The difficulty in healing structural damage is that most existing schemes need external help to bring the fractured surfaces in contact before healing can occur. To facilitate the existing schemes to heal macroscopic cracks, we envision that the cracked surfaces can be brought in contact through constrained shape recovery of a shape memory polymer (SMP) fibre-reinforced grid skeleton that is embedded in thermoset polymer matrix, similar to stitch a cut in the human skin by suture. In this study, we show that polyurethane SMP fibres can be hardened through cyclic cold-drawing programming, which makes them suitable for reinforcement and healing in thermoset polymer composites. We characterized the microstructure of the SMP fibres, which provides fundamental understanding of the effect of programming on the degree of crystallinity and molecular orientation. Then, a micromechanical multiscale viscoplastic theory is developed to predict the thermomechanical behaviours of the SMP fibres, including the cyclic hardening and stress recovery responses. The proposed theory takes into account the stress-induced crystallization process and the evolution of the morphological texture based on the applied stresses. The cyclic loading and the thermomechanical responses of the SMP fibres confirm the capabilities of the proposed model in capturing these phenomena.
Damage healing in thermoset polymer composite structures has become a popular topic recently (Voyiadjis et al. 2011). Several healing schemes have been reported in the literature primarily for healing microcracks with narrow opening, including incorporation of external healing agents such as liquid healing agent by microcapsules (White et al. 2001), hollow fibres (Pang & Bond 2005) and microvascular networks (Toohey et al. 2007), and solid healing agent such as embedded thermoplastic particles (TPs; Zako & Takano 1999; Voyiadjis et al. 2012a). Some polymers such as ionomers (Plaisted & Nemat-Nasser 2007; Varley & van der Zwaag 2008) and thermally reversible covalent bonds (TRCBs; Liu & Chen 2007) by themselves possess healing capabilities. A combination of microcapsule and shape memory alloy (SMA) wire has also been studied (Kirkby et al. 2009). Because some damages, such as impact damages, are usually in the structural-length scale, the main challenge to heal these macrocracks remains the closure of such a wide opening. The existing healing systems are unable to very effectively heal macroscopic cracks. For instance, in the case of the microencapsulated liquid healing agent, a large amount of healing agent is needed in order to heal macrocracks. However, incorporation of a large amount of healing agent will significantly alter the physical/mechanical properties of the host structure. In addition, large capsules/thick hollow fibres themselves may become potential defects when the encased healing agent is released. For ionomers and TRCB polymers, they need external help to bring the fractured surfaces in contact before chemical bonds can be established. While Kirkby et al. (2009) proposed a very smart idea, one limitation is that the SMA is expensive, heavy weight and most critically, its recovery force cannot be effectively transferred because the polymer matrix becomes soft at the SMA recovery temperature while the SMA is very stiff. In addition, the ‘run-off’ of the liquid monomer in wide-opened crack is another challenge before polymerization occurs. Therefore, the grand challenge facing the scientific community is how to heal structural-length scale damage such as impact damage repeatedly, efficiently and molecularly.
Recently, a biomimetic healing scheme has been proposed and validated to repeatedly and molecularly heal macroscopic crack (Li & Uppu 2010; Nji & Li 2010a; Xu & Li 2010). It uses the confined shape recovery of the shape memory polymer (SMP) matrix for the purpose of sealing or closing cracks and the incorporated TPs for molecular length-scale healing. SMP is a fast-growing area. The shape memory (SM) mechanisms, applications and latest advancements in SMPs can be found in several recent review papers, see for example Behl & Lendlein (2007), Ratna (2009), Huang et al. (2010), Leng et al. (2011) and Ratna & Karger-Kocsis (2011). The key for the success of this self-healing scheme is that the volume of the SMP matrix must be reduced during programming and external confinement must be provided during shape recovery (Li & Uppu 2010). It has been proved that volume reduction can be realized through compression programming (Li & Nettles 2010; Li & Uppu 2010) and external confinement can be provided through architectural design of composite structures such as grid-stiffened SMP-cored sandwich (John & Li 2010), three-dimensional woven-fabric-reinforced SMP composite (Nji & Li 2010b), or even the sandwich skin (Li & John 2008). In order to speed up the programming process, cold-compression programming of thermoset SMP has also been proposed, tested and modelled (Li & Xu 2011). However, this bio-inspired scheme cannot be extended to healing of conventional thermoset polymers because regular thermoset polymers do not have the SM capabilities.
In order to repeatedly and molecularly heal macroscopic cracks in conventional thermoset polymers, we here propose a new biomimetic scheme: an SMP fibre z-pinned, continuous SMP-fibre-reinforced polymer grid skeleton that is filled in with conventional thermosetting polymer dispersed with TPs; see a unit cell representation in figure 1. We envision that the proposed composite will work similar to the two-step healing of human skin: close then heal, i.e. close the wound by bleeding and clotting or surgery (suture or sew the skin together) before new cells gradually grow. Particularly, we propose to use the constrained shape recovery of SMP ribs and z-pins for the purpose of narrowing/closing the macroscopic crack (step 1) and molten TPs for healing molecularly (step 2). The basic idea is that the SMP fibres are strain-hardened through cold-drawing programming before fabrication. When an impact is identified (by surface indentation, portable non-destructive testing such as ultrasound, etc.), localized heating surrounding the impact damaged bay(s) or cell(s) will be conducted using contact or non-contact heating such as infrared light. Once the local temperature in the SMP ribs and z-pins is higher than the transition temperature, the SMP ribs and z-pins surrounding the damaged bay(s) remember their original shape and tend to shrink. Owing to the elastic constraints by the neighbouring ‘cold’ bays (which have high stiffness), the shrinkage of the SMP ribs and z-pins is not free. A three-dimensional compressive force will be applied to the boundary of the damaged bay(s), leading to narrowing or closing of the cracks with wide opening. This is step 1 of the biomimetic scheme. Further heating leads to melting of the embedded TPs, and the molten thermoplastic will be sucked into the narrowed crack through capillary force and diffused into the fractured thermoset polymer matrix owing to concentration gradient and recovery compressive force. When cooling down, the thermoplastic hardens and glues the crack molecularly. This completes one molecular damage-healing cycle.
It is noted that the wide-opened crack must first be narrowed or closed by the shape recovery of the embedded SMP fibres in order for the capillary force to take effect; in addition, the thermoplastic must be miscible with the thermosetting matrix in order for diffusion to occur. It is also noted that one potential problem of using TPs is that the molten thermoplastic may diffuse onto the surface during the healing process. It is found that by ensuring a certain chemical miscibility between the TPs and the thermosetting polymer matrix, which can be evaluated through a dynamic mechanical analysis (DMA) test, this problem can be minimized (Nji & Li 2010a).
As indicated by Li & Uppu (2010), each constrained shape recovery process also represents a new training cycle to the SMP ribs and z-pins, suggesting that SMP fibres only need to be programmed one time before fabrication. Subsequent programming will be autonomous by coupling with shape recovery (healing) of the composite. Together with the fact that the TPs can also be repeatedly melted and hardened, the damage-healing cycle is repeatable. The working principle can be visualized by figure 1. It is noted that while the bio-inspired healing scheme is cited as two steps, it actually just needs one step in practice—heating up all the way to the melting temperature of the TPs; the SMP ribs and z-pins will shrink during the course of heating. In figure 1, Tg denotes the glass transition temperature of the SMP fibre; Tm represents the melting temperature of TPs; Tgp indicates the glass transition temperature of the thermoset polymer matrix; and Tc denotes the curing temperature of the thermoset polymer matrix. The basic requirement for the designed healing system is: Tgp>Tm>Tg>Tc (colour change visualizes temperature change in the figure).
It is noted that, previously, we have demonstrated that orthogrid- or isogrid-stiffened syntactic-foam-cored sandwiches have a considerably higher impact tolerance and residual load-carrying capacity than laminated composites with the same fibre volume fraction (Li & Muthyala 2008; Li & Chakka 2010). We have also demonstrated that this type of grid-stiffened SMP-based syntactic-foam-cored sandwich can inherently provide the required external confinement and facilitate the confined shape recovery for self-healing (John & Li 2010). Because of this, it is believed that grid-stiffened sandwich structures can be used in large structures such as plates or shells. In other words, the SMP-grid-stiffened sandwich proposed in this study has a potential to be used in large structures. It is believed that such structures can be optimized through mathematical modelling so that the grid pattern, rib thickness, bay area, etc., can be optimized (Li & Cheng 2007). Because localized heating is a key step to trigger the biomimetic healing process in actual structures, the temperature needs to be carefully monitored and controlled, which may need heat-transfer analysis of the structures and surface temperature measurement using devices such as thermocouple or infrared camera.
In order to validate the healing scheme in figure 1, a finite element analysis was conducted on an SMP-orthogrid-stiffened thermoset polymer composite; see a schematic in figure 2a. The central bay in figure 2a contains a macroscale crack which is an ellipsoidal hole with major diameter of 20 mm, minor diameter of 5 mm and height of 5 mm. Heating is conducted locally on the central bay and its surrounding four ribs. The following parameters were assumed: the SMP-fibre-reinforced rib has a modulus of 600 MPa and maximum recovery stress of 10 MPa, and the polymer composite in the bay has a modulus of 1000 MPa. It is believed that, through localized heating, the elastic modulus of the thermosetting matrix is reduced. On the other hand, the programmed SMP fibre exhibits excellent thermal stability. As shown in §2, the elastic modulus of the SMP fibre does not vary significantly within the temperature range investigated. Hence, the macrocrack in the softened thermosetting matrix can be narrowed or closed by the shrinkage of the SMP fibre at the healing temperature.
Solid tetrahedral elements with 10 nodes are used. Once the localized recovery stresses overcome the stiffness of the bay, the walls of the crack are collapsed together, as shown in the magnified view in figure 2b. In addition, the displacement field in y-direction is shown in figure 2b, which confirms that the walls of the crack march towards each other owing to the applied localized stresses from the SMP ribs.
It is noted that when compared with the previous biomimetic healing scheme proposed by Li & Uppu (2010), the scheme illustrated in figure 1 has fundamental differences and is considered to be more realistic and feasible. In the previous studies, the purpose is to heal macrocracks in the SMP matrix through confined shape recovery of the SMP matrix (Li & Uppu 2010; Nji & Li 2010a; Xu & Li 2010). However, in practice, conventional thermosetting polymers, such as epoxy, vinyl ester, polyester, etc., which do not have SM capability, are usually used as the matrix in load-bearing structures, such as in fibre-reinforced polymer composite structures. Therefore, it is more desired to endow conventional thermosetting polymer matrix with self-healing capability so that a large variety of engineering structures can be benefited. In this study, we aim at using a small amount of SMP fibres to bestow conventional thermosetting polymer matrix with self-healing capability, which can close macrocracks in the polymer matrix through constrained shape recovery (shrinkage) of the embedded cold-drawn SMP fibres, as discussed in figure 1. Therefore, it is believed that this study opens up new opportunities to heal conventional thermosetting polymer composite structures by using the proposed two-step biomimetic self-healing scheme. The comparisons of the proposed SMP-fibre-based self-healing scheme with the previous SMP-matrix-based self-sealing scheme are summarized in table 1.
Currently, SMP fibres are primarily made of thermoplastic SMPs, particularly polyurethane for non-structural applications (Meng & Hu 2008). The polyurethane semicrystalline SMP fibres are constituted from the crystalline hard phase and amorphous soft phase and they show excellent solution ability, melting, diffusion, processability and repeatability of the SM cycle (Ping et al. 2005). Soft segment may consist of the amorphous (e.g. polyester and polyether) or the semicrystalline (e.g. poly(ϵ-caprolactone), PCL) structures while the hard segments (e.g. diisocyanate (TDI), aromatic urethane or aramid) may be dispersed over the soft segment to form thermally stable chemical or physical cross-links. Existence of the distinguishable hard phases in the polyurethane elastomers provides the physical basis for a micromechanics approach towards the multiscale analysis of these materials (Clough & Schneider 1968; Clough et al. 1968; Shojaei & Li submitted). The stress-induced crystallization (SIC) process in the semicrystalline polymers has been well studied in the literature (Bowden & Young 1974; Guo & Narh 2002). Accordingly during this process, the spherulite morphology is changed upon stretching, and the crystalline molecular chains align in the direction of the applied macroscopic loads. In the case of polyurethane SMP fibres, this process results in the enhanced mechanical properties along the fibre direction.
In this work, we will experimentally investigate the strain hardening of polyurethane fibres through cold-drawing programming, in order to achieve the required recovery stress in figure 2. We will then investigate the thermomechanical cycle (programming and shape recovery) of the strain-hardened SMP fibres. Microstructure change and anisotropic behaviour owing to programming will be examined by polarized optical microscope, Fourier transform infrared spectroscopy (FTIR), small-angle X-ray scattering (SAXS) and DMA. After that, a representative volume element (RVE) is used to correlate the microstructure of the SMP fibre with the macroscopic loading conditions. The soft and hard segments are assumed to follow, respectively, the amorphous and crystalline constitutive relations. The well-established micromechanics averaging techniques are then incorporated to average the micro-stress and micro-strain fields in these sub-phases and the macroscale mechanical response of the SMP fibre is then estimated. This approach was proposed formerly by Eshelby and later it has been developed to the Mori-Tanaka and self-consistent methods (Eshelby 1957; Nemat-Nasser & Hori 1993). The local–global relations between the applied macroscale and the resulting microscale mechanical fields can be established analytically when the medium behaves elastically. Once the nonlinearity is introduced in one of the sub-phases, a history-dependent solution algorithm should be enforced in order to update these relations. Shojaei & Li (submitted) discussed in detail the solution algorithm for such a multiscale analysis incrementally. In §2, experimental characterizations of the SMP fibres are discussed. In §3, the constitutive relations for the amorphous and crystalline phases together with the texture updates are elaborated. Furthermore, the constitutive relations for the stress recovery and the SIC process are proposed. In §4, the kinematics of the finite deformation together with the required numerical algorithms are elaborated. In §5, simulation results are presented.
2. Experimental characterization of shape memory polymer fibres
In this study, polyurethane was synthesized from poly(butylene adipate)-600 (Mn) (PBA), 4′4-diphenylmethane diisocyanate (MDI) and 1,4-butanediol (BDO). On average, the molar ratio of (MDI+BDO):PBA=3:1. The average formula weight ratio of (MDI+BDO):PBA=1021:650. The hard segment, soft segment and their contents were selected to prepare polyurethane with amorphous soft segment phase and crystalline hard segment phase. The polyurethane fibre was spun by melt spinning. The fibre passed three pairs of rollers with the same rotation speed before being wound up. In this work, two types of single SMP fibres (filaments) are characterized in which their microstructure changes upon cold-drawing process are evaluated. Figure 3 shows the optical microscopy picture for a single SMP fibre with an initial diameter of 0.04 mm, which is called sample no. 1 hereinafter. The non-stretched SMP fibre no. 1 is shown in figure 3a in which the microstructure of the fibre is almost random; when the fibre is highly stretched, the microstructure aligns along the fibre (loading) direction (figure 3b). Figure 4 shows similar pictures for the SMP fibre with an initial diameter of 0.002 mm, which is denoted as sample no. 2 hereinafter. The oriented microstructures along the fibre direction are obvious when the fibre is cold drawn as shown in figure 4b. The samples no. 1 and no. 2 are cyclically stretched up to 350 per cent and 200 per cent level of strains before taking the pictures in figures 3b and 4b, respectively.
Using an MTS Alliance RT/5 machine, which is specified for fibre tension tests, the SMP fibre no. 1 and fibre no. 2 are cyclically stretched, and the results are shown in figure 5a,b, respectively. The elastic stiffness of the SMP fibre is increased gradually, whereas the strength of the fibre is increased significantly. Obviously, cyclic cold-drawing leads to strain hardening of the SMP fibres. The reason for this is the alignment of the amorphous phase and crystalline phase along the loading direction upon cold-tension, as evidenced in figures 3 and 4.
The fully constrained (zero strain) stress recovery response of these fibres with respect to time is depicted in figure 6. In this process, the programmed fibre is gripped by the fixture while the heating process is controlled by a digital furnace. An initial stretch within the elastic region, e.g. ϵ=10 per cent, is applied to ensure that the fibres remain zero strain before the shape recovery process starts, which needs a temperature-rising process and causes thermal expansion and thus looseness of the fibre between the grips. It is worthwhile to note that this pre-stretch is determined in a trial-and-error process in which the stress recovery is set to start from almost zero stress. In fact, both of these fibres show a sudden stress relaxation before activation of the shape recovery process. In other words, the pre-tension is fully consumed by the stress relaxation and thermal expansion before the stress recovery starts. Therefore, the pre-stretch does not affect the stress recovery. In this study, the heating rate is 0.35°C s−1 and the final temperature is set to be 90°C in the digital furnace. From figure 6, the pre-tension stress is relaxed to zero at about 20 s. Because the starting temperature of the fibre is about 23°C, the temperature becomes 30°C after 20 s of heating. As shown in figure 7, this is the temperature when the glass transition starts. Therefore, as expected, the 10 per cent pre-tension is fully used up by stress relaxation and thermal expansion of the fibre before stress recovery starts. The pre-stretch is a technique to compensate for the inability for the machine to measure the stress when the fibre is loose. From figure 6, the stabilized recovery stress is about 16 MPa. On the basis of figure 2, it is seen that the fibre after cyclic cold-drawing programming is able to provide the required recovery stress (10 MPa in figure 2) for closing the macroscopic crack. We also tested the stress recovery of non-stretched (as received) SMP fibre using the same procedure. The stress recovery is almost zero.
Figure 7 shows the DMA experiments (with heating rate of 5°C min−1 and frequency of 1 Hz) in which both the non-stretched and cold-drawn SMP fibres no. 1 are tested to investigate the changes in their glass transition temperature, Tg, upon work-hardening process. From figure 7a, the cold-drawn SMP fibre shows a small shift in its Tg towards higher temperature. The glass transition starts at about 30°C, which echoes the claim in figure 6. From figure 7b, it is clear that the storage modulus of the strain-hardened SMP fibres is much higher than that of the as received counterparts, and plateaus in a wide range of temperature, suggesting a significant increase in stiffness and thermal stability.
In order to better understand the SM mechanism owing to strain hardening by cold-drawing, the change in the microstructure of the SMP fibre is further investigated. Using TENSOR 27, the FTIR test is implemented on two samples to check the microstructural changes during cold-stretching process of SMP fibres. Figure 8 shows the FTIR test results for the sample no. 2 in which the blue line shows the non-stretched SMP fibre no. 2, and the red line represents the SMP fibre no. 2 which is stretched up to 180 per cent strain level prior to the FTIR test. The synchronized peaks confirm that the chemical compositions of the two fibres are the same, and there are no new chemical bonds upon cold-drawing. The change in the intensity after strain hardening by cold-drawing programming may indicate the change in density and molecular alignment.
Using the SAXS facilities in the Center for Advanced Microstructures and Devices at Louisiana State University, the microstructure of the non-stretched and stretched SMP fibres (sample no. 1) is investigated. As shown in figure 9a, the SAXS image for the non-stretched fibre shows a non-oriented microstructure, whereas in the case of the stretched fibre, the SAXS image shows orientational changes in the microstructure in figure 9b.
The crystallization process in polymers may occur during the cooling-down process from the melting point along the direction of the largest temperature gradient or may occur by external loading which is called SIC. In many cases, the crystallization process does not develop fully and consequently the resulting microstructure contains dispersed crystalline phases within an amorphous matrix, leading to a polymeric system that is called semicrystalline.
On the basis of the above test results and microstructural examination, it is evident that both the soft segments and hard segments in the semicrystalline SMP fibre align along the loading direction after cyclic cold-drawing programming, which also leads to considerable increase in stress recovery. This suggests that the polyurethane fibres, after strain hardening through cold-drawing programming, may have a potential to be used in load-bearing structural applications, particularly in the biomimetic healing scheme proposed in figure 1. However, an in-depth understanding of the thermomechanical behaviour of the semicrystalline polyurethane fibre needs constitutive modelling.
3. Constitutive behaviours of semicrystalline shape memory polymer fibres
The proposed viscoplastic theory in this work considers the governing relations for each of the individual micro-constituents, and establishes the microscale state of the stress and strain in each of the sub-phases. These microscale fields are then averaged through the micromechanics framework to demonstrate the macroscale constitutive mechanical behaviours. This multiscale approach incorporates more realistic material inputs when compared with the pure phenomenological models. Speaking in general, the individual micro-constituents mechanical behaviours may vary when they are packed in a multiphase material system, and a certain deviation in their mechanical responses may exist between the individual and their assembled configurations. In the following, two well-established viscoplastic theories for the amorphous and crystalline polymers are presented. These theories are based on certain physical description of the viscoplastic deformation mechanisms in the glassy polymers. It is noted that the performances of these two viscoplastic theories are well-established in the literature for both the amorphous and crystalline polymers. In this work, these two viscoplastic theories are linked together, within the micromechanics framework, in order to develop a physically consistent framework for semicrystalline polymers and it is applied to the viscoplastic deformation of the semicrystalline SMP fibres.
(a) Amorphous phase constitutive relation
The well-established Boyce model for the inelastic deformation of the amorphous glassy polymers is assumed to be held for the amorphous phase of the semicrystalline SMP fibre. This theory is formulated based on the underlying deformation mechanism of amorphous polymers, previously proposed by Argon (1973), which considers a cubic unit cell consisting of eight polymeric chains. It is proved that when compared with other models such as three-chain model or five-chain model, the mechanical responses of this eight-chain unit cell to external loading represent the global deformation better (Arruda & Boyce 1993a,b). In comparison with phenomenological models, the main advantage of the Boyce model is that it incorporates physical parameters which are measurable from simple mechanical testing. The plastic multiplier in this model is introduced as follows (Boyce et al. 1989): 3.1where a is a material parameter, p is pressure and s is the athermal shear strength. The material parameter A and the evolution law for s are defined as follows: 3.2where ω and are material parameters, and k is the Boltzmann's constant (Argon 1973). The material parameter h shows the rate of the strain softening and sss represents the asymptotic preferred structure. The initial value of s for the annealed material is s0=0.077μ/(1−ν), where μ is the elastic shear modulus and ν is the Poisson's ratio.
The material constant , in equation (3.1), is called amorphous pre-exponential inelastic strain rate, and is the effective equivalent inelastic deformation rate of a glassy polymer subjected to the effective equivalent shear stress, |τ|, which is defined at the absolute temperature, θ, as follows: 3.3where in which sij=σij−1/3σkkδij is the deviatoric Cauchy stress and Xij=αij−1/3αkkδij is the deviatoric back stress tensor. The back stress tensor, αij, is defined as follows (Arruda & Boyce 1993a,b; Arruda et al. 1993): 3.4where n is the number of chains per unit volume, k is the Boltzmann's constant and λL is the limit of chain extensibility and ζ is a viscoplastic-related material constant which controls the magnitude of the hardening with respect to the inelastic stretches (Ahzi et al. 1995; Shojaei & Li submitted). In limit analysis, Langevin function, , is used extensively and it imposes a limiting case in the evolution of the back stress tensor. Equation (3.3) represents the magnitude of the amorphous inelastic strain rate; whereas the direction of the amorphous inelastic flow rate, , is governed by the deviatoric driving stress, . The following flow rule is then proposed for the inelastic deformation in the amorphous phase (Argon 1973): 3.5
(b) Crystalline phase constitutive relation
Inelastic deformation of the crystalline phase in semicrystalline polymers includes three different mechanisms: (i) crystallographic slip, (ii) twining and (iii) Martensite transformations (Stevenson 1995). In this work, the slippage mechanism is assumed to be the dominant influencing mechanism. A reference vector, ct, is aligned with the crystallographic texture to show its evolution with deformation while two active slip mechanisms in the polymeric slippage system have been taken into account which are (i) chain slip: the burgers vector is aligned with ct and (b) transverse slip: the burgers vector is perpendicular to ct (Ahzi et al. 1995; Shojaei & Li submitted). There are only four linearly independent crystalline slip systems, which are indicated by the unit vectors in the direction of the slip and normal to the slip planes, including (i) chain slip: (100) and (010), and (ii) transverse slip: (100) and (010) (Lee et al. 1993; Shojaei & Li submitted). Vector denotes the slip direction and represents the unit normal vector to the slip plane, where α=1–4 is the number of the slip systems. The inelastic crystalline stretch rate tensor, , is introduced by the following relation (Lee et al. 1993): 3.6where is the symmetric part of the Schmid tensor as defined by: , and it represents the αth crystalline slip plane and the shear rate, , is defined as follows (Asaro 1979; Asaro & Needleman 1985; Hutchinson 1976; Lee et al. 1993): 3.7where is the crystalline reference inelastic strain rate and g(α) is the shear strength for the αth slip system and nc is the rate sensitivity factor. The effective shear stress, τ(α), at the αth slippage system, is given by (Shojaei & Li submitted): 3.8where is the projected deviatoric Cauchy stress, i.e. sij=σij−1/3σkkδij, in the direction of the deviatoric part of the dyadic cicj, i.e. , where Iij is the unity second-order tensor. This constraint is enforced based on the inextensibility of the crystalline chain together with the incompressibility assumption (Ahzi et al. 1995). The lattice spin which controls the rate of change of the direction of ci is introduced as follows (Asaro & Rice 1977; Lee et al. 1993): 3.9where is the skew part of the Schmid tensor and is the skew part of the velocity gradient in which .
(c) Cyclic texture update
It is experimentally confirmed that the amorphous and crystalline polymers undergo morphological texture changes in their polymeric networks upon stretching (Arruda & Boyce 1993a,b; Arruda et al. 1993; Lee et al. 1995). Then, the strain-hardening phenomenon in the SMP fibre is influenced by the texture changes in the crystalline and amorphous phases. In the following, the governing relations for texture updates in each of these sub-phases are brought forward.
(i) Crystalline phase texture update
The crystallographic texture is updated based on the applied lattice spins as discussed in equation (3.8) and it is expressed in the following form (Asaro & Rice 1977): 3.10where 3.11On the basis of Cayley–Hamilton expression for exponential term, one may find (Ahzi et al. 1995; Shojaei & Li submitted): 3.12with .
(ii) Amorphous phase texture update
In the case of amorphous phase, one may only take into account the influence of the inelastic deformation on molecular chain rotations and the subsequent strain-hardening effects. Boyce et al. relate the initial values of the back stress tensor, αij, athermal shear resistance, s, network stretch vector, Λi and residual stress tensor, σij, to the strain-hardening effect in the amorphous phase and it is shown that setting these parameters can effectively monitor the hardening phenomenon in the amorphous phase (Boyce et al. 1989; Dupaix & Boyce 2007).
(d) Recoverable stresses
One of the vital parameters in designing the bio-inspired close–then–heal healing systems is the available amount of the crack closure force which is necessary to close the macroscale cracks. As shown in figure 6, the SMP fibres show considerable recovery stresses which are, in general, dependent on the amount of the induced strain hardening during the cold-drawing process. Basically upon the cold-drawing process, the crystalline phase of the semicrystalline polyurethane SMP fibres undergoes the SIC process, and it stores the applied deformations through the entropy changes in the crystalline network. Once the temperature exceeds the glass transition temperature of the semicrystalline polymer, the viscosity of the polymeric network drops, and the frozen crystalline network is allowed to release the stored energy and achieve its minimum energy level. In other words upon heating, the stored energy in the crystalline phase is released, and the polymeric network returns to its minimum level of internal energy. As shown in figure 6 after a few cold-drawing cycles, the amount of the recoverable stresses is considerably aggrandized. Then, one may relate the amount of the recoverable stresses to the loading history and SIC process. In this study, the recoverable stress is assumed to be a function of the SIC process and the accumulated inelastic strains in the amorphous phase. Basically, a portion of these induced inelastic entropic and energetic changes in the SMP molecular network is recoverable upon heating where the viscosity of the frozen network drops, and the recoverable inelastic strains are restored. Then, the proposed evolution law for the stress recovery takes into account the history of the loading, including the inelastic strains in the sub-phases. The kinematic and isotropic hardening relations in the classical continuum plasticity context provide a suitable governing equation form to explain the stress recovery process (Chaboche 1991; Shojaei et al. 2010; Voyiadjis et al. 2012b). The stress recovery evolution relation is then proposed as follows: 3.13where η is a material constant that controls the rate of saturation of the recovery stress to its final value which is R, and T and Troom are, respectively, elevated and room temperatures. Parameter R takes into account the history of the loading which includes the inelastic deformation, texture updates and residual stresses owing to the cyclic hardening. Taking time derivative of equation (3.12) results in the following incremental relation for the stress recovery computations: 3.14where shows the rate of the heating process. As shown in figure 6, the maximum stress recovery is achieved after reaching Tg, and the rate of heating controls this peak time. The heating process is controlled by time integration of the heating rate as: , and once the temperature reaches its final value during the simulation, the heating rate is set to zero. The saturation limit, R, for the stress recovery is related to the loading history and the amount of the plastic strain by the following expression: 3.15where ω and Ξ are two material parameters for controlling the saturation parameter, R, and is the equivalent plastic strain rate for the crystalline phase. Then during each of the cyclic loadings, the magnitude of R is updated incrementally and its final value is introduced in equation (3.12) for the stress recovery computation. Equation (3.14) represents a monotonically increasing value for the parameter R up to a certain saturation limit which is enforced by Ξ. The underlying physics for equations (3.12) and (3.14) is that the recoverable stress in an SMP fibre is a function of the recoverable microstructural changes during the cold-drawing process and this stress recovery should saturates to a certain limit owing to the limit in reversibility in these microstructural changes. In other words, certain amounts of the microstructural changes are reversible in an SMP fibre; and after certain limit, these microstructural changes may result in failure of the polymeric networks and produce non-reversible defects. In this study, it is assumed that the recovery stress saturates to a certain limit as a function of the microstructural changes through equations (3.12) and (3.14), and the physical parameter to control these changes is the accumulated inelastic strain and its rate.
(e) Stress-induced crystallization
The SIC is a phenomenon which is experimentally investigated by many researchers (Bowden & Raha 1970; Bowden & Young 1974). Here, a phenomenological constitutive relation for the SIC process is introduced. Figure 10a shows the RVE of the microstructure of the manufactured SMP fibre where a thin layer of initially formed crystalline phase is parallelized with the amorphous phase. Figure 10b shows the enlarged cross section of the crystalline phase owing to the applied stretches. Then, one may assume that the microstructural changes are governed by the crystalline phase formation upon stretching. In order to take into account this fact in the governing relations of the semicrystalline polymers, one may propose the following relations which relate the initial inner, and outer, , radii, and the final inner, and outer, , radii of the crystalline phase to the magnitude of the inelastic strains as follows: 3.16where q and q′ are material parameters and is the effective accumulated crystalline plastic strains and the outer, roc, and inner, ric, crystalline radii start from their initial values and they converge to their final values. Equation (3.16) provides a smooth transition between highly amorphous to a highly crystalline microstructure while the history of loading is incorporated by formulating this process based on the inelastic strains. Then, crystalline inner and outer radii change rates are given by the time derivation of equation (3.16) as follows: 3.17The volume fraction of the crystalline, ccry, and amorphous phases is then given by the following relations: 3.18where ϵz is the macroscopic strain in the fibre direction and and are, respectively, the crystalline and amorphous phases Poison's ratio. The fibre is assumed to be transversely isotropic, i.e. .
It is experimentally confirmed in §2 that the crystalline phase of the semicrystalline polyurethane in an SMP fibre is aligned with the loading direction after a few rounds of cyclic hardening. Due to the fact that the manufactured fibres are assumed to be transversely isotropic, one may assume that the amorphous and crystalline phases in the SMP fibres are assembled in a parallel configuration. While other micromechanics configurations, such as series or rule of mixtures, are easily applicable without affecting the theory, the parallel configuration simplifies the numerical simulations significantly. Then, the states of the stress and strain are given by the following basic micromechanics relations: 3.19where ϵz and are, respectively, the longitudinal macroscopic strain and stress and and are the microscale strain fields in the amorphous and crystalline phases, respectively; and and are, respectively, the local stresses in the amorphous and crystalline phases. One may assume that the loading condition is strain-controlled, then the strains in all phases are equal to the macroscale applied strain while the local stresses are computed from the respective microscale constitutive relations. Instead of simply assuming parallel amorphous and crystalline phases, one may consider a more general configuration through the rule of mixtures (Shojaei et al. submitted).
The RVE is chosen based on the experimental observations of the microstructure of the SMP fibres. As shown in §2, the non-stretched fibre is initially non-oriented and upon cold-drawing the microstructure of the SMP fibre varies. In this work, an asymmetric mechanical property is assumed for both of the stretched and non-stretched fibres while the microstructural changes are assumed to be SIC and morphological texture changes upon loading. Figure 10a shows the status of the microstructure for the manufactured SMP fibre. Upon loading, the SIC process results in larger volume fraction of the oriented crystalline phase as shown in figure 10b. The morphological texture updates for the amorphous and crystalline phases, however, are implicitly accounted in their respective governing constitutive relations. In this way, the simulations can monitor two significant microstructural changes which are texture updates in the amorphous and crystalline phases and the SIC process. The SAXS picture for the cold-drawn SMP fibre no. 1 shown in figure 9b confirms the microstructural alignment after a few cycles of the cold-drawing process.
4. Kinematic and computational aspect of finite stain
In the case of the finite deformation, the strain description based on the displacement gradients becomes nonlinear which may results in some computational difficulties for elasto-plastic analysis. To avoid such phenomenon, the gradients of the elastic and plastic deformations are decomposed multiplicatively. This is called multiplicative decomposition proposed previously by Lee and co-workers (Lee & Liu 1967; Lee 1970): 4.1where Fij is the total deformation gradient which correlates the material and deformed configurations, is the elastic deformation gradient and it is obtained by elastically unloading the deformed configuration to stress-free status; this unloaded state is called intermediate configuration, and is the plastic deformation gradient. Additive decomposition of the Lagrangian strain tensor, dij, which is measured in the material configuration, was proposed by Green & Naghdi (1965, 1971) as follows: 4.2where and are Lagrangian elastic and plastic strain tensors, respectively. This additive assumption which follows the basic thermodynamic principals (Naghdi & Trapp 1975) of solids provides the basis for the elastic predictor–plastic corrector elasto-plastic solutions as discussed by Simo and co-workers (Simo & Pister 1984; Simo & Ortiz 1985; Simo & Taylor 1986). The resulting return mapping algorithms are the direct consequence of the additive decomposition, equation (4.2). Let the second-order displacement tensor, uij, describe an incremental deformation field and its second-order gradient tensor, ∇uij, shows the deformation rate. The outline for the return mapping algorithm is as follows: (i) an increment of the deformation gradient is introduced such as , where superscripts n and n+1 indicate, respectively, to the previous and current load steps, (ii) a ‘Trial-Elastic’ deformation gradient is introduced subsequently to elastically stretch the material configuration with , where superscripts ‘e’ and ‘p’ denote the ‘elastic’ and ‘plastic’ components, respectively; and shows the frozen inelastic deformation gradient, (iii) the elastically stretched configuration is then relaxed until the state of the stress returns to the yield surface. The return mapping relaxes the stresses along the steepest descent path which is defined based on the yield function (associated flow rules) or potential functions (non-associated flow rules; Voyiadjis et al. 2012b). In this work, an isochoric condition is assumed for the large deformation process which is stated by λ1λ2λ3=1, where λi denotes the principal stretches. Then, the volume-preserving part of the deformation gradient, Fij, the elastic deformation gradient, , right, and left, , Cauchy–Green tensors are given as follows (Simo & Ortiz 1985; Shojaei & Li submitted): 4.3where ‘hat’ symbol represents the volume-preserving components for its respective tensor parameter and J=det(Fij). This kinematic decomposition approach results in the isochoric inelastic deformation in which . The isochoric assumption should be extended for all three steps of the return mapping algorithm as discussed by Shojaei & Li (submitted). To declare the stress–strain relation in the finite deformation process, the proper stress definition should be chosen. Accordingly, the Cauchy stress tensor, σij, is assigned to the deformed body and it is defined as force per unit deformed area. To obtain a more convenient description of the stress, the first Piola–Kirchhoff (PK) stress, , is computed which is the force per unit undeformed area that is still allocated in the deformed configuration. If the first PK stress is pulled-back to the material configuration, then the second PK stress tensor, , is achieved (Simo & Ortiz 1985; Shojaei & Li submitted). Anand (1979) proposed a relation between the Cauchy stress and the Hencky strain as follows: 4.4where and is the fourth-order elastic stiffness tensor. The corresponding elastic stiffness tensors for the amorphous, , and crystalline, , sub-phases are used in equation (4.3) to establish the local state of stress and strain.
The computation algorithm for a micromechanics-based multiscale analysis (Shojaei & Li submitted) is provided in the electronic supplementary material.
5. Results and discussion
The stress recovery test result is demonstrated in figure 11. The evolution of the parameter R controls the saturation limit of the recoverable stress based on the loading history, and basically this parameter identifies the amount of the recoverable stress based on the cyclic strain hardening and accumulated inelastic strains. The established stress recovery results in figure 11 take into account the heating rate, initial and final temperatures and history of the loading. Table 2 denotes the necessary material constants for the stress recovery and the SIC simulations.
It is noted that, during testing, the programmed SMP fibre is held within a fibre gripping device in an MTS machine, and the heat is applied by a digital furnace with constant heating rate of 0.35°C s−1. The stress–time curve is captured by the MTS machine while thermocouples pick up the temperature in the furnace. Because the heating rate of our furnace cannot be changed, only one set of experimental results is available in figure 11. To demonstrate the capability of our model in simulating the effect of heating rate, two more heating rates, 0.7°C s−1 and 0.17°C s−1, are simulated, and the results are also shown in figure 11. The simulations can conceptually describe the role of the heating rates on stress recovery. It is seen that the higher heating rates result in faster stress recoveries, whereas the lower heating rates result in slower stress recoveries, which is understandable because higher heating rate needs less time to reach the transition (stress recovery) temperature. However, it seems that the final recovery stress is independent of the heating rates.
Due to the fact that the proposed multiscale viscoplastic analysis incorporates the individual constitutive equations for each of the sub-phases, it can easily capture a vast variety of the microstructural configurations, such as 100 wt% amorphous or crystalline fibres or any microstructure configuration which lies between these two extreme cases. In figure 12, three cyclic tension test results for the single SMP fibre no. 1 together with the simulation results are depicted. The simulation for the 10 wt% crystalline phase volume fraction shows perfect correlation with the experimental results in the first cycle. In the second and third cyclic tensions, the SMP fibre is work hardened and shows stiffer mechanical responses. The proposed theory can capture the second cycle, whereas in the third cycle it shows some deviation from the experiments. The changes in mechanical responses of the SMP fibre may rely on the fact that the fibre undergoes large deformation and more accurate strain measurements are required to take into account the transverse deformation effects which narrows the fibres at high strain levels. The theory may then still hold its genuineness for larger cycles if the higher order accuracy stress–strain measurements, such as image processing measurement techniques (G'Sell et al. 2002), are used.
In reality, the fibre structure is a non-homogenous state of material, and this fact should be incorporated during assembling the stiffness matrix for these fibres. Here, it is assumed that the fibre is transversely isotropic, and the primarily non-hardened elastic properties are assumed for the transverse direction, whereas the properties along the fibre direction are updated based on the strain-hardening process. These transient changes in the elastic stiffness at the fibre direction are clearly depicted in figure 12, where the elastic modulus varies gradually from a pure amorphous to a crystalline dominant phase. Once the related material parameters for the amorphous and the crystalline phases are established, the volume fraction of the crystalline phase, which is controlled by the SIC governing equations, is introduced to the computational module and the resultant macroscale mechanical responses are captured subsequently. Table 3 shows the amorphous- and crystalline-related material parameters. The crystalline slippage system for the polymeric network is given in table 4. Because of experimental difficulties in the measurement of the exact volume fraction of the crystalline and amorphous phases, the volume fractions used in figure 12 are only a rough estimation of these values. In other words, the introduced amorphous, c(0), crystalline, c(1) and volume fractions in figure 12 are hypothetical values to obtain the best match with the experiments. These hypothetical values can be easily mapped to the actual values upon establishment of the actual volume fraction of the crystalline and amorphous phases.
The SIC process is simulated in figure 13 based on the proposed governing relations. The inner and outer radii of the crystalline annulus are expanded and shrunk, respectively, during the cold-drawing process to show the SIC microstructural changes. Then upon loading, the volume fraction of the oriented crystalline phase is gradually increased, and the fibre becomes stiffer.
6. Concluding remarks
A bio-inspired healing scheme is proposed in this study through architectural design of a composite structure, which is constructed by an SMP-fibre z-pinned, continuous SMP-fibre-grid-reinforced thermoset polymer embedded with TPs. The scheme is demonstrated through experimental testing and finite element modelling. It is found that the polyurethane thermoplastic fibres, upon strain hardening by cold-drawing programming, can achieve the required recovery stress to close macroscopic cracks. Further, the microstructure changes owing to the cold-drawing programming are characterized by instrumented microstructural analysis, which provides fundamental understanding and parameters for the constitutive modelling. Due to the fact that the polyurethane SMP fibres are categorized in the class of the semicrystalline polymers, the enhanced mechanical responses of the cold-drawn fibres are correlated to the SIC process and the morphological texture changes in the amorphous and crystalline phases in this work. A micromechanical multiscale viscoplastic theory is developed to link the microscale mechanical responses of the amorphous and crystalline sub-phases to the macroscale mechanical behaviours of the SMP fibres, including the cyclic hardening, and stress recovery responses. The proposed theory takes into account the SIC process and the initial morphological texture while the polymeric texture is updated based on the applied stresses. The cyclic loading and the thermomechanical responses of the SMP fibres are experimentally investigated in which the proposed theory is used to capture these phenomena. The proposed viscoplastic theory together with the material characterizations of the SMP fibres assists designers to predict the strength, stress recovery and life of the self-healing structures made from the semicrystalline SMP fibres. This study may open up new opportunities for the application of SMP fibres in load-bearing and self-healing composite structures.
This study was financially supported by the LONI Institute, under the Louisiana Board of Regents grant no. LEQSF(2007-12)-ENH-PKSFI-PRS-01, the Center for Computation and Technology (CCT) at Louisiana State University, the US National Science Foundation under grant no. CMMI 0900064 and the Cooperative Agreement NNX11AM17A between NASA and the Louisiana Board of Regents under contract NASA/LEQSF(2011-14)-Phase3-05. The financial support by LONI, CCT, NSF, NASA and LEQSF are greatly appreciated. The assistance by Dr Harper Meng in manufacturing SMP fibres is also appreciated.
- Received October 17, 2011.
- Accepted March 13, 2012.
- This journal is © 2012 The Royal Society