Elastic Cauchy Research Articles

A consistent algorithm for finite-strain visco-hyperelasticity and visco-plasticity of amorphous polymers

Shared rheology elements of soft tissues and additive-manufactured amorphous thermoplastics benefit from a common visco-hyperelastic and visco-plastic constitutive framework. For finite-strain and general visco/elastic/plastic constitutive laws, allowing full anisotropy, we use the Kröner–Lee decomposition of the deformation gradient combined with Mandel stress-based yield function. Relatively weak conditions are necessary for specific laws to be incorporated in this framework. For example, stress-like variables present in visco-elasticity are removed from the list of constitutive unknowns. Constitutive iteration is performed for the elastic Cauchy–Green tensor Ce and the plastic multiplier increment Δλ. The source is here the right Cauchy–Green tensor provided by any discretization. For the integration of the flow law we adopt a scaled/squared series approximation of the matrix exponential. The exact Jacobian of the second Piola–Kirchhoff stress is determined with respect to this source, consistent with the integrator. The resulting system is produced by symbolic source-code generation for each yield function and hyperelastic strain-energy density function with each of the viscous terms added. The constitutive system is solved by a damped Newton–Raphson algorithm with the corresponding stress error being extensively assessed with stress error maps. A cellular beam structure is analyzed and results compared with experiments from our group. Some of the symbolic calculations and closed-form solutions are beyond what is manually achievable and therefore the symbolic sources for this work are made available in a source code repository.

Anisotropic hyperelastic/plastic behavior on stress-constrained thin structures by iterating on the elastic Cauchy–Green tensor

Thin-walled structures impose zero-stress constraints at free faces. These are in general nonlinear functions of strain and, in the elasto-plastic case, non-smooth. Under these circumstances, the conjugate strain components become constitutive unknowns. By iterating for the elastic Cauchy–Green tensor in the 6−dimensional space, we directly obtain a system that includes the zero-stress constraints. Coupling of the stress and strain constitutive unknowns is possible due to the use of a convenient equation arrangement. This contrasts with the standard decoupling schemes. Herein, we use the Kröner–Lee decomposition of the deformation gradient to obtain a differential–algebraic system which includes stress constraints in analogous form to additional yield conditions. The source is the right Cauchy–Green tensor. The main contribution is the development of a thin-walled implicit integrator for any hyperelastic case and any yield function. Integration is based on a backward-Euler method for the flow law complemented by the solution of a yield condition and the stress constraints. We make use of the elastic Mandel stress construction, which is power-consistent with the plastic strain rate. Two complete fully orthotropic worked examples are presented.

A finite strain incremental-secant homogenization model for elasto-plastic composites

This paper presents a finite strain extension of the incremental-secant mean-field homogenization (MFH) formulation for two-phase elasto-plastic composites. The formulation of the local finite strain elasto-plastic constitutive equations of each phase is based on a multiplicative decomposition of the deformation gradient as suggested by Simo in (Computer Methods in Applied Mechanics and Engineering, 99(1):61–112, 1992.). The latter has proposed algorithms which preserve the classical return mapping schemes of the infinitesimal theory by using principal Kirchhoff stresses and logarithmic eigenvalues of the left elastic Cauchy–Green strain. Relying on this property, we show that, by considering a quadratic logarithmic free energy and J2-flow theory at the local level, infinitesimal strain incremental-secant MFH is readily extended to finite strains. The proposed formulation and corresponding numerical algorithms are then presented. Finally, the predictions are illustrated with several numerical simulations which are verified against full-field finite element simulations of composite cells, demonstrating that the micro-mechanically based approach is able to predict the influence of the micro-structure and of its evolution on the macroscopic properties in a very cost-effective manner.

Numerical Simulation of the Viscohyperelastic Behaviour of PET near the Glass Transition Temperature

The presentation deals with the non linear strongly elastic and viscous behaviour of poly ethylene terephthalate near the glass transition temperature and biaxially stretched at high strain rates representative of the injection stretch blow moulding process. A non linear visco-hyperelastic model inspired from [1] and identified from the experimental results of the equi-biaxial tension test [2], have been developed and presented in [3] is implemented into a finite element code developed with Matlab. The thermal behaviour modelling, identification and simulation has also been managed. First, a numerical simulation of 2D plane stress case has been performed involving 2 fields (global velocity V and elastic Cauchy Green tensor Be). Rectangular finite elements with quadratic and linear interpolations have been employed for velocity and the elastic left Cauchy Green tensor. Second, an axi symmetric formulation involving 4 fields (global velocity V, lagrange multiplier p associated with the global incompressibility condition, and multiplier q associated with the incompressibility of the elastic part) has been performed using rectangular elements. Degree of interpolation have been tested for all possible combinations to test the LBB like condition. Both simulations are compared with equi biaxial or sequential biaxial testing in order to reproduce the strain hardening effect and the self-heating observed. The final goal of this work is to perform the free blowing simulation to compare with experimental data. Therefore, we should solve an iterative procedure for a thermo-mechanical equation. At each time step, a four-field approach is adopted for the mechanical part, and a classical heat transfer equation is discretised for the thermal part.

Identification of internal cracks in a three-dimensional solid body via Steklov–Poincaré approaches

This Note deals with the identification of internal planar cracks inside a three-dimensional elastic body via two approaches relying on domain decomposition using elastostatic measurements. These approaches consist in recasting the problem in terms of primal or dual Steklov–Poincaré equations. The primal approach is a straightforward continuation to the elastic Cauchy problem of the work presented in J. Ben Abdallah (2007) [1] which is devoted to the Cauchy problem for the scalar Laplace equation. The numerical performances of these formulations are compared.

Generalized inner bending continua for linear fiber reinforced materials

This paper deals with the effective behaviour of elastic materials periodically reinforced by linear slender elastic inclusions. Assuming a small scale ratio ε between the cell size and the characteristic size of the macroscopic deformation, the macro-behaviour at the leading order is derived by the homogenization method of periodic media. Different orders of magnitude of the contrast between the shear modulus of the material μ m and of the reinforcement μ p are considered. A contrast μ m / μ p of the order of ε 2 leads to a full coupling between the beam behaviour of the inclusions and the elastic behaviour of the matrix. Under transverse motions, the medium behaves at the leading order as a generalized continuum that accounts for the inner bending introduced by the reinforcements and the shear of the matrix. Instead of the second degree balance equation of elastic Cauchy continua usually obtained for homogenized composites, the governing equation is of the fourth degree and the description differs from that of a Cosserat media. This description degenerates into, (i) the usual continua behaviour of elastic composite materials when O( μ m / μ p ) ⩾ ε, (ii) the usual Euler–Bernoulli beam behaviour when O( μ m / μ p ) ⩽ ε 3. The constitutive parameters are derived and can be computed or estimated from simplified geometries. Simple criteria are given to identify the appropriate model for real reinforcements under given loadings.

Geometry and balance of hyperstresses

Surface hyperstress is introduced along a layer with vanishing thichness, a layer across which two second-grade elastic Cauchy bodies are glued. Surface balances are obtained by making use of the relative power of actions and of invariance requirements for the Lagrangian. The action of bulk hyperstress on vacancies, linear inclusions and cracks is also evaluated.

Objective constitutive relations in elasticity and viscoelasticity

The compatibility between the objectivity principle and affine constitutive equations for the elastic Cauchy and Piola-Kirchhoff stress tensors with non-zero residual stress is examined. It is found that the Cauchy stress is allowed to be only a constant tensor, proportional to the identity tensor, while the Piola-Kirchhoff stress may be a linear function on the deformation gradient thus generalizing previous results by Fosdick and Serrin. The same conclusions are arrived at also by starting from viscoelasticity. Finally, in the case of Maxwell-like materials, the solutions to the objective evolution equations are shown to be objective functionals.

The elastic Cauchy relation and ionicity

The elastic Cauchy relation and ionicity