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In this work, stress and displacement fields were computed around a crack tip in the case of nearly incompressible and isotropic neo-Hookean material. The constitutive equation was linearized, so that the Cauchy stress tensor could be written as a sum of two components: the linear response in term of elastic Hooke’s law and the nonlinear one. Based on this decomposition, an asymptotic analysis has been developed, the fields of linear elastic fracture mechanics (LEFM-theory) are the zero-order terms of the asymptotic expansion. The validity of the proposed theory has been checked in the case of a mode-I crack problem. A numerical model was constructed using a finite element method. It has shown that the computed fields arising from this theory are qualitatively in agreement with those of the finite element simulations.

We analyze the compression of a right cylinder made of an elastomeric material, sliding on a rigid plate in the framework of linear elasticity theory. Lubrication seems to reduce the heterogeneous effects of lateral bulging so that the deformations can be considered as “quasi-homogeneous.” As a result, we show that the ratio between the transversal deformation and axial strain is not a Poisson’s ratio but a Poisson’s function.

In the present paper, a study of the weak shock waves propagation into thermohyperelastic solids is presented using the theory of singular surfaces. The equation of motion of the weak shock waves, which is obtained by the later theory, reduces to an eigenvalue problem. The propagation speeds of weak shock waves are obtained from the eigenvalues of the isentropic acoustic tensor for a particular state of deformation. The constitutive equation of a near-incompressible thermohyperelastic solid is considered. This constitutive model derives from a Helmholtz free energy which does not seem to be used in published scientific literature. The shock waves speeds are expressed in the framework of standard loadings: simple uniaxial tension, biaxial stretching, and uniform dilatation. The material seems to be always stable, and no singularity of the isentropic acoustic tensor has been found which corresponds to a critical state.

We propose a new hyper-elastic model that is based on the standard invariants of Green–Cauchy. Experimental data reported by Treloar (Trans. Faraday Soc. 40:59, 1944) are used to identify the model parameters. To this end, the data of uni-axial tension and equi-bi-axial tension are used simultaneously. The new model has four material parameters, their identification leads to linear optimisation problem and it is able to predict multi-axial behaviour of rubber-like materials. We show that the response quality of the new model is equivalent to that of the well-known Ogden six parameters model. Thereafter, the new model is implemented in FE code. Then, we investigate the inflation of a rubber balloon with the new model and Ogden models. We compare both the analytic and numerical solutions derived from these models.

The purpose of this paper is to investigate rheology and hydrodynamic behavior of different mass concentration of bentonite suspensions in pipe flow. Bentonite suspensions exhibited shear thinning and yield stress non-Newtonian rheological behavior. Most of the data reported are for the laminar and turbulent flow, data are also included the start-up flow development. The axial velocity distribution was determined using a Ultrasonic Pulsed Doppler Velocimetry technique. The experimental results show a progressive destructuration of the suspensions in the entrance region due to the segregation of the structural elements of the fluid and tend to a steady state and a fully developed laminar pipe flow. In the majority of cases, axisymmetric flow is observed for both laminar and turbulent flow conditions. Also an asymmetry characterizes the axial velocity profiles in the laminar-turbulent transitional flow. In the laminar regime, the flow parameters were fitted by using the Herschel–Bulkley model. For the turbulent flow, the Dodge and Metzner model was applied to fit the experimental data.

The rheological equation of a standard linear solid, i.e., the Zener model, is thermodynamically consistent. Thus, it was often used as a starting point for the development of nonlinear viscoelastic models, especially for elastomers. The basic idea of this paper is a generalization of the one-dimensional fractional constitutive equation of the Zener model to large strains. To reduce the number of material parameters of differential models based on the concept of the internal variables and to avoid integral constitutive equations, we develop a differential model based on the concept of dual stress and strain tensors and their derivatives. To this end, we select two couples of dual stress and strain tensors that have been used in finite elasticity. Then we obtain two constitutive models of incompressible isotropic materials called M1 and M2. We show that the M1 model is not suitable for describing the viscoelastic behavior of elastomers. To improve the predictions of the M2 model, we assume that the material is thixotropic. Therefore, the ratio of the relaxation and creep time depends on deformation. Experimental results show that this ratio may be represented as a function of the first invariant of the Cauchy–Green strain tensor. This yields a new constitutive equation whose material parameters were identified using experimental data on relaxation loadings in the literature. Next, we show that the model is able to predict the experimental data for combined loads of tension–torsion. Consequently, the model seems to be efficient at predicting the multiaxial visco-hyperelastic behavior of elastomers. The main advantage of the current model is that it has a differential form with relatively few parameters and is mathematically convenient.

The purpose of this paper is to compute the relaxation and creep functions from the data of shear complex modulus, G∗(iν). The experimental data are available in the frequency window ν∈[νmin ,νmax ] in terms of the storage G′(ν) and loss G″(ν) moduli. The loss factor \(\eta( \nu) = \frac{G''( \nu )}{G'(\nu )}\) is asymmetrical function. Therefore, a five-parameter fractional derivative model is used to predict the complex shear modulus, G∗(iν). The corresponding relaxation spectrum is evaluated numerically because the analytical solution does not exist. Thereby, the fractional model is approximated by a generalized Maxwell model and its rheological parameters (Gk,τk,N) are determined leading to the discrete relaxation spectrum G(t) valid in time interval corresponding to the frequency window of the input experimental data. Based on the deterministic approach, the creep compliance J(t) is computed on inversing the relaxation function G(t).

The hyperelastic behavior of rubber-like materials can be derived from the strain energy density function W which is determined from experimental results obtained in homogeneous strain, such as uniaxial tension, pure shear and equibiaxial tension. In the particular case of carbon filled reinforced rubber, the shape of the W function can be identified from a minimum number of experimental tests. The form of the W function has been proposed in the literature only for large strain domain. We present in this paper a new strain energy density function W for carbon black reinforced rubber which can predict the mechanical behavior of the material at small and large strain.

We propose an identification method for the determination of elastomer behaviour which combines strain field obtained by digital image analysis and the optimisation of an aproximated stress field adapted to the specimen geometry. The chosen loading test is a biaxial elongation performed on a cross specimen. In doing so, we turn the problem of heterogeneity into an advantage: it allows the simultaneous identification of several states of deformation: uni‐axial, planar and biaxial tension. The stress field is approximated by the sum of a homogeneous field and an additional field. The latter satisfies the boundary conditions on the free edge and decreases when entering the sample. The characteristic decreasing length is optimized so that the approximate field globally verifies the momentum equation. It is then possible to identify the hyperelastic potential by calculating explicitly the two derivatives ƒ = ∂W/∂I1 and g = ∂W/∂I2

International audience; We present a micro-mechanical model based on the network theory for the description of the elastic response of rubber-like materials at large strains. The material microstructure is characterized by chain-like macromolecules linked together at certain points; therefore an irregular three-dimensional network is formed. The material behaviour at the micro-level is usually described by means of statistical mechanics. Using certain assumptions for the certain distributions, one arrives at a continuum mechanical model of finite elasticity. However, the macromolecules interactions are neglected usually in these approaches. In the present contribution, we propose to add the effect of the interactions between chains of the cross-linked network. Following Arruda and Boyce (1993, 2000) [31,2], a cubic unit cell is defined where the entanglements fluctuations are localised in the corners of the cubic sub unit cell. These entanglements are linked by chains which ensure the interactions between the chains of idealized network (without interactions). These interactions can be represented by chains which are located in the principal directions of the cubic sub unit cell in undeformed state. We assume the probability densities which describe the free chain response of idealized network, and, the chain of constraints networks are independent. Then, the free-energy of the entire network is obtained by adding the free-energies of the free idealized (without interactions) and constraints (due to the chains interactions) networks. The constraint network reduces to four of the three-chain model of James and Guth (1943) [4] in undeformed state. Therefore, the free-energy of constraint network is obtained using the standard three-chain model, and, the free-energy of the free idealized network is constructed by means of the eight-chain model. The constitutive model involves five physical material parameters, namely, the shear modulus at small strains (μ0), the numbers of links that form the macromolecular chain of the eight-chain, and three-chain models (N8,N3) respectively, a micro-macro variable Ki, and, non-dimensional parameters (η,ρ). In order to determine the material parameters, the Langevin function in the single chain configuration is replaced by its first order Padé approximant [see, Cohen (1991) [5]; Perrin (2000) [6]], and, the material parameters are identified. The excellent predictive performance of the proposed model is shown by comparative to various available experimental data of homogeneous tests. However, the present model requires a validation because the relationship between the micro and macro levels needs to be clarified. Indeed, the identification of the physical parameters (μf,μc,N8,N3) from experimental results data at micro is hoped in order to simulate the macroscopic (i.e. bulk) behaviour of the material.