A. Dorfmann

A pseudo-elastic model for loading, partial unloading and reloading of particle-reinforced rubber

Particle-reinforced rubbers exhibit a marked stress softening during unloading after loading in uniaxial tension tests, i.e. the stress on unloading is significantly less than that on loading at the same stretch. This hysteretic behaviour is not accounted for when the mechanical properties are represented in terms of a strain-energy function, i.e. if the material is modelled as hyperelastic. In this paper a theory of pseudo-elasticity is used to model loading, partial or complete unloading and the subsequent reloading and unloading of reinforced rubber. The basis of the model is the inclusion in the energy function of a variable that enables the energy function to be changed as the deformation path changes between loading, partial unloading, reloading and any further unloading. The dissipation of energy, i.e. the difference between the energy input during loading and the energy returned on unloading is accounted for in the model by the use of a dissipation function, the form of which changes between unloading, reloading and subsequent unloading.

On Variational Formulations in Nonlinear Magnetoelastostatics

Two new variational principles for nonlinear magnetoelastostatics are derived. Each is based on use of two independent variables: the deformation function and, in one case the scalar magnetostatic potential, in the other the magnetostatic vector potential. The derivations are facilitated by use of Lagrangian magnetic field variables and constitutive laws expressed in terms of these variables. In each case all the relevant governing equations, boundary and continuity conditions emerge. These principles have a relatively simple structure and therefore offer the prospect of leading to finite-element formulations that can be used in the solution of realistic boundary-value problems.

A constitutive model for muscle properties in a soft-bodied arthropod

In this paper, we examine the mechanical properties of muscles in a soft-bodied arthropod under both passive and stimulated conditions. In particular, we examine the ventral interior lateral muscle of the tobacco hornworm caterpillar, Manduca sexta, and show that its response is qualitatively similar to the behaviour of particle-reinforced rubber. Both materials are capable of large nonlinear elastic deformations, show a hysteretic behaviour and display stress softening during the first few cycles of repeated loading. The Manduca muscle can therefore be considered as different elastic materials during loading and unloading and is best described using the theory of pseudo-elasticity. We summarize the basic equations for transversely isotropic pseudo-elastic materials, first for general deformations and then for the appropriate uniaxial specialization. The constitutive relation proposed is in good agreement with the experimental data for both the passive and the stimulated conditions.

A micro-mechanical model for the response of filled elastomers at finite strains

Abstract Constitutive equations are developed for the isothermal response of particle-reinforced elastomers at finite strains. A rubbery polymer is treated as a network of chains bridged by junctions. A strand between two junctions is thought of as a series of inextensible segments linked by bonds. Two stable conformations are ascribed to a bond: flexed and extended. Deformation of a specimen induces transition of bonds from their flexed conformation to the extended conformation. A concept of trapped entanglements is adopted, according to which not all junctions are active in the stress-free state. Under straining, some entanglements are transformed from their passive (dangling) state to the active state, which results in a decrease in the average length of a strand. Stress–strain relations for an elastomer and kinetic equations for the rate of transition of bonds from their flexed conformation to the extended conformation are derived by using the laws of thermodynamics. Simple phenomenological equations are suggested for the evolution of the number of active entanglements. The model is determined by five adjustable parameters which are found by fitting experimental data in uniaxial tensile tests. Fair agreement is demonstrated between the results of numerical simulation and observations for a polysulfide elastomer reinforced with polystyrene particles and two natural rubber vulcanizates with different cross-linkers.

Shear, compressive and dilatational response of rubberlike solids subject to cavitation damage

In this paper we examine the change in material response, in particular the dilatational response, due to cavitation damage arising from tensile hydrostatic stresses of sufficient magnitude. A general discussion of stress softening and cavitation is followed by a description of some new experimental results concerning the change in response in hydrostatic tension or compression or in shear due to cavitation damage. In hydrostatic tension there is a progressive reduction in the value of the tensile bulk modulus of the material during loading and significant stress softening on unloading. As a result of the cavitation damage the tensile bulk modulus in the natural configuration is reduced. Ultimately, failure of the material occurs at sufficiently large hydrostatic tension, typically when the volume increase locally exceeds a critical value, of the order of 2–3%. However, the compressive bulk modulus is unaffected by the cavitation damage. Moreover, it is also found that the shear modulus is likewise unchanged by cavitation. The experimental data are used to develop a theoretical model, based on the concept of pseudo-elasticity, to describe these phenomena. Specifically, the dilatational part of the strain-energy function of an elastic material depends on a damage parameter which provides a means for switching the form of the strain-energy function, thereby reflecting the stress softening associated with unloading. A good correspondence between the theory and the data is obtained.

Scaling of caterpillar body properties and its biomechanical implications for the use of a hydrostatic skeleton

SUMMARY Caterpillars can increase their body mass 10,000-fold in 2 weeks. It is therefore remarkable that most caterpillars appear to maintain the same locomotion kinematics throughout their entire larval stage. This study examined how the body properties of a caterpillar might change to accommodate such dramatic changes in body load. Using Manduca sexta as a model system, we measured changes in body volume, tissue density and baseline body pressure, and the dimensions of load-bearing tissues (the cuticle and muscles) over a body mass range from milligrams to several grams. All Manduca biometrics relevant to the hydrostatic skeleton scaled allometrically but close to the isometric predictions. Body density and pressure were almost constant. We next investigated the effects of scaling on the bending stiffness of the caterpillar hydrostatic skeleton. The anisotropic non-linear mechanical response of Manduca muscles and soft cuticle has previously been quantified and modeled with constitutive equations. Using biometric data and these material laws, we constructed finite element models to simulate a hydrostatic skeleton under different conditions. The results show that increasing the internal pressure leads to a non-linear increase in bending stiffness. Increasing the body size results in a decrease in the normalized bending stiffness. Muscle activation can double this stiffness in the physiological pressure range, but thickening the cuticle or increasing the muscle area reduces the structural stiffness. These non-linear effects may dictate the effectiveness of a hydrostatic skeleton at different sizes. Given the shared anatomy and size variation in Lepidoptera larvae, these mechanical scaling constraints may implicate the diverse locomotion strategies in different species.

The stress-strain response and ultimate strength of filled elastomers

Abstract A constitutive model is derived for the mechanical behavior of reinforced elastomers at finite strains. A polymer is treated as a rigid-rod network, whose rupture is tantamount to breakage of chains treated as bond scission. Adjustable parameters in the stress–strain relations are found by fitting observations in tensile tests for filled and unfilled ethylene–octene copolymers. It is demonstrated that the model correctly describes stress–strain curves up to the break points. We analyze the effects of temperature, the degree of crystallinity and the filler content on Youngs modulus and the ultimate strain per bond. It is shown that the dependences of material constants on the volume fraction of carbon black are substantially altered at the critical filler contents which correspond to the percolation thresholds found by dc conductivity measurements.

The nonlinear viscoelastic response of carbon black-filled natural rubbers

Abstract Three series of tensile relaxation tests are performed on natural rubber filled with various amounts of carbon black. The elongation ratio varies in the range from λ =2.0 to 3.5. Constitutive equations are derived for the nonlinear viscoelastic behavior of filled elastomers. Applying a homogenization method, we model a particle-reinforced rubber as a transient network of macromolecules bridged by junctions (physical and chemical cross-links, entanglements and filler clusters). The network is assumed to be strongly heterogeneous at the meso-level: it consists of passive regions, where rearrangement of chains is prevented by surrounding macromolecules and filler particles, and active domains, where active chains separate from temporary nodes and dangling chains merge with the network as they are thermally agitated. The rate of rearrangement obeys the Eyring equation, where different active meso-domains are characterized by different activation energies. Stress–strain relations for a particle-reinforced elastomer are derived by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data. It is demonstrated that the filler content strongly affects the rearrangement process: the attempt rate for separation of strands from temporary nodes increases with elongation ratio at low fractions of carbon black (below the percolation threshold) and decreases with λ at high concentrations of filler.

Experimental and computational aspects of cavitation in natural rubber

Abstract General constitutive equations for hyperelastic materials are based on the first law of thermodynamics whereby the total strain energy function is expressed either in terms of strain invariants or principal stretches. For most applications the strain energy functional does not need to include dilatational components. However, the pressure–volume relationship for nearly incompressible materials must be explicitly accounted for when rubber components are highly constrained. Thus, the hyperplastic response needs to be expressed in terms of dilatational and deviatoric components. Experimental evidence has been reviewed to show that rubber is subjected to a loss of stiffness attributed to cavitation damage when subjected to a hydrostatic tensile stress state. The critical pressure is identified for which microscopic material imperfections will tear open to form internal bubbles and cracks. Cavitation damage in rubber is associated with a significant reduction in the bulk modulus. Thus, a variable bulk modulus can best be used to describe the behaviour of rubber when cavitation damage occurs. The introduction of a cavitation damage modulus is suggested as a simple approach to represent realistically the mechanics of cavitation in rubber solids.

Finite viscoelasticity of filled rubbers: the effects of pre-loading and thermal recovery

Constitutive equations are derived for the viscoelastic behavior of filled elastomers at isothermal loading with finite strains. A particle-reinforced rubber is thought of as a composite where regions with low concentrations of junctions between chains are randomly distributed in the bulk material. The onset of these inclusions is associated with the inhomogeneity in spatial distribution of a cross-linker during the mixing process. With reference to the theory of transient networks, the time-dependent response of an elastomer is modelled as thermally activated processes of breakage and reformation of chains in domains with low concentrations of junctions, whereas junctions in the bulk medium are treated as permanent. Stress-strain relations are developed by using the laws of thermodynamics. Adjustable parameters in the constitutive equations are found by fitting experimental data in tensile relaxation tests for several grades of unfilled and carbon black filled natural rubber. It is demonstrated that (i) the average relaxation time noticeably grows with the elongation ratio, which is explained by mechanically-induced crystallization of strands, and (ii) the relaxation spectrum of a filled elastomer is not affected by mechanical pre-loading and thermal recovery at elevated temperatures.