occurs in the polymer. For example, for HDPE, the temperature should be more than 120°C, typically for a certain time in the order of hours.

Experiments have been performed at temperatures between 110°C and 130°C with very good results especially at an annealing temperature in the order of 130°C for more than one hour, for example two hours, for the polymer HDPE, if the nano- material is Montmorrillonite and the nanoclay is exfoliated with a surfactant before moulding.

The method in a further embodiment implies constructing a product, preferably a biomedical joint implant, with a high degree of resistance against stress softening, creep or of resistance against plastic deformation due to cyclic loads, the method comprising the steps of

- predicting a range of magnitude of load on the implant,

- determining an acceptable long term deformation of the implant due to this load, - selecting a nanocomposite material comprising nano-material and a polymer for at least part of the structure of the implant,

- identifying the strain in different parts of the implant in dependence of the load stress, - determining that part or those parts of the implant that are to be composed of the nanocomposite,

- determining whether the selected nanocomposite when incorporated in the implant can withstand the load with a long term deformation of the implant less than or equal to the acceptable deformation, - incorporating the nanocomposite material in the implant.

The determined part or parts of the product may comprise the surface of the product, and in some instance be limited to the surface of the product. For example, the product may have a different bulk material coated with the nanocomposite. In another em- bodiment, the determined part or parts of the product include the bulk material of the product. In the case of nanocomposite as the bulk product, the surface may be a nanocomposite as well but could also comprise a different material.

The preferred polymers are from the group of polyolefins, for example High Density Polyethylene (HDPE) or Ultra High Molecular Weight Polyethylene (UHMWPE). The preferred clay is smectite type clay, for example Montmorillonite. However, other clays may be used.

It should be mentioned that the method primarily is intended for products being bio- medical implant, especially biomedical implants or parts of a biomedical implants, for example biomedical joint implants. For such a biomedical joint implant or bone implant, the described method and product are especially suited to achieve resistance against stress softening, creep or for resistance against total plastic deformation due to cyclic loads.

However, the method may be used more general and may as well apply to the design of other types of products that are exposed to cyclic load and where a long term deformation of the products is to be minimised. An UHMWPE or HDPE or LDPE or LLDPE or PP nanocomposite could also be useful for other products that are exposed to cyclic loads or long term loads, for example earth crake resistant pipes. The method according to the invention may as well be used for other applications with static or cyclic loads including pressurized pipes, for example used for transport of water, gas or chemicals and furthermore, bottles, loudspeaker membranes geomembranes, insti- tutional and container can liners, grocery sacks and merchandise bags, large blow moulded industrial containers.

It is likely, that the observed effects on the significant effect of nanoclay filled poly- olefmes can be enhanced by adding other nanoflllers, like nanoparticles or nanofibres thus obtaining a nanohybrid composite.

Though not strictly within the sense of the claims, the path leading to the invention implied a surprising discovery that annealing of polyethylene - and possibly valid for polyolefins in general - also led to an improvement in the strain-stress behaviour. An- nealed HDPE showed an effect very much like the nanocomposite according to the invention and may be used for substitution of the polyolein/clay nanocomposite above. A reduction in the strain ε by a factor of 2 was observed for the higher loads when the HDPE without clay was annealed for 2 hours at 110°C. A more pronounced effect with about a factor of 3 for the reduction was achieved when the annealing tem- perature was 12O⁰C, and an even stronger effect was observed for annealing at 130°C. Thus, a polyolefm, for example HDPE, annealed at a certain temperature below the temperature for thermal disintegration of the polymer, but preferably above 100⁰C or rather above HO⁰C, may successfully be used for a moulded product, for example for a biomedical joint implant or bone implant, to achieve a high resistance against stress softening, creep or resistance against total plastic deformation due to cyclic loads.

Annealing times are in the order of a larger fraction of an hour and some hours, for example between half an hour and 3 hours.

SHORT DESCRIPTION OF THE DRAWINGS

The invention will be explained in more detail with reference to the drawing, where FIG. 1 shows measurements of the engineering stress σ versus the tensile strain ε during the first two cycles of tensile deformation of neat polymer with a cross head speed of lOmm/min. The minimum stress σ_min= 0.0 MPa, and the maxi- mum strains are ε_max = 0.05 (unfilled circles , 0.10 (filled circles), 0.15 (diamonds) and 0.20 (stars);

FIG. 2 shows measurements of the engineering stress σ versus the tensile strain ε during the first two cycles of tensile deformation of a hybrid nanocomposite with a cross head speed of lOmm/min. The minimum stress c_min = 0.0 MPa, and the maximum strains are ε_max = 0.05 (unfilled circles , 0.10 (filled circles),

0.15 (diamonds) and 0.20 (stars);

FIG. 3 shows measurements of the dimensionless stress σ/σ_max versus time t in tensile relaxation tests at various strains ε for neat polymer; FIG. 4 shows measurements of the dimensionless stress σ/σ_max versus time t in tensile relaxation tests at various strains ε for hybrid nanocomposite;

FIG. 5 shows measurements of strain ε versus time t in creep tests with the engineering stress σ = 5.6 MPa;

FIG. 6 shows measurements of strain ε versus time t in creep tests with the engineer- ing stress σ = 18.8 MPa;

FIG. 7 shows measurements of the engineering stress σ versus the tensile strain ε during 10 cycles of tensile deformation of a neat polymer with a cross head speed of 2 mm/min. The maximum stress was 18.8 MPa and the minimum stress σ_min= 5.6 MPa; FIG. 8 shows measurements of the engineering stress σ versus the tensile strain ε during 10 cycles of tensile deformation of a hybrid nanocomposite with a cross head speed of 2 mm/min. The maximum stress was 18.8 MPa and the minimum stress σ_min= 5.6 MPa;

FIG. 9 shows measurements of the engineering stress σ versus strain ε for neat poly- mer at tensile deformations with various cross-head speeds. Symbols are experimental data. Solid lines are results of numerical simulation;

FIG. 10 shows measurements of the engineering stress σ versus strain ε for nanocomposite at tensile deformations with various cross-head speeds. Symbols are experimental data. Solid lines are results of numerical simulation; FIG. 11 shows measurements of the engineering strain ε versus time t in tensile creep tests with various engineering stresses σ (MPa). Symbols: experimental data on non-annealed HDPE; FIG. 12 shows measurements of the engineering strain ε versus time t in tensile creep tests with various engineering stresses σ (MPa). Symbols: experimental data on non-annealed HDPE + MMT nanocomposite;

FIG. 13 shows measurements of the engineering strain ε versus time t in tensile creep tests with various engineering stresses σ (MPa). Symbols: a) experimental data on HDPE annealed at T=IlO⁰C for 2 h. and b) experimental data on HDPE annealed at T=120°C for 2 h.;

FIG. 14 shows measurements of the engineering strain ε versus time t in tensile creep tests with various engineering stresses σ (MPa). Symbols: experimental data on HDPE+MMT nanocomposite annealed at T= 13 O⁰C for 2 h. The vector indicates the breakage point;

FIG. 15 shows measurements of the engineering stress σ versus engineering strain ε in a cyclic tensile test with HDPE (10 cycles of loading-retraction with the cross- head speed 10 mm/min. Symbols are experimental data and solid line is a re- suit of a theoretical simulation, a) Virgin HDPE, the maximum stress σ_max=19.0 and the minimum stress σ_min=2.6 (MPa), b) HDPE annealed T=I lO⁰C for 2 h, the maximum stress σ_max=18.9 and the minimum stress σ_min=2.3 (MPa), c) HDPE annealed at T=I 2O⁰C for 2 h, the maximum stress σ_max=18.7 and the minimum stress σ_min=2.3 (MPa), d) HDPE annealed at T=I 30⁰C for 2 h, the maximum stress σ_max=18.7 and the minimum stress σ_min=2.3 (MPa).

FIG. 16 shows measurements comparable to FIG. 15 of the ^"engineering stress σ versus engineering strain ε in a cyclic tensile test with HDPE + MMT (10 cycles of loading-retraction with the cross-head speed 10 mm/min. Symbols are ex- perimental data and solid line is a result of a theoretical simulation, a) Non- annealed HDPE+MMT, the maximum stress σ_max=17.9 and the minimum stress σ_min=2.3 (MPa), b) annealed HDPE+MMT T=I lO⁰C for 2 h, the maximum stress σ_max=19.6 and the minimum stress σ_min=2.2 (MPa), c) annealed HDPE+MMT at T=I 20⁰C for 2 h, the maximum stress σ_max=18.2 and the minimum stress σ_min=2.2 (MPa), d) annealed HDPE+MMT at T=I 30⁰C for 2 h, the maximum stress σ_max=19.6 and the minimum stress σ_min=2.4 (MPa).; FIG. 17 shows the engineering stress σ versus engineering strain ε in a tensile test with strain rate dε/dt=2- 10^"3 s^'1. Symbols are experimental data and solid line is a result of a theoretical simulation, a) Virgin HDPE, b) HDPE annealed T=I 10°C for 2 h, c) HDPE annealed at T=120°C for 2 h, d) HDPE annealed at T=130°C for 2 h.

FIG. 18 shows the engineering stress σ versus engineering strain ε in a tensile test with strain rate dε/dt=2- 10^"3 s^'1. Symbols are experimental data and solid line is a result of a theoretical simulation, a) Non-annealed HDPE+MMT, b)

HDPE+MMT annealed T=I lO⁰C for 2 h, c) HDPE+MMT annealed at

T=120°C for 2 h, d) HDPE+MMT annealed at T=I 3O⁰C for 2 h.

FIG. 19 The adjustable parameters Ic₄. and k. versus increment of elastic strain

Δε_e.Symbols are treatment of observatios in a cyclic tensile test and solid lines are approximations of the experimental data for a) Virgin HDPE, b) HDPE annealed T=IlO⁰C for 2 h, c) HDPE annealed at T=120°C for 2 h, d) HDPE annealed at T=I 3 O⁰C for 2 h.

FIG. 20 The adjustable parameters Ic₊ and k. versus increment of elastic strain

Δε_e.Symbols are treatment of observatios in a cyclic tensile test and solid lines are approximations of the experimental data for a) Non-annealed

HDPE+MMT, b) HDPE+MMT annealed T=I lO⁰C for 2 h, c) HDPE+MMT annealed at T=120°C for 2 h, d) HDPE+MMT annealed at T=130°C for 2 h.

FIG. 21 The Young's modulus of neat HDPE (left) and HDPE/MMT nanocomposite

(right) annealed for 2 h at various temperatures, FIG. 22 The adjustable parameter α for neat HDPE (left) and HDPE/MMT nanocomposite (right) annealed for 2 h at various temperatures,

FIG. 23 The adjustable parameter β for neat HDPE (left) and HDPE/MMT nanocomposite (right) annealed for 2 h at various temperatures,

FIG. 24 The adjustable parameter a°₊ for neat HDPE (left) and HDPE/MMT nano- composite (right) annealed for 2 h at various temperatures,

FIG. 25 The adjustable parameter b°₊ for neat HDPE (left) and HDPE/MMT nanocomposite (right) annealed for 2 h at various temperatures,

FIG. 26 The adjustable parameter a⁰, for neat HDPE (left) and HDPE/MMT nanocomposite (right) annealed for 2 h at various temperatures, FIG. 27 The adjustable parameter b°_ for neat HDPE (left) and HDPE/MMT nanocomposite (right) annealed for 2 h at various temperatures,

FIG. 28 The adjustable parameter K_. for neat HDPE (left) and HDPE/MMT nanocomposite (right) annealed for 2 h at various temperatures, FIG. 29 The adjustable parameter K for neat HDPE (left) and HDPE/MMT nanocom- posite (right) annealed for 2 h at various temperatures.

DETAILED DESCRIPTION / PREFERRED EMBODIMENT' Clay is known as filler in polymers in order to improve the stability of polymers. Clay consists of a stack of platelets. This stack has a surface area more or less corresponding to the surface of the stack. If the plates in the clay can be separated some distance from each other, which is known as exfoliation, a polymer will start interacting with the individual platelets, and hence the effective surface area of the clay dramatically changes.

However, the surface of the clays normally does not tend to bind to the polymers. Therefore, in the preparation of clay, different steps are followed to chemically modify the surface in order to make it compatible with a given polymer, and to achieve the desired exfoliation.

In order to characterise clay containing polymers, work has been done and disclosed in the field regarding properties of these materials. A mechanical test, where the polyam- ide Nylon 6 has been compared with a nanocomposite containing the commercially available product Cloisite® revealed that the tensile strength, the tensile modulus, and the flexural modulus were higher for the nanocomposite. These findings are more or less state of the art, and numerous tests on a large range of materials exist. The modifications are large enough to be industrially interesting as it can lead to improved properties or cheaper materials.

Fig. 1 illustrates the complications that have been reported in the above mentioned articles by Krzypow D. J. and Rimnac CM in "Cyclic steady state stress-strain behaviour of UHMWPE" published in Biomaterials 21 (2000) 2081-2087; and by Bergstrόm J.S., Rimnac CM. and Kurtz S.M. in "An augmented hybrid constitutive model for simulation of unloading and cyclic loading behaviour of conventional and highly cross linked UHMWPE" published in Biomaterials 25 (2004) 2171-2178; and by Meyer R.W. and Pruitt, L. A. in "The effect of cyclic true strain on the morphology, structure and relaxation behaviour of ultra high molecular weight polyethylene." published in Polymer 43 (2001) 5293-5306, namely the fact that hysteresis in the elastic properties of the material leads to continuing deformation of the material after exposure to stress. The loading and unloading leads to a permanent plastic deformation of the material after exposure as well as to cyclic softening.

Fig. 1 shows measurements of the engineering stress σ versus the tensile strain ε during the first two cycles of tensile deformation of neat polymer - where neat polymer means polymer without fill material and without previous load - with a cross head speed of lOmm/min. The minimum stress σ_min = 0.0 MPa, and the maximum strains were ε_roax = 0.05 (unfilled circles, 0.10 (filled circles), 0.15 (diamonds) and 0.20 (stars). Not surprising, it appears that the continuing deformation (ε values near the abscissa) is larger, the larger the initial deformation has been.

When comparing with the corresponding measurements for a hybrid nano-composite based on polypropelene mixed with nanoclay (commercial product from the company Polyone), see FIG. 2, only moderate differences are observed. These only moderate differences in the mechanical properties and the higher production costs are the reasons why nanocomposites often are not preferred over polymers.

Comparing the dimensionless stress σ/σ_max versus time t in tensile relaxation tests at various strains ε between neat polymer, FIG. 3, and the hybrid nanocomposite, FIG. 4, as well, only moderate differences are observed.

Experiments on PP

The investigation leading to the invention has been made on polypropylene (PP), after which also HDPE has been subject to experiments. Polypropylene differs from polyethylene in the composing units of propylene instead of ethylene. However, the chemical nature is very similar, despite the fact that the introduction of a more bulky side group provides a steric effect. Polyethylene and polypropylene are often given a joint name to indicate this similarity in nature: They are called polyolefmes. They only consist of carbon and hydrogen and are aliphatic (nonaromatic) groups. It is therefore fair to assume that results from the measurements on polypropelene are similar in nature to results that can be obtained from UHMWPE. With the standard measurements of FIG. 1 through FIG. 4, no indication is given that polymers with nanoclay should have any profound advantage over UHMWPE, which traditionally is used, especially not with regard to damage of the material due to cyclic load. On the contrary, the moderate differences shown in FIG. 1 through 4 do not seem to justify the enhanced production difficulties and higher production costs that polymers with nanoclay imply as compared to UHMWPE when used in connection with artificial joints or bone implants.

In contrast to the conclusion based on the foregoing figures, a more thorough analysis did surprisingly reveal a very distinct behaviour of the nanocomposite relative to the neat polymer. When measuring the strain ε versus time t in creep tests with the engineering stress σ = 5.6 MPa, a slight difference was observed for the clay filled material, which is illustrated in FIG. 5. But it was not until the engineering stress was increased, such as to a high level of σ = 18.8 MPa as in FIG. 6, that a distinct and sur- prising difference was revealed. Suddenly, it became very clear that clay loaded material had highly improved properties in creep at large stress levels.

The difference in the above mentioned properties of the tested nanocomposite relative to neat polymer is clearly illustrated in figure 7 and 8 demonstrating the behaviour in cyclic loading, where FIG. 7 corresponds to the unfilled PP and figure 8 corresponds to the nanoclay filled PP. Though the stress loadings are not exactly the same in the two plots, the figures clearly demonstrate that stress softening has been significantly improved by the addition of nanoclay. In addition, an improvement has been achieved concerning the lifetime to fracture, because the permanent deformation after 10 cycles of load has been significantly reduced when adding the nanoclay.

The engineering stress σ versus strain ε at tensile deformations with various cross- head speeds is illustrated in FIG. 9 for neat polymer and in FIG. 10 for nanocomposite polymer. Symbols indicate experimental data and solid lines are results of numerical simulation.

To summarize we obtained:

• Pronounced improvement of total plastic deformation in loading-unloading experiments both for single and multiple loading cycles • Improved creep properties

• Reduced cyclic softening

• Slight increase in modulus and strength

The results shown above are thrilling as nanoclays apparently show a large impact on certain properties and only modest influence on other properties. This finding adds additional parameters to take into regard in industrial applications when selecting material that is exposed to cyclic stress.

One of these areas is for the UHMWPE total implants, such as but not limited to joint replacements and bone implants, where at least a number of references attribute the failure with the cyclic loading and softening. The idea is to make such kind of implant of a polyolefin, for example HDPE or a UHMWPE, with a percentage of chemically prepared nanoclay. Experiments have shown good results with an amount of clay in the order of 1-8%. In certain cases 10% clay may be useful. However, often it is not necessary with such high amounts, especially not, if the exfoliation is efficient.

An UHMWPE or HDPE or LDPE or LLDPE or PP nanocomposite could also be useful for other products that are exposed to cyclic loads or long term loads, for example earth crake resistant pipes. Other applications with static or cyclic loads involve pressurized pipes, for example used for transport of water, gas or chemical and further, bottles, loudspeaker membranes geomembranes, institutional and container can liners, grocery sacks and merchandise bags, large blow moulded industrial containers.

In experiments, we used a commercially available nanoclay master batch from the material supplier "Polyone": Nanoblend™ concentrates are high-performance materials based on nanocomposite technology. Nanoclays make up approximately 40% (weight) of the Nanoblend™ concentrates and mixed the Nanoblend™ concentrate with a commercial grade of polypropylene in an injection moulding machine. The same nanoclay master batch can be mixed with e.g. UHMWPE or HDPE. We injection moulded bars for mechanical testing, but we could also have moulded the shape of an implant that can be further tested or placed in a patient. Instead of using injection moulding as the forming process, other industrial processes can be used like compression moulding, extrusion, foaming, blow moulding, fibre spinning and more. In order to replace the nanoclay mastebatch (Nanoblend™), one could use other types of commercially available clays, for example Closite 2OA from Southern Clay Inc. mixed with the polymer and around 1% of a maleate polypropylene.

There are many known ways of preparing nanoclay and a surfactant and the method should be selected in dependence of the specific product and application.

The clay

Concerning the clay, many types can be used. The improved properties of nanocom- posites depend on the high surface area of the individual platelets of the clay, for example montmorillonite (MMT). Both the compatibility of the clay chemical treatment with the resin matrix and the melt blending conditions determine the degree of de- lamination and dispersion.

For the first preparation, sometimes the natural cation is replaced by a Na+ exchange. Such prepared clay is commercially available from e.g. Southern Clay Inc. When water is added, the clay exfoliates and a surfactant can be added. A surfactant is molecules with affinity to the clay surface by certain group(s) and affinity to the polymer system by another group. The polymer can then be added after drying or dissolved or in situ polymerized — there are many ways of bringing polymer and clay together:

Melt intercalation is a method where clay is treated such that a surfactant compatible with the polymer is inserted between the clay platelets. Intercalation implies that the distance between clay platelets is increased slightly, but not more than that they still have an attraction keeping the clay platelets sticking together. Adding the clay to a compatible polymer and applying heat mechanical deformation by mixing for a certain time causes the system to exfoliate, and nanoclay is obtained.

Exfoliation and adsorption occurs when the clay is exfoliated in a solvent, and the polymer also exists dissolved in the same system such that it can be attracted to the surface of the clay. When the solvent is removed, for example by evaporation, an exfoliated nanoclay loaded polymer is obtained. In-situ polymerization is a variant where the polymer is not dissolved but where the solvent itself is the monomers of the polymer. In this process, the growth of polymer chains can push clay platelets apart if they are only intercalated.

5 Nanoclay could also be made by use of template synthesis, so that the polymer solution contains silicates that will grow to obtain a mixture of silicate crystals and polymer.

The surfactant 0 The surfactant can be formulated in a large number of different ways. The bonding to the clay can be weak secondary forces, ion bonds or covalent bonds. In one embodiment of the invention for use in implants, ammonium surfactants employed at the intercalation stage are replaced by polyethylene oxide (PEO) or polymer chains also involving polypropylene oxides (PPO) such that PEO-PPO-PEO copolymers and 5 polysaccharides. The latter substances are biocompatible and do not have negative effects on living cells. For instance, PEO is used in medical applications.

Additional aspects

In order to achieve optimum effects in connection with the medical application of the 0 invention, nanoclay should be used which has a biocompatible and non-toxic effect when mixed with a polyolefrn, such as PE. It should be prepared in a way so the exfoliated polymer nanoclay is readily formed. In addition, the use of a biocompatible surfactant is advantageous.

5 Experiments with PE

As mentioned before, our preliminary observations on isotactic polypropylene (iPP) and iPP/MMT nanocomposites revealed that reinforcement of semicrystalline polymers with nanoclay noticeably reduces plastic creep (the growth of average strain per O cycle of deformation with number of cycles).

The objective of this further study on polyethylene is four-fold:

1. To demonstrate that the same effect is observed on HDPE and HDPE/MMT nanocomposites. 2. To analyze the influence of annealing in the sub-melting region of temperatures on cyclic viscoplasticity of HDPE and its nanocomposites.

3. To derive constitutive equations in viscoplasticity of semicrystalline polymers and hybrid nanocomposites and to find adjustable parameters in the stress-strain relations by fitting observations.

4. To discuss the effects of annealing and reinforcement with MMT nanoclay on the mechanical response of HDPE in terms of the constitutive model.

High-density polyethylene Eraclene MM 95 (density 0.953 g/cm³, melt flow index 11 g/10 min) was supplied by Polimeri Europa SpA (Italy). Dumbbell specimens (ASTM standard D638) with length 148 mm, width 9.8 mm and thickness 3.8 mm were prepared by using the injection-molding machine Ferromatic Kl 10/S60-2K.

Masterbatch MB 1001 E was purchased from PolyOne Inc. (nanocomposite with poly- propylene matrix filled with 40 wt.-% of Nanoblend concentrate, chemically modifed MMT nanoclay). Pellets of polyethylene and masterbatch were carefully mixed in proportion 80:20 by weight, which corresponded to approximately 8 wt.-% of the modifed nanoclay in the nanocomposite. Nanocomposite samples for testing were prepared in the same way as specimens of neat HDPE.

The concentration of nanoclay in the hybrid nanocomposite was chosen for two reasons:

1. this amount is close to the upper bound of the interval, within which mechanical properties of hybrid nanocomposites monotonically depend on concentration of clay platelets,

2. this concentration is relatively high in order to reveal substantial differences between the material parameters of HDPE and HDPE/MMT nanocomposite.

To evaluate the effect of annealing in the sub-melting region of temperatures, some specimens were annealed for 2 h at the temperatures T = 110, 120 and 130 ⁰C and slowly cooled by air. Mechanical experiments were performed at least 24 h after preparation of samples. Mechanical tests were performed at room temperature with the help of a universal testing machine Instron-5568 equipped with electro-mechanical sensors for the control of longitudinal strains in the active zone of samples. The tensile force was measured by a standard load cell. The engineering stress σ was determined as the ratio of the axial force to the cross-sectional area of specimens in the stress-free state.

In FIG. 11, experimental data on HDPE (high density polyethylene) are shown, illustrating the engineering strain ε versus time t in tensile creep tests with various engineering stresses σ (MPa) as indicated in the upper left corner of the figure. As a com- parison, corresponding data are shown in FIG. 12 for a nanocomposite comprising HDPE + MMT (montmorrillonite). It is apparent that also for HDPE, dramatic changes are observed, especially at higher stresses, if it is part of a nanocomposite. At a stress of σ=20 MPa, an improvement in the order of a factor of 4.5 is observed.

Temperature dependent effects

Surprisingly, an improvement in the strain-stress behaviour was observed when HDPE was annealed. This is illustrated in FIG. 13 a, where a reduction in the strain ε by a factor of 2 is observed for the higher loads when the HDPE is annealed for 2 hours at HO⁰C. A more pronounced effect with about a factor of 3 for the reduction was achieved when the annealing temperature was 12O⁰C, which is illustrated in FIG. 13b. It is believed that the annealing increases the degree of crystallisation of the polymer and the type of crystals that are formed. It is further believed that such crystals may act as interconnections in a way similar to nano-platelets from the MMT, such that a like- wise effect is observed.

The effect of a nanocomposite HDPE + MMT was demonstrated in connection with FIG. 12, and an improvement of a factor of about 4.5 was observed. This implies that the nanocomposite yields a higher influence on the physical properties, especially stress, of HDPE than the annealing. The question was, whether annealing would improve the properties further. This was tested and experiments revealed that an improvement of a factor of an order of magnitude could be achieved by the nanocomposite annealed for 2 hours at 13O⁰C, see FIG. 14, as compared to non-annealed HDPE, see FIG. 11. When it comes to annealing, the temperature should not be so high that the polymer decomposes.

As indicated in FIG. 14, at some strain, the sample breaks, which was the case as illus- trated for a stress σ=25 MPa.

For each test related to the following figures, samples were deformed with a constant cross-head speed of 10 mm/min (which corresponded to the strain rate έ-2 ^■ 1(T³,?^""1). This cross-head speed was chosen as a compromise between two contradictory re- quirements: (i) the strain rate should be relatively high to disregard the viscoelastic phenomena (stress relaxation), and (ii) it should be relatively low in order to reproduce accurately the loading program.

Two uniaxial tests were carried out on each group of samples. In the first test, speci- mens were loaded with the strain rate έ up to necking (or up to the maximum strain ε_max = 1.0). In the other test, specimens were loaded up to the maximum stress σ _max with the strain rate έ , unloaded down to the minimum stress σ _min with the strain rate - έ , reloaded up to the maximum stress with the strain rate έ , etc. Each cyclic test consisted of N = 10 cycles of deformation.

Each tensile and cyclic test was performed on at least two different specimens.

FIG. 15 shows measurements of the engineering stress σ versus engineering strain ε in a cyclic tensile test with HDPE of the type Eraclene MM95-3616800 (10 cycles of loading-retraction with the cross-head speed 10 mm/min, a) Virgin HDPE, the maximum stress σ_roax=19.0 and the minimum stress σ_min=2.6 (MPa). b) HDPE annealed T=I 10°C for 2 h, the maximum stress σ_max=18.9 and the minimum stress σ_min=2.3 (MPa). c) annealed at T=120°C for 2 h, the maximum stress σ_max=18.7 and the minimum stress σ_min=2.3 (MPa). d) annealed at T=130°C for 2 h, the maximum stress σ_max=18.7 and the minimum stress σ_min=2.3 (MPa). It is observed by comparing FIG. 15a and FIG. 15b that for the first cycles, there is a distinct difference in behaviour dependent on the annealing at 110°C. This is also expected from the tendency as illustrated in FIG. 13a when compared to FIG. 11. After 10 cycles, the endpoint of strain ε is around 0.12 for virgin HDPE dropping to 0.07 for HDPE annealed at 110°C, which was a surprising effect. The behaviour is improved, if HDPE is annealed at 120 degrees, where the endpoint of strain ε is around 0.05, or if HDPE is annealed at 130 degrees, where the endpoint of strain ε is below 0.03.

FIG. 16 illustrates the corresponding shows measurements for HDPE with 8 wt.-% of the modifed nanoclay in the nanocomposite. Likewise as in connection with the data of FIG. 15, 10 cycles of loading-retraction with the cross-head speed 10 mm/min were performed with a) Non-annealed HDPE+MMT, the maximum stress σ_max=17.9 and the minimum stress σ_min=2.3 (MPa), b) annealed T=I 10°C for 2 h, the maximum stress σ_max=19.6 and the minimum stress σ_min=2.2 (MPa), c) annealed at T=120°C for 2 h, the maximum stress σ_max=18.2 and the minimum stress σ_min=2.2 (MPa), d) annealed at T=130°C for 2 h, the maximum stress σ_max=19.6 and the minimum stress σ_min=2.4 (MPa).

When comparing data from non annealed HDPE + MMT in FIG. 16a with data from HDPE + MMT annealed at 110°C in 16b, almost no difference is observed for the endpoint at 0.03. Thus, annealing at 110⁰C seems not to lead to any improvement. Only a slight improvement is seen with annealing at 120°C, whereas an effect occurs at annealing with 130°C, where the endpoint is 0.02, a factor of 1.5 better than for annealed HDPE without MMT.

The corresponding engineering stress σ versus engineering strain ε in a tensile test with strain rate dε/dt=2- 10^"3 s^.are shown in FIG. 17 for a) Virgin HDPE, b) HDPE annealed T=I 10°C for 2 h, c) HDPE annealed at T=120°C for 2 h, d) HDPE annealed at T=130°C for 2 h, and in FIG. 18 for a) Non-annealed HDPE+MMT, b) HDPE+MMT annealed T=I 10°C for 2 h, c) HDPE+MMT annealed at T=120°C for 2 h, d) HDPE+MMT annealed at T=130°C for 2 h. Symbols are experimental data and solid line is a result of a theoretical simulation.

The following conclusions may be drawn: 1. The loading paths of the stress-strain curves for HDPE and HDPE/MMT nanocomposite demonstrate pronounced yield points. The apparent yield strain is reduced drastically by the annealing process, but more pronounced for HDPE than for HDPE with nanocomposites. Hybrid nanocomposite annealed at 13O⁰C demonstrates brittle fracture before the stress reaches the yield point.

2. The yield stress of HDPE and HDPE/MMT specimens strongly increases with temperature of annealing. The yield stress of nanocomposite exceeds that of neat polymer for each temperature of annealing.

3. The experimental stress-strain diagrams at cyclic deformation are strongly nonlinear both at loading and retraction.

4. The upper creep strain ε ₊ (the maximum strain per cycle) grows with number of cycles. This growth is the most pronounced for virgin HDPE. Annealing of HDPE results in a decrease in the rate of increase in ε ₊. For example, after 10 cycles of deformation, £^• ₊ = 0.121 for virgin HDPE, ε₊ = 0.069 for HDPE annealed at HO⁰C, ε₊ = 0.052 for HDPE annealed at

120⁰C and ε₊ = 0.028 for HDPE annealed at 13O⁰C (due to annealing, this parameter is reduced by 4.3 times).

5. The rate of growth of the upper creep strain of HDPE/MMT nanocomposite is reduced after annealing as well, but the decay occurs relatively weakly. After 10 cycles of deformation, ε₊ = 0.032 for virgin HDPE/MMT nanocomposite, ε₊ = 0.032 for nanocomposite annealed at HO⁰C, ε₊ = 0.025 for nanocomposite annealed at 12O⁰C and ε₊ - 0.018 for nanocomposite annealed at 13O⁰C (combination of reinforcement with nanoclay and annealing results in a decrease in the upper creep strain by 6.7 times). Numerical model

Our aim now is to derive stress-strain relations that can describe the experimental data depicted in the Figures 15 to 18. For the sake of generality, a constitutive model is developed for arbitrary three-dimensional cyclic deformations with small strains.

According to the conventional standpoint, a semicrystalline polymer is treated as a two-phase composite consisting of amorphous and crystalline domains. It is convenient to consider the crystalline phase as a skeleton composed of mutually connected spherulites. Each spherulite consists of a number of crystalline lamellae organized in a rather complicated manner. The amorphous phase is thought of as a polymer network in the rubbery state. This network is located between crystallites and between lamellae in spherulites.

Deformation of a semicrystalline polymer (relatively large in order to observe residual strains at unloading and relatively small in order to neglect changes in the degree of crystallinity) leads to mechanically-induced transformations in the amorphous and crystalline regions.

Transformations in the amorphous phase are observed as

1. orientation of macromolecules,

2. changes in the concentration of entanglements between chains (junctions of the polymer network),

3. formation and growth of micro-voids.

It is believed that transformations in the crystalline phase reflect

1. formation and motion of dislocations in crystallites,

2. inter-lamellar separation and shear,

3. rotation of lamellar stacks, 4. micro-necking of lamellae,

5. rearrangement of the spherulitic structure into the fiber structure, etc.

Interphase transformations include

1. slippage of chains along crystallites, 2. diffusion of micro-cavities from the amorphous phase into the crystalline phase,

3. creation of dislocations at the lamellae surfaces, etc.

A shortcoming of this list (where not all possible phenomena are included) is that it involves too many micro-mechanisms in order to formulate a physical model for plastic deformations of a semicrystalline polymer. Thus, we adopt a merely phenomenol- ogical approach, and treat a semicrystalline polymer as an isotropic homogeneous medium. This allows the number of material constants to be reduced compared to other (more physical) models, where the internal structure of crystalline and amorphous regions, as well as interactions between chains in the rubbery state and crystalline lamellae are taken into account.

To simplify the derivation, we accept the incompressibility condition, which reflects the fact that high-density polyethylene is a weakly compressible polymer (its Poisson's ratio υ is about 0.42).

At deformation of a viscoplastic medium with small strains, the strain tensor for macro-deformation ε is split into the sum of strain tensors for elastic, έ _e, and plastic, έ_v, deformations, ε = ε_e + ε_p . (1.1)

To ensure the incompressibility condition, we treat the tensors ε _e and ε _p as traceless. The rate-of-strain tensor for plastic deformation is assumed to be proportional to the rate-of-strain tensor for macro-deformation,

where t stands for time, and φ is a scalar function to be described in what follows.

The initial condition in Eq. (1.2) (where 0 denotes the zero tensor) expresses the fact that plastic strain vanishes at the initial instant.

The strain energy (per unit volume) of an incompressible medium is given by

W = →ε_e : ε_e , (1.3) where μ stands for an elastic modulus, and colon denotes convolution of tensors. Differentiating Eq. (1.3) with respect to time and using Eqs. (1.1) and (1.2), we find that dW (_Λ ,\_Λ dε ,_Λ ..

At isothermal deformation of an incompressible medium with small strains, the Clau- sius-Duhem inequality reads dW dp dt dt where Q stands for energy dissipation per unit time and unit volume, σ is the stress tensor, and prime denotes the deviatoric component of a tensor.

Inserting Eq. (1.4) into this equality and disregarding dissipation of energy, we arrive at the stress-strain relation σ = -pϊ + μ(l-φiβ-8_p), (1.5) where p is an unknown pressure, and 1 stands for the unit tensor.

Equations (1.2) and (1.5) are satisfied for an arbitrary deformation program ε{t) . These relations involve the function φ(t) that characterizes the viscoplastic response of a solid polymer. To describe this function, we distinguish the first loading path for a virgin specimen and all subsequent paths of retraction and reloading.

At first loading of a semicrystalline polymer, the coefficient φ in Eq. (1.2) is determined by the formula

where

is the intensity of elastic strain, and a and β are positive parameters. Our choice of the stretched exponential function (1.6) is explained by the following reasons:

1. Eq. (1.6) involves only two material constants, a and β , to be found by fitting observations.

2. This relation is rather flexible to ensure good agreement with observations for a number of solid polymers. 3. According to Eqs. (1.2) and (1.6), the rate of plastic strain equals zero at the initial instant, it monotonically increases with elastic deformation, and it coincides with the rate of strain for macro-deformation when the elastic strain becomes relatively large. All these conclusions appear to be physically plausible.

At all subsequent paths of reloading and retraction, the function φ{t) is governed by the first-order nonlinear evolution equation

% + B? - Λ&-, ,S(O)= O, (1.7) at at where A and B are positive quantities. Equation (1.7) is chosen for the following rea- sons:

1. It contains only two adjustable parameters A and B [the exponent β is the same measure of nonlinearity of the mechanical response as the corresponding parameter in Eq. (1.6)].

2. Eq. (1.7) involves the strain energy W as an input, which seems physically plausible: evolution of the rate of plastic strain is determined by the mechanical energy stored under deformation.

3. Formula (1.7) ensures that the current rate of plastic strain is affected by the entire history of deformation, but the memory about the deformation history decays algebraically (when β < 1 ) with time.

As we concentrate on cyclic deformation with a constant strain-rate intensity έ\ , it is convenient to combine Eqs. (1.1)-(1.3) and (1.7) and to re- write the result in the form

where

a = μA , b = γ-τ.

Following the concept of pseudo-elasticity, a and b in Eq. (1.8) are treated as constants along each path of a stress-strain diagram. This means that a and b remain constant during each subsequent retraction and reloading, but they alter their values at the instants when the strain rate changes its sign. To complete description of the model, it is necessary to determine

1. a = α° and b=b° for the first retraction path, that is, the quantities a and b at the first instant t when the strain rate changes its sign,

2. a = a_° and b = b°_ for the first reloading path, that is, the quantities a and b at the first instant t when the stress tensor σ reaches its "minimum value,"

3. a rule that allows the parameters α° , b₊ and a_° , έ>° to be transformed into the coefficients a and b in Eq. (1.8) for each subsequent cycle of deformation.

The following phenomenological relations are proposed for retraction:

and reloading: a = k_a_^o , b = k_b° . (1.10)

Equations (1.9) and (1.10) mean that the coefficients a and b in Eq. (1.8) are affected in the same way by deformation history. As a measure of the deformation history, we introduce the parameter

ΔJ_e = J_e -J_e° , where J_e is the current intensity of elastic strain when the strain rate changes its sign, and J° denotes the intensity of elastic strain at the instant when the first retraction (reloading) starts.

We postulate that evolution of Jc₊ with ΔJ_e is described by the zero-order kinetic equation

%- = -K, (1.11) dΔJ_e where K is an adjustable parameter. Integration of Eq. (1.11) results in k₊ = \ -KhJ_e . (1.12)

Strain-induced changes in the coefficient k_ are determined by the first-order kinetic equation

where κ > 0 is an adjustable parameter. Integration of Eq. (1.13) yields

Given a strain-rate intensity and parameters of a deformation program (e.g., σ_max and <τ_min for a stress-controlled uniaxial tensile test), constitutive equations (1.2), (1.5), (1.6), (1.8), (1.9), (1.10), (1.12) and (1.14) involve 9 adjustable parameters μ , a , β , al, bl, a_^o , b_°, K , κ .

This number is substantially lower than the number of material constants in other models for cyclic viscoplasticity of polymers.

Material parameters

Our aim now is two-fold:

1. to demonstrate that the constitutive equations correctly describe stress-strain diagrams at cyclic deformation depicted in Figures 15 and 16,

2. to show how the material parameters can be found one after another by matching each subsequent part (reloading and retraction) of a stress-strain curve.

Uniaxial deformation

We begin with simplification of the constitutive equations for uniaxial tension of a specimen. For uniaxial deformation of an incompressible medium, the strain tensor reads

S = S(A e₁ Q e₁ — (e- ® e₂ + e₃ ® e₃) , . (1.15)

where ε(i) stands for longitudinal engineering strain, e_k (k = 1, 2, 3) are unit vectors of a Cartesian coordinate frame, whose vector e_x coincides with the direction of de- formation, and ® denotes tensor product. We suppose that the plastic strain tensor ε_p can be presented in the form (1.15),

= ε_p(t) e, ® e, -- ₂ (we₂ ® - e-₂, +- e-,, ®- e-₃„) (1-16)

where ε P (t) is a function to be found, substitute Eqs. (1.15), (1.16) into Eqs. (1.2) and find that dp

-£ = φ, *,(0) = 0. (1.17)

Inserting expressions (1.15) and (1.16) into Eq. (1.5) and excluding the unknown pressure p from the boundary condition on the lateral surface of the specimen, we calculate the tensile stress, σ = E{l -φ)(ε -ε_p), (1.18)

3 where E - —μ stands for an analog of the Young's modulus. It follows from Eqs.

(1.6) and (1.15) that along the first loading path of the stress-strain diagram, the function φ reads φ = l-exp[-a(ε-ε_pY}. (^U9)

First loading

The parameters E , a and β in Eqs. (1.17)-(1.19) are found by matching the stress- strain curves depicted in Figures 17 and 18 with the help of the following algorithm. We fix some intervals [0,a_Q] and [θ,/?_o], where the best-fit parameters a and β are assumed to be located, and divide these intervals by the points a_p= iAa and β_j = jAβ with Aa = a_o /J and Aβ = β_o /j(i,j = l,...,J-l). For each pair ψ^β_j], Eqs. (1.17)-(1.19) are integrated numerically (by the Runge-Kutta method with the step Aε = 1.0 • lO^""5 ) from ε = 0 to ≤r, max '

We choose £_max = 0.4 for the data depicted in Figures 17a, 17, b, and 17c, and ^_max = 0.2 for the observations reported in Figure 17d. For the experimental stress- strain curve plotted in Figures 18a, 18b, 18c, and 18d, ε_max is determined by the last experimental point presented in these figures (this point corresponds to the beginning of necking of specimens).

The pre-factor E in Eq. (1.18) is found by the least-squares method from the condition of minimum of the functional

F

(1.20) where the sum is calculated over all points ε_n at which observations are reported, <τ_exp is the stress measured in an appropriate test, and σ_mm is given by Eq. (1.18). The best- fit parameters a and β are chosen from the condition of minimum of Eq. (1.20) on the set of pairs

After finding these quantities, the initial intervals [θ,o:₀] and [θ,β₀] are replaced with the new intervals [a -Aa,cc +Aa], \β - Δβ, β + Aβ] , and the above calculations are repeated. To ensure an acceptable quality of fitting observations, this procedure is repeated 3 times with J = IO. The best-fit material constants E , a and β are presented in Figures 20 to 22.

The following conclusions are drawn from these figures:

1. Reinforcement with MMT nanoclay and annealing induce a pronounced increase in the elastic modulus E . For example, the presence of nanofiller in a non-annealed HDPE results in the growth of its modulus by 33%. Combination of reinforcement and annealing leads to an increase in the modulus by 76%.

2. Annealing of neat HDPE does not affect practically the parameter a (except for annealing at the highest temperature T = 13O⁰C). On the contrary, reinforcement with nanoclay induces a substantial growth of this parameter.

3. The exponent β is weakly affected by reinforcement and annealing. For all samples (except for those annealed at the highest temperature), the exponent β of HDPE/MMT nanocomposite slightly exceeds that of neat HDPE.

It is worth noting that the modulus E is found by fitting the entire loading path of the stress-strain diagram. This implies that E does not necessary coincide with the Young's modulus determined by the conventional technique (a linear approximation of an initial part of the stress-strain curve).

First retraction It follows from Eqs. (1.15) and (1.16) that the evolution equation (1.8) at the first retraction reads & = a{l -φls -ε_p)+bφP . (1.21) dε

The initial condition for Eq. (1.21) refects the continuity condition for the function φ at the point where the strain rate changes its sign.

To find the coefficients a = Ω° and b = b₊° in Eq. (1.21), we match the first retraction paths of the stress-strain diagrams depicted in Figures 15 and 16. Each curve is approximated separately according to the following algorithm. We fix some intervals [θ,α_o] and [θ,b_o], where the best-fit values of a and b are assumed to be located, and divide these intervals by the points α_; = iAa and b_j = jAb with Aa

Eqs. (1.17), (1.18) and (1-21) are integrated numerically (by the Runge-Kutta method with the step Aε = 1.0 - 1(T⁵ from σ = σ_max to σ = σ_min . The best-fit parameters a and b are determined from the condition of minimum of functional (1.20) on the set of pairs ψ^b_j). When these values are found, the initial intervals [θ,α_o] and [θ,b_o] are replaced with the new intervals [α — Aa,a + Aa] and p — Ab,b + Ab\, and the above calculations are repeated. This procedure is repeated 3 times with J = 10. The best-fit values of α° and Z>° are presented in Figures 24 and 25.

The following conclusions are drawn from these figures:

1. The adjustable parameter strongly increases with temperature of anneal- ing both for neat HDPE and HDPE/MMT nanocomposite. The growth is monotonic for HDPE, whereas for the nanocomposite, annealing at 110 and 12O⁰C results in practically the same values of this parameter. For each program of thermal treatment, the coefficient α° of nanocomposite exceeds that of HDPE. 2. The adjustable parameter b\ grows with temperature of annealing both for neat HDPE and HDPE/MMT nanocomposite. The increase is stronger for HDPE and appears to be less pronounced for the nanocomposite. The parameter

of the nanocomposite substantially (by several times) exceeds that of neat HDPE. Second cycle of deformation

The quantities a = a_° and b = b° are determined by fitting the experimental data at the second cycle of deformation.

For the first reloading path of each stress-strain diagram, Eq. (1.8) reads

whereas for the second retraction path, the evolution equation is presented in the form (1.8) and (1.9),

& = k₊[ιl(l-φiε-s_p)₊by]. (1.23)

To find a = α° and b = b° , we approximate the observations depicted in Figures 15 and 16 with the help of the following algorithm. We fix some intervals [θ,α_o] and [θ,b_o], where the best-fit parameters a = a° and b = b°_ in Eq. (1.22) are assumed to be located, and an appropriate interval [θ,k_o] for the parameter k = Jc₊ in Eq. (1.23). These intervals are divided ^• by the points a_t = iAa , b_}. - jAb and k, = IAk with Aa = a_o /J,Ab = b_o /J and Ak = k₀ / j{i,j,l ~ l,..., J - 1) . For any triplet [a^b_j, k_t), the governing equations are integrated numerically (by the Runge-Kutta method with the step \Aε\ = 1.0 • 1 (T⁵ ). First, Eqs. (1.17), (1.18) and (1.22) are integrated from σ = σ_min to σ = σ_max with a = α_; and b = b_j . Afterwards, Eqs. (1.17), (1.18) and (1.23) are integrated from σ = σ_max to σ = σ_min with k₊ = k_l . The best-fit parameters a , b and k are determined from the condition of minimum of functional (1.20) on the set [a^b_j^k,]. Then, the initial intervals [θ,α₀], [θ,Z?_o] and [θ,A:_o] are replaced with the new intervals [a -Aa,a + Aa], [b -Ab,b + Ab\ and \k ~Ak,k + MJ, and the above calculations are repeated. This procedure is repeated 3 times with J = 10 for each stress-strain diagram separately. The quantities a_° and b° are reported in Figures 26 and 27.

The following conclusions are drawn from these figures:

1. The adjustable parameter α° strongly increases with temperature of annealing for neat HDPE. The influence of temperature of annealing on a° for HDPE/MMT nanocomposite appears to be non-monotonic. For each program of thermal treatment, the coefficient α° of nanocomposite exceeds that of HDPE.

2. The adjustable parameter έ>° grows with temperature of annealing both for neat HDPE and HDPE/MMT nanocomposite. For each program of thermal treatment, the coefficient b°_ of nanocomposite exceeds that of HDPE.

Other cycles of deformation

We proceed with fitting other cycles of deformation of the experimental stress-strain diagrams depicted in Figures 15 and 16. Each stress-strain curve is approximated separately. A loading-retraction path of a stress-strain diagram corresponding to the nth cycle of deformation (n = 3, ,10) is matched by using a version of the above algorithm with only two adjustable parameters k_ and k₊ , where k_ and k₊ are coefficients in Eqs. (1.8)-(1.10). According to these relations, the evolution equation for the function φ reads

for reloading and

for retraction.

To find the best-fit values of k_ and Ar₊ , we fix some intervals [0,A₀], where these parameters are assumed to be located, and divide these intervals by the points k_t = iAk and k_j = jΔk with Ak = k_Q /j(i,j = l,...,J-l) . For each pair {&_/,£,■}, Eqs. (1.17), (1.18) and (1.24) are integrated numerically from σ = σ_min to σ = σ_max with k_ = &,. and, afterwards, Eqs. (1.17), (1.18) and (1.25) are integrated from σ = σ_max to σ = σ_min with Ar₊ = k.. Numerical simulation is performed by the Runge-Kutta method with the step \Aε = 1.0 - 10^~5. The best-fit parameters k_ and k^~ ₊ are determined from the condition of minimum of functional (1.20) on the set {&,-, Ar₇/. When these quantities are found, the initial intervals are replaced with ψ_ - Δk,k_ + Ak] and [&₊ -M,&₊ + M], respectively, and the above calculations are repeated. This procedure is repeated 3 times with J = IO. The best-fit values of Jc₊ and k_ are plotted versus increment of elastic strain Aε_e [the difference between the current value of ε_e and its value

for the first retraction (re- loading)] in Figures 17 and 18. The experimental data are approximated by Eq. (1.12) and (1.14),

/c₊ = l - KAε _e , k_ = exp(- κAε_e ) , (1.26) where the coefficient K and K ^• are determined by the last-squares algorithm. The quantities K and K are reported in Figures 28 and 29.

The following conclusions are drawn from these figures:

1. Given a thermal program, the values of K and K for the nanocomposite exceed those for neat polymer.

2. The parameters K and K monotonically increase with temperature of an- nealing both for neat HDPE and HDPE/MMT nanocomposite.

3. The rate of growth of K and K is more pronounced for neat HDPE than for the nanocomposite.

General conclusions 1. Reinforcement of HDPE with MMT nanoclay and its annealing in the sub- melting region of temperatures induce substantial reduction in plastic creep.

2. This decrease in plastic creep may be associated with changes in crystalline morphology of HDPE driven by annealing.

3. An apparent similarity between the effects of reinforcement and annealing leads to a conclusion that the presence of clay platelets causes transformations of the crystalline morphology of HDPE analogous to those induced by thermal treatment.

4. A constitutive model is developed that allows experimental data in cyclic tensile tests to be adequately described. An algorithm is proposed that al- lows adjustable parameters in the stress-strain relations to be found with the help of a relatively simple procedure by fitting observations in uniaxial cyclic tensile tests. The constitutive equations may be applied for the analysis of cyclic deformations of nanocomposite structures with complicated geometry.

Claims

1. A product for a biomedical joint implant or bone implant, the product comprising nanocomposite material containing polymer and nano-material in the form of nano- platelets with a ratio between an average height of the platelets and an average width of the platelets of at least 1 :20.

2. A product according to claim 1, wherein the ratio is at least 1 :200.

3. A product according to any preceding claim, wherein the ratio is at least 1 :1000

4. A product according to any preceding claim, wherein the nano-material is nano- clay.

5. A product according to claim 4, wherein the clay is smectite type clay.

6. A product according to claim 5, wherein clay is Montmorillonite.

7. A product according to any preceding claim, wherein the polymer is a polyolefin.

8. A product according to claim 7, wherein the polymer is High Density Polyethylene.

9. A product according to claim 7, wherein the polymer is Ultra High Molecular Weight Poyethylene, Low Density Polyethylene or Linear Low Density Polyethylene.

10. A product according to any preceding claim, wherein the material of the product comprises a ceramic filler material.

11. A product according to any preceding claim, wherein the implant comprises the nanocomposite material as part or whole of its surface.

12. A product according to any preceding claim, wherein the implant comprises the nanocomposite material as part or whole of its bulk material.

13. A product according to any preceding claim, wherein the implant is part of a hip joint or a knee joint.

14. Method for minimising stress softening, creep or plastic deformation due to cyclic loads in a product for a biomedical joint implant or bone implant, the method comprising the steps of - providing a thermoplastic polymer

- mixing nano-material in the form of nano-platelets into the polymer, the nano platelets having a ratio between the average height of the platelets and the average width of the platelets of at least 1:20,

- heat moulding the mixture.

15. Method according to claim 14, wherein ratio between the average height of the platelets and an average width of the platelets is at least 1 :200 and the average height of the platelets is in the range of between one nanometer and a few nanometers.

16. Method according to claim 15, wherein the method comprises annealing the moulded product at a temperature of more than 100°C for a predetermined time.

17. Method according to claim 16, wherein the predetermined time is in the range of one hour or longer.

18. Method according to claim 16 or 17, wherein the temperature is at least 110°C.

19. Method according to claim 18, wherein the polymer is HDPE.

20. Method according to any one of the claims 14-19, wherein the nano-material is nanoclay.

21. Method according to claim 20, wherein the polymer is HDPE, the nano-material is Montmorrillonite, the nanoclay is exfoliated with a surfactant before moulding, the moulded product is annealed at a temperature in the order of magnitude of 130°C for more than one hour.

22. Method according to any one of the claims 14-21, wherein the method comprises - predicting a range of magnitude of load on the implant,

- determining an acceptable long term deformation of the implant due to this load,

- selecting a nanocomposite material comprising nano-material and a polymer for at least part of the structure of the implant,

- identifying the strain in different parts of the implant in dependence of the load stress,

- determining that part or those parts of the implant that are to be composed of the nanocomposite,

- determining whether the selected nanocomposite when incorporated in the implant can withstand the load with a long term deformation of the implant less than, or equal to the acceptable deformation,

- incorporating the nanocomposite material in the implant.

23. Method according to claim 22, wherein the determined part or parts comprise the surface of the product.

24. Method according to claim 22 or 24, wherein the determined part or parts include the bulk material of the product.

25. Method according to claim 22, 23, or 24, wherein the product is a biomedical implant.

26. Method according to claim 25, wherein the product is a biomedical joint implant.

27. Use of a product for a biomedical joint implant or bone implant according to any claim 1-13 or a method according to any claim 14-26 for resistance against stress softening, creep or for resistance against total plastic deformation due to cyclic loads