Constitutive Description Of Material Research Articles

Incorporating uncertainty in stress‐strain data acquisition: extended model‐free data‐driven identification

AbstractBased on the increasing ability to collect and store large amounts of data and information, data‐driven methods have recently gained importance in the context of computational mechanics. Compared to the traditional approach of defining a suitable constitutive model and fit the parameters to experimental data, these methods aim to capture complex material behavior without the assumption of a certain material formulation. Distinguished are model‐based data‐driven methods, leading to an approximation of the constitutive material description for example, by neural networks, and model‐free data‐driven methods. The approach of data‐driven computational mechanics (DDCM), introduced by Kirchdoerfer and Ortiz (2016), enables to circumvent any material modeling step by directly incorporating material data into the structural analysis. A basic prerequisite for both types of data‐driven methods is a large amount of data representing the material behavior, in solid mechanics consisting of stresses and strains. Obtaining these databases numerically by multiscale approaches is computationally expensive and requires the definition of lower scale models. In case of an experimental characterization, constitutive descriptions are generally required to compute the stress states corresponding to displacement fields, for example, identified by full‐field measurement techniques, such as digital image correlation (DIC). The method of data‐driven identification (DDI), introduced in Leygue et al. (2018) based on the principles of DDCM, enables the determination of detailed information about the constitutive behavior based on displacement fields and applied boundary conditions without a specific material model. Stresses corresponding to given strains are identified by iteratively clustering and enforcing equilibrium. The algorithm has shown to be applicable to synthetic as well as real data taking linear and non‐linear material behavior into account. Generalized polymorphic uncertainty models, resulting as a combination of aleatoric and epistemic uncertainty models, are utilized to take variability, imprecision, inaccuracy and incompleteness of data into account. The consideration of uncertain material properties by data‐driven simulation approaches leads to the requirement of data sets representing uncertain material behavior. In this contribution, different sources of uncertainty occurring within DDI of stress‐strain relations are addressed and an efficient method for the identification of data sets representing uncertain material behavior based on the concept of DDI is proposed. In order to demonstrate the developed methods, numerical examples are carried out.

Experimental characterization of cyclic behaviour of pure lead: Temperature sensitivity and strain-rate effects

Proton beam pulses with an energy of 20 GeV/c collide with a pure-lead based target installed in the neutron Time-Of-Flight facility (n_TOF) at the European Laboratory for Particle Physics (CERN). The interaction between the proton beam and lead produces neutrons via spallation mechanism and results in a rapid temperature increase and propagation of stress waves. To evaluate the material response in such challenging conditions, a reliable thermo-mechanical characterization is necessary for the calibration of an appropriate constitutive model for pure lead that is valid under cyclic plasticity and high temperature. In this work, the experimental bases for the development of such constitutive material description are lied. Starting with metallurgical characterization, the typical grain size of the material was initially investigated as well as any variations in the metallurgical features. The grains appeared to have an equivalent size ranging from 2 to 6 mm. Then, static tensile tests were conducted at room temperature and different strain-rates from 10−1 to 10−4 s−1. The obtained results were crucial for optimizing the specimen geometry and test setup for the subsequent cyclic tests. Tension-compression cyclic tests were performed at different strain amplitudes from 0.1 to 1.5%, and at three different temperatures (room temperature, 90 °C and 150 °C). The strain amplitudes were controlled by an extensometer and the strain field evolution during the test was recorded by means of 2D DIC.

Comparison of Flow Behaviors at High Temperature of Two Press Hardening Boron Steels with Different Hardenability

The deformation behavior and the constitutive description of materials are important in the design and numerical simulation of manufacturing processes. In this work, the rheological behavior of two quenchable boron steels was derived in the range of interest for the Press Hardening (PH) process. For this study, the following steps were performed: (i) Design of the specimen geometry adopted during the hot tensile tests in the Gleeble®3180 physical simulator. (ii) Performance of hot tensile tests at temperatures between 750 and 850 °C and strain rates of 0.01–1 s−1. (iii) The description of the material constitutive model by means of the Arrhenius-type equation and the comparison between the experimental and predicted data. Firstly, the method was validated on USIBOR®1500 boron steel by means of a comparison with existing literature data, and good agreement with other scientific works was found. Finally, this method was also adopted on USIBOR®2000, a steel that belongs to the new generation of Ultra-High-Strength Steels (UHSS). A comparison between the rheological behavior of these two steels was carried out. The results show that USIBOR®2000 steel exhibits greater strength compared to USIBOR®1500 steel. The Arrhenius constitutive model well predicts the flow behavior of both steels.

Virgin Passive Colon Biomechanics and a Literature Review of Active Contraction Constitutive Models

The objective of this paper is to present our findings on the biomechanical aspects of the virgin passive anisotropic hyperelasticity of the porcine colon based on equibiaxial tensile experiments. Firstly, the characterization of the intestine tissues is discussed for a nearly incompressible hyperelastic fiber-reinforced Holzapfel–Gasser–Ogden constitutive model in virgin passive loading conditions. The stability of the evaluated material parameters is checked for the polyconvexity of the adopted strain energy function using positive eigenvalue constraints of the Hessian matrix with MATLAB. The constitutive material description of the intestine with two collagen fibers in the submucosal and muscular layer each has been implemented in the FORTRAN platform of the commercial finite element software LS-DYNA, and two equibiaxial tensile simulations are presented to validate the results with the optical strain images obtained from the experiments. Furthermore, this paper also reviews the existing models of the active smooth muscle cells, but these models have not been computationally studied here. The review part shows that the constitutive models originally developed for the active contraction of skeletal muscle based on Hill’s three-element model, Murphy’s four-state cross-bridge chemical kinetic model and Huxley’s sliding-filament hypothesis, which are mainly used for arteries, are appropriate for numerical contraction numerical analysis of the large intestine.

Assessment of surface and bulk-dominated methodologies to measure critical resolved shear stresses in hexagonal materials

Crystallographic slip in hexagonal metals involves a number of geometrically distinct slip families characterized by their slip direction and slip plane (basal, prismatic, and pyramidal). Owing to the low symmetry of hexagonal lattices, each of these slip families only have a few symmetrically equivalent slip systems (family members). Furthermore, different slip families become active at different resolved shear stress, i.e., they have different critical resolved shear stress values (CRSS). The plastic anisotropy of hexagonal materials makes the numerical prediction of their plastic behavior challenging and depends critically on the knowledge of CRSS values. The present contribution assesses the reliability of three proposed methods (with additional variations) to quantify CRSS values of the different hexagonal slip families. Those methods (a to c) rely on: (a) the statistics of observed surface slip traces in a (slightly) deformed polycrystal; (b) an iterative adjustment of CRSS values until a simulated single crystal indentation matches the corresponding experiment in terms of load–displacement response and residual surface topography of the indent; (c) in-situ high-energy X-ray diffraction to measure the evolution of resolved stress (from lattice strains) in grains for which single-family slip is probable based upon specific lattice reorientation conditions. Virtual experiments are performed on synthetic microstructures such that the (extracted) CRSS values resulting from simulating the different methodologies can be rigorously compared against the (target) CRSS values that are installed in the phenomenological constitutive material description used in the simulations. The resulting CRSS values of methods (a) exhibit a strong dependence on, and deterioration with, decreasing level of slip trace observability, which is an uncertain quantity in experimental measurements. For the inverse indentation method (b), the predicted CRSS values are within 8 % of their reference CRSS values for the two investigated cases. The high-energy X-ray diffraction method (c) most reliably determines CRSS values for basal and prism slip, but lacks a strict grain selection criterion to assess pyramidal slip.

Assessment of Surface and Bulk-Dominated Methodologies to Measure Critical Resolved Shear Stresses in Hexagonal Materials

Crystallographic slip in hexagonal metals involves a number of geometrically distinct slip families characterized by their slip direction and slip plane (basal, prismatic, and pyramidal). Owing to the low symmetry of hexagonal lattices, each of these slip families only have a few symmetrically equivalent slip systems (family members). Furthermore, different slip families become active at different resolved shear stress, i. e., they have different critical resolved shear stress values (CRSS). The plastic anisotropy of hexagonal materials renders the numerical prediction of their plastic behavior challenging and depends critically on the knowledge of CRSS values. The present contribution assesses the reliability of three proposed methods (with additional variations) to quantify CRSS values of the different hexagonal slip families. Those methods (a to c) rely on: (a) the statistics of observed surface slip traces in a (slightly) deformed polycrystal; (b) an iterative adjustment of CRSS values until a simulated single crystal indentation matches the corresponding experiment in terms of load-displacement response and residual surface topography of the indent; (c) in-situ high-energy X-ray diffraction to measure the evolution of resolved stress (from lattice strains) in grains for which single family slip can be deduced from specific lattice reorientation conditions. Virtual experiments are performed on synthetic microstructures to assess how reliably each method recovers prescribed CRSS values using a phenomenological constitutive material description. The resulting CRSS values of methods (a) exhibit a strong dependence on, and deteriorates with decreasing level of slip trace observability, which is an uncertain quantity in experimental measurements. For the inverse indentation method (b), the predicted CRSS values are within 8% of their reference CRSS values for the two investigated cases. The high-energy X-ray diffraction method (c) most reliably determines CRSS values for basal and prism slip, but lacks a strict grain selection criterion to assess pyramidal slip.

Numerical characterisation of uncured elastomers by a neural network based approach

This research paper contributes to the model-free characterisation of elastic and inelastic materials. The introduced material description is based on neural networks for the constitutive stress-strain-relationship and is an alternative to a classical constitutive material description. This approach is an efficient method to represent the material behaviour numerically. The performance of the novel model-free formulation is exemplary shown for uncured natural rubber. One major advantage of the presented approach is the capability to use it for the representation of a wide range of materials.The novel model-free characterisation consists of a neural network that is coupled to the so-called micro-sphere approach. This formulation was developed for the characterisation of rubber like materials and takes the micro-structure of the material into account. As a benefit of this coupling, the neural networks, representing the stress-stretch-dependency, have to be exploited analysed only in onedimensional direction. In the first instance, the derivation is introduced for an Artificial Neural Network to yield a pure elastic description. Subsequently, the model-free approach is expanded in order to represent inelastic material behaviour as well. This extended formulation is obtained via a Recurrent Neural Network.Finally, uncured natural rubber material is described by the derived numerical approach. The model-free characterisation is validated by the finite element simulation of material tests. A complex forming of a rubber block into a mould is basis for a final validation of the model-free description.

DAMASK: the Düsseldorf Advanced MAterial Simulation Kit for studying crystal plasticity using an FE based or a spectral numerical solver

The solution of a continuum mechanical boundary value problem requires a constitutive response that connects deformation and stress at each material point. Such connection can be regarded as three separate hierarchical problems. At the top-most level, partitioning of the (mean) boundary values of the material point among its microstructural constituents and the associated homogenization of their response is required, provided there is more than one constituent present. Second, based on an elastoplastic decomposition of (finite strain) deformation, these responses follow from explicit or implicit time integration of the plastic deformation rate per constituent. Third, to establish the latter, a state variable-based constitutive law needs to be interrogated and its state updated.The Düsseldorf Advanced MAterial Simulation Kit (DAMASK) reflects this hierarchy as it is built in a strictly modular way. This modular structure makes it easy to add additional constitutive models as well as homogenization schemes. Moreover it interfaces with a number of FE solvers as well as a spectral solver using an FFT.We demonstrate the versatility of such a modular framework by considering three scenarios: Selective refinement of the constitutive material description within a single geometry, component-scale forming simulations comparing di_erent homogenization schemes, and comparison of representative volume element simulations based on the FEM and the spectral solver.

A note on the role of dilatation in the fracture of viscoelastic elastomer

Fracture models addressing crack propagation in viscoelastic materials typically draw on cohesive force distributions coupled with a linearly viscoelastic constitutive material description. While the resulting formulation of the crack speed as a function of applied loads provides good agreement with measurements on polyurethane rubber as well as three other rubbery solids studied by A. N. Gent on the more global scale, the size of the cohesive zone required to match theory and measurements is unreasonably small. Although this glaring discrepancy has not deterred the use of the theory for viscoelastic fracture, it has remained a troublesome question on the wider applications to polymer fracture. The present note revisits this issue and draws on results developed during the past decade to explain this discrepancy via the effect of stress-induced dilatation on the relaxation or retardation times of viscoelastic solids.

A micro‐macro approach to the failure of rubber‐like materials

AbstractRubbery polymers are modelled with hyperelastic constitutive laws by which the material can be stretched indefinitely or up to a certain locking stretch value. However, no material can sustain indefinite stretch and store infinite energy. Failure occurs after a threshold value for stress and/or energy is attained. Within this context, an approach for failure analysis of rubber–like materials based on the constitutive description of material is presented. The model is based on a hyperelastic material law representing the energy stored in rubber network due to the chain conformations connected in series to an interatomic bond potential representing the energy stored in the polymer chain due to the interatomic displacement. For the representation of the micro–macro transition in terms of non–affine kinematics, micro–sphere model is utilized. Morse potential is used for the description of the interatomic bond, which describes the energetic contribution to rubber–like materials and governs the failure of the polymer chain. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Cracks in rubber

The onset of crack propagation in rubber is studied computationally by using the softening hyperelasticity approach. The basic idea underlying the approach is to limit the capability of a material model to accumulate energy without failure. The latter is done by introducing a limiter for the strain energy density, which results from atomic/molecular considerations and can be interpreted as the average bond energy or the failure energy. Including the energy limiter in a constitutive description of material it is possible to enforce softening and, consequently, allow tracking the onset of structural instability corresponding to the onset of material failure. Specifically, initiation of crack propagation is studied in the case of a thin sheet of a rubber-like solid under the hydrostatic tension. The large deformation neo-Hookean material model enhanced with the energy limiter is used for finding the critical tension corresponding to the onset of static instability of the sheet, i.e. the onset of fracture propagation. The influence of the crack sharpness and length on the critical load is analyzed. It is found that material is sensitive to the crack sharpness when the shear modulus is significantly greater than the average bond energy. The sensitivity declines when the value of the shear modulus approaches the value of the failure energy. Roughly speaking, softer materials are less sensitive to cracks than more brittle materials where the brittleness is defined as a ratio of the shear modulus to the failure energy. It is also found that the critical tension is proportional to the inverse square root of the crack length for more brittle materials. The latter means that the Griffith theory based on the linearized elasticity is also applicable to softer materials undergoing large deformations. Unfortunately, the applicability of the Griffith theory is restricted to cracks with equivalent sharpness only.

A Superposition Framework for Discrete Dislocation Plasticity

A superposition technique is introduced that allows for the application of discrete dislocation (DD) plasticity to a wide range of thermomechanical problems with reduced computational effort. Problems involving regions of differing elastic and/or plastic behavior are solved by superposing the solutions to i) DD models only for those regions of the structure where dislocation phenomena are permitted subject to either zero traction or displacement at every point on the boundary and ii) an elastic (EL) (or elastic/cohesive-zone) model of the entire structure subject to all desired loading and boundary conditions. The DD subproblem is solved with standard DD machinery for an elastically homogeneous material. The EL subproblem requires only a standard elastic or elastic/cohesive-zone finite element (FE) calculation. The subproblems are coupled: the negative of the tractions developed at the boundaries of the DD subproblem are applied as body forces in the EL subproblem, while the stress field of the EL subproblem contributes a driving force to the dislocations in the DD subproblem structure. This decomposition and the generic boundary conditions of the DD subproblem permit the DD machinery to be easily applied as a “black-box” constitutive material description in an otherwise elastic FE formulation and to be used in a broader scope of applications due to the overall enhanced computational efficiency. The method is validated against prior results for crack growth along a plastic/rigid bimaterial interface. Preliminary results for crack growth along a metal/ceramic bimaterial interface are presented.

Microdeformation analysis of packages and interconnects to improve finite element models for reliability assessments

The quality of mechanical modeling is of essential influence on the success of finite element analysis (FEA) and of subsequent studies of failure mechanisms. Missing knowledge about time and temperature dependent material behavior often leads to uncertainties in constitutive material description of rather complex structures like filled underfills or organic substrates. Micro deformation measurement methods help to overcome the problem by supplying displacement and strain fields comparable with FEA results. MicroDAC is an established versatile measurement tool for local and global deformation analyzes on thermally or mechanically stressed specimens. The paper presents the basics of the microDAC concept and different kinds of measurements on chip scale packages (CSP) and on flip chip assemblies to illustrate the application to integrated circuit (IC) packaging. A modified microDAC algorithm has been used to determine coefficients of thermal expansion (CTE) at small sized material samples and materials of anisotropic CTEs.

On the Formability/Instability of Stretch-Forming Sheet Metals

Manufacturing processes of aircraft, automobile and electronic components involve a lot of stretch-forming of sheet metals. The processes contain a large amount of straining which, in turn, may lead to plastic instabilities such as necking, slip formation or shear banding. Understanding the instability phenomenon requires the development of constitutive description of material characterizing the plastic instability and post uniform deformation behavior. The previous work on plastic instability usually lacks the experimental verification of the theoretical models. This may be owing to the complexity involved in designing a suitable experimental methodology for accurate measurements of strain in the locally deforming zone as the deformation progresses rather rapidly once the local neck sets in. A review of the important methodologies and analyses that were and are used in investigating instability of sheet metal in uniaxial tension is presented in this paper. Some supporting results are also presented.