Multi-scale computational homogenization of heterogeneous materials

The mechanical response of a heterogeneous medium results from the interactions of mechanisms spanning several length scales. The computational homogenization method captures direct influence of underlying constituents and morphology with a numerically efficient framework. This study reviews the performance of first order computational homogenization technique with a flat punch indentation problem. Results obtained are benchmarked against those using direct numerical simulations (DNS) with full microstructural details. It is shown that the computational homogenization method is able to capture structural response adequately, even for constituent materials with nonlinear behavior. However, the first order computational homogenization method becomes problematic when localized macroscopic deformation occurs. In this context, some re- cent trends addressing the issues are discussed.

On the Behavior of Mechanical Stress Fields at Indentation of Materials with Residual Stresses

It is an obvious fact that residual stresses can have a detrimental effect on the mechanical integrity of structures. Measuring such stresses can often be a tedious task and for that reason sharp indentation testing has been proposed as an alternative for this purpose. Correlation between global indentation properties and residual stresses has been studied quite frequently, and a solid foundation has been laid down concerning this issue. Empirical, or semi-empirical, relations have been proposed yielding results of quite good accuracy. Further progress and mechanical understanding regarding this matter will require a more in-depth understanding of the field variables at this particular indentation problem and this is the subject of the present study. In doing so, finite element simulations are performed of sharp indentation of materials with and without residual stresses. Classical Mises plasticity and conical indentation are considered. The main conclusion from this study is that the development of stresses in materials with high or medium-sized compressive residual stresses differs substantially from a situation with tensile residual stresses, both as regards the level of elastic deformation in the contact region and the sensitivity of such stresses. Any attempt to include such stress states in a general correlation effort of indentation quantities is therefore highly unlikely to be successful.

Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response

The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.

Simulation of constrained elastic curves and application to a conical sheet indentation problem

We consider variational problems that model the bending behavior of curves that are constrained to belong to given hypersurfaces. Finite element discretizations of corresponding functionals are justified rigorously via $\varGamma $-convergence. The stability of semi-implicit discretizations of gradient flows is investigated, which provide a practical method to determine stationary configurations. A particular application of the considered models arises in the description of conical sheet deformations.

Indentation of a prestretched strain-stiffening elastomer

Strain-stiffening behavior of materials such as rubberlike materials and biological soft tissues is an important phenomenon. In this paper, we proposed a surface Green’s function tensor to describe the strain-stiffening behavior of the stretchable elastomer based on the Gent constitutive model. The surface Green’s function tensor of the Gent constitutive model can be recovered to that of neo-Hookean model, and applied to the indentation problem with a flat-ended cylindrical indenter. The relation between the indentation force and strain-stiffening parameter is analytically derived for equi-biaxial prestretched elastomers. The study shows that the strain-stiffening of the elastomer has a great impact on indentation behaviors, especially for the cases of large prestretches. For a given indentation depth, the indentation force decreases with the increase of the strain-stiffening parameter. For a given stiffening parameter, the indentation force increases with the increase of the prestretches. The proposed surface Green’s function tensor has also potential applications in other fields, such as wetting, cell migration, self-assembly on strain-stiffening materials, etc.

The Effect of the Size of the Sample on Results of Indentation Tests

The results of the numerical solution of the problem about interaction between spherical stamp and weakly compressible elastic specimen are investigated. The nonlinear generalization of linear elastic Hencky model is used as a constitutive relation. The results of the indentation problem solution are in good agreement with experimental data. The tests were performed on the kinematical loading fixture. The influence of geometrical parameters of specimen during indentation test on stress strain state and macro response are investigated.

Extraction of mechanical properties of materials through deep learning from instrumented indentation

Instrumented indentation has been developed and widely utilized as one of the most versatile and practical means of extracting mechanical properties of materials. This method is particularly desirable for those applications where it is difficult to experimentally determine the mechanical properties using stress–strain data obtained from coupon specimens. Such applications include material processing and manufacturing of small and large engineering components and structures involving the following: three-dimensional (3D) printing, thin-film and multilayered structures, and integrated manufacturing of materials for coupled mechanical and functional properties. Here, we utilize the latest developments in neural networks, including a multifidelity approach whereby deep-learning algorithms are trained to extract elastoplastic properties of metals and alloys from instrumented indentation results using multiple datasets for desired levels of improved accuracy. We have established algorithms for solving inverse problems by recourse to single, dual, and multiple indentation and demonstrate that these algorithms significantly outperform traditional brute force computations and function-fitting methods. Moreover, we present several multifidelity approaches specifically for solving the inverse indentation problem which 1) significantly reduce the number of high-fidelity datasets required to achieve a given level of accuracy, 2) utilize known physical and scaling laws to improve training efficiency and accuracy, and 3) integrate simulation and experimental data for training disparate datasets to learn and minimize systematic errors. The predictive capabilities and advantages of these multifidelity methods have been assessed by direct comparisons with experimental results for indentation for different commercial alloys, including two wrought aluminum alloys and several 3D printed titanium alloys.

Expanding Cavity Model Combined With Johnson–Cook Constitutive Equation for the Dynamic Indentation Problem

The indentation formed on a metallic component by the high-velocity impingement of a small object can fracture the component, and this is known as foreign object damage. In this type of dynamic indentation, it is necessary to consider the effects of work hardening, strain rate hardening, and thermal softening in the impinged material. In this study, in order to consider these effects, the expanding cavity model based on a spherical formulation is modified via the Johnson–Cook constitutive equation for the dynamic indentation problem. Additionally, an equation is developed based on energy conservation and the modified expanding cavity model to predict the size of the indentation formed by an impingement of a solid sphere (EPIS). The distributions of equivalent plastic strain, equivalent plastic strain rate, temperature, and equivalent von Mises stress obtained via the expanding cavity model were in good agreement with the data obtained from the finite element analysis (FEA). Furthermore, it was demonstrated that EPIS accurately predicted the indentation size formed on various metallic materials at several impingement velocities in the range of 50–300 m/s. Consequently, EPIS can be effectively applied to an impingement problem of a hard sphere onto a sufficiently thick ductile material within 300 m/s without any help of FEA.