Stress-strain Constitutive Relation Research Articles

Impact of FRP and FRCM on the ductility of strengthened masonry members

Strengthening strategies have become extremely efficient as demonstrated in recent scientific works and real field applications. The recent calamitous events focused the interest on existing masonry structures. In the common practice the strengthening strategies are performed improving the load capacity of existing structural elements. However, especially for slender masonry elements like as arches and barrel vaults, the increase of load capacity may not be the only and optimal approach. The ductility capacity represents an important aspect that should be taken into account in a strengthening strategy. A great number of applications is performed without relying on and assessing the ductility of the strengthened elements. This approach could promote deleterious effects on the structural behavior due to the brittle behavior provided by excessive amount of strengthening. Furthermore, a strengthening strategy must respect the compatibility with the masonry substrate especially for heritage applications.The present paper focuses on the out-of-plane behavior of slender masonry elements strengthened with FRP or FRCM systems. A parametrical analysis was performed and results, in terms of bending moment–curvature diagrams and ultimate curvatures were analyzed in dimensionless form. The behavior of strengthened masonry is independent on the type of stress–strain constitutive relationship of composite, depending on mechanical fiber reinforcement ratio, ω. It is very important for practical applications when an overestimation of the effective amount of reinforcement is designed. In this sense the type of composite can be optimized, especially for poor masonry, where higher ω could promote brittle behaviors. At low ω, the type of the stress–strain constitutive relationship becomes a key aspect (e.g. linear or bi-linear strongly correlated to the type of composite: FRP or FRCM). Furthermore, the impact of the axial load both on the ductility capacity and on the load capacity becomes negligible at high values of ω. The results, provided in a dimensionless form, constitute the basis for a valid support to the design of interventions using composites on masonry structures.This work represents a preliminary study to highlight some issues often overlooked in the strengthening of masonry structures, based on direct application of available design guidelines. It is the basis for future development of normalized approaches to perform targeted experimental validations and optimize the design of strengthening systems.

Modeling Tensile Damage and Fracture Behavior of Fiber-Reinforced Ceramic-Matrix Minicomposites.

Evolution of damage and fracture behavior of fiber-reinforced mini ceramic-matrix composites (mini-CMCs) under tensile load are related to internal multiple damage mechanisms, i.e., fragmentation of the brittle matrix, crack defection, and fibers fracture and pullout. In this paper, considering multiple micro internal damage mechanisms and related models, a micromechanical constitutive stress–strain relationship model is developed to predict the nonlinear mechanical behavior of mini-CMCs under tensile load corresponding to different damage domains. Relationships between multiple micro internal damage mechanisms mentioned above and tensile micromechanical multiple damage parameters are established. Experimental tensile nonlinear behavior, internal damage evolution, and micromechanical tensile damage parameters corresponding to different damage domains of two different types of mini-CMCs are predicted. The effects of constitutive properties and damage-related parameters on nonlinear behavior of mini-CMCs are discussed.

Characterization of Microcrack Orientation Using the Directivity of Secondary Sound Source Induced by an Incident Ultrasonic Transverse Wave

In this paper, characterization of the orientation of a microcrack is quantitatively investigated using the directivity of second harmonic radiated by the secondary sound source (SSS) induced by the nonlinear interaction between an incident ultrasonic transverse wave (UTW) and a microcrack. To this end, a two-dimensional finite element (FE) model is established based on the bilinear stress–strain constitutive relation. Under the modulation of contact acoustic nonlinearity (CAN) to the incident UTW impinging on the microcrack examined, the microcrack itself is treated as a SSS radiating the second harmonic. Thus, the directivity of the second harmonic radiated by the SSS is inherently related to the microcrack itself, including its orientation. Furthermore, the effects of the stiffness difference between the compressive and tensile phases in the bilinear stress–strain model, and the UTW driving frequency, as well as the radius of the sensing circle on the SSS directivity are discussed. The FE results show that the directivity pattern of the second harmonic radiated by the SSS is closely associated with the microcrack orientation, through which the microcrack orientation can be characterized without requiring a baseline signal. It is also found that the SSS directivity varies sensitively with the driving frequency of the incident UTW, while it is insensitive to the stiffness difference between the compressive and tensile phases in the bilinear stress–strain model and the radius of the sensing circle. The results obtained here demonstrate that the orientation of a microcrack can be characterized using the directivity of the SSS induced by the interaction between the incident UTW and the microcrack.

Constitutive stress–strain law for FRP-confined tuff masonry

The beneficial effect provided by external Fiber-Reinforced Polymers (FRP) confinement on the compressive behaviour of masonry columns has been widely demonstrated by experimental results. Conversely to FRP-confined concrete, for which some reliable stress–strain theoretical models have been developed, in case of confined masonry any unified model has been yet provided. In this paper, after the elaboration of a database containing experimental results on FRP-wrapped masonry columns tested under axial loads, new design-oriented stress–strain constitutive relationships for FRP-confined masonry are proposed. These laws—which have general validity for whatever confined masonry—were derived from the existing Lam and Teng’s model originally developed for FRP-confined concrete, and here adapted to confined tuff masonry. Two functions rule the shape of theoretical stress–strain curve: a second order parabola defines the elastic behaviour, while a straight-line describes the post-elastic behaviour. The entire axial stress–strain response is completely defined by unconfined masonry mechanical properties, composite material characteristics and by a single unknown parameter. This latter takes into account of the dispersion of experimental data and has been evaluated by means of the regression analyses procedure based on the least-squares optimization criteria. The outputs of the proposed constitutive laws were compared with the experimental results currently available in literature. A good matching is observed between the experimental results and the theoretical predictions in terms of axial stress–strain curves.

Influence of pore water pressure change on consolidation behavior of saturated poroelastic medium

Abstract. There are many factors causing land subsidence, and groundwater extraction is one of the most important causes of subsidence. A set of coupled partial differential equations are derived in this study by using the poro-elasticity theory and linear stress-strain constitutive relation to describe the one-dimensional consolidation in a saturated porous medium subjected to pore water pressure change due to groundwater table depression. Simultaneously, the closed-form analytical solutions for excess pore water pressure and total settlement are obtained. To illustrate the consolidation behavior of the poroelastic medium, the saturated layer of clay sandwiched between two sand layers is simulated, and the dimensionless pore water pressure changes with depths and the dimensionless total settlement as function of time in the clay layer are examined. The results show that the greater the water level change in the upper and lower sand layers, the greater the pore water pressure change and the total settlement of the clay layer, and the more time it takes to reach the steady state. If the amount of groundwater replenishment is increased, the soil layer will rebound.

A Three-Variable Geometrically Nonlinear New First-Order Shear Deformation Theory for Isotropic Plates: Formulation and Buckling Analysis

This paper presents a displacement-based geometrically nonlinear first-order shear deformation theory for the analysis of shear deformable isotropic plates. Nonlinear strain–displacement relations as utilized by the von Karman plate theory and linear stress–strain constitutive relations are used to formulate this theory. Governing equations of this theory are derived by utilizing equilibrium equations for an infinitesimal plate element. Commonly occurring plate edge boundary conditions are described based on the physical understanding of the plate deformation. As against other geometrically nonlinear first-order shear deformation plate theories reported in the literature, main contributions of this theory are that (1) it incorporates the rotation-free shear deformation plate kinematics of the first order; (2) this theory involves only three governing equations involving only three unknown functions; (3) expressions of governing equations of this theory have a striking resemblance to corresponding expressions of the von Karman plate theory; (4) this theory describes two unique, physically meaningful plate clamped edge boundary conditions. Illustrative examples pertaining to the buckling of shear deformable isotropic Levy-type plates and the comparison of the results obtained with the corresponding results reported in the literature demonstrate the efficacy of the proposed theory.

A Coupled Elastoplastic Damage Model for Clayey Rock and Its Numerical Implementation and Validation

This paper presents a new constitutive model for describing the strain-hardening and strain-softening behaviors of clayey rock. As the conventional Mohr-Coulomb (CMC) criterion has its limitation in the tensile shear region, a modified Mohr-Coulomb (MMC) criterion is proposed for clayey rock by considering the maximal tensile stress criterion. Based on the results of triaxial tests, a coupled elastoplastic damage (EPD) model, in which the elastic and plastic damage laws are introduced to describe the nonlinear hardening and softening behaviors, respectively, is developed so as to fully describe the mechanical behavior of clayey rock. Starting from the implicit Euler integration algorithm, the stress-strain constitutive relationships and their numerical formulations are deduced for finite element implementation in the commercial package ABAQUS where a user-defined material subroutine (UMAT) is provided for clayey rock. Finally, the proposed model is used to simulate the triaxial tests and the results validate the proposed model and numerical implementation.

Application of adaptive neuro-fuzzy inference system for numerical interpretation of soil torsional shear test results

The aim of this study is to replace the stress-strain constitutive relation within a finite element method (FEM) implementation with an adaptive neuro-fuzzy inference system (ANFIS). The computation is supported by the Compute Unified Device Architecture (CUDA) technology available on modern Nvidia graphics processing units (GPUs). The multi-threaded multiple input / multiple output (MIMO) CUDA-ANFIS model is compared with experimental results of torsional shear (TS) tests. The cylindrical non-cohesive soil samples were tested within the small strain range (0.001–0.1%). Particular attention was paid to a good reflection of the results of laboratory tests in numerical simulations based on FEM with embedded ANFIS. The obtained results confirm the possibility of replacing explicit constitutive relations with a neuro-fuzzy inference system and using this model to determine the soil mechanical properties in engineering practice. The most important finding is that the decrease in the discretization mesh density as well as the decrease in the number of linguistic labels result in better overall system performance. In addition, the created ANFIS algorithm adapted to the GK110 graphics processor architecture speeds up calculations 27 times compared to single-threaded implementation. The presented approach can be used to construct implicit constitutive models based on the numerical interpretation of the results obtained from laboratory and/or field tests.

Data-driven ultimate conditions prediction and stress-strain model for FRP-confined concrete

Fiber Reinforced Polymer has been widely used in the retrofit of existing structures and the construction of new structures. The ultimate conditions and stress–strain model of FRP-confined composites are critical to structural design and prediction of structural response, especially under extreme loads such as earthquakes. In this paper, a data-driven neural network prediction model for ultimate conditions and stress–strain constitutive relation of FRP-confined concrete is proposed, and the validity and accuracy of the model are verified. A uniaxial compression database containing 169 FRP-confined normal concrete cylinders is collected from the open literature, and the quality of the database is examined and evaluated in detail. Based on the feed forward neural network technology, a prediction model for the ultimate conditions of FRP-confined normal concrete cylinders is established. Configurations and hyper-parameters of the network are carefully analyzed, and the optimal model is used for prediction and comparison. Besides, a uniaxial stress–strain model for FRP-confined concrete is established using a neural network with a recursive structure. The prediction accuracy of the proposed model is proven to be superior to the existing design-oriented models. The data-driven neural network prediction models developed in this paper can provide a rapid prediction and design for FRP-confined composites.

Multiscale ceramic matrix composite thermomechanical damage model with fracture mechanics and internal state variables

This work presents a multiscale damage model that accounts for ceramic matrix composite (CMC) length scale dependent behavior, including scale-specific brittle matrix damage initiation and propagation. Internal state variable theory is implemented to obtain the stress–strain constitutive relationship for damaged ceramic matrix material using damage variables which capture the effects of matrix cracking and nucleation and growth of matrix porosity. Matrix cracking is modeled using a damage variable determined using fracture mechanics and the self-consistent scheme. This methodology provides an effective way to model the effects of matrix cracks which initiate when stress intensity factors exceed the fracture toughness of the material, and whose growth is governed by crack growth kinetics. An additional source of material nonlinearity, matrix porosity, occurs as a result of material diffusion around grain boundaries and is related to the material entropy dissipation. These effects are captured using a porosity state variable that is explicitly related to the volumetric strain. The nonlinear predictive capabilities of the material model are demonstrated for monolithic silicon carbide, unidirectional (UD) carbon fiber/silicon carbide matrix (C/SiC) CMC, and five-harness satin (5HS) woven C/SiC CMC. The model predictions are in excellent agreement with experimental observations for 2D woven C/SiC CMCs from literature.

Deformation of a Three-Layer Rod with a Compressible Core in a Neutron Flow*

We studied the deformation of a three-layer elastoplastic rod with a compressible core in a neutron flow. To describe the kinematics of the sandwich asymmetric across the thickness, the broken line hypothesis are adopted: Bernoulli’s hypothesis for the thin face layers and Tymoshenko’s hypothesis with linear approximation of displacements over the thickness for the compressible core. The resistance of the core in the tangential direction is taken into account. The constitutive stress–strain relations correspond to the theory of small elastoplastic deformations. The system of differential equilibrium equations is derived using the variational method. The kinematic boundary conditions of the rod is free support of the ends by fixed rigid supports. The boundary-value problem is solved by determining four functions of deflections and longitudinal displacements of the midsurfaces of the face layers. The analytical solution is obtained on the basis of the method of elastic solutions. Its numerical analysis is carried out in the case of an evenly distributed load.

A single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams

In this paper, a displacement-based single-variable first-order shear deformation nonlocal theory for the flexure of isotropic nanobeams is presented. In the present theory, the beam axial displacement consists of a bending component, whereas the beam transverse displacement consists of a bending component and a shearing component. Bending components do not take part in the cross-sectional shearing force, and the shearing component does not take part in the cross-sectional bending moment. The present theory utilizes linear strain–displacement relations. The displacement functions of the present theory give rise to the constant transverse shear strain through the beam thickness. Hence as is the case with the nonlocal Timoshenko beam theory, the present theory also requires a shear correction factor. In the present theory, nonlocal differential stress–strain constitutive relations of Eringen are utilized in order to take into account the size-dependent behaviour of nanobeams. In the present theory, beam gross equilibrium equations are utilized to derive the governing differential equation. As against other first-order shear deformation nonlocal beam theories, the present theory involves only one governing differential equation of fourth order and has only one unknown function. There exists a strong resemblance between expressions of the cross-sectional shearing force, cross-sectional bending moment and governing differential equation of the present theory and corresponding expressions of the nonlocal Bernoulli–Euler beam theory based on the nonlocal elasticity theory of Eringen. For the present theory, boundary conditions are described. The present theory also describes two distinct and physically meaningful clamped boundary conditions. Illustrative examples pertaining to the flexure of shear deformable isotropic nanobeams demonstrate the efficacy of the present theory.

Improved Junction Body Flow Modeling Through Data-Driven Symbolic Regression

A novel data-driven turbulence modeling framework is presented and applied to the problem of junction body flow. In particular, a symbolic regression approach is used to find nonlinear analytical expressions of the turbulent stress-strain coupling that are ready for implementation in computational fluid dynamics (CFD) solvers using Reynolds-averaged Navier-Stokes (RANS) closures. Results from baseline linear RANS closure calculations of a finite square-mounted cylinder with a Reynolds number of 11; 000, based on diameter and freestream velocity, are shown to considerably overpredict the separated flow region downstream of the square cylinder, mainly because of the failure of the model to accurately represent the complex vortex structure generated by the junction flow. In the present study, a symbolic regression tool built on a gene expression programming technique is used to find a nonlinear constitutive stress-strain relationship. In short, the algorithm finds the most appropriate linear combination of basis functions and spatially varying coefficients that approximate the turbulent stress tensor from high-fidelity data. Here, the high-fidelity data, or the so-called training data, were obtained from a hybrid RANS/Large Eddy Simulation (LES) calculation also developed with symbolic regression that showed excellent agreement with direct numerical simulation data. The present study, therefore, also demonstrates that training data required for RANS closure development can be obtained using computationally more affordable approaches, such as hybrid RANS/LES. A procedure is presented to evaluate which of the individual basis functions that are available for model development are most likely to produce a successful nonlinear closure. A new model is built using those basis functions only. This new model is then tested, i.e., an actual CFD calculation is performed, on the well-known periodic hills case and produces significantly better results than the linear baseline model, despite this test case being fundamentally different from the training case. Finally, the new model is shown to also improve predictive accuracy for a surface-mounted cube placed in a channel at a cube height Reynolds number of 40; 000 over traditional linear RANS closures.

Crack velocity- and strain rate- dependent dynamic compressive responses in brittle solids

Crack velocity and strain rate have an important influence on the dynamic fracture in brittle solids. However, the correspondence between crack velocity- and strain rate- dependent dynamic compressive responses is hardly established. In this study, a dynamic compression-based model triggered by microcrack growth of brittle solids is proposed. This dynamic model consists of the improved model of wing crack extension containing crack inclination effect, the correlation of crack velocity and strain rate, and the crack velocity-related dynamic fracture toughness. This correlation of crack velocity and strain rate is obtained by the time derivative of crack extension relating to axial strain. Crack extension relating to axial strain is obtained by use of correlation of damage relating to crack and strain. The whole variations of strain rate- (or crack velocity-) dependent stress-strain constitutive relationships from low strain rate (or crack velocity) to high strain rate (or crack velocity) are studied. The micro-mechanisms corresponding to the dynamic stress-strain curves also are explained. Moreover, a unified conclusion revealing the sensitivities of strain rate on peak axial strain ε1peak appearing at peak strength and initial tangent modulus Eini of stress-strain curves is proposed. Influences of crack velocity and strain rate on crack initiation stress and dynamic strength are analyzed in compression. Effects of the size, inclination and friction of microcracks on dynamic strength also are illustrated. The reasonability of these proposed theoretical results is verified via the published experimental results.

Modeling strategy for dynamic-modal mechanophore in double-network hydrogel composites with self-growing and tailorable mechanical strength

Smart materials with self-growing and tailorable mechanical strength have wide-range potential applications in self-healing, self-repairing, self-assembly, artificial muscle, soft robots and intelligent devices. However, their working mechanisms and principles are not fully understood yet and mathematically and physical modeling is a huge challenge, as traditionally synthesized materials cannot self-grow and reconstruct themselves once formed or deformed. In this study, a phenomenological constitutive model was developed to investigate the working mechanisms of self-growing and tailorable mechanical strength in double-network (DN) hydrogel composites, induced by mechanochemical transduction of dynamic-modal mechanophore. An extended Maxwell model was firstly employed to characterize the mechanical unzipping of hydrogel composites, and then mechanochemically induced destruction and reconstruction processes of brittle network in the hydrogel composite were formulated. The enhanced mechanical strength of brittle network has been identified as the key driving force to generate self-growing and tailorable mechanical strength in the hydrogel composite. Finally, a stress-strain constitutive relationship was developed for the dynamic-modal mechanophorein the hydrogel composite. Simulation results obtained from the proposed model were compared with the experimental data, and a good agreement has been achieved. This study provides an effective strategy for modelling and exploring the working mechanism in the mechanoresponsive DN hydrogel composites with self-growing and tailorable mechanical strength.

Study on microcrack growth-based dynamic compressive mechanical properties in brittle rocks

The dynamic microcrack growth has a great influence on the macroscopic dynamic mechanical properties of brittle rocks under dynamic compressive loadings. However, the relationships between dynamic microcrack growth and macroscopic dynamic mechanical properties are rarely studied. Based on the microcrack growth-induced stress-strain constitutive relationship, the relationship between quasi-static and dynamic fracture toughness, the relationship between crack velocity and strain rate, the relationship between strain rate and dynamic fracture toughness, a micromechanics-based stress-strain constitutive model is proposed. Furthermore, the relationship between crack velocity and strain rate is derived by the time derivative of equation describing the relationship between crack length and axial strain. The relationship between strain rate and dynamic fracture toughness is obtained by coupling the suggested relationship between crack velocity and strain rate, and the crack velocity and fracture toughness. Effects of strain rate on stress-strain relation and dynamic compressive strength are studied. The reasonability of the proposed dynamic constitutive model is verified by the experimental results. Sensitivities of initial damage, confining pressure, parameters m, e0 and R on stress-strain relation, dynamic compressive strength and dynamic elastic modulus are discussed. These studies will provide a certain theoretical help for analyzing the stability of brittle surrounding rocks in deep undergrounding engineering.

Effect of temperature gradient on transient thermal creep of heated and stressed concrete in transient state tests

The stress-strain constitutive relation of concrete subjected to thermomechanical loads is extremely complicated since its deformation relies on the sequences in which the heat and loads are applied. Existing experiments demonstrated that the development of quasi-instantaneous transient thermal creep, generally referred to as TTC, was dependent on the virgin heating of a mechanically pre-loaded concrete specimen. However, the mechanism behind it is still not fully understood. In this paper, a numerical method is developed to analyze the stress fluctuation and corresponding plastic strain caused by the combination of pre-load and temperature gradient. The results show that the TTC calculated using existing semi-implicit methods is reasonable only at early stage but could be wrong at high temperatures. The sharp increase of TTC beyond around 500 °C observed in many tests could be attributed to the implicit inclusion of the extra plastic strain resulted from temperature gradient. By excluding the extra plastic strain generated at different temperature levels, a more accurate formula for calculating TTC is proposed.

Study on Stress-strain Constitutive Relationship of Super-long-age Plastic Concrete under Triaxial Compression

The triaxial stress-strain relationship of super long-aged (540d) plastic concrete mixed with clay and bentonite was studied. According to the test, the stress-strain curves of plastic concrete under different confining pressures are obtained. It is found that the stress-strain curves under the lateral confining pressure are basically the same (0.4 MPa, 0.6 MPa), (0.4 MPa, 0.8 MPa), and the slope of the rising section is basically the same. Compared with the peak pressure ratio (0.2 MPa, 0.4 MPa), the lateral confining pressure is larger. In addition, the calculation formula of the peak secant elastic modulus and the triaxial compressive stress-strain relationship in accordance with the lateral pressure of plastic concrete are also proposed. The fourth-order polynomial mathematical model. The establishment of the mathematical model provides a basis for the development of plastic concrete specifications and triaxial compression performance numerical analysis, and further promotes the engineering application of plastic concrete.

Second strain gradient theory in orthogonal curvilinear coordinates: Prediction of the relaxation of a solid nanosphere and embedded spherical nanocavity

In this paper, Mindlin’s second strain gradient theory is formulated and presented in an arbitrary orthogonal curvilinear coordinate system. Equilibrium equations, generalized stress-strain constitutive relations, components of the strain tensor and their first and second gradients, and the expressions for three different types of traction boundary conditions are derived in any orthogonal curvilinear coordinate system. Subsequently, for demonstration, Mindlin’s second strain gradient theory is represented in the spherical coordinate system as a highly-practical coordinate system in nanomechanics. Second strain gradient elasticity have been developed mainly for its ability to capture the surface effects in the presence of micro-/nano- structures. As a numeric illustration of the theory, the surface relaxation of spherical domains in Mindlin’s second strain gradient theory is considered and compared with that in the framework of Gurtin–Murdoch surface elasticity. It is observed that Mindlin’s second strain gradient theory predicts much larger value for the radial displacement just near the surface in comparison to Gurtin–Murdoch surface elasticity.

Stress-strain constitutive model of concrete corroded by saline soil under uniaxial compression

Abstract Stress-strain constitutive relationship is one of the basic mechanical properties of concrete materials, and the stress-strain relationship of concrete subjected to salt corrosion has changed significantly. Saline soil in Western China has the characteristics of many kinds of salt and high concentration, which makes the damage to concrete more serious. In this paper, the uniaxial compressive stress-strain constitutive relation of corroded concrete in saline soil area of Western China is studied, and the constitutive model based on ion diffusion, chemical reaction and mechanical analysis was also given. Firstly, the causes and mechanisms of the change of stress-strain curve of corroded concrete were analyzed via the stress-strain curve, scanning electron microscopy with energy dispersive spectroscopy (SEM-EDS), X-ray diffraction (XRD), and porosity. Then, the corroded concrete specimens were divided into three parts from the outside to the inside, namely, the corrosive damage part, the corrosive enhancement part and the non-corrosive part. Combined Fick's diffusion law and chemical reaction, the formulas for calculate the critical failure depth and the internal expansion stress were given. The corrosion effects of the corrosive damage part and the corrosive enhancement part were expressed by the critical failure depth and the internal expansion stress respectively. Finally, based on the meso-statistical parallel bar constitutive model of non-corroded concrete, a stress-strain constitutive model of corroded concrete in saline soil region of Western China was established considered the corrosion effect.