The aim of this contribution is to explain in a straightforward manner how Bayesian inference can be used to identify material parameters of material models for solids. Bayesian approaches have ...already been used for this purpose, but most of the literature is not necessarily easy to understand for those new to the field. The reason for this is that most literature focuses either on complex statistical and machine learning concepts and/or on relatively complex mechanical models. In order to introduce the approach as gently as possible, we only focus on stress–strain measurements coming from uniaxial tensile tests and we only treat elastic and elastoplastic material models. Furthermore, the stress–strain measurements are created artificially in order to allow a one-to-one comparison between the true parameter values and the identified parameter distributions.
We discuss Bayesian inference for the identification of elastoplastic material parameters. In addition to errors in the stress measurements, which are commonly considered, we furthermore consider ...errors in the strain measurements. Since a difference between the model and the experimental data may still be present if the data is not contaminated by noise, we also incorporate the possible error of the model itself. The three formulations to describe model uncertainty in this contribution are: (1) a random variable which is taken from a normal distribution with constant parameters, (2) a random variable which is taken from a normal distribution with an input-dependent mean, and (3) a Gaussian random process with a stationary covariance function. Our results show that incorporating model uncertainty often, but not always, improves the results. If the error in the strain is considered as well, the results improve even more.
This contribution discusses surrogate models that emulate the solution field(s) in the entire simulation domain. The surrogate uses the most characteristic modes of the solution field(s), in ...combination with neural networks to emulate the coefficients of each mode. This type of surrogate is well known to rapidly emulate flow simulations, but rather new for simulations of elastoplastic solids. The surrogate avoids the iterative process of constructing and solving the linearized governing equations of rate-independent elastoplasticity, as necessary for direct numerical simulations or (hyper-)reduced-order-models. Instead, the new plastic variables are computed only once per increment, resulting in substantial time savings. The surrogate uses a recurrent neural network to treat the path dependency of rate-independent elastoplasticity within the neural network itself. Because only a few of these surrogates have been developed for elastoplastic simulations, their potential and limitations are not yet well studied. The aim of this contribution is to shed more light on their numerical capabilities in the context of elastoplasticity. Although more widely applicable, the investigation focuses on a representative volume element, because these surrogates have the ability to both emulate the macroscale stress-deformation relation (which drives the multiscale simulation), as well as to recover all microstructural quantities within each representative volume element.
Numerous materials are essentially structures of discrete fibres, yarns or struts. Considering these materials at their discrete scale, one may distinguish two types of intrinsic randomness that ...affect the structural behaviours of these discrete structures: geometrical randomness and material randomness. Identifying the material randomness is an experimentally demanding task, because many small fibres, yarns or struts need to be tested, which are not easy to handle. To avoid the testing of hundreds of constituents, this contribution proposes an identification approach that only requires a few dozen of constituents to be tested (we use twenty to be exact). The identification approach is applied to artificially generated measurements, so that the identified values can be compared to the true values. Another question this contribution aims to answer is how precise the material randomness needs to be identified, if the geometrical randomness will also influence the macroscale behaviour of these discrete networks. We therefore also study the effect of the identified material randomness to that of the actual material randomness for three types of structures; each with an increasing level of geometrical randomness.
•The material parameters of fibres in stochastic networks are considered to be realisations from some multivariate PDF.•An identification scheme is proposed that only requires the testing of a few fibres in order to identify the parameters of the multivariate material parameter PDF.•This is achieved by Bayes' theorem which provides a regularisation.•The approach is applied to damageable and elastoplastic fibers.•Numerous forward studies are included that assess the required accuracy of the identification with respect to the amount of geometrical randomness of fibre networks.
•Strain energy densities employing the Seth–Hill strain tensors can be used to describe isotropic and anisotropic compressible elasticity for moderately large deformations.•One parameter ...distinguishes which member of the Seth–Hill family is employed, but in practice this parameter is not used for fitting in the nonlinear regime.•Three generalizations based on the Seth–Hill strain tensors, are proposed to enable a true fitting of the material response in the nonlinear regime.•The generalisations reduce to the conventional hyperelasticity if a single strain tensor is used.•The generalisations reduce to the isotropic and anisotropic infinitesimal theory for small deformations, meaning that the identification of the Young’s moduli, Poisson’s ratios and shear moduli remains unaffecteed.
Strain energy densities based on the Seth–Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for relatively large deformations. Since one parameter distinguishes which strain tensor of the Seth–Hill family is used, one has in theory the possibility to fit the material response in the nonlinear regime. Most often for compressible deformations however, this parameter is selected such that the Hencky strain tensor is recovered, because it yields rather physical stress-strain responses. Hence, the response in the nonlinear regime is in practise not often tailored to match experimental data. To ensure that elastic responses in the nonlinear regime can more accurately be controlled, this contribution proposes three generalisations that combine several Seth–Hill strain tensors. The generalisations are formulated such that the stress-strain responses for infinitesimal deformations remain unchanged. Consequently, the identification of the Young’s moduli, Poisson’s ratios and shear moduli is not affected. 3D finite element simulations are performed for isotropy and orthotropy, with an emphasis on the identification of the new material parameters.
This contribution discusses Bayesian inference (BI) as an approach to identify parameters in viscoelasticity. The aims are: (i) to show that the prior has a substantial influence for viscoelasticity, ...(ii) to show that this influence decreases for an increasing number of measurements and (iii) to show how different types of experiments influence the identified parameters and their uncertainties. The standard linear solid model is the material description of interest and a relaxation test, a constant strain-rate test and a creep test are the tensile experiments focused on. The experimental data are artificially created, allowing us to make a one-to-one comparison between the input parameters and the identified parameter values. Besides dealing with the aforementioned issues, we believe that this contribution forms a comprehensible start for those interested in applying BI in viscoelasticity.
Lattice models employing trusses and beams are suitable to investigate the mechanical behavior of woven fabrics. The discrete features of the mesostructures of woven fabrics are naturally ...incorporated by the discrete elements of lattice models. In this paper, a lattice model for woven materials is adopted which consists of a network of trusses in warp and weft direction, which represent the response of the yarns. Additional diagonal trusses are included that provide a resistance against relative rotation of the yarns. The parameters of these families of discrete elements can be separately identified from tensile experiments in three in-plane directions which correspond with the orientations of the discrete elements. The lattice model and the identification approach are applied to electronic textile. This is a fabric in which conductive wires are incorporated to allow the embedment of electronic components such as light-emitting diodes. The model parameters are established based on tensile tests on samples of the electronic textile. A comparison between the experimental results of an out-of-plane punch test and the simulation results shows that the lattice model and its characterization procedure are accurate until extensive biaxial tensile deformation occurs.
Slender constituents are present in many structures and materials. In associated mechanical models, each slender constituent is often described as a beam. Contact between beams is essential to ...incorporate in mechanical models, but associated contact frameworks are only demonstrated to work for beams with circular cross sections. Only two studies have shown the ability to treat contact between beams with elliptical cross sections, but those frameworks are limited to point-wise contact, which narrows their applicability. This contribution presents initial results of a framework for shear-deformable beams with elliptical cross sections if contact occurs along a line or at an area (instead of at a point). This is achieved by integrating a penalty potential over one of the beams’ surfaces. Simo–Reissner geometrically exact beam elements are employed to discretize each beam. As the surface of an assembly of such beam elements is discontinuous, a smoothed surface is introduced to formulate the contact kinematics. This enables the treatment of contact for large sliding displacements and substantial deformations.
Lattice models and discrete networks naturally describe mechanical phenomena at the mesoscale of fibrous materials. A disadvantage of lattice models is their computational cost. The quasicontinuum ...(QC) method is a suitable multiscale approach that reduces the computational cost of lattice models and allows the incorporation of local lattice defects in large-scale problems. So far, all QC methods are formulated for conservative (mostly atomistic) lattice models. Lattice models of fibrous materials however, often require non-conservative interactions. In this paper, a QC formulation is derived based on the virtual-power of a non-conservative lattice model. By using the virtual-power statement instead of force-equilibrium, errors in the governing equations of the force-based QC formulations are avoided. Nevertheless, the non-conservative interaction forces can still be directly inserted in the virtual-power QC framework. The summation rules for energy-based QC methods can still be used in the proposed framework as shown by two multiscale examples.
Laminated paperboard is often used as a packaging material for products such as toys, tea and frozenfoods. To make the paperboard packages appealing for consumers, the fold lines must be both neat ...and undamaged. The quality of the folds depends on two converting processes: the manufacture of fold lines (creasing) and the subsequent folding. A good crease contains some delamination, initiated during creasing, to reduce the bending stiffness and to prevent the board from breaking during folding. However, for boards of high grammage breaking of the top layer is nevertheless a frequent problem. The mechanisms that operate in the creasing zone during creasing and folding, and that may thus result in breaking of the top layer, are studied in this contribution on the basis of idealized small-scale creasing and folding experiments. However, since experimental observations are only limited means to study the paperboard’s behavior, a mechanical model is proposed to obtain more detailed insight. Although the material and delamination descriptions used in the mechanical model are both relatively straightforward, comparisons between the model and the experimental data show that the model predicts the paperboard’s response well. The mechanical model shows – in combination with experimental strain fields – that multiple delaminations are initiated in the shear regions. Moreover, only the mechanical model reveals the mechanism that is responsible for the failure of the top layer if a crease is too shallow. Finally, the model also demonstrates that not only delamination but also plastic behavior must occur during creasing if breaking of the top layer is to be avoided.
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