Constitutive Behavior Research Articles

Semi-analytical approach for curved masonry pillars reinforced with FRCM

The paper presents a semi-analytical approach for the study of the debonding phenomenon of Fiber Reinforced Cementitious Matrix (FRCM) systems externally applied to curved masonry pillars. One of the main features of the approach consists of considering the strengthening system composed by three separated components: external mortar layer, central fiber net and internal mortar layer. These components, assumed subjected to a longitudinal state of stress, interact one each other through tangential stresses developing at the level of zero-thickness interfaces. Regarding the latter, a tri-linear shear stress-slip relationship is assumed to account for a first elastic phase, a second phase exhibiting linear softening and a third phase with a possible non-null residual strength. Since in case of curved substrates, by equilibrium, normal stresses at the interfaces between fiber and matrix arise, which modify the peak tangential resistance and the ductility, a classic Mohr-Coulomb criterion is introduced in the approach. Additionally, the central fiber net progressively transfers along the bond length the force applied at its loaded end by means of an elastic interface interposed between the internal mortar layer and the substrate, the latter assumed rigid and infinitely resistant. The longitudinal equilibrium equations written for the two mortar layers, suitably re-arranged considering the constitutive behavior of the layers, allow to deduce a field problem governed by six first order differential equations into six unknowns. The non-linearity is tackled by means of a recursive elastic numerical algorithm where the elastic modulus of the damaged materials is progressively dropped down, subdividing the bonded length into small portions where the material properties are assumed constant. For each element the solution of the field problem is known in closed form and the only variables to determine are the integration constants coming from the solution of the differential equation system. After a standard assemblage, all constants are derived imposing the boundary conditions at the extremes of the elements, which depend on the state of cracking of the matrix layer. The validation of the proposed approach is carried out with reference to recent experimental tests carried out by the Authors. The obtained results show the reliability of the approach to account for the influence of the curvature of the substrate on the debonding process of FRCM systems.

Hydraulic fracturing phase-field model in porous viscoelastic media

The mechanical behavior of shale exhibits pronounced time-dependent nonlinearity. Existing phase-field models are inadequate for evaluating the fluid-driven fracture propagation process in porous viscoelastic media. In this study, a new viscoelastic phase-field model for fluid-driven fracture propagation was introduced based on a thermodynamically consistent framework using the Maxwell–Wiechert model to describe viscoelastic constitutive behavior, with the bulk and shear moduli expanded using the Prony series. Volumetric deviatoric splits were used for both elastic and viscous energies. The fluid flow in both the fracture and the matrix adhered to Darcy's law. Additionally, the phase-field driving force considers the combined influence of rock elastic, viscous, and fluid energies. The numerical calculation iteration format was constructed using finite element discretization and the Newton–Raphson (NR) iterative method. A staggered iteration algorithm was applied to address the displacement, pressure, and phase-field problems. This method not only accurately simulates mechanical cyclic loading but also models the extension of multiple hydraulic fractures in homogeneous reservoirs and hydraulic fracture propagation in fractured reservoirs.

A New Macro-Element for Predicting the Behavior of Masonry Structures under In-Plane Cyclic Loading

A new macro model for the finite element modeling of unreinforced masonry (URM) exhibiting in-plane nonlinear cyclic behavior is proposed. The ultimate objective is to predict the seismic response of multi-story URM buildings. The macro model enables the modeling of URM shear walls with a limited number of degrees of freedom (DOF) at low computation times. The macro model consists of a deformable elastic frame supported by diagonal struts with nonlinear behavior aiming to capture all dissipative phenomena occurring during seismic events. The nonlinear constitutive behavior of diagonal struts is inspired by models documented in the literature, ensuring a robust foundation for the proposed approach. This paper first provides a comprehensive review of the principal models currently available for URM analysis. It then articulates the rationale behind the development of this new numerical model, aiming to address the limitations encountered in existing methodologies and to offer a simple and fast tool for predicting the seismic behavior of URM buildings. Afterward, the new model is presented and tested with the simulations of two experimental campaigns performed on different URM walls. The comparison between experimental and numerical results shows that with a limited number of DOF and parameters, it is possible to obtain a prediction of the experimental results with satisfying accuracy.

Configurational force method enables fracture assessment in soft materials

Configurational mechanics offers a framework for quantifying the tendency of defects to alter the material configuration. When applied to fracture mechanics, configurational forces can be used to quantify the propensity of cracks to propagate. An alternative, well-established approach involves analytical solutions for crack tip displacement fields. However, these solutions typically apply to a limited range of constitutive behaviors and oftentimes to the linear small strain regime. The ease of calculating configurational forces in a numerical Finite Element implementation, along with their applicability to soft fracture at large strains, motivates the study of their performance as a standalone fracture framework. In contrast to the majority of works that remain theoretical and numerical, our study includes a robust experimental approach to configurational forces at finite strains. We report tensile experiments on a soft elastomer with pre-cuts ante fracture initiation. In a first attempt to approach the J-integral via configurational forces, we explore the performance of the linear elastic fracture mechanics solutions on Pacman-shaped domains that reproduce the crack tip vicinity. Then, we implement the entire boundary value problem with three-dimensional simulations that replicate the empirical tensile deformation of the soft elastomer samples. Subsequently, the results are benchmarked against estimations of the J-integral obtained through a bespoke finite strain analytical crack tip solution. With the successful validation of the configurational force method at finite strains, we aim to establish a pipeline for the calculation of configurational forces in a standalone manner and circumventing the need for close-form analytical solutions.

A high fidelity multiscale approach for compression-after-impact behavior of 2D triaxially braided composites

In the current study, limited research has been conducted on modeling high-velocity impacts and their effect on residual strength. Though the simulation of compression after low-velocity impact are mature, these methods still have some numerical issues when utilized in the case of high-velocity impacts. Thus, a brand-new method suitable for continuous simulation of compression after high-velocity impact was implemented. As for the modeling techniques for 2DTBC, both the research on the model structure and constitutive behavior is still crude and imprecise. To address this situation, a high fidelity multiscale approach which contains a more accurate RUC model, a strain rate sensitive viscoelastic constitutive model and a novel subcell model was established. The HVI and CAI examples were conducted based on this framework and most results showed an error of less than 5% in both the simulation of impact resistance and residual performance. Also, the damage morphologies of simulated results demonstrated its capability and effectiveness.

Mechanical performance analysis of prestressed socket assembled concrete frame structure relying on computer simulation analysis

In order to improve the mechanical performance analysis of assembled concrete frame structure, this paper combines with computer simulation technology to analyze the mechanical performance of concrete frame structure, and uses the concrete damage plastic constitutive model provided by ABAQUS to define the nonlinear constitutive behavior of concrete. Based on the elastic-plastic model, the damage variables are added to the model, which can better reflect the performance of concrete under cyclic loading, such as stiffness degradation. In addition, this paper uses ABAQUS finite element analysis software to further analyze the structural parameters of this form of joints, and design related prestressed socket assembled concrete frame to study the seismic performance of prestressed socket assembled frame structure. Finally, the effectiveness of this model is verified by computer simulation.

Constitutive behavior of semi-solid Al80Mg5Li5Zn5Cu5 light-weight high entropy alloy

The constitutive behavior of engineering materials can predict the change of flow stress with processing parameters, which is of great significance for the selection of processing parameters. In the present work, the constitutive behavior of semi-solid Al80Mg5Li5Zn5Cu5 light-weight high entropy alloy was investigated by using thixotropic compression tests performed at semi-solid temperatures ranging from 500 °C–530 °C with strain rates ranging from 0.01 s−1–1 s−1, and the microstructure evolution was observed. The results indicate that the thixotropic compression deformation process of semi-solid Al80Mg5Li5Zn5Cu5 light-weight high entropy alloy can be divided into three stages: the flow stress increasing stage, the flow stress decreasing stage and the steady-state flowing stage. The peak stress of semi-solid Al80Mg5Li5Zn5Cu5 light-weight high entropy alloy increases with the increase of strain rate. With the increase of deformation temperature, the peak stress decreases obviously. A reasonable constitutive model of the peak stress of semi-solid Al80Mg5Li5Zn5Cu5 light-weight high entropy alloy as a function of strain rate, deformation and liquid fraction was obtained. The average relative error between the calculated and experimental values is 6.35%.

Plasticity-Induced Heating: Revisiting the Energy-Based Variational Model.

Temperature evolution during plastic deformation is of great importance for the design of manufacturing processes, as well as for the analysis and prediction of tool wear. However, the results from experimental- and numerical-type research are still often contradictory. In this paper, we analyze methods for estimating plasticity-induced heating directly from displacement fields that can be recorded during experiments or extracted from simulation results. In terms of computational methodology, the thermodynamically motivated energy-based variational formulation of the coupled thermo-mechanical boundary-value problem is adapted to the problem at hand. Since an analysis of this variational formulation exhibits challenges and distinct inconsistencies with respect to the problem at hand, an alternative approach is proposed. This alternative approach is essentially a purely thermal finite element simulation, and it is conducted using a heat source term that is empirically based on the fraction of irreversible deformation work converted to heat. Our approach estimates plasticity-induced heating based on the strain and strain rate data derived from displacement fields. We therefore incorporate thermo-visco-plastic constitutive behavior (Johnson-Cook) with a thermodynamically motivated model that specifies the fraction of plastic work converted to heat (the Taylor-Quinney coefficient).

Description of the constitutive behaviour of stainless steel reinforcement

This paper presents a comprehensive analysis of the constitutive relationship of stainless steel reinforcement and proposes new material models for both austenitic and duplex stainless steel bars. These are an advancement on existing models which have largely been developed for structural stainless steel plate, rather than for reinforcement bars. Current design guidance for material modelling of reinforcing bars does not include representative stress-strain relationships which capture the unique mechanical properties of stainless steel reinforcement. Codes include idealised elastic-plastic material models which are inappropriate and inefficient for the highly nonlinear and ductile material response of stainless steel. The present study aims to address this issue by first conducting a series of tensile tests to ascertain the stress-strain material responses and then employing this data to examine the validity of existing approaches and propose new material models where required. It is shown that new material models are required and those that are developed are able to accurately capture the stress-strain response of stainless steel reinforcement, and provide a better, more accurate, representation than existing methods.

Tensile Tests for Material Characterisation of High- and Ultra-High-Strength Steels S690QL and S960QL under Natural Fire Conditions

The mechanical material behavior of mild steels is reversible in the cooling phase of natural fires, which is proven by experimental evidence. For the material behavior of high-strength steels during cooling, no results are yet available. The paper provides the first comprehensive test program on the constitutive material behavior of high-strength steels S690QL and S960QL as well as mild steel S355 J2 + N in the case of natural fires. It is elaborated that the mechanical material behavior of high-strength steels in the cooling phase differs from the behavior in the heating phase and is not reversible due to phase changes of the microstructure. A constitutive material model for structural fire design purposes is developed on the basis of experimental data and the soundness and reliability of the model are proven by a statistical study that systematically evaluates the deviation of the model prediction from the test data.

Experimental characterization and constitutive modeling of thermoplastic polyurethane under complex uniaxial loading

This paper presents the testing and constitutive modeling of a Thermoplastic Polyurethane (TPU) compound used in commercial applications. The tested specimens were extracted directly from a TPU sphere used in check valves through water-jet cutting. The tests included tensile and compression tests under complex uniaxial loading protocols to capture different nonlinear phenomena, such as stress softening, hysteresis, relaxation, creep, and rate dependence. The material is modeled assuming a nonlinear elastic equilibrium path that may exhibit stress softening (i.e., Mullins effect), and a hysteretic viscoplastic response that presents rate dependence at three different time scales. To achieve this constitutive behavior, a Parallel Rheological Framework model is used. The nonlinear elastic equilibrium path is modeled using the generalized Yeoh hyperelastic model. The stress softening of the equilibrium path is modeled using the Ogden-Roxburgh damage model on the deviatoric response. The hysteretic viscous response is further split into three viscoplastic chains to represent time dependence at three different time scales in a decoupled way. Each viscoplastic chain is modeled using the Bergstrom-Boyce model with its standard evolution law of the creep strain. The model parameters were found using a stochastic optimization scheme to simultaneously fit all the considered tests. The outstanding agreement between the model and the experimental data across a wide range of loading scenarios provides additional insight into the time-dependent behavior and deformation mechanism of TPUs. Moreover, it shows that the mechanical behavior of these materials can be represented by decoupling the nonlinear viscoplastic behavior in different time scales.

Using spherical indentation to determine creep behavior with considering empirical friction coefficient

Indentation testing is a common technique for characterizing the mechanical properties of materials. When it comes to assessing the yield behavior of metals, conical indentation is often favored due to its ability to induce significant plastic deformation at relatively shallow indentation depths. On the other hand, spherical indentation is typically chosen for evaluating metal creep behavior, as it helps minimize the influence of plastic deformation on creep measurements. In spherical indentation tests, the friction at the contact interface is particularly crucial, given the larger contact area and more pronounced slip zones. However, accurately determining the friction coefficient at the contact interface is a complex multi-scale challenge, with the exact coefficient varying based on specific testing conditions and surface treatments. Consequently, empirical friction coefficients are frequently employed, often without thorough justification. In this work, we proposed Bayesian inference method for obtaining the creep constitutive behavior of alloys through spherical indentation tests. Our model considers not only the complexity of creep law, but also the indentation friction, which is unable to consider in most spherical indentation tests with traditional treatment. By using experimental data of spherical indentation tests on 310S stainless steel and pure nickel alloy from literature, present study demonstrates the effectiveness of the Bayesian inference approach, with the inferred constitutive models exhibiting well agreement with uniaxial creep behavior. Furthermore, the method considers the friction coefficient as an extra factor in inferring creep constitutive behavior from indentation tests.

Non-linear analysis of rectangular cross-sections with different reinforcements: dimensionless closed form solution

This paper presents an algorithm to solve a non-linear rectangular cross-section analysis by means of closed form equations. The equations are provided in a dimensionless form in order to provide generalizable results. The algorithm is based on the direct integration of the non-linear constitutive behaviour of the materials into the stress-block constitutive behaviour. The algorithm is based on masonry rectangular cross-sections characterized by an external axial load similar to Reinforced-Concrete cross-sections. The failure of the strengthening system and the masonry are analysed in the proposed algorithm, in an unconventional manner, analysing the behaviour even after the attainment of the ultimate strains, accounting for progressive damage. The algorithm allows to prevent the cross-section fiber analysis and the consequential iterative procedure to retrieve the neutral axis depth. Furthermore, the direct integration allows to evaluate the exact solution of the problem without numerical approximations. The results are discussed by means of several parametric analyses in terms of P–M domains and bending moment–curvature diagrams in dimensionless form. The proposed approach aims to simplify the cross-section analysis process in complex algorithms as in the risk analysis based on a large scale approach, where a large number of structures and structural members are analysed in the vulnerability process.

A multistable composite hinge structure

A composite tape-spring (CTS) structure is a thin-walled open slit tube with fibres oriented at ±45°, which is stable in both extended and coiled configurations. The governing factors of its bistability include composite constitutive behaviour, initial geometrical proportions, and geometrically non-linear structural behaviour. Its bistable principle can be employed to produce a flexible multistable hinge structure with tailorable stability. This is achieved by introducing variable stiffness design within a cylindrical shell structure, where folding stability is dependent on central functional patch region, and then connected to linking ploy regions. Thus, a novel multistable composite hinge structure can be designed with positive Gaussian curvature deformation, and its multistability is highly tailorable: a lengthy one-dimensional mechanical arm can be designed to coil and fold multiple times to enable large folding ratio. An analytical model was established based on the strain energy principle, in order to determine effects from functional tape length; the typical structural stability and stable configurations were then predicted with respect to regional length of the functional layer. It is found that the stability of a multistable composite hinge structure is dependent on geometry and combination of both the functional patch region, and connecting ploy region; the stable criteria are then proposed and show good agreement with experimental observations and FE analysis. These enrich the diversities of functional deployable structures to benefit novel requirements for various deployable mechanisms, and enable customised design, as well as smart driving for flexible and multifunctional mechanical composite hinge applications.

Physics-based neural network as constitutive law for finite element analysis of sintering

In the sintering of advanced ceramics, a digital twin consisted of a finite element model can predict the final shape and microstructure of a sintered ceramic. However, to minimise uncertainties, it is essential to continually update the mechanical properties in the digital twin using data collected from the manufacturing process. One promising approach for achieving this is through artificial neural networks (ANN). This study introduces a machine learning strategy to update the constitutive behaviour of advanced ceramics in finite element analysis of sintering deformation. A major challenge in implementing machine learning in material processing is the huge amount of data required by the training and validation of an ANN, which are often unavailable or incomplete in a real manufacturing process. This study demonstrates that the data requirement can be reduced by employing a two-step training technique. Firstly, the ANN is trained using a nonlinear constitutive law, which describes a general relationship between the strain rates and stresses. Subsequently, the weights and biases of the ANN are transferred for the retraining using limited experimental data for an actual ceramic. It is shown that such approach can successfully capture the nonlinear constitutive behaviour of fine-grained alumina without demanding for large amount of experimental data. A case study is provided, highlighting the feasibility of implementing the ANN in a commercial finite element package, to replace the constitutive law and predict the shrinkage and distortion of a sintering part. In particular, the sintered dumb-bell shape part, simulated using the retrained ANN, showed grain size and relative density that markedly different from those using the nonlinear constitutive law. It is important to note that the proposed methodology is generic and can be used to create ANNs to replace constitutive laws in finite element analysis in digital twins for a wide range of other engineering processes.

Numerical Prediction of Ballistic Performance of Thin Concrete Plate

This research explores the ability of 60[Formula: see text]mm thick reinforced concrete (RC) plates to resist impacts from ogive-nosed hard steel projectiles. The projectiles used in the present study have a diameter of 19[Formula: see text]mm, a length of 200[Formula: see text]mm, and a mass of 0.4[Formula: see text]kg impacted on the RC plates with incident velocities ranging from 92[Formula: see text]m/s to 161[Formula: see text]m/s. Numerical simulations were conducted using ABAQUS/Explicit finite element software to validate the experimental results. The Holmquist–Johnson–Cook (HJC) material model was employed to simulate the constitutive behavior of concrete, while the Johnson–Cook (JC) material model was used to simulate the material response of reinforcing steel bars. The residual velocities obtained from the simulations closely matched the actual experimental results, showing a polynomial correlation with the incidence velocities of the projectiles. Moreover, the experimentally and numerically determined ballistic limit for the 60[Formula: see text]mm thick RC plate was found to be 108[Formula: see text]m/s and 109.5[Formula: see text]m/s, respectively. In contrast, the ballistic limit calculated using empirical mathematical expressions was 107.4[Formula: see text]m/s. This alignment between predicted, calculated, and actual ballistic limits underscores the reliability and accuracy of both numerical and empirical approaches.

Elastic properties and constitutive behaviour of graphene at finite temperature and large deformation

In the earlier reported theoretical studies, the bending modulus of the graphene sheet predicted with the inclusion of the dihedral energy term is observed to be almost double as compared to those reported without considering the dihedral energy term. The bending modulus reported considering dihedral energy term is observed to be more accurate when compared with the quantum mechanical calculations. The findings of these studies for bending modulus (with/without considering dihedral energy terms) are reported without considering the effect of finite temperature. In the present work, a multiscale computational framework incorporating the dihedral energy term and finite temperature in the constitutive model is presented for the first time. A temperature-dependent bond length is expressed in terms of strain and curvature tensor through the temperature-related quadratic-type Cauchy-Born rule. The total Helmholtz free energy of the unit cell is expressed as a sum of the potential energy of the unit cell and thermal energy due to the thermal vibrations of atoms arising from the finite temperature. The atomic interactions are modelled through the Tersoff-Brenner potential considering different sets of empirical parameters, including the authors’ recently proposed set of empirical parameters. The stress/moment tensors and the tangent constitutive matrix are obtained from the derivatives of the Helmholtz free energy density with strains and curvatures. The thermal properties, elastic properties, and tangent stiffness coefficients for armchair and zigzag graphene with/without dihedral energy term at finite strain, curvature, and temperature are investigated and discussed in detail. The proposed multiscale model facilitates the accurate prediction of the constitutive behaviour of the graphene in a thermal environment with remarkable improvements for the prediction of bending modulus due to the inclusion of the dihedral energy term at finite temperature.

High temperature effect on mechanical properties of autoclaved aerated concrete

In this paper, an experimental study on mechanical properties at high temperature of a typical autoclaved aerated concrete (AAC) tested in compression is presented (compressive strength = 5 MPa and density = 600 kg/m3 at 20 °C). Parameters such as the compressive strength, strain at the peak stress, and elastic modulus are investigated, along with the stress-strain response in the thermal range 20–700 °C. The behavior of the AAC tested in this research project is different from that of normal concrete, because of a series of chemo-physical processes occurring at high temperature. Between 100 and 600 °C, the compressive strength increases and then starts decreasing (+51% at 600 °C and +10% at 700 °C compared to ambient conditions). The strain at the peak stress increases as well with the temperature similar to – but more than in – normal concrete. The stress-strain curves and the failure modes show that AAC’s behavior is brittle at any temperature and that the damage inside the material increases with the temperature in such a way that the existing models for normal concrete exposed to high temperatures cannot be used in the case of AACs. Hence, a new polynomial model is proposed to properly describe the constitutive behavior of AACs at high temperature.

An adaptive Ritz formulation for progressive damage modelling in variable angle tow composite plates

In this work, an adaptive Ritz model for the analysis of variable angle tow composite plates featuring damage initiation and evolution under progressive loading is proposed, developed, implemented and tested. The plate kinematics is represented employing a first-order shear deformation theory, while the plate equilibrium equations at a given load step are obtained by minimizing the structure potential energy. The constitutive behaviour is modelled within the framework of continuum damage mechanics. In particular the initiation and evolution of damage, up to failure, are tracked by defining irreversible damage indices related to both fibres and matrix, both in tensile or compression loading. The discrete equations are then obtained by assuming a polynomial Ritz approximation of the primary kinematic variables in the energy minimization. Preliminary tests show how the application of the method as a single-domain approach induces the emergence of problematic spurious effects, related to Gibbs artefacts due to the inability of the selected polynomial basis to represent damage localization. An adaptive multi-domain technique is thus proposed to circumvent such issues, which has been successfully validated by benchmark tests. Eventually, original results about variable angle tow plates featuring damage evolution under progressive loading are presented.

Methodological Aspects and Mesh Convergence in Numerical Analysis of Athermal Fiber Network Material Deformation

A balance between model complexity, accuracy, and computational cost is a central concern in numerical simulations. In particular, for stochastic fiber networks, the non-affine deformation of fibers, related non-linear geometric features due to large global deformation, and size effects can significantly affect the accuracy of the computer experiment outputs and increase the computational cost. In this work, we systematically investigate methodological aspects of fiber network simulations with a focus on the output accuracy and computational cost in models with cellular (Voronoi) and fibrous (Mikado) network architecture. We study both p and h-refinement of the discretizations in finite element solution procedure, with uniform and length-based adaptive h-refinement strategies. The analysis is conducted for linear elastic and viscoelastic constitutive behavior of the fibers, as well as for networks with initially straight and crimped fibers. With relative error as the determining criterion, we provide recommendations for mesh refinement, comment on the necessity of multiple realizations, and give an overview of associated computational cost that will serve as guidance toward minimizing the computational cost while maintaining a desired level of solution accuracy.