Mechanical dissipation in random fiberglass networks under small strains

A numerical study on elastic properties of low-density two-dimensional networks of crosslinked long fibers

This paper presents a nonlinear micropolar nonclassical mathematical continuum theory for finite deformation/finite strain deformation physics of compressible thermoviscoelastic solids based on classical rotations c Θ and its rates. Stress and moment measures for finite deformation/finite strain physics are utilized in conjunction with the finite deformation/finite strain measures presented in ref. [1] to derive conservation and the balance law as well as the constitutive theories using conjugate pairs in the entropy inequality and the representation theorem. The nonlinear micropolar nonclassical continuum theory presented in this paper for thermoviscoelastic solid: (1) incorporates nonlinear ordered rate dissipation mechanism for the viscous medium based on rates of Green’s strain tensor up to order n . This is usual viscous dissipation (macrodissipation) in the solid medium due to the viscosity of the medium. (2) Also incorporates additional ordered rate dissipation mechanism due to microconstituents and the viscosity of medium, which depends upon rates of the symmetric part of the rotation gradient tensor up to order n ˜ . We refer to this dissipation mechanism as microdissipation or microviscous dissipation. This dissipation mechanism is consistent with the deformation measure derived in ref. [1] for nonlinear micropolar nonclassical continuum theory. (3) With the assumption of small deformation, small strain, the nonlinear micropolar nonclassical continuum theory presented here reduces to a consistent linear micropolar nonclassical continuum theory with both mechanisms of dissipation. (4) In the absence of micropolar physics, the theory reduces to finite deformation/finite strain classical continuum theory for compressible thermoviscoelastic solid medium. The complete mathematical model consisting of the conservation and balance laws and the constitutive theories has closure without the conservation of micro inertia law needed in the micropolar theories of Eringen for closure. It has been shown that the balance of moment of moments is an essential balance law in all micropolar theories for achieving thermodynamic and mathematical consistency of the resulting linear micropolar theory. The balance of moment of moments balance law is necessary and has been successfully used in ref. [2] to derive a nonlinear micropolar theory for thermoelastic solid and is essential in this nonlinear micropolar nonclassical continuum theory for thermoviscoelastic solid based on classical rotations c Θ presented in this paper. The nonlinear micropolar nonclassical continuum theory based on rotations Θ c , Θ α and α Θ (neglecting c Θ ) is not considered in the present work due to the fact that the linear micropolar nonclassical continuum theory based on these rotations is thermodynamically and mathematically inconsistent [1].

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler's equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic as-How to cite this paper:

This paper presents a thermodynamically consistent and kinematic assumption free formulation for dynamics of thermoviscoelastic plates/shells based on the conservation and balance laws of classical continuum mechanics (CCM) in which dissipation mechanism has been incorporated through ordered rate constitutive theory for deviatoric stress tensor. In this paper, we consider small deformation, small strain. The conservation and balance laws of CCM in [Formula: see text] in Lagrangian description using Cauchy stress tensor ([Formula: see text]) and linearized Green strain tensor ([Formula: see text]) constitute the mathematical model. The constitutive theory for the deviatoric Cauchy stress ([Formula: see text]) is derived using conjugate pairs in entropy inequality in conjunction with representation theorem. The argument tensors of [Formula: see text] are [Formula: see text] and rates of [Formula: see text] up to order [Formula: see text]. This yields a constitutive theory with dissipation mechanism based on rates of strain up to order [Formula: see text]. Constitutive theory for heat vector is also derived using the conjugate pairs in the entropy inequality and representation theorem. Finite element method is used to obtain solutions of the initial value problems descried by the balance of linear momenta (BLM), energy equation and the constitutive theories. The shell element geometry is described by the middle surface and the nodal vectors at the middle surface defining bottom and top surfaces of the element. The local approximation for the displacement field is [Formula: see text] - version hierarchical in the plane of the element as well as in the transverse direction. A space-time decoupled finite element formulation using Galerkin Method with Weak Form (GM/WF) in space is constructed for BLM as well as energy equation, both resulting in ordinary differential equations (ODEs) in time. The ordinary differential equations (ODEs) in time resulting from the finite element formulation of BLM are used to study: (i) natural undamped modes of vibration (ii) the transient dynamic response using the ODEs in time recast in modal basis: (a) using Rayleigh damping (b) using the ordered rate damping proposed in this paper. Time response is calculated using modal damping based on Rayleigh damping as well as using the proposed ordered rate damping mechanism. Model problem studies are presented to demonstrate: (1) accuracy of the natural frequencies obtained from the present formulation for thin and thick plates/shells (in which shear deformation is significant) and the results are compared with the currently used plate formulations (2) accuracy of damped transient response using proposed damping mechanism is compared with time response using Rayleigh damping (3) it is shown that Rayleigh damping has no physical basis and leads to spurious stationary states. The proposed damping yields accurate stationary states that are in exact agreement with the solution of corresponding BVP. A single formulation presented in this paper remains valid and accurate for very thin as well as very thick plates/shells and correctly simulates 3D state of deformation regardless of plate/shell thickness and is free of shear locking problems as well as need for shear corrections. When obtaining the time response, solution for an increment of time alternates between the solution of BLM followed by the solution of the energy equation. Details are presented in the paper.

This paper considers conservation and balance laws for non-classical solid continua in the presence of internal rotations ( $${}_i \pmb {\varvec{\varTheta }}$$ ) due to the Jacobian of deformation and Cosserat rotations ( $${}_e \pmb {\varvec{\varTheta }}$$ ) at each material point. In these balance laws, internal rotations are completely defined as functions of the displacement gradient tensor, but Cosserat rotations are additional three degrees of freedom at each material point. When these rotations are resisted by the deforming matter, conjugate moments are created. For thermoviscoelastic solids with memory, these result in additional energy storage, dissipation mechanism, and rheology. This paper presents a thermodynamically consistent derivation of constitutive theories for such solids based on the entropy inequality in conjunction with representation theorem. Material coefficients are derived and discussed. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as additional balance law for non-classical continuum theories, and the resulting theories are compared with classical continuum theories in which this balance law is not needed. Retardation moduli corresponding to the Cauchy stress tensor as well as the Cauchy moment tensor are derived. In this paper we only consider small strain, small deformation physics.

The work presented in this paper extends the kinematic assumption free and thermodynamically consistent formulation for bending of thermoelastic beams presented by Surana et al. for bending of thermoviscoelastic beams with dissipation mechanism without memory. We consider small strain, small deformation physics in Lagrangian description. Conservation and balance laws of classical continuum mechanics (CCM) constitute the mathematical model for the physics considered in this paper. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and the principle of equipresence. Cauchy stress tensor is decomposed into equilibrium stress tensor () and deviatoric stress tensor (). Constitutive theory for is derived using Helmholtz free energy density in conjunction with incompressibility condition. The constitutive theory for is derived in by first establishing its argument tensors using conjugate pairs in the entropy inequality and other desired physics and then using the representation theorem with complete basis (integrity). The constitutive theory for is a nonlinear constitutive theory in terms of strain tensor containing up to fifth degree terms in the components of the strain tensor and an ordered rate theory in the strain rate tensors up to order n. Simplified linear ordered rate theory for is also presented. The formulation presented here for thermoviscoelastic beams is based on the conservation and balance laws of classical continuum mechanics and construction of beam finite element formulation using hpk framework with variationally consistent integral form. In this approach the mathematical model consist of true conservation and balance laws and the choice of local approximations for the beam finite elements facilitates incorporation of required kinematic description based on the application. This approach is free of the a priori assumptions of kinematic relations, computations are unconditionally stable and the local approximations can be of higher degree (p) and of higher order (k). This approach addresses slender as well as deep beam physics and can be used to measure error in the computed solution through residual functional. The rate constitutive theory is based on the second law thermodynamics consistent with physics of deformation. Mathematical details of new formulation and the model problem studies and comparisons with currently used beam models are presented for slender as well as deep beams. This approach permits consideration of reversible (thermoelastic) as well as irreversible (thermoviscoelastic) processes. It is shown that Rayleigh damping used currently to derive modal damping has no physical basis and can lead to spurious solution. The dissipation mechanism presented here has physical basis and yields non-spurious and valid solutions Model problem studies are presented for undamped natural modes of vibration, damped and undamped transient dynamic response. The results obtained from the formulation presented here are compared with published works.

ContentsPageTable 61. Internal friction in single crystals892. Internal friction in polycrystalline metals903. Internal friction in glass914. Internal friction in rocks92 The vibrations of solid bodies are accompanied by dissipation of energy attributable to their “ internal friction”; this loss of energy is additional to whatever external losses may occur. There are various ways of specifying the internal friction, and few of the published researches have adopted a common terminology. In this section the results have been reduced to show a dimensionless quantity, 1/Q, which may be called the “dissipation function” or the “internal friction.” The logarithmic decrement Δ of free vibrations is related to this function by Δ = π/Q.If dEis the loss of energy per cycle, and Ethe total energy, then 1/Q = dE/(2πE),Thus the internal friction is small when 1/Qis small. Only in the last few years has the technique of measuring internal friction reached a point where the losses in single crystals can be studied [ 4, 16, 17]. The mechanism of internal dissipation of energy in single crystals probably involves plastic flow and strain hardening even for very small strains. The internal friction is sensitive to the condition of the surface and to annealing [ 16]. Unannealed single crystals may show losses approaching those of polycrystalline material. In the latter, internal friction arises from a number of distinct sources, including (1) losses within the individual crystals, (2) losses at the surfaces of the . . .

Connective tissue mechanics is highly nonlinear, exhibits a strong Poisson's effect, and is associated with significant collagen fiber re-arrangement. Although the general features of the stress-strain behavior have been discussed extensively, the Poisson's effect received less attention. In general, the relationship between the microscopic fiber network mechanics and the macroscopic experimental observations remains poorly defined. The objective of the present work is to provide additional insight into this relationship. To this end, results from models of random collagen networks are compared with experimental data on reconstructed collagen gels, mouse skin dermis, and the human amnion. Attention is devoted to the mechanism leading to the large Poisson's effect observed in experiments. The results indicate that the incremental Poisson's contraction is directly related to preferential collagen orientation. The experimentally observed downturn of the incremental Poisson's ratio at larger strains is associated with the confining effect of fibers transverse to the loading direction and contributing little to load bearing. The rate of collagen orientation increases at small strains, reaches a maximum, and decreases at larger strains. The peak in this curve is associated with the transition of the network deformation from bending dominated, at small strains, to axially dominated, at larger strains. The effect of fiber tortuosity on network mechanics is also discussed, and a comparison of biaxial and uniaxial loading responses is performed.

Presently, experimental evidence for extremely high strain-sensitivity of dissipation in rocks and similar microstructured materials is obtained both in laboratory and field conditions, in particular observations of pronounced amplitude modulation of the radiation of high-stability seismo-acoustic sources by tidal deformations of rocks with typical strains ~ 10−8. Such data indicate the presence of some thresholdless in amplitude and very efficient mechanism of strain-dependent dissipation. Conventionally, its origin is discussed in the context of frictional or adhesion-hysteretic loss at cracks in rocks. However, such dissipation mechanisms are not relevant to weak perturbations with displacements smaller than atomic size. Here, we revise thresholdless thermoelastic loss in dry cracks and viscous loss in saturated cracks taking into account wavy asperities typical of real cracks, which can create elongated (strip-like) contacts or almost closed "waists" in cracks. Thermoelastic loss at these contacts can be very efficient. Besides, the state of such contacts can already be strongly perturbed by the average strain which yet practically does not change the mean opening of the entire crack. Thus the dissipation localized at such contacts can be significantly affected by quite small average strain (e.g., 10−8), which is usually believed to be unable to produce any appreciable effect on the dissipation. Next, for liquid-saturated cracks, the presence of inner elongated asperities also drastically changes the character of squirt-type viscous dissipation. Velocity gradients and consequently the dissipation are localized in the vicinity of the nearly-closed waists which almost harness the liquid flow in the crack. This dissipation can be comparable in magnitude with viscous dissipation at the entire crack with smooth interface, but the decrement maximum is strongly shifted downwards on the frequency axis. Since near the waist the gap is much smaller than the average crack opening, this local dissipation can also exhibit giant strain sensitivity. These mechanisms of near-contact loss suggest interpretation for observations of pronounced amplitude modulation of seismo-acoustic waves by tidal strains.

Interferometric Gravitational Wave Detectors, coming online lin late 2000, look for small space strains, leading to apparent motions of test masses of 10-19 m or less; isolation from other forces is crucial. They require a formidable vibration isolation level in a frequency range between few Hz and few kHz. The off-band residual motion must be kept below 10-12 m not to saturate the phase sensors. These exceptional requirements are met, in all degrees of freedom, with a chain of active and passive filters. The key isolation mechanism is the use of mechanical oscillators above their resonant frequencies, pendula horizontally, springs vertically. Very high quality pendular suspensions are needed at the mirror level to limit the thermal noise from fluctuations in the dissipation mechanisms. Off-band electromagnetic actuators on or near the mirror keep its magnitude of attenuation in the longitudinal direction. To provide the bulk of the attenuation, virtually all in the vertical direction, they are suspended from Seismic Noise Attenuation Systems. Attenuation filters, either active or passive, are chained, each providing 2 or 3 orders of magnitude of attenuation. Passive attenuation is obtained with springs and pendula. The vertical is the toughest direction to deal with because the oscillators also fight against gravity. The vertical attenuation requirements, although orthogonal to the beam direction, are only slightly less stringent than the vertical ones due to cross-couplings (Earth curvature is the source of one of them). High internal damping springs organized in hierarchical stacks are used in most early designs. More advanced designs increasingly rely on chains of filters equipped with high quality cantilever springs driven to low resonant frequencies by different mechanisms. The Quality Factors of each resonance are actively and/or passively spoiled at the chain suspension point. IN the latest designs, Ultra Low Frequency Oscillators filter out the microseismic and other low frequency perturbations. This paper addresses one approach to achieving the required seismic isolation level.

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

Biological soft tissues are intrinsically viscoelastic materials which play a significant role in affecting the activity of cells. As potential artificial alternatives, double-network (DN) gels, however, are pure elastic and mechanically time independent. The viscoelasticization of DN gels is an urgent challenge in enabling DN gels to be used for advanced development of biomaterial applications. Herein, we demonstrate a simple approach to regulate the viscoelasticity of tough double-network (DN) hydrogels by forming sulfonate-metal coordination. Owing to the dynamic nature of the coordination bonds, the resultant hydrogels possess highly viscoelastic, mechanical time-dependent, and self-recovery properties. Rheological measurements are performed to investigate the linear dynamic mechanical behavior at small strains. The tensile tests and cyclic tensile tests are also systematically performed to evaluate the rate-dependent large deformation mechanical behaviors and energy dissipation behaviors of various ion-loaded DN hydrogels. It has been revealed based on the systematic analysis that robust strong sulfonate-Zr4+ coordination interactions not only serve as dynamic crosslinks imparting viscoelastic rate-dependent mechanical performances, but also strongly affect the relative strength of the first PAMPS network, thereby increasing the yielding stress σy and the fracture stress at break σb and reducing the stretch ratio at break λb. It is envisioned that the viscoelasticization of DN gels enables versatile applications in the biomedical and engineering fields.

Sustainable adhesives with simultaneous high strength and excellent ductility are important but still challenging. Here, an epoxy adhesive with simultaneously strong, tough, and sustainable features by designing a new hierarchical energy‐dissipated cross‐linking structure is reported: an annular structure containing quadruple internal B─N coordination boronic esters (AQBN). Under small strain, the AQBN structure is rigid to enhance the tensile strength of the adhesive. With medium strain applied, its non‐coplanar ring and internal quadruple B─N coordination can orderly open, providing mechanical energy dissipation and excellent ductility. Therefore, this multilevel structure can endow epoxy adhesives with both high strength and toughness. Besides, the rapid dynamic exchange and internal B─N coordination in boronic esters enable the adhesives to have good sustainability and environmental tolerance, respectively. As a result, the epoxy adhesives exhibit simultaneous high adhesive strength (18.5 MPa), outstanding toughness (work of debonding of 29692 N m−1), good chemical/physical recyclability, and excellent environmental reliability (humidity, water, high/ultralow temperature, and organic solvents). This work provides a new strategy for balancing the strength, toughness, and sustainability of adhesives.

Biological hydrogels can become many times stiffer under deformation. This unique ability has only recently been realised in fully synthetic gels. Typically, these networks are composed of semi-flexible polymers and bundles and show such large mechanical responses at very small strains, which makes them particularly suitable for application as strain-responsive materials. In this work, we introduced strain-responsiveness by crosslinking the architecture with a multi-functional virus-like particle. At high stresses, we find that the virus particles disintegrate, which creates an (irreversible) mechanical energy dissipation pathway, analogous to the high stress response of fibrin networks. A cooling-heating cycle allows for re-crosslinking at the damaged site, which gives rise to much stronger hydrogels. Virus particles and capsids are promising drug delivery vehicles and our approach offers an effective strategy to trigger the release mechanically without compromising the mechanical integrity of the host material.

Czochralski-grown single crystals of stoiehiometric spinel (MgAl2O4) have been deformed at 1780–1980°C along several different directions. Crystals deformed parallel to [111], [112], and [123] slip on the {111} 01 system, while {101} 01 slip is observed for the specimens deformed parallel to 001. There is thus little plastic anisotropy between these two systems. Work softening occurs at large strains (γ = 0·015) for both types of specimens; the resolved steady-state flow stress for the specimens with {111} slip planes is independent of the number of slip systems being activated and is 7sim;50% higher than for {101} slip planes. Dislocation structures on the primary and cross-slip planes have been studied as a function of strain for specimens deformed at 1800°C. At small strains (σ ∼0·005), a high density of long straight edge dislocations belonging to the expected primary slip system is observed. At higher strains (γ ∼0·01), a random three-dimensional dislocation network has evolved, correspondin...