共查询到20条相似文献,搜索用时 31 毫秒
1.
Linear viscoelastic properties of SiO2/(AP/EP) suspension with various SiO2 volume fractions (ϕ) in a blend of acrylic polymer (AP) and epoxy (EP) were investigated at various temperatures (T). The AP/EP contained 70 vol.% of EP. The SiO2 particles were treated with epoxy silane coupling agent. The effects of the SiO2 particles are more pronounced in the terminal zone: a transition from viscoelastic liquid (ϕ ≤ 30 vol.%) to viscoelastic solid (ϕ ≥ 40 vol.%) was observed which can be interpreted as a critical gelation occurring at a critical particle content and critical
gel temperature. The SiO2/(AP/EP) systems exhibited a critical gel behavior at ϕ ≅ 35 vol.% and T ≅ 100°C characterized with a power–law relationship between the storage and loss moduli (G
′ and G
″) and frequency (ω); G
′ = G
″/tan(nπ/2) ∝ ω
n
. The critical gel exponent (n) was estimated to be about 0.45. The gelation occurred with increasing T. 相似文献
2.
Saint-Venant's torsion of symmetric cylindrical bars consisting of two or four homogeneous phases is studied. A symmetric
section is meant that the cross section of the cylindrical bar possesses reflectional symmetry with respect to one or more
axes. Each constituent region may have different shear modulus. The idea of the analysis is to superimpose suitably reflected
potentials to obtain the torsion solution of the same composite section but with different moduli. For two-phase sections,
we show that, if the warping fields for a given symmetric section with phase shear moduli μ1 and μ2 are known a priori, then the warping fields for the same configuration but with a different set of constituent moduli μ1
′ and μ2
′ are readily found through simple linear superpositions. Further, suppose that the torsional rigidities T(μ1,μ2) and T(μ1
′,μ2
′) for any two sets of phase moduli can be measured by some experimental tests or evaluated through numerical procedures, then
the torsional rigidity for any other combinations of constituent moduli T(μ1
′′,μ2
′′) can be exactly determined without any recourse to the field solutions of governing differential equations. Similar procedures
can be applied to a 4-phase symmetric section. But the coefficients of superposition are only found for a few branches. Specifically,
we find that depending on the conditions of μ and μ′, admissible solutions can be divided into three categories. When the correspondence between the warping field is known to
exist, a link between the torsional rigidities can be established as well.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
3.
Relaxation patterns of long,linear, flexible,monodisperse polymers: BSW spectrum revisited 总被引:1,自引:1,他引:0
Theoretical predictions for the dynamic moduli of long, linear, flexible, monodisperse polymers are summarized and compared
with experimental observations. Surprisingly, the predicted 1/2 power scaling of the long-time modes of the relaxation spectrum
is not found in the experiments. Instead, scaling with a power of about 1/4 extends all the way up to the longest relaxation
times near τ/τ
max = 1. This is expressed in the empirical relaxation time spectrum of Baumgaertel-Schausberger-Winter, denoted as “BSW spectrum,”
and justifies a closer look at the properties of the BSW spectrum. Working with the BSW spectrum, however, is made difficult
by the fact that hypergeometric functions occur naturally in BSW-based rheological material functions. BSW provides no explicit
solutions for the dynamic moduli, G
′(ω), G
″(ω), or the relaxation modulus G(t). To overcome this problem, close approximations of simple analytical form are shown for these moduli. With these approximations,
analysis of linear viscoelastic data allows the direct determination of BSW parameters. 相似文献
4.
Christian Sailer Martin Weber Helmut Steininger Ulrich A. Handge 《Rheologica Acta》2009,48(5):579-588
In this work, we studied the melt rheology of multigraft copolymers with a styrene–acrylonitrile maleic anhydride (SANMA)
terpolymer backbone and randomly grafted polyamide 6 (PA 6) chains. The multi-grafted chains were formed by interfacial reactions
between the maleic anhydride groups of SANMA and the amino end groups of PA 6 during melt blending. Because of the phase separation
of SANMA and PA 6, the grafted SANMA backbones formed nearly circular domains which were embedded in the PA 6 melt with a
diameter in the order of 20 to 40 nm. The linear viscoelastic behaviour of PA 6/SANMA blends at a sufficiently large SANMA
concentration displayed the characteristics of the critical gel state, i.e. the power relations G′ ∝ G′′ ∝ ω
0.5. In elongation, the PA 6/SANMA blend at the critical gel state showed a non-linear strain hardening behaviour already at
a very small Hencky strain. In contrast to neat PA 6, the elasticity of the PA 6/SANMA blends was strongly pronounced, which
was demonstrated by recovery experiments. Rheotens tests agreed with the linear viscoelastic shear oscillations and the measurements
using the elongational rheometer RME. Increasing the SANMA concentration led to a larger melt strength and a reduced drawability.
The occurrence of the critical gel state can be interpreted by the cooperative motion of molecules which develops between
the grafted PA 6 chains of neighbouring micelle-like SANMA domains. 相似文献
5.
V. F. Shumsky I. Getmanchuk T. Ignatova Yu. Maslak P. Cassagnau G. Boiteux F. Melis 《Rheologica Acta》2010,49(8):827-836
The viscoelastic characteristics of the blends of poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) (PMMA/SAN) were investigated at various temperatures below, near, and above the phase separation temperature.
The investigated polymer system is characterized by a lower critical solution temperature. Rheological behavior of the blends
in the region of a phase separation was compared with change of the light scattering intensity. The presence of nanofillers
in the blend results in that the phase separation occurs at a higher temperature. At the isothermal conditions, the phase
separation begins earlier and proceeds with a higher rate as compared with the same blend without filler. The results of the
study show the considerable change of the viscoelastic characteristics of PMMA/SAN when the polymer system passes from the
homogeneous state to the heterogeneous one. Such characteristics as the dependence of the storage modulus (G
′) on the loss modulus (G
″), the dependence of the loss viscosity (η
″) on the dynamic viscosity (η
′), the dependences of the complex viscosity (η*), and the free volume fraction (f) on the blend composition are the most sensitive to the phase separation. The phase separation affects the characteristics
G
′(ω), where ω is the frequency only in a low-frequency range. Temperatures of phase separation were estimated using dependence G
′(T) at ω, which is the constant in the range of low frequencies. 相似文献
6.
Pierre Terech Séverine Friol Neralagatta Sangeetha Yeshayahu Talmon Uday Maitra 《Rheologica Acta》2006,45(4):435-443
We present a comparative investigation of two opposite classes of self-assembled fibrillar networks. Ribbons and tubes having cross-sectional dimensions in the nanoscale can be formed in aqueous solutions of steroids derived, respectively, from deoxycholic (DC) and lithocholic (LC) acids. Rheological features distinguish energetic networks of DC ribbons rigidly fixed in cylindrical bundles and entropic transient networks of LC tubes weakly interacting in shear-sensitive suspensions. The two classes are characterized by their frequency sweep profiles, viscoelastic linear domains, scaling laws of the elastic shear modulus vs concentration, kinetics of formation of the networks, and their optical birefringence aspects. A theoretical context for networks of rigid fibers is used to account for the scaling exponents α in the G’ (and σ*) ∝C
α
laws (α=2.0 and 1.0, respectively, for DC and LC). The evolution observed in DC gels from ribbons to cylindrical fibers with monodisperse sections made up with four ribbons is an indication of an equilibrated balance between face-to-face attractions and untwisting elastic processes of the constitutive ribbons.Paper presented at the Annual Meeting of the European Society of Rheology, Grenoble, April 2005 相似文献
7.
Giovanni Romeo Giovanni Filippone Alberto Fernández-Nieves Pietro Russo Domenico Acierno 《Rheologica Acta》2008,47(9):989-997
We investigate the relation between the structure and the viscoelastic behavior of a model polymer nanocomposite system based
on a mixture of titanium dioxide (TiO2) nanoparticles and polypropylene. Above a critical volume fraction, Φ
c, the elasticity of the hybrids dramatically increases, and the frequency dependence of the elastic and viscous moduli reflects
the superposition of the independent responses of the suspending polymer melt and of an elastic particle network. In addition,
the elasticity of the hybrids shows critical behavior around Φ
c. We interpret these observations by hypothesizing the formation of a transient network, which forms due to crowding of particle
clusters. Consistent with this interpretation, we find a long-time, Φ-dependent, structural relaxation, which emphasizes the transient character of the structure formed by the particle clusters.
For times below this characteristic relaxation time, the elasticity of the network is Φ-independent and reminiscent of glassy behavior, with the elastic modulus, G′, scaling with frequency, ω, as G′∼ω
0.3. We expect that our analysis will be useful for understanding the behavior of other complex fluids where the elasticity of
the components could be superimposed. 相似文献
8.
The influence of preshearing on the rheological behaviour of model suspensions was investigated with a stress-controlled
cone-and-plate rheometer. The used matrix fluids showed Newtonian behaviour over the whole range of applied shear stresses.
Highly monodisperse spherical glass spheres with various particle diameters were used as fillers. By applying steady preshearing
at a low preshear stress, where a diffusion of particles can be expected, it was found for all model suspensions investigated
at volume fractions ranging from 0.20 to 0.35 that the time-temperature superposition in the steady shear and in the dynamic
mode holds within the chosen temperature range. Furthermore, all presheared model suspensions displayed a high and a low frequency
range which are either separated by a shoulder or by a plateau value of G′ at intermediate frequencies. It could clearly be
demonstrated that the low frequency range strongly depends on the preshear conditions. Hence, the features observed in the
low frequency range can be attributed to a structure formation of a particulate network. In the high frequency range a frequency-dependent
behaviour was observed which obeys the classical behaviour of Newtonian fluids (G′∝ω2, G′′∝ω). The resulting temperature shift factors from the dynamic and the steady shear mode are identical and independent
of the volume fraction and the particle size of the filler.
Received: 29 November 2000 Accepted: 12 July 2001 相似文献
9.
H. Watanabe Tomohiro Sato Motoyuki Hirose Kunihiro Osaki Ming-Long Yao 《Rheologica Acta》1998,37(6):519-527
Dielectric relaxation behavior was examined for 4-4′-n-pentyl-cyanobiphenyl (5CB) and 4-4′-n-heptyl-cyanobiphenyl (7CB) under flow. In quiescent states at all temperatures examined, both 5CB and 7CB exhibited dispersions
in their complex dielectric constant ε*(ω) at characteristic frequencies ω
c
above 106 rad s–1. This dispersion reflected orientational fluctuation of individual 5CB and 7CB molecules having large dipoles parallel to
their principal axis (in the direction of C≡N bond). In the isotropic state at high temperatures, these molecules exhibited no detectable changes of ε*(ω) under flow at
shear rates . In contrast, in the nematic state at lower temperatures the terminal relaxation intensity of ε*(ω) as well as the static
dielectric constant ε′(0) decreased under flow at . This rheo-dielectric change was discussed in relation to the flow effects on the nematic texture (director distribution)
and anisotropy in motion of individual molecules with respect to the director.
Received: 14 April 1998 Accepted: 29 July 1998 相似文献
10.
We monitor the time-dependent shear compliance of a solution of semi-flexible polymers, using diffusing wave spectroscopy (DWS) and video-enhanced single-particle-tracking (SPT) microrheology. These two techniques use the small thermally excited motion of probing microspheres to interrogate the local properties of polymer solutions. The solutions consist of networks of actin filaments which are long semi-flexible polymers. We establish a relationship between the mean square displacement (MSD) of microspheres imbedded in the solution and the time-dependent creep compliance of the solution, <Δr 2(t)>=(k B T/πa)J(t). Here, J(t) is the creep compliance, <Δr 2(t)> is the mean-square displacement, and a is the radius of the microsphere chosen to be larger than the mesh size of the polymer network. DWS allows us to measure mean square displacements with microsecond temporal resolution and Ångström spatial resolution. At short times, the mean square displacement of a 0.96μm diameter sphere in a concentrated actin solution displays sub-diffusion. <Δr 2(t)>∝t , with a characteristic exponent =0.78±0.05, which reflects the finite rigidity of actin. At long times, the MSD reaches a plateau, with a magnitude that decreases with concentration. The creep compliance is shown to be a weak function of polymer concentration and scales as J p ∝c –1.2±0.3. This exponent is correctly described by a recent model describing the viscoelasticity of semi-flexible polymer solutions. The DWS and video-enhanced SPT measurements of the compliance plateau agree quantitatively with compliance measured independently using classical mechanical rheometry for a viscous oil and for a solution of flexible polymers. This paper extends the use of DWS and single-particle-tracking to probe the local mechanical properties of polymer networks, shows for the first time the proportionality between mean square displacement and local creep compliance, and therefore presents a new, direct way to extract the viscoelastic properties of polymer systems and complex fluids. 相似文献
11.
The rheology of F-actin networks has attracted a great attention during the last years. In order to gain a complete understanding of the rheological properties of these novel materials, it is necessary the study in a large deformations regime to alter their internal structure. In this sense, Schmoller et al. (2010) showed that the reconstituted networks of F-actin crosslinked with α-actinin unexpectedly harden when they are subjected to a cyclical shear. This observation contradicts the expected Mullins effect observed in most soft materials, such as rubber and living tissues, where a pronounced softening is observed when they are cyclically deformed.We think that the key to understand this stunning effect is the gelation process. To define it, the most relevant constituents are the chemical crosslinks – α-actinin –, the physical crosslinks – introduced by the entanglement of the semiflexible network – and the interaction between them. As a consequence of this interaction, a pre-stressed network emerges and introduces a feedback effect, where the pre-stress also regulates the adhesion energy of the α-actinin, setting the structure in a metastable reference configuration. Therefore, the external loads and the evolvement of the trapped stress drive the microstructural changes during the cyclic loading protocol. In this work, we propose a micromechanical model into the framework of nonlinear continuum mechanics. The mechanics of the F-actin filaments is modelled using the wormlike chain model for semiflexible filaments and the gelation process is modelled as mesoscale dynamics for the α-actinin and physical crosslink. The model has been validated with reported experimental results. 相似文献
12.
Thomas G. Mason 《Rheologica Acta》2000,39(4):371-378
We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr
2(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different
representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G
*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G
r
(t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr
2(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the
frequency extremes, we estimate the transforms algebraically by describing <Δr
2(t)> as a local power law. If the logarithmic slope of <Δr
2(t)> can be accurately determined, these estimates generally perform well at the frequency extremes.
Received: 8 September 2000/Accepted: 9 March 2000 相似文献
13.
Paulo S. Santos Mohamad G. Abiad Marcelo A. Carignano Osvaldo H. Campanella 《Rheologica Acta》2012,51(1):3-11
The rheological behavior of Dibenzylidene D-Sorbitol (DBS) gels formed in ethylene glycol, glycerol, mineral oil, ethanol,
and chlorobenzene was studied using oscillatory squeezing flow viscometry. The storage (G
′) and loss (G
″) moduli were measured as a function of gellant concentration (0.5–2 w/w) and type of solvent. As expected greater values of gel strength (G
′) were observed for gels containing higher concentrations of DBS. In addition, both storage and loss moduli of 2% systems
were mostly frequency independent over the studied range, whereas 0.5% gels did exhibit some degree of dependence. We also
found that the solvent plays an important role in the properties of the gels. Among the parameters that affect the viscoelastic
properties of DBS gels, the solvent polarity and its ability to form hydrogen bonding may have significant effects on the
gel rheology. 相似文献
14.
Soumitra Choudhary Joseph C. White Whitney L. Stoppel Susan C. Roberts Surita R. Bhatia 《Rheologica Acta》2011,50(1):39-52
We report the preparation and rheological characterization of interpenetrating polymer network (IPN) hydrogels made from alginate
and hydrophobically modified ethyl hydroxyl ethyl cellulose (HMEHEC). To our knowledge, there have been no studies of the
gelation behavior of IPNs. We found that the rheology of these systems can be easily tuned, with the elastic modulus of the
IPN strongly dependent on the relative ratio of HMEHEC to alginate. The sol–gel transition of these systems was found to satisfy
the Winter–Chambon criterion for gelation at various crosslinker densities. From the power law relationship of the dynamic
moduli (G
′ ~G
″ ~ω
n), the exponent n appears to be dependent on both the crosslinker density and relative amount of two polymers. The value of n was found to be ~0.5 for all samples for stoichiometric amounts of crosslinker. The effect of molecular weight of HMEHEC
on the gel point and viscoelastic exponent has also been reported. Alginate seems to dominate the kinetics of the process
but the effect of high molecular weight HMEHEC on the gel point, especially at lower proportion was also evident. 相似文献
15.
Bas van der Vorst E. Matthijs Toose Dirk van den Ende Rob J. J. Jongschaap Jorrit Mellema 《Rheologica Acta》1995,34(3):274-280
A theory analogue to tha of Rouse is given, to describe the rheological behavior of dilute solutions consisting of clusters of crosslinked polymers. The frequency-dependent behavior of the dynamic moduli of these fluids differs substantially from that of the well-known Rouse-like fluid (GG1/2). In our case the storage modulus becomes proportional to 3/2, while the loss modulus is proportional to . The loss modulus dominates the dynamic behavior for frequencies smaller than the largest normal frequency of the clusters. 相似文献
16.
Hiroshi Watanabe Tomohiro Sato Motoyuki Hirose Kunihiro Osaki Ming-Long Yao 《Rheologica Acta》1999,38(2):100-107
Rheo-dielectric behavior was examined for 4−4′−n-octyl-cyanobiphenyl (8CB) having large dipoles parallel to its principal axis (in the direction of the C≡N bond). In the
quiescent state at all temperatures (T) examined, orientational fluctuation of the 8CB molecules was observed as dielectric dispersions at characteristic frequencies
ωc>106 s−1. In the isotropic state at high T, no detectable changes of the complex dielectric constant ɛ*(ω) were found under slow flow at shear rates ˙γ≫ωc. In the nematic state at intermediate T, the terminal relaxation intensity of ɛ*(ω) was decreased under such slow flow. In the smectic state at lower T, the flow effect became much less significant. These results were related to the flow-induced changes of the liquid crystalline
textures in the nematic and smectic states, and the differences of the rheo-dielectric behavior in these states are discussed
in relation to a difference of the symmetry of molecular arrangements in the nematic and smectic textures.
Received: 1 October 1998 Accepted: 13 January 1999 相似文献
17.
We have investigated the rheological changes in two particulate-filled dental composite resin cements during the curing process
using a Micro-Fourier Rheometer (MFR). In the MFR, the sample was sandwiched between two parallel plates, and pseudorandom
small amplitude squeezing was applied by oscillating the upper plate over a range of frequencies. Fourier transforms of the
displacement signal and the resulting time dependent force signal enabled the rapid determination of the dynamic properties
G′ and G′′ over the frequency range 2π–200π rad/s . This technique permitted us to follow changes in the rheological properties of
the resin cements through the setting period. A typical result was that G′ increased from 2×103 Pa to 2×105 Pa after about 120 s, and that G′′ changed from 4×103 Pa to 4×104 Pa over the same period at frequency 40π rad/s. We also found that the dental composite resin cements show linear viscoelastic
behaviour over a range of strain amplitudes before curing, but the response becomes distinctly non-linear at the later stages
of curing for strain amplitudes γ>0.067%. 相似文献
18.
The dynamic flow behavior of polyamide-6 (PA-6) and a nanocomposite (PNC) based on it was studied. The latter resin contained
2 wt% of organoclay. The two materials were blended in proportions of 0, 25, 50, 75, and 100 wt% PNC. The dynamic shear rheological
properties of well-dried specimens were measured under N2 at T=240 °C, frequency ω=0.1–100 rad/s, and strains γ=10 and 40%. At constant T, γ, and ω the time sweeps resulted in significant
increases of the shear moduli. The γ and ω scans showed a complex rheological behavior of all clay-containing specimens. At
γ=10% the linear viscoelasticity was observed for all compositions only at ω>1 rad/s, while at γ=40% only for 0 and 25 wt%
of PNC. However, the effect was moderate, namely decreasing G′ and G′′ (at ω=6.28 rad/s; γ=50%) by 15 and 7.5%, respectively.
For compositions containing >25 wt% PNC two types of non-linearity were detected. At ω≤ωc=1.4 ± 0.2 rad/s yield stress provided evidence of a 3-D structure. At ω > ωc, G′ and G′′ were sensitive to shear history – the effect was reversible. From the frequency scans at ω > ωc the zero-shear relative viscosity vs concentration plot was constructed. The initial slope gave the intrinsic viscosity from
which the aspect ratio of organoclay particles, p=287 ± 9 was calculated, in agreement with the value calculated from the
reduced permeability data, p=286.
Received: 24 May 2001 Accepted: 27 August 2001 相似文献
19.
S. K. Ahuja 《Rheologica Acta》1980,19(3):299-306
Summary The viscoelastic properties of 65/35 styrenen-butyl methacrylate random copolymers were determined using the Eccentric Rotating Disks device of the Rheometrics Mechanical Spectrometer. Similar to the behavior observed in homopolymers, an increase in the molecular weight of the copolymer resulted in extension of the rubbery plateau and in a reduction in the terminal region. The dynamic complex viscosity showed onset of non-Newtonian flow at higher frequencies, with the non-Newtonian region increasing with increasing molecular weight.The elastic modulus,G, was dependent upon the frequency,, asG
1.5 in the terminal region, rather than asG
2 observed for polystyrene. The viscous modulus,G, was proportional to the frequency,, asG , similar to what is observed for polystyrene. The dynamic viscosity |
*| at high frequencies showed a region independent of molecular weight where a power law of |
*|
0.9 is applicable, consistent with entanglement models. Thy dynamic viscosity at low frequencies in the Newtonian region is related to molecular weight as |*|
. Using WLF equations, the coefficient of expansion,
f
, was obtained that, together with glass transition, showed a negative deviation from the Fox-Flory relationship.
With 10 figures and 1 table 相似文献
Zusammenfassung Die viskoelastischen Eigenschaften von statistischen 65/35-Styrol/n-Butyl-Methacrylat-Kopolymeren wurden mit Hilfe einer Maxwell-Rheometer-Anordnung in Verbindung mit dem Mechanischen Spektrometer der Fa. Rheometrics bestimmt. Ähnlich dem bei Homopolymeren beobachteten Verhalten ergab sich auch hier mit wachsendem Molekulargewicht eine Verbreiterung des Kautschuk-Plateaus und eine Verkleinerung des Endbereichs. Die komplexe Viskosität zeigte erst bei höheren Frequenzen das Einsetzen nicht-newtonschen Fließens an, wobei der nichtnewtonsche Bereich mit steigendem Molekulargewicht größer wurde.Der SpeichermodulG ergab sich im Endbereich als proportional zu 1,5, im Unterschied zu der bei Polystyrol beobachteten Proportionalität mit 2. Dagegen war der VerlustmodulG der Frequenz direkt proportional, ähnlich wie es auch bei Polystyrol beobachtet worden war. Die dynamische Viskosität | *| zeigte unabhängig vom Molekulargewicht bei hohen Frequenzen einen Bereich, in dem eine Potenz-Beziehung | *| ~ 0,9 herrschte, was auf die Wirkung von Verzweigungen hindeutet. Dagegen galt bei den niedrigen Frequenzen des newtonschen Bereichs|*| ~ . Mit Hilfe der WLF-Gleichung wurde der Ausdehnungskoeffizient f bestimmt, der ebenso wie der Glasübergang eine negative Abweichung von der Fox-Flory-Beziehung zeigte.
With 10 figures and 1 table 相似文献
20.
T.C.T. Ting 《Journal of Elasticity》1997,47(1):23-50
The three Barnett-Lothe tensors H, L, S appear often in the Stroth formalism of two-dimensional deformations of anisotropic
elastic materials [1–3]. They also appear in certain three-dimensional problems [4, 5]. The algebraic representation of H,
L, S requires computation of the eigenvalues pv(v=1,2,3) and the normalized eigenvectors (a, b). The integral representation of H, L, S circumvents the need for computing
p
v(v=1,2,3) and (a, b), but it is not simple to integrate the integrals except for special materials. Ting and Lee [6] have recently
obtained an explicit expression of H for general anisotropic materials. We present here the remaining tensors L, S using the
algebraic representation. They key to our success is the obtaining of the normalization factor for (a, b) in a simple form.
The derivation of L and S then makes use of (a, b) but the final result does not require computation of (a, b), which makes
the result attractive to numerical computation. Even though the tensor H given in [6] is in terms of the elastic stiffnesses
Cμ v while the tensors L, S presented here are in terms of the reduced elastic compliances s′
μv
, the structure of L, S is similar to that of H. Following the derivation of H, we also present alternate expressions of
L, S that remain valid for the degenerate cases p
1
p
2 and p1=p2 = p
3. One may want to compute H, L, S using either C
μv
or s′
μv
v, but not both. We show how an expression in Cμ v can be converted to an expression in s′
μv
v, and vice versa. As an application of the conversion, we present explicit expressions of the extic equation for p in Cμ v and s′
μv
v.
This revised version was published online in August 2006 with corrections to the Cover Date. 相似文献