Construction of anisotropic polyconvex energies and applications to thin shells

In computer simulations where constitutive equations are considered anisotropic polyconvex energies can preferably be used because the existence of minimizers is then automatically guaranteed. In this work we investigate the capability to simulate anisotropy effects of anisotropic thin shells using polyconvex anisotropic energies. The construction of the considered polyconvex transversely isotropic energy is based on specific structural tensors. The iterative enforcement of the zero normal stress condition at the integration points allows the consideration of arbitrary three-dimensional constitutive equations. As a representative example we compare results for isotropic and anisotropic plates.     

Mixed-mode K-calculations in anisotropic materials

The crack tip stress singularities for anisotropic materials are derived using the finite difference method and they are compared to other derivations. Subsequently, these asymptotic fields are implemented into a local stress method, enabling the determination of the mixed-mode stress intensity factor distribution along an arbitrary crack in an anisotropic material. Comparison with the interaction integral method shows that the local method yields excellent results and exhibits a better robustness with respect to irregular meshes.     

Fracture analysis of anisotropic materials using enriched crack tip elements

Enriched finite element methodology, which employs special crack tip elements, is extended for cracks in anisotropic materials. Enrichment formulation is described briefly and three validation examples using single crystal, directionally solidified, and orthotropic material properties are presented to demonstrate the accuracy and effectiveness of the methodology. In addition to validation examples, the effect of material anisotropy on stress intensity factors is investigated using the common compact tension specimen and the results are compared to the ASTM solution for isotropic materials. It is shown that the effect of anisotropy on the computed stress intensity factors can be significant, depending on the degree of anisotropy, material orientation, and a/W ratio in the compact tension specimen geometry.     

On evolution equations for elastic deformation and the notion of hyperelasticity

For elastic-plastic and elastic-viscoplastic materials it is possible to introduce an evolution equation for an elastic deformation measure. Also, it is possible to develop constitutive equations for which the stress and strain energy are functions of elastic deformation only, the stress is determined by a derivative of the strain energy function and the associated material response is rate-independent and non-dissipative in the absence of the rate of inelasticity. Yet, these equations do not necessarily exhibit hyperelastic response in the elastic range. The objective of this paper is to emphasize the importance of satisfying an additional condition that requires the work done between two configurations to be insensitive to the history and rate of total deformation. This work condition places restrictions on the evolution equation which ensure that the integrated elastic deformation measure is a function of total deformation only. Also, it is argued that there is no need to complicate the evolution equation for elastic deformation to accommodate alternative strain measures since the nonlinearities of these strain measures can be absorbed into the form of the strain energy function.     

Shear induced loss of saturation in a fluid infused swollen hyperelastic cylinder

We discuss a continuum model for the absorption and redistribution of fluid in swollen elastomers due to mechanical loading; and in particular distinguish between systems that are saturated with liquid and those that are not. To this end we consider a boundary value problem of radial displacement combined with azimuthal shear for an annular cylinder consisting of a fluid infused hyperelastic media. In the absence of load the elastomer deforms by free swelling, giving a homogeneous expansion in which the imbibed fluid is uniformly distributed. This free swelling is described by the theory of Flory and Rehner. We then consider the effect of various boundary displacements and tractions so as to study how this alters the uniform fluid distribution. This problem was previously considered by Wineman and Rajagopal for the case in which the lateral surfaces maintained the radius associated with free swelling. Here we consider certain generalizations in which the lateral surfaces may undergo not only a prescribed relative twist, but also radial displacement. This permits fluid to either enter or exit the cylinder. A numerical method is invoked to solve these boundary value problems using representative material parameters. Certain boundary tractions generate an overall volume increase after the free swelling. If the amount of available fluid is limited, this gives rise to the possibility of complete fluid absorption, whereupon the system is no longer saturated. It is found that the overall mechanical response after loss of saturation is stiffer than it would be if the system had remained saturated.     

Torsion of a fiber reinforced hyperelastic cylinder for which the fibers can undergo dissolution and reassembly

In previous work, the authors have developed a theory for treating microstructural changes in fiber reinforced hyperelastic materials. In this theory, fibers undergo dissolution as a result of increasing elongation and then reassemble in a direction defined as part of the model. Processes in which the fibers reassemble in the direction of maximum principal stretch of the matrix were specifically considered. This model was previously illustrated for various cases of homogeneous deformation. The present work studies the implications of the model during the non-homogeneous deformation of axial stretch and torsion of a circular solid cylinder composed of an isotropic matrix and families of helically wound fibers. It is shown that the process of fiber dissolution and reassembly produces complex morphological changes in the fibrous structure and hence, in the response of the cylinder. Such events can give rise to an outer layer of material in which the fibers have undergone dissolution and reassembly. The interface between this region and the as yet unaltered core material can then move radially inward as axial stretch and/or twist increase. Gradual reassembly of the fibers with increasing stretch and twist changes their contributions to the torque and axial force and their helical orientation. Different sequences of axial stretch and twist result in different morphologies in the fibrous structure.     

Limit analysis of composite materials with anisotropic microstructures: A homogenization approach

A nonlinear mathematical programming approach together with the finite element method and homogenization technique is developed to implement kinematic limit analysis for a microstructure and the macroscopic strength of a composite with anisotropic constituents can be directly calculated. By means of the homogenization theory, the classical kinematic theorem of limit analysis is generalized to incorporate the microstructure - Representative Volume Element (RVE) chosen from a periodic composite/heterogeneous material. Then, using an associated plastic flow rule, a general yield function is directly introduced into limit analysis and a purely-kinematic formulation is obtained. Based on the mathematical programming technique, the finite element model of microstructure is finally formulated as a nonlinear programming problem subject to only one equality constraint, which is solved by a direct iterative algorithm. The calculation is entirely based on a purely-kinematical velocity field without calculation of stress fields. Meanwhile, only one equality constraint is introduced into the nonlinear programming problem. So the computational cost is very modest. Both anisotropy and pressure-dependence of material yielding behavior are considered in the general form of kinematic limit analysis. The developed method provides a direct approach for determining the macroscopic strength domain of anisotropic composites and can serve as a powerful tool for microstructure design of composites.     

An anisotropic elastic formulation for configurational forces in stress space

A new variational principle for an anisotropic elastic formulation in stress space is constructed, the Euler–Lagrange equations of which are the equations of compatibility (in terms of stress), the equilibrium equations and the traction boundary condition. Such a principle can be used to extend recently obtained configurational balance laws in stress space to the case of anisotropy.     

An approach to finite static axially symmetric deformation of a hyperelastic membrane

Summary A simple approach to the problem of finite quasi-static axially symmetric deformation of an isotropic incompressible hyperelastic membrane is presented. Lagrangian type equilibrium equations, expressed in terms of the Biot stresses, are used along with constitutive relations expressing the principal components of Biot stress in terms of the principal stretches. Numerical results, obtained from the application of a finite element method to the governing equilibrium equations and constitutive relations, are presented for two problems, and for two different strain energy functions. It is shown how the proposed equilibrium equations can be obtained from a variational principle, and the variational principle is also used to obtain approximate solutions.     

Laser-induced breakdown spectroscopy analysis of energetic materials

A number of energetic materials and explosives have been studied by laser-induced breakdown spectroscopy (LIBS). They include black powder, neat explosives such as TNT, PETN, HMX, and RDX (in various forms), propellants such as M43 and JA2, and military explosives such as C4 and LX-14. Each of these materials gives a unique spectrum, and generally the spectra are reproducible shot to shot. We observed that the laser-produced microplasma did not initiate any of the energetic materials studied. Extensive studies of black powder and its ingredients by use of a reference spectral library have demonstrated excellent accuracy for unknown identification. Finally, we observed that these nitrogen- and oxygen-rich materials yield LIBS spectra in air that have correspondingly different O:N peak ratios compared with air. This difference can help in the detection and identification of such energetic materials.     

Tangent modulus matrix for finite element analysis of hyperelastic materials

Summary In this study, using the VEC operator [1], compact expressions are formulated for the tangent modulus matrix of hyperelastic materials, in particular elastomers, using Lagrangian coordinates. Compressible, incompressible, and near-compressible materials are considered. Expressions are obtained for the corresponding finite element tangent stiffness matrices. It is observed that the incremental stress-strain relations should be considered anisotropic. Numerical procedures based on Newton iteration are sketched. The limiting case of small strain is developed. Finally, the tangent modulus matrix is presented for the Mooney-Rivlin material, with application to the rubber rod element.     

An alternative model for anisotropic elasticity based on fabric tensors

Motivated by the mechanical analysis of multiphase or damaged materials, a general approach relating fabric tensors characterizing microstructure to the fourth rank elasticity tensor is proposed. Using a Fourier expansion in spherical harmonics, the orientation distribution function of a positive, radially symmetric microstructural property is approximated by a scalar and a symmetric, traceless second rank tensor. Following this approximation, a general expression of the elastic free energy potential is derived from representation theorems for anisotropic scalar functions. Based on a homogeneity assumption for the elastic constitutive law with respect to the microstructural property, a particular elasticity model is developed that involves three independent constants beside the fabric tensors. Strict positive definiteness of the corresponding elasticity tensor is ensured under explicit conditions on the independent constants for arbitrary fabric tensors.     

An approach to systematic materials selection

An approach to systematic materials selection is believed to be suitable for both manual and computer aided materials selection. The procedure is divided into two stages; the discriminating and the optimising parts. In the former, all requirements on properties which do not influence the final sizing of the component in question are considered. From this stage a number of materials survives which are used in the optimisation stage. To rank the materials, merit parameters are introduced. This is exemplified for pressurized containers. With the help of the merit parameters it is possible to decide which material is the most suitable in a given situation. A procedure is also proposed from which the influence of the design parameters on the selection can be evaluated. This is exemplified by analysing how the pressure in the containers affects the choice of optimum material.     

Computational up-scaling of anisotropic swelling and mechanical behavior of hierarchical cellular materials

The hygro-mechanical behavior of a hierarchical cellular material, i.e. growth rings of softwood is investigated using a two-scale micro-mechanics model based on a computational homogenization technique. The lower scale considers the individual wood cells of varying geometry and dimensions. Honeycomb unit cells with periodic boundary conditions are utilized to calculate the mechanical properties and swelling coefficients of wood cells. Using the cellular scale results, the anisotropy in mechanical and swelling behavior of a growth ring in transverse directions is investigated. Predicted results are found to be comparable to experimental data. It is found that the orthotropic swelling properties of the cell wall in thin-walled earlywood cells produce anisotropic swelling behavior while, in thick latewood cells, this anisotropy vanishes. The proposed approach provides the ability to consider the complex microstructure when predicting the effective mechanical and swelling properties of softwood.     

The adhesive contact of a flat punch on a hyperelastic substrate subject to a pull-out force or a bending moment

In the last decade, instrumented macro- and nano-indentation tests have become useful tools for probing mechanical properties of materials. In this context, little in-depth work has been done for soft bio-materials like biofilms, tissues, gels, cells, etc. Such materials show large elastic strains when subjected to relatively fast loading (excluding viscoelastic phenomena). The present work focuses on the adhesive contact of such materials. Flat punches of circular imprint are used and are subjected to pullout normal forces or bending moments. The materials are modeled as hyperelastic (fast-testing conditions), using the Mooney-Rivlin (M-R) strain energy density function. We examine both incompressibility and compressibility issues. The contact problem of half-space materials interacting with rigid indenters is solved explicitly for moderate deformations and is compared with finite element results. Experiments are conducted with an artificial material (gel, reinforced with talc powder) that is modeled as a hyperelastic material. The present work is expected to extend indentation testing to important technologies like medical applications (health monitoring of tissues) and food industries (quality control of various production stages). It is concluded that the adhesive contact can be used to estimate the initial elastic modulus of soft substrates, but not all the material constants required by the hyperelastic models.     

A method of two-scale analysis with micro-macro decoupling scheme: application to hyperelastic composite materials

The aim of this study is to propose a strategy for performing nonlinear two-scale analysis for composite materials with periodic microstructures (unit cells), based on the assumption that a functional form of the macroscopic constitutive equation is available. In order to solve the two-scale boundary value problems (BVP) derived within the framework of the homogenization theory, we employ a class of the micro-macro decoupling scheme, in which a series of numerical material tests (NMTs) is conducted on the unit cell model to obtain the data used for the identification of the material parameters in the assumed constitutive model. For the NMTs with arbitrary patterns of macro-scale loading, we propose an extended system of the governing equations for the micro-scale BVP, which is equipped with the external material points or, in the FEM, control nodes. Taking an anisotropic hyperelastic constitutive model for fiber-reinforced composites as an example of the assumed macroscopic material behavior, we introduce a tensor-based method of parameter identification with the ‘measured’ data in the NMTs. Once the macro-scale material behavior is successfully fitted with the identified parameters, the macro-scale analysis can be performed, and, as may be necessary, the macro-scale deformation history at any point in the macro-structure can be applied to the unit cell to evaluate the actual micro-scale response.     

Electro-analysis of energetic materials

Cyclic voltammetric studies of triaminoguanidine nitrate (TAGN), 3,3'-hydrazino bis(bis[6,6'-(3,5-dimethylpyrazol-lyl])-1,2,4,5-tetrazine (HBPT), 4,6-dinitrobenzofuroxan (DNBF) and 3,3'-diamino-4,4'-azoxyfurazan (DAAF) were carried out at different pH conditions in 50% aqueous acetonitrile using glassy carbon electrode. Optimum pH was selected for individual compounds. Influence of scan rate and concentration on the voltammetric response were studied in optimum pH. The number of electron transferred was determined by controlled potential coulometry. All compounds undergo diffusion controlled electrochemical reaction. Based on cyclic voltammetric results, differential pulse and square wave voltammetric methods have been developed for the analytical determination. Instrumental parameters such as initial scan potential, amplitude, pulse increment, pulse period, pulse width and frequency were studied. Optimum experimental conditions for each compound were obtained. After fixing optimum conditions, the effect of concentration was studied and calibration plot was arrived. These plots can be used to determine the traces of the above said four energetic materials.     

A numerical approach to the analysis of failure modes in anisotropic plates

This study is motivated by the attempt to characterize failure modes of silicon chips commonly used in electronic industries. Previous experimental investigations provided the failure probability of dies made of a single-crystal and produced a large variety of crack patterns, but were not able to elucidate the link between defect distributions and crack initiation and propagation. To get some insight in the fracture activation and propagation mechanisms, we resort to finite element analyses and adopt an explicit methodology for crack tracking, based on the self-adaptive insertion of cohesive elements into a coherent mesh of solid elements. Finite kinematics material models with anisotropic features for both bulk and cohesive surfaces are employed to describe the behavior of single-crystal silicon plates undergoing a particular bending test up to failure. The cohesive model adopted in the calculation is fully anisotropic and newly formulated to accomplish the present study. Numerical simulations considering different material properties were able to ascertain the effects of particular flaws on failure modes of brittle silicon plates.     

An approach to micro-macro modeling of heterogeneous materials

 A micro-macro strategy suitable for modeling the mechanical response of heterogeneous materials at large deformations and non-linear history dependent material behaviour is presented. When using this micro-macro approach within the context of finite element implementation there is no need to specify the homogenized constitutive behaviour at the macroscopic integration points. Instead, this behaviour is determined through the detailed modeling of the microstructure. The performance of the method is illustrated by the simulation of pure bending of porous aluminum. The influence of the spatial distribution of heterogeneities on the overall macroscopic behaviour is discussed by comparing the results of micro-macro modeling for regular and random structures. Received 9 July 2000