Nonlinear Constitutive Law Research Articles

Nonlinear Finite Element Modelling and Parametric Analysis of Shear Strengthening RC T-Beams with NSM CFRP Technique

In this study, a robust three-dimensional finite element (FE) model has been developed for reinforced concrete beams strengthened in shear with near surface mounted (NSM) carbon fibre reinforced polymer (CFRP) rods. The FE models were developed and validated against existing experiments and presented various nonlinear constitutive material laws and interfacial relations. A detailed parametric study was performed to investigate the effects of various parameters on the performance of strengthened member. It was shown that increasing the concrete compressive strength ( $${f_{\text{c}}}^{\prime }$$ ) from 20 to 50 MPa, leads to an increase in the beam’s ultimate load and contribution of NSM CFRP reinforcement. While for NSM reinforcement ratio ( $${\rho _{\text{f}}}$$ ), the ultimate load slightly increased when $${\rho _{\text{f}}}$$ is 0.14–0.22%, and then increased by 11% in average when $${\rho _{\text{f}}}$$ increased to 0.28%. Varying the percentage of existing steel stirrups ( $${\rho _{{\text{sw}}}}$$ ) from 0.11 to 0.36%, leads to an increase in ultimate load from 8 to 15% compared to the control un-strengthened specimen. However, the further increase in $${\rho _{{\text{sw}}}}$$ (more than 0.36%) caused a reduction in the contribution of NSM CFRP technique because of the changing in the failure mode. The distance between existing steel stirrups and NSM reinforcement does not affect the behaviour. In addition, the model predictions were used to evaluate several design formulas available in the literature for this technique. It was found that some theoretical equations were conservative as long as the governing failure mode is shear.

Class A prediction of mechanised tunnelling in Rome

Contract T3 of Line C of Rome underground, currently under construction, crosses the archaeological area of the historical centre, with significant interferences with the existing monumental built environment. A fully instrumented green field control section was established at the beginning of this contract, in representative ground conditions. This paper presents a thorough Class A prediction of the passage of the tunnels through the control section, obtained using a recently developed advanced numerical procedure. The ground was modelled with a non-linear constitutive law, calibrated with all the available data from the geotechnical investigation. The main physical processes occurring around the shield, including cutter-head overcut, shield tapering and tail void grouting were modelled in detail. The numerical results agree qualitatively with the findings from well documented case histories and results from physical models. The installed instrumentation will provide an opportunity to test the ability of the adopted procedure to reproduce quantitatively the measured performance, once the tunnels will cross the control sections and the field data will become available.

Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity

This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients.

Shear-shear-torsional homogenous beam models for nonlinear periodic beam-like structures

In this paper, the proper modeling of a class of nonlinear periodic beam-like structures, behaving as shear beams in the 3D environment, is critically considered. Multi-story frame buildings are taken as an example of these structures and the suitability of the (unflexurable) shear-shear-torsional beam model in describing the behavior of the building, is investigated. In this context, the range of applicability of the equivalent beam model is detected through an order-of-magnitude analysis, which is developed in linear regime and corroborated by means of finite-element analyses. A new continuous shear-shear-torsional beam model, here referred to as ‘complete model’, is formulated in the framework of the direct one-dimensional approach, while its nonlinear hyperelastic constitutive law is obtained by identification with a three-dimensional body, through an energy equivalence. In addition, a ‘minimal model’ is derived from the complete one for which only the cubic terms generated by the stretch of the columns are retained, while the remaining cubic terms and all the quadratic terms, generated by the finite twist angle, are neglected. Perturbation solutions for both the complete and the minimal model are obtained. The limits of applicability of the minimal model are discussed in terms of nonlinear static responses of some structures taken as case-studies.

A nonlinear approach to the wind strength of Gothic Cathedrals: The case of Notre Dame of Paris

The problem of assessing the strength to wind actions of Gothic Cathedrals is addressed in this paper. A nonlinear approach, based upon a large-strain, large displacements formulation and using a nonlinear constitutive law modeling the no-tension behavior of the material as a damage law is proposed. The method is applied to the study of the Cathedral Notre Dame of Paris.

Fully coupled thermo-viscoplastic analysis of composite structures by means of multi-scale three-dimensional finite element computations

The current paper presents a two scale Finite Element approach (FE2), adopting the periodic homogenization method, for fully coupled thermo-mechanical processes. The aim of this work is to predict the overall response of rate-dependent, non-linear, thermo-mechanically coupled problems of 3D periodic composite structures. The material constituents implicated in the analyses obey generalized standard materials laws, while the characteristic equations of the problem (balance law, first law of thermodynamics) are expressed and satisfied in both microscopic and macroscopic scales. For the numerical implementation in both scales, the finite element commercial software ABAQUS is utilized in the framework of small strains and rotations. A set of dedicated scripts and a specially designed Meta-UMAT subroutine allow the connection between the macroscopic structure and the microscopic unit cells attached to every macroscopic integration point. The two-scale finite element framework is applied to simulate thermoelastic-viscoplastic materials of complex 3D composite structures, and its capabilities are demonstrated with proper numerical examples. It is worth mentioning that the proposed computational strategy can be applied for any kind of 3D periodic microstructure and non-linear constitutive law.

Wall structure finite-element by BEM coupling

The calculation of structures made of the assembly of several walls requires finite element modelling, henceforth, it gives raise to a large number of degrees of freedom. The actual work shows that the number of parameters is reduced by using boundary elements to derive the stiffness matrix of the element. The proposed technique gives origin to positive definite and symmetrical matrix due to the use of constant boundary elements. Moreover, the resulting matrix can be assembled together with finite element ones in order to obtain the description of complex three dimensional structures. Sub-structuring and multi region modelling are possible and continuity relaxation between elements as well as interface and nonlinear constitutive laws can be comprised. Some examples are reported in order to verify the accuracy and the feasibility of the procedure for the elastic case.

Nonlinear Adaptive Magneto-Thermal Analysis at Bushing Regions of a Transformers Cover Using Finite Difference Method

Abstract In this study, losses analysis at bushing regions of a transformer covers is done using finite difference method (FDM), considering that FDM being more flexible to deal with the nonlinear constitutive law and easier to be implemented than finite element (FE) and analytical methods. The analysis is performed based on a 2-level adaptive mesh solution of Maxwell equations and Ohm law at the cross section area in the axial symmetry page of a steel disk, taking account the nonlinear magnetic permeability of the steel. The losses density obtained, as a heat source, is imported into an alternating direction implicit (ADI) approach of heat conduction equation. Therefore, a finite difference (FD) solution algorithm for magneto-thermal analysis on cover plate is obtained by combination of adaptive mesh refinement and ADI-FDM, which improves the accuracy and decreases the computational time without losing accuracy. The reliability of the proposed technique is confirmed by experimental and FE method (FEM) results, considering the temperature distribution of the cover. The comparison of the results with those obtained from FEM and experiments shows the efficiency and capability of the method.

A data-driven computational homogenization method based on neural networks for the nonlinear anisotropic electrical response of graphene/polymer nanocomposites

In this paper, a data-driven-based computational homogenization method based on neural networks is proposed to describe the nonlinear electric conduction in random graphene-polymer nanocomposites. In the proposed technique, the nonlinear effective electric constitutive law is provided by a neural network surrogate model constructed through a learning phase on a set of RVE nonlinear computations. In contrast to multilevel (FE $$^2$$ ) methods where each integration point is associated with a full nonlinear RVE calculation, the nonlinear macroscopic electric field-electric flux relationship is efficiently evaluated by the surrogate neural network model, reducing drastically (by several order of magnitudes) the computational times in multilevel calculations. Several examples are presented, where the RVE contains aligned graphene sheets embedded in a polymer matrix. The nonlinear behavior is due to the modeling of the tunelling effect at the scale of graphene sheets.

Thermodynamic framework for non-local transport-damage modeling of fluid driven fracture in porous media

We present a novel non-local damage transport model for hydraulic fracture in porous media. The new model is sought to provide an enhanced description for the long-range damage and transport interactions in the fracture process zone. The non-local model is derived from thermodynamic principles, where a new thermodynamic potential expression for non-local transport is introduced. The thermodynamically consistent model is reduced to a gradient non-local permeability relationship that simultaneously provides a non-local transport description and regularized damage growth via a permeability-stress relationship. The analogy of the proposed model to Darcy-Brinkman fluid flow is demonstrated and discussed. A monolithic 3-field (u−P−κ˜) mixed finite element framework is then used to discretize and solve the nonlinear coupled physics poromechanics problem. The hydraulic fracture continuum models previously presented in the literature rely on using different fluid flow laws inside and outside the fracture zone, which introduces a discontinuity in the continuum model. Our model prescribes Darcy type fluid flow all over the domain with a non-linear permeability constitutive law that allows for elevated fluid velocity in the crack zone, which preserves the continuity of all quantities within the domain. The numerical examples confirm the capability of the proposed model in capturing the essential features of hydraulic fracture simulation. The significance of the non-local transport modeling is demonstrated through modeling hydraulic fracturing in materials with pre-existing high permeability zones. The incorporation of non-local transport effects in hydraulic fracture modeling is proved to uncover possible flow paths through pre-existing high permeability zones. The proposed model can act as a platform for quantifying the fluid leak-off from hydraulic fracture and can be further used to analyze the hydraulic fracture interaction with neighboring geological material.

Simulation of spatially non-uniform frost damage in RC beams under various exposure and confining conditions

This paper aims to investigate the spatially non-uniform frost damage process in RC beams during large numbers of freeze/thaw cycles (FTC). The material-based thermodynamic and poromechanical frost damage models have been up-scaled to the structural level considering the anisotropic stress-strain conditions and the condensed water movement in the crack system. The ice-induced pore pressures are also integrated in the smeared RC element in consideration of the path-dependent nonlinear constitutive laws. The frost damage accumulation processes (referred as the expansive strain) are simulated with different depths of heat and moisture exposure, as well as various arrangements of reinforcing bars. Based on the proposed empirical model which links the expansive strain with the reduction in ultrasonic velocity, the 3D frost damage profiles are compared with 2D ultrasonic velocity data, which show a satisfactory agreement. Finally, the effects of different steel and structural confinements on the frost damage profile are numerically investigated and discussed in detail.

A three-dimensional visco-hyperelastic FE model for simulating the mechanical dynamic response of preloaded phalanges

This study lays the groundwork for a multi-scale strategy that will lead to a better understanding and prediction of the effects of vibration on the digital arterial network. This is accomplished by modelling the mechanical and biological factors that could disturb the basal vasoconstriction balance in the fingertip. The first stage of this novel approach involved building and validating an original dissipative constitutive law for the fingertip soft tissue for the purpose of finite element modelling of the mechanical response of preloaded phalanges in vibration. This visco-hyperelastic constitutive law was established by means of a two-stage procedure for combining a classical pure static nonlinear law with an original dissipative model. First, the parameters of an Ogden–Hill pure static nonlinear constitutive law were identified using a constrained optimisation algorithm. Second, an original viscous dissipation model was proposed in the spectral domain. This model is based on the linearization of the nonlinear quasi-linear viscoelasticity law and the use of a viscoelastic relaxation modulus, expressed as a continuous distribution of relaxation spectra suitable for living tissues. The experimental data used to fit this model were the static and dynamic stiffnesses of preloaded fingertips acquired from a group of 20 subjects. The relative errors between the measured and simulated stiffnesses were less than 5% in the static procedure and approximately 8% using dynamic analysis. The computed mechanical pressure and maximal tangential stress within the fingertip were high in the soft tissues close to the vibration excitation and also in the bones and interphalangeal cartilages far from the vibration source. Mechanical power was only dissipated significantly in the immediate vicinity of the contact area between the probe and the finger. The main contribution of this study was to implement and identify the parameters of a new spectral dissipative law for fingertip soft tissues. This work may apply in occupational health for modifying the vibration dose assessment or for the follow-up and screening of connective tissue diseases.

Resolving force indeterminacy in contact dynamics using compatibility conditions

Contact dynamics (CD) is a powerful method to solve the dynamics of large systems of colliding rigid bodies. CD can be computationally more efficient than classical penalty-based discrete element methods (DEM) for simulating contact between stiff materials such as rock, glass, or engineering metals. However, by idealizing bodies as perfectly rigid, contact forces computed by CD can be non-unique due to indeterminacy in the contact network, which is a common occurence in dense granular flows. We propose a frictionless CD method that is designed to identify only the unique set of contact forces that would be predicted by a soft particle method, such as DEM, in the limit of large stiffness. The method involves applying an elastic compatibility condition to the contact forces, which maintains no-penetration constraints but filters out force distributions that could not have arisen from stiff elastic contacts. The method can be used as a post-processing step that could be integrated into existing CD codes with minimal effort. We demonstrate its efficacy in a variety of indeterminate problems, including some involving multiple materials, non-spherical shapes, and nonlinear contact constitutive laws.

Modeling, simulation, and experiments of high velocity impact on laminated composites

High velocity impact on laminated composite panels is investigated by modeling, simulation, and experiments. Impact velocities of 100, 200, and 300 m/s are considered and normal as well as oblique impacts are studied. FEM simulations are conducted to design three different laminate configurations to achieve the cases of a reflecting, almost stucking, and fully perforating impactor, respectively. Within a domain where material nonlinearities are expected the shell-based ply-scale approach is used. Every ply is modeled by a layer of shell elements which are connected via cohesive zone elements. Both element types are assigned nonlinear constitutive laws to capture their mechanical behavior appropriately. Outside this domain, a single layer of composite shell elements is used. Corresponding laminated plates are produced and tested using a gas gun setup, where energy absorption and damage patterns are assessed.The comparison of simulations and experiments shows excellent agreement. The three cases are predicted well, and also the damage patterns and the energy absorptions match very well.

Computational Rod Model With User-Defined Nonlinear Constitutive Laws

Computational rod models have emerged as efficient tools to simulate the bending and twisting deformations of a variety of slender structures in engineering and biological applications. The dynamics of such deformations, however, strongly depends on the constitutive law in bending and torsion that, in general, may be nonlinear, and vary from material to material. Jacobian-based computational rod models require users to change the Jacobian if the functional form of the constitutive law is changed, and hence are not user-friendly. This paper presents a scheme that automatically modifies the Jacobian based on any user-defined constitutive law without requiring symbolic differentiation. The scheme is then used to simulate force-extension behavior of a coiled spring with a softening constitutive law.

Numerical coarsening using discontinuous shape functions

In this paper, an efficient and scalable approach for simulating inhomogeneous and non-linear elastic objects is introduced. Our numerical coarsening approach consists in optimizing non-conforming and matrix-valued shape functions to allow for predictive simulation of heterogeneous materials with non-linear constitutive laws even on coarse grids, thus saving orders of magnitude in computational time compared to traditional finite element computations. The set of local shape functions over coarse elements is carefully tailored in a preprocessing step to balance geometric continuity and local material stiffness. In particular, we do not impose continuity of our material-aware shape functions between neighboring elements to significantly reduce the fictitious numerical stiffness that conforming bases induce; however, we enforce crucial geometric and physical properties such as partition of unity and exact reproduction of representative fine displacements to eschew the use of discontinuous Galerkin methods. We demonstrate that we can simulate, with no parameter tuning, inhomogeneous and non-linear materials significantly better than previous approaches that traditionally try to homogenize the constitutive model instead.

Computational modeling of finite deformation piezoelectric material behavior coupling transient electrical and mechanical fields

This paper develops a finite element formulation and computational model for coupling electric and finite strain dynamic fields with widely discrepant frequencies, governing the behavior of piezoelectric materials. The piezoelectric materials are defined by time-dependent nonlinear constitutive laws. A fully coupled, total Lagrangian finite element formulation is developed for modeling the electric and mechanical fields. A challenge in computational modeling of piezoelectric devices arise due to large discrepancy in the frequencies of the electrical signal and mechanical vibrations, especially when a large number of mechanical cycles need to be simulated. A wavelet transformation induced multi-time scaling (WATMUS) algorithm is developed for dynamic piezoelectric simulations. The WATMUS algorithm projects the high frequency electric fields on to the lower frequency of displacement and velocity fields, on which time integration is performed. The method significantly enhances the computational efficiency in comparison with single time scale integration methods. The accuracy and efficiency of the WATMUS algorithm are validated through piezo-electric applications.

A Non-Linear Stress-Strain Relation Endowed With Fractional Derivative Elements

In this paper a non-linear stress-strain relation based on an integral formulation with a power-law kernel is proposed. This constitutive law is able to reproduce both the viscoelastic behavior and the inelastic irreversible phenomenon. It is shown how the proposed stress-strain law is capable to fit experimental data obtained from tensile tests on two kind of metal alloys. Such best-fitting procedure have shown the accuracy of the proposed model and its results are compared to other ones obtained with the aid of classical non-linear constitutive law.

Numerical analysis on seismic resistance of a two-story timber-framed structure with stone and earth infill

ABSTRACTDue to their seismic resistance, traditional timber-framed structures with masonry infill suffered little damage during recent earthquakes. Moreover, timber-framed structures can be built with reduced costs thanks to the use of locally available materials such as wood, stone, and earth. Based on an experimentally validated numerical simulation for a one-story house, the seismic resistance of a similar two-story house is investigated. A simplified Finite Element Model with linear and nonlinear truss elements is proposed to analyze the seismic resistance of a two-story building. Nonlinear hysteresis constitutive laws are defined only for two majorcomponents of the structure which are assumed to be representative of the global structure behavior: diagonal X-crosses (concentrating the interaction with the infill material) and steel strip connections. These kinds of structures have been overlooked due to a lack of knowledge of their potential behavior in seismic prone area and a lack of building codes and standards for their own design. To promote them, a failure criterion, that might easily be used in engineering studies, is required. This article proposes a simple criterion based on Eurocode 8 to quantify the seismic resistance of one- and two-story houses. The simulation shows that, even in case of high intensity ground motion, the two-story building should not be collapsed. This study may help at designing two-story timber-framed structures in seismic prone areas for (re)construction projects.

Quantitative measures of deformed rocks: The links to dynamics

One aim of structural geology is to understand the processes that operate during deformation and metamorphism so that the conditions (P, T, strain-rates, stresses, rates of other processes such as grain-size reduction) that control these processes can be identified and quantified. In this paper we explore ways of quantifying the geometry of deformed metamorphic rocks in ways that are directly related to the processes that operated. We propose that structures we observe in nature are the result of coupling between nonlinear processes and hence deforming metamorphic systems behave as nonlinear dynamical systems. As such the descriptions of such systems should utilise the toolbox of methods that have been developed for dynamic systems over the past 50 years or so. These methods include the construction of attractors, multifractal analysis, recurrence plots, recurrence quantification analysis and dynamical network analysis. We illustrate the approach first using numerical models of fold systems, with linear and nonlinear constitutive relations and with and without initially imposed noise (imperfections). We then extend the illustrations to consider natural fold systems. Our conclusions are that the irregularity we see in natural fold systems is the result of nonlinear constitutive laws, not that of initial imperfections and that the number of processes operating to produce the observed fold shapes is relatively small.