Macroscopic Stretch Research Articles

New two-parameter constitutive models for rubber-like materials: Revisiting the relationship between single chain stretch and continuum deformation

The connection between macroscopic deformation and microscopic chain stretch is a key element in constitutive models for rubber-like materials that are based on the statistical mechanics of polymer chains. A new micro-macro chain stretch relation is proposed, using the Irving–Kirkwood–Noll procedure to construct a Cauchy stress tensor from forces along polymer chains. This construction assumes that the deformed polymer network remains approximately isotropic for low to moderate macroscopic stretches, a starting point recently adopted in the literature to propose a non-affine micro-macro chain stretch relation (Amores et al., 2021). Requiring the constructed Cauchy stress to be consistent with the stress tensor derived from the strain energy density results in a new chain stretch relation involving the exponential function. A hybrid chain stretch relation combining the new chain stretch with the well-known affine relation is then proposed to account for the whole range of stretches in experimental datasets. Comparison of the model predictions to experimental data in the literature shows that the two new micro-macro chain stretch relations in this work result in two-parameter constitutive models that outperform those based on existing chain stretches with no increase in the number of fitting parameters used.

3D in situ study of damage during a ‘shear to tension’ load path change in an aluminium alloy

A load path change (LPC) from shear to tension has been studied for a recrystallized 2198 T8 aluminium alloy sheet material by three-dimensional (3D) X-ray imaging combined with image correlation and interpreted by complementary 3D finite element (FE) simulations. A cross-shaped specimen was designed for the non-proportional loading and multiscale study. The effect of the LPC on the formability and related strain localisation, damage and failure was investigated and damage mechanisms could be clearly identified. The macroscopic tension stretch to fracture, measured by an optical extensometer during the shear to tension test, was reduced by about 20% compared to the proportional tension test. Damage, measured by in situ laminography imaging at μm-scale resolution, has interestingly already been found under shear, and was quantified as surface void fraction during the LPC. Strain was measured inside the material and at the mesoscale by 2D digital image correlation (DIC) on projected volume data, using the (natural) intermetallic particle contrast present in the 3D laminographic data. An accumulated equivalent strain definition, suited for the description of non-proportional loading, has been applied to the DIC data and FE simulations, indicating good agreement between both. On the microscopic scale, damage was seen to nucleate under shear load in the form of flat cracks, of similar width as the grain size, and in the form of cracks inside intermetallic particles. This damage subsequently grew and coalesced during tensile loading which in turn led to final fracture.

Abstract P1-06-04: SimBioSys TumorScope: Spatio-temporal modeling of the breast tumor microenvironment accurately predicts chemotherapeutic response

Abstract Background A single neuron cannot think, just as a single hepatocyte cannot filter blood. The functions of the tissues that make up our bodies are emergent, depending on both their constituent cells' internal biology, as well as how they interact with the other cells around them. Tumors are no different. They interact with nearby cells through both mechanical and molecular mechanisms, adapting their own behaviors to suit local environmental conditions, and even hijacking the functions of nearby tissues for tumor-promoting purposes. As such, predicting how a given tumor will grow or respond to treatment requires more than just an understanding of the tumor's intrinsic characteristics, it requires a spatially- and temporally-resolved description of the tumor's shape, its surrounding tissues, and the milieu of diffusible molecules that drive its behavior and interactions. Methods SimBioSys TumorScope for Breast Cancer implements a multi-scale simulation methodology that couples genome-scale metabolic modeling with reaction-diffusion and material mechanics models in order to predict how individual patients' breast tumors will respond to different chemotherapeutic regimens. The simulations explicitly track the 3D morphology of the tumor and surrounding adipose, fibroglandular, and vascular tissues (initialized using pre-treatment MRI images), as well as the concentrations of key nutrients, metabolic byproducts, and drugs as they change over time. The vasculature serves as both source and sink for the modeled chemicals; nutrients and drugs are able to diffuse out of the blood vessels, while metabolic byproducts can diffuse in. The vascular concentrations of the drugs are assumed to vary in accordance with their administration schedules and known pharmacokinetics. At each location within the simulation, the local metabolic response of the tissues to the available nutrients is modeled using flux balance analysis, yielding uptake and efflux rates for each chemical species, as well as local growth rates for the tissues. At the same time, drugs may also be taken up by the cells, leading to cell death at rates that depend on the local intracellular drug concentrations, as well as their pharmacodynamic properties. As different regions of the tissues grow or shrink in response to their microenvironmental conditions, the macroscopic tissues stretch and deform. Results SimBioSys TumorScope for Breast Cancer was retrospectively applied to over 100 patients that received neoadjuvant chemotherapy (NACT) culled from publicly-available datasets (via the TCIA). Simulations were initialized with pre-treatment MRI data, and run through the entirety of each patient's specified treatment regimen. Predicted changes in tumor volume and longest dimension were then compared against measured values at several time-points after initiation of therapy, yielding Pearson correlations of over 0.93 for both. Conclusions Through accurate spatio-temporal modeling of drug and nutrient perfusion, metabolic behavior, and the physico-chemical interactions that arise between tissues, the SimBioSys TumorScope for Breast Cancer can accurately predict the response of patients treated with NACT. Citation Format: John A Cole Jr., Joseph R. Peterson, Tyler M. Earnest, Michael J. Hallock, Eduardo Braun. SimBioSys TumorScope: Spatio-temporal modeling of the breast tumor microenvironment accurately predicts chemotherapeutic response [abstract]. In: Proceedings of the 2019 San Antonio Breast Cancer Symposium; 2019 Dec 10-14; San Antonio, TX. Philadelphia (PA): AACR; Cancer Res 2020;80(4 Suppl):Abstract nr P1-06-04.

Extensional rheology of highly-entangled α-olefin molecular bottlebrushes

The transient uniaxial extensional response of a series of ultrahigh molecular weight poly(α-olefins) (PαO) with densely grafted bottlebrush architectures was measured as a function of side chain length, N sc. The PαO bottlebrushes studied here have N sc values ranging from 6 carbons [poly(1-octene)] to 16 carbons [poly(1-octadecene)] (POD). To understand the effect of bottlebrush architecture on the nonlinear response, a linear polyolefin (polypropylene), with N sc = 1, was included in this study. All the PαO bottlebrushes show strong extensional strain hardening (SH) at Weissenberg numbers, W i R, greater than 1. Strain-hardening ratios, S H R = η E + ( t ) / η E , LVE +, show increasing dependence on the rate of extension, described by power laws, with exponents dependent on the N s c values. Moreover, the maximum SHR values, S H R max, are increasing function of the side chain length, reaching values more than one order of magnitude larger for the POD, compared to atactic polypropylene (aPP, e.g., at W i R ∼ 200, S H R max = 3.5 and 75 for PP and POD, respectively). Another remarkable feature of these PαO bottlebrushes is their large extensibility, which is also an increasing function of both the extension rate and the side chain length. For instance, the maximum macroscopic stretch ratio values (before breakup), λ max, reach values of ∼400 (for POD), which is more than an order of magnitude larger than that for aPP ( λ max ∼ 25 ). A mechanism to explain the strong SH and stretchability in the PαO bottlebrushes, based on flow-induced side chain reorientation and interdigitation, is discussed.

Finite Strain Homogenization Using a Reduced Basis and Efficient Sampling

The computational homogenization of hyperelastic solids in the geometrically nonlinear context has yet to be treated with sufficient efficiency in order to allow for real-world applications in true multiscale settings. This problem is addressed by a problem-specific surrogate model founded on a reduced basis approximation of the deformation gradient on the microscale. The setup phase is based upon a snapshot POD on deformation gradient fluctuations, in contrast to the widespread displacement-based approach. In order to reduce the computational offline costs, the space of relevant macroscopic stretch tensors is sampled efficiently by employing the Hencky strain. Numerical results show speed-up factors in the order of 5–100 and significantly improved robustness while retaining good accuracy. An open-source demonstrator tool with 50 lines of code emphasizes the simplicity and efficiency of the method.

Disarrangements and instabilities in augmented one-dimensional hyperelasticity

In the present work, the overall nonlinear elastic behaviour of a one-dimensional multi-modular structure incorporating possible imperfections at the discrete (microscale) level is derived with respect to both tensile and compressive applied loads. The model is built up through the repetition of n units, each one comprising two rigid rods having equal lengths, linked by means of pointwise constraints capable of elastically limiting motions in terms of relative translations (sliders) and rotations (hinges). The mechanical response of the structure is analysed by varying the number n of the elemental moduli, as well as in the limit case of an infinite number of infinitesimal constituents, in light of the theory of (first-order) structured deformations (SDs), which interprets the deformation of any continuum body as the projection, at the macroscopic scale, of geometrical changes occurring at the level of its sub-macroscopic elements. In this way, a wide family of nonlinear elastic behaviours is generated by tuning internal microstructural parameters, the tensile buckling and the classical Euler's elastica under compressive loads resulting as special cases in the so-called continuum limit —say when n → ∞ . Finally, by plotting the results in terms of the first Piola–Kirchhoff stress versus macroscopic stretch, it is for the first time demonstrated that such SD-based one-dimensional models can be used to generalize some standard hyperelastic behaviours by additionally taking into account instability phenomena and concealed defects.

Multiscale mechanics of the cervical facet capsular ligament, with particular emphasis on anomalous fiber realignment prior to tissue failure.

The facet capsular ligaments encapsulate the bilateral spinal facet joints and are common sources of painful injury due to afferent innervation. These ligaments exhibit architectural complexity, which is suspected to contribute to the experimentally observed lack of co-localization between macroscopic strain and microstructural tissue damage. The heterogeneous and multiscale nature of this ligament, combined with challenges in experimentally measuring its microscale mechanics, hinders the ability to understand sensory mechanisms under normal or injurious loading. Therefore, image-based, subject-specific, multiscale finite-element models were constructed to predict the mechanical responses of the human cervical facet capsular ligament under uniaxial tensile stretch. The models precisely simulated the force-displacement responses for all samples ([Formula: see text]) and showed promise in predicting the magnitude and location of peak regional strains at two different displacements. Yet, there was a loss of agreement between the model and experiment in terms of fiber organization at large tissue stretch, possibly due to a lack of accounting for tissue failure. The mean fiber stretch ratio predicted by the models was found to be significantly higher in regions that exhibited anomalous fiber realignment experimentally than in regions with normal realignment ([Formula: see text]). The development of microstructural abnormalities was associated with the predicted fiber-level stretch ([Formula: see text]), but not with the elemental maximum principal stress or maximum principal strain by logistic regression. The multiscale models elucidate a potential mechanical basis for predicting injury-prone tissue domains and for defining the relationships between macroscopic ligament stretch and microscale pathophysiology in the subfailure regime.

Collagen structural alterations contribute to stiffening of tissue after split-thickness skin grafting.

The gold standard treatment for full thickness injuries of the skin is autologous split-thickness skin grafting. This involves harvesting the epidermis and superficial dermis from healthy skin and transplanting it onto the prepared wound bed. The donor site regenerates spontaneously, but the appendages and cellular components from the dermal layer are excluded from the graft. As a result, the new tissue is inferior; the healed graft site is dry/itchy, has decreased elasticity, increased fragility, and altered sensory function. Because this dermal layer is composed of collagen and other extracellular matrix proteins, the aim was to characterize the changes in the dermal collagen after split thickness grafting that could contribute to a deficit in functionality. This will serve as a baseline for future studies designed to improve skin function using pharmacological or cell-based therapies for skin repair. A xenograft model whereby human split-thickness grafts were implanted into full-thickness defects on immunocompromised (athymic Nu/Nu) mice was used. The grafts were harvested 4 and 8 weeks later. The collagen microstructure was assessed with second harmonic generation with dual-photon microscopy and light polarization analysis. Collagen fiber stiffness and engagement stretch were estimated by fitting the results of biaxial mechanical tensile tests to a histo-mechanical constitutive model. The stiffness of the collagen fibril-proteoglycan complex increased from 682 ± 226 kPa/sr to 1016 ± 324 kPa/sr between 4 and 8 weeks postgrafting. At the microstructural level there were significant decreases in both thickness of collagen fibers (3.60 ± 0.34 μm vs. 2.10 ± 0.27 μm) and waviness ratio (2.04 ± 0.17 vs. 1.43 ± 0.08) of the collagen fibers postgrafting. The decrease of the macroscopic engagement stretch from 1.19 ± 0.11 to 1.09 ± 0.08 over time postgrafting mirrored the decrease in waviness measured at the microscopic level. This suggested that the integrity of the collagen fibers was compromised and contributed to the functional deficit of the skin postgrafting.

Heterogeneous response of traction force at focal adhesions of vascular smooth muscle cells subjected to macroscopic stretch on a micropillar substrate

Traction force generated at focal adhesions (FAs) of cells plays an essential role in regulating cellular functions. However, little is known about how the traction force at each FA changes during cell stretching. Here we investigated dynamic changes in traction force at FAs during macroscopic stretching of porcine aortic smooth muscle cells (SMCs) cultured on elastic micropillar substrates. SMCs were cultured on polydimethylsiloxane (PDMS)-based substrates with a micropillar array, and stretched approximately in the direction of their major axis and then released by stretching and relaxing the substrates. This stretch–release cycle was repeated twice with cell strain rates of 0.3%/15 s up to a 3% strain, and the deflection of the PDMS micropillars was measured simultaneously to obtain the traction force at each FA F, total force in the cell's major axis direction F all, and whole-cell strain ε cell. Traction forces of SMCs during stretching varied widely with location: their changes at some pillars synchronized well with the applied strain ε cell, but others did not synchronized. Whole-cell stiffness estimated as the slope of the loading limb of the F all– ε cell curves was ∼10 nN/%, which was the same order of magnitude of the reported stiffness of cultured SMCs obtained in a tensile test. Interestingly, F all at a zero-strain state (pretension at the whole-cell level) actively increased in some cells following the loading/unloading process, as did whole-cell stiffness. Such a change did not occur in cultured SMCs in the tensile test in which cells were held with a pair of micropipettes coated with nonspecific adhesive. These results indicate that SMCs showed a myogenic response when stretched through their multiple FAs, but not through nonspecific adhesions on their membrane. SMCs may behave differently depending on the sites through which they are stretched.

T0201-3-6 円周方向引張負荷に伴うブタ胸大動脈壁内微視的変形挙動の観察([T0201-3]細胞の構造と流れにおけるマイクロ・ナノスケール解析(3))

The artery walls consist of structural components that have various mechanical properties: elastin, collagen and smooth muscle cells. Such variety may cause heterogeneous deformation of the components at a microscopic level when arterial tissue is subjected to macroscopic uniform stretch. To examine this hypothesis, we sliced 15-μm-thick specimens from porcine thoracic aorta with cryotome, and stretched them under a transmitted light microscope to observe their microscopic deformation caused by macroscopic stretch. Microscopic strain was found to be highly heterogeneous: stretch ratio in the elastic lamina (EL) and smooth muscle layer (SML) was 1.1-2.3 and 1.1-2.1, respectively, at circumferential stretch ratio of 1.5. Strain difference between an EL and a SML adjacent to each other could be as high as 6 times. These results suggest that the microscopic deformation in the artery wall is highly heterogeneous. Mechanical environment of the artery wall might not be so homogeneous at a cellular level.

Examination of the effects of computationally determined network topology on an analytical constitutive model for bimodal elastomers

Bimodal elastomer networks, so called due to their nominal bimodal molecular weight distribution of starting oligomers, are of continued interest due to the enhanced strength and toughness seen in certain mixtures. Researchers have suggested that the enhanced properties stem from the particular micromechanics of the networks formed by these systems at these optimal compositions. This work extends an existing analytical constitutive model for bimodal elastomer networks by incorporating aspects of network topology, including network connectivity patterns and realistic chain length distributions, determined through computational simulations of the formation of the network structure. These factors are included as functions of bimodal composition and are shown to affect the predicted mechanical, optical, and orientation responses of the network. The extended model elucidates how the naturally occurring doubled connection topology creates a micro-mechanism that lowers overall chain orientation in the lower molecular weight component and recreates experimentally observed optical response phenomena. Specifically, the model predicts that the presence of the stiff, contractile, doubled connections forces the rest of the network to conform more to the macroscopic stretch ratio while reducing the measured average orientation of the lower molecular weight component in the system; the effect diminishes as the composition-dependent population of doubled connections in the system decreases.

Strain hardening of polymer glasses: Entanglements, energetics, and plasticity

Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While stress-strain curves for a wide range of temperature can be fit to the functional form predicted by entropic network models, many other results are fundamentally inconsistent with the physical picture underlying these models. Stresses are too large to be entropic and have the wrong trend with temperature. The most dramatic hardening at large strains reflects increases in energy as chains are pulled taut between entanglements rather than a change in entropy. A weak entropic stress is only observed in shape recovery of deformed samples when heated above the glass transition. While short chains do not form an entangled network, they exhibit partial shape recovery, orientation, and strain hardening. Stresses for all chain lengths collapse when plotted against a microscopic measure of chain stretching rather than the macroscopic stretch. The thermal contribution to the stress is directly proportional to the rate of plasticity as measured by breaking and reforming of interchain bonds. These observations suggest that the correct microscopic theory of strain hardening should be based on glassy state physics rather than rubber elasticity.

Constitutive Modeling of the Finite Deformation Behavior of Membranes Possessing a Triangulated Network Microstructure

The mechanical behavior of the membrane of the red blood cell is governed by two primary microstructural features: the lipid bilayer and the underlying spectrin network. The lipid bilayer is analogous to a two-dimensional fluid in that it resists changes to its surface area, yet poses little resistance to shear. A skeletal network of spectrin molecules is cross-linked to the lipid bilayer and provides the shear stiffness of the membrane. Here, a general continuum level constitutive model of the large stretch behavior of the red blood cell membrane that directly incorporates the microstructure of the spectrin network is developed. The triangulated structure of the spectrin network is used to identify a representative volume element (RVE) for the model. A strain energy density function is constructed using the RVE together with various representations of the underlying molecular chain force-extension behaviors where the chain extensions are kinematically determined by the macroscopic deformation gradient. Expressions for the nonlinear finite deformation stress-strain behavior of the membrane are obtained by proper differentiation of the strain energy function. The stress-strain behaviors of the membrane when subjected to tensile and simple shear loading in different directions are obtained, demonstrating the capabilities of the proposed microstructurally detailed constitutive modeling approach in capturing the small to large strain nonlinear, anisotropic mechanical behavior. The sources of nonlinearity and evolving anisotropy are delineated by simultaneous monitoring of the evolution in microstructure including chain extensions, forces and orientations as a function of macroscopic stretch. The model captures the effect of pretension on the mechanical response where pretension is found to increase the initial modulus and decrease the limiting extensibility of the networked membrane.

Constitutive Modeling of the Stress-Stretch Behavior of Biological Membranes Containing Folded Domains

AbstractThe mechanical behavior of the red blood cell membrane is governed by the lipid bilayer which resists changes in surface area and the underlying spectrin network which resists changes in shape. The spectrin network can be modeled as an idealized triangulated network. Each spectrin chain consists of folded domains along the length of the chain which can unfold during stretching of the chain. A domain will completely unfold under the application of a chain force of20to35 pNdepending on the rate of imposed chain stretch. During macroscopic stretch of a network, individual chains within the network will experience different levels of chain stretch since the network chains will collectively stretch and rotate to accommodate the imposed stretch. Hence, the stretch on any individual chain will depend on the magnitude and state of macroscopic strain. A microstructurally informed continuum level constitutive model is developed which tracks individual chain deformation behavior as a function of macroscopic strain and also determines the overall macroscopic network stress-strain behavior. Using the introduced continuum approach and statistical mechanics based models of individual chain force-extension behavior, the stress-stretch behavior of the membrane under uniaxial tension is simulated at large stretches and the behavior of the constituent chains is monitored. Domain unfolding occurs within constituent chains during network deformation and the effects of this domain unfolding on the overall macroscopic stress-strain behavior of the network subject to deformation at different strain rates is revealed.

Elasticity theory of a twisted stack of plates

We present an elastic model of B-form DNA as a stack of thin, rigid plates or base pairs that are not permitted to deform. The symmetry of DNA and the constraint of plate rigidity limit the number of bulk elastic constants contributing to a macroscopic elasticity theory of DNA to four. We derive an effective twist-stretch energy in terms of the macroscopic stretch epsilon along and relative excess twist sigma about the DNA molecular axis. In addition to the bulk stretch and twist moduli found previously, we obtain a twist-stretch modulus with the following remarkable properties: 1) it vanishes when the radius of the helical curve following the geometric center of each plate is zero, 2) it vanishes with the elastic constant K_{23} that couples compression normal to the plates to a shear strain, if the plates are perpendicular to the molecular axis, and 3) it is nonzero if the plates are tilted relative to the molecular axis. This implies that a laminated helical structure carved out of an isotropic elastic medium will not twist in response to a stretching force, but an isotropic material will twist if it is bent into the shape of a helix.

Hole structures of polymers and zeolites probed by two-dimensional angular correlation of annihilation radiation

The two-dimensional angular correlation of annihilation radiation (2D-ACAR) technique has been developed to measure the hole structure of stretched and oriented polymers. The determination of polymer hole structure is demonstrated in a stretched semi-crystalline polyaryl-ether-ether-ketone (PEEK) sample. The atomic scale anisotropy is found to be consistent with the macroscopic scale stretch ratio. Applications of the 2D-ACAR method to image the three-dimensional hole, free-volume, and cavity structures of molecular systems are discussed.

Microscopic vs. macroscopic deformation of the pulmonary alveolar duct.

The stretch of the perimeters of alveolar ducts was measured at the surface of saline-filled specimens of human and dog lung parenchyma that were stretched biaxially. The microscopic stretch of these ducts was measured at several levels of isotropic biaxial macroscopic stretch of the parenchyma with stretch ratio (lambda x = lambda y) in the range of 1.20-1.40, which roughly corresponds to tidal breathing in humans and dogs. Alveolar walls were found to be load-carrying elements in the saline-filled lung, as seen by their straightness at all levels of stretch. Quantitatively, let l, A, L, and S denote, respectively, the duct perimeter length and area and the parenchymal target perimeter and area in the deformed state and lo, Ao, Lo, and So the corresponding variables in the undeformed state. The microscopic stretch ratio of the ducts (l/lo) was found to be approximately 4% larger than the macroscopic stretch ratio (L/Lo) in human lung and approximately 10% larger in dog lung. The microscopic area ratio of the ducts (A/Ao) was found to be approximately 10% larger than the macroscopic area ratio (S/So) in human lung and approximately 22% larger in dog lung. Ducts within human parenchyma were seen to be about twice as stiff as ducts within dog parenchyma over the range of macroscopic stretch studied. This correlates with the volume fractions of collagen and elastin being higher in the human lung than in dog lung. The observed nonuniformity in strain field at the microstructural level suggests the need to include a force balance between alveolar ducts and septal walls when modeling the mechanics of saline-filled parenchyma.

Studies of fibrin film. II. Small-angle x-ray scattering.

AbstractMeasurements of small‐angle x‐ray scattering have been made on bovine fibrin film, prepared by gentle compaction of coarse fibrin clots, containing 12–15% fibrin plasticized with either aqueous buffer or glycerol. Most of the experiments were made with water‐plasticized films because of the better x‐ray contrast. Both unligated and ligated (i.e., with α‐α and λ‐λ ligation by fibrinoligase, factor XIIIa) films were studied. Theoretical angular scattering profiles were calculated for a simplified model of the fibrin protofibril as a function of stretch ratio in uniaxial elongation based on an orientation distribution derived in the preceding paper. Measurements were made on unstretched films both with slit geometry and with pinhole geometry and two‐dimensional position‐sensitive detection. From the former, which had better resolution, the fundamental peak corresponded to a repeat spacing of 230 ± 5 Å, in agreement with early results of Stryer et al. [(1963) Nature 197, 793–794] and electron micrographs of stained fibrin, and with the half‐staggered overlapping model of the protofibril. Measurements were made with pinhole geometry on film stretched up to a stretch ratio λ = 1.59 at different times after imposition of strain and after release and recovery. The changes in meridional and equatorial scattering profiles, including the more prominent appearance of a harmonic in the former, show the orientation of the protofibrils as previously deduced from birefringence and related quantitatively to stretch ratio. In addition, the original repeat spacing is gradually replaced by one of about 290 Å. The “internal” stretch ratio λx, averaging about 1.28, is independent of the macroscopic stretch ratio λ from 1.14 to 1.59. The change, which is more nearly complete for unligated film and is reversible on release and retraction of the stretched strips, is attributed to an internal transition in the fibrin monomer units, probably involving extension of either the helical connectors or the terminal nodules. The results of stress and birefringence relaxation reported in the preceding paper of this series are interpreted on this basis.

An approach to rubber elasticity with internal constraints

The effect of configurational hindrances in unswollen rubberlike networks is taken into account by assuming the existence of a proper correlation between the macroscopic deformation and the rms fluctuations of the network junctions along the various directions of space. Following James and Guth’s approach, some junctions have been fixed at the surface of a proper parallelepiped in order to avoid collapse of the network, and the above correlation is expressed through the introduction of appropriate δ functions into the configurational partition function. Results identical with those previously arrived at by Flory–Hoeve–Ciferri are obtained by assuming that the rms fluctuation components transform as the macroscopic stretch ratios. This assumption cannot hold under high deformations because it implies that the junction fluctuations increase without limit with the deformation. Therefore an intermediate model is proposed where (i) the average ms fluctuations of the junctions are independent of strain, as in James–Guth theory, and (ii) the ratios between the rms (mean square) fluctuations are equal to those between the corresponding stretch ratios as in Flory–Hoeve–Ciferri theory. The intermediate model can explain, at least to some extent, the so-called Mooney effect. It is also shown that the relation −[∂ln(σ/T)/∂T]L,V=dln〈r2〉0/dT is valid for James–Guth theory only.