Stress-strain Constitutive Equation Research Articles

A New Theory of Non-Linear Thermo-Elastic Constitutive Equation of Isotropic Hyperelastic Materials

Considering the influence of temperature and strain variables on materials. According to the relationship of conjugate stress-strain, a complete and irreducible non-linear constitutive equation of isotropic hyperelastic materials is derived and the constitutive equations of 16 types of isotropic hyperelastic materials are given we study the transformation methods and routes of 16 kinds of constitutive equations and the study proves that transformation of two forms of constitutive equation. As an example of application, the non-linear thermo-elastic constitutive equation of isotropic hyperelastic materials is combined with the natural vulcanized rubber experimental data in the existing literature base on MATLAB, The results show that the fitting accuracy is satisfactory.

Hot deformation behavior of AZ40 magnesium alloy at elevated temperatures

Hot compression tests on AZ40 magnesium alloy were conducted on a Gleeble 1500d hot simulation testing machine in a deformation temperature range of 330 °C-420 °C and a strain rate range of 0.002-2 s-1. Hot deformation behaviors were investigated on the basis of the analysis of the flow stressstrain curves, constitutive equation, and processing map. The stress exponent and apparent activation energy were calculated to be 5.821 and 173.96 kJ/mol, respectively. Deformation twins and cracks located in grain boundaries were generated at 330 °C and 0.02 s-1, which are associated with a high strain rate and a limited number of available slip systems. With increasing temperature and decreasing strain rate, the twins disappeared and the degree of dynamic recrystallization increased. The alloy was completely dynamically recrystallized at 420 °C and 0.002 s-1, with a homogenous grain size of approximately 13.7 μm. The instability domains of the deformation behavior can be recognized by processing maps. By considering the processing maps and characterizing the microstructure, the optimum hot deformation parameters in this experiment were determined to be 420 °C and 0.002 s-1.

A contact problem in thermoviscoelastic diffusion theory with second sound

We consider a contact problem in thermoviscoelastic diffusion theory in one space dimension with second sound. The contact is modeled by the Signorini’s condition and the stress-strain constitutive equation is of Kelvin−Voigt type. The thermal and diffusion disturbances are modeled by Cattaneo’s law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier’s law. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions and the lack of regularity. We prove that the viscoelastic term provides additional regularity leading to the existence of weak solutions. Then, fully discrete approximations to a penalized problem are considered by using the finite element method. A stability property is shown, which leads to a discrete version of the energy decay property. A priori error analysis is then provided, from which the linear convergence of the algorithm is derived. Finally, we give some computational results.

Reduced order modeling strategies for computational multiscale fracture

The paper proposes some new computational strategies for affordably solving multiscale fracture problems through a FE2 approach. To take into account the mechanical effects induced by fracture at the microstructure level the Representative Volume Element (RVE), assumed constituted by an elastic matrix and inclusions, is endowed with a large set of cohesive softening bands providing a good representation of the possible microstructure crack paths. The RVE response is then homogenized in accordance with a model previously developed by the authors and upscaled to the macro-scale level as a continuum stress–strain constitutive equation, which is then used in a conventional framework of a finite element modeling of propagating fracture.For reduced order modeling (ROM) purposes, the RVE boundary value problem is first formulated in displacement fluctuations and used, via the Proper Orthogonal Decomposition (POD), to find a low-dimension space for solving the reduced problem. A domain separation strategy is proposed as a first technique for model order reduction: unconventionally, the low-dimension space is spanned by a basis in terms of fluctuating strains, as primitive kinematic variables, instead of the conventional formulation in terms of displacement fluctuations. The RVE spatial domain is then decomposed into a regular domain (made of the matrix and the inclusions) and a singular domain (constituted by cohesive bands), the required RVE boundary conditions are rephrased in terms of strains and imposed via Lagrange multipliers in the corresponding variational problem. Specific low-dimensional strain basis is then derived, independently for each domain, via the POD of the corresponding strain snapshots.Next step consists of developing a hyper-reduced model (HPROM). It is based on a second proposed technique, the Reduced Optimal Quadrature (ROQ) which, again unconventionally, is determined through optimization of the numerical integration of the primitive saddle-point problem arising from the RVE problem, rather than its derived variational equations, and substitutes the conventional Gauss quadrature. The ROQ utilizes a very reduced number of, optimally placed, sampling points, the corresponding weights and placements being evaluated through a greedy algorithm. The resulting low-dimensional and reduced-quadrature variational problem translates into very relevant savings on the computational cost and high computational speed-ups.Particular attention is additionally given to numerical tests and performance evaluations of the new hyper-reduced methodology, by “a-priori” and “a-posteriori” error assessments. Moreover, for the purposes of validation of the present techniques, a real structural problem exhibiting propagating fracture at two-scales is modeled on the basis of the strain injection-based multiscale approach previously developed by the authors. The performance of the proposed strategy, in terms of speed-up vs. error, is deeply analyzed and reported.

Shear strength prediction for steel fiber reinforced concrete beams without stirrups

This study proposes a theoretical approach based on modified compression field theory to predict the shear strength of steel fiber-reinforced concrete (SFRC) beams without stirrups. The tensile contribution of the steel fibers is considered in the tensile stress–strain constitutive equations, which take into account the distribution of the fibers. The proposed strength model is verified by comparing it with 139 shear failure tests previously conducted on SFRC and reinforced concrete beams without stirrups. The influences of concrete strength, fiber volume, shear span-to-depth ratio, and longitudinal steel ratio on predicted shear strength are also discussed. Comparisons between the predicted and experimental results show that the proposed model can estimate shear strength accurately.

A quasi-static contact problem in thermoviscoelastic diffusion theory

The problem of thermoviscoelastic quasi-static contact between a rod and a rigid obstacle, when the diffusion effect is taken into account, is modeled and analyzed. The contact is modeled by the Signorini's condition and the stress–strain constitutive equation is of the Kelvin–Voigt type. In the quasi-static case, the governing equations correspond to the coupling of an elliptic and two parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions. The existence of solutions is proved as the limit of solutions to a penalized problem. Moreover, we show that the weak solution converges to zero exponentially as time goes to infinity. Finally, we give some computational results where the influence of diffusion and viscosity are illustrated in contact.

Deformation and strength behaviors of frozen clay with thermal gradient under uniaxial compression

Thermal gradient is one of the main features in artificial freezing ground and permafrost regions. The deformation and strength behavior of frozen soils with thermal gradient is of utmost importance for stabilities analysis of frozen engineering. A series of uniaxial compression tests were carried out on frozen saturated clay at various average temperatures and thermal gradients. The experimental results indicated that the uniaxial stress–strain curve for frozen clay with thermal gradient presents elastic-linear strain hardening characteristics, both the hardening modulus and uniaxial compression strength increase as the thermal gradient and average temperature decreased, but the elastic modulus varies little as the thermal gradient increased. The polygonal stress–strain constitutive equations were implemented into the FEM (Finite Element Method) and the local axial strain distribution of frozen clay with thermal gradient was studied. Two stages are observed from the curves describing the relationship between local axial strain and specimen height, i.e., increase stage and decrease stage for the slope of d. Characteristic parameters such as the demarcation height of Hd, the maximum slope of dmax are defined and analyzed. It is found that the characteristic parameters are all dependent on the thermal gradient and average temperature. The thermal gradient presents weakening effects to the local axial strain rate at middle specimen height of δεa,H=10′/δt in frozen clay with identical average temperature, and that further lead to the decrease of hardening modulus and uniaxial compression strength.

An experimental investigation of the behaviour of steels over large temperature and strain rate ranges

During forging and machining manufacturing processes, the material is subject to large strains at high strain rates which provoke local heating and microstructural changes. Modelling of these phenomena requires precise knowledge of the stress–strain constitutive equations for a large range of strains, strain rates and temperatures. An experimental study of the rheology of both hyper- and hypo-eutectoid steels (with different microstructures) over a temperature range from 20°C to 1000°C and with strain rates from 10−2 to 105s−1 has been undertaken. These tests were performed in compression on cylindrical specimens and in shear using hat-shaped specimens. Both a GLEEBLE 3500 thermomechanical testing machine and a Split-Hopkinson Pressure Bar apparatus were used. From these tests, three deformation domains have been identified as a function of the material behaviour and of the changes in the deformed microstructure. Each domain was characterized by its behaviour, including the competition between hardening and softening, strain rate sensitivity on the flow stress and the softening phenomenon (i.e. recrystallisation or recovery, etc.). Finally, based on thermodynamical considerations, the conditions of thermoplastic instability (i.e. shear bands, twinning, heterogeneities, etc.) and microstructural changes are highlighted using process maps of the dissipated power repartition.

Efficient Solution Techniques for Multiscale Structural Optimization in Materials Science

ABSTRACTWe consider the modeling and simulation of multiscale phenomena which arise in finding the optimal shape design of microcellular composite materials with heterogeneous microstructures. The paper focuses on the solution of the resulting partial differential equation (PDE) constrained structural optimization problem and development of efficient multiscale numerical algorithms which are challenging tools in reducing the computational complexity. The modeling strategy is applied in materials science for microstructural ceramic materials of multiple constituents. Our multiscale method is based on the efficient combination of both macroscopic and microscopic models. The homogenization technique based on the concept of strong separation of scales and the asymptotic expansion of the unknown displacements is applied to extract the macroscopic information from the microscale model.In the framework of all-at-once approach we find a proper combination of the iterative procedure for the nonlinear problem arising from the first order necessary optimality conditions, also known as Karush-Kuhn-Tucker (KKT) conditions, and efficient large-scale solvers for the stress-strain constitutive equation. We use the path-following predictor-corrector schemes by means of Newton's method and fast multigrid (MG) solution techniques. The performance of two preconditioners, incomplete Cholesky (IC) and algebraic multigrid (AMG), for the resulting homogenized state equation is studied. The comparative analysis for both preconditioners in terms of number of iterations and computing times is presented and discussed. Our interests focus on the parallel implementation of the preconditioning techniques and the use of BoomerAMG as a part of the free software library Hypre developed at the Center for Applied Scientific Computing (CASC), Lawrence Livermore National Laboratory (LLNL).

Compressive stress–strain relationship of steel fiber-reinforced reactive powder concrete after exposure to elevated temperatures

An experimental research is performed on the complete compressive stress–strain relationships for reactive powder concrete (RPC) with various steel fiber contents after exposure to 20–900°C. The steel fiber volume dosage is 1%, 2% and 3%. The results indicate that the compressive strength and elastic modulus of RPC increase at first, then decrease with the increasing temperature, and the loss of elastic modulus is quicker than the compressive strength. The peak strain and ultimate strain reach peaks at 600°C and 700°C respectively, and they increase exponentially before the peak points, but decrease linearly after the peak points. Based on the experiment results, the compressive stress–strain constitutive equations applicable to the unheated and heated RPC are proposed, which provide a valuable reference for future industrial applications and design.

Shear behavior model for steel fiber-reinforced concrete members without transverse reinforcements

Due to the complex shear mechanism of steel fiber-reinforced concrete (SFRC) members, there is lack of comprehensive shear behavior models for SFRC members. The shear behavior model, based on a smeared crack model, requires the tensile stress–strain constitutive equation of SFRC membrane subjected to biaxial stresses. After SFRC panel tests under biaxial stresses were recently conducted, it has been possible to create a more complete smeared crack model for estimating the shear behavior of SFRC members. It is, however, very difficult to conduct such experiments for different types of steel fibers, various amount of steel fibers, different ranges of concrete strengths, etc. Thus, in this study, steel fibers are modeled as average direct tensile contribution elements in a modified smeared crack truss model, considering directionality and distribution of fibers. In this way, only simple bond tests are required to reflect the effects of different characteristics of SFRC. In addition, the shear contribution of steel fibers can be obtained considering the bond failure of steel fibers. The proposed model was compared to the test results of 8 SFRC panels and 80 SFRC beams, and the shear behavior of the SFRC members was well estimated.

Hybrid mixture theory based moisture transport and stress development in corn kernels during drying: Coupled fluid transport and stress equations

In first part, a solution scheme for the multiscale fluid transport equation of Singh et al. (2003a) was developed, which could be easily implemented in a commercial finite elements package for solving swelling/shrinking problems. The solution scheme was applied to drying of corn kernels. Laplace transformation was used to convert the integral part of the fluid transport equation to a set of differential equations. In materials undergoing volume change during drying, a moving mesh is needed in Eulerian coordinates, which is tedious to implement. An alternate method was used that involved transforming the equation to stationary Lagrangian coordinates. Once the equations were solved, the data was transformed back to the moving Eulerian coordinates during post-processing. Corn heterogeneity was taken into account by using different coefficient of diffusivity values for various corn components. To predict stresses, it was assumed that the shape of a kernel remains geometrically similar to its initial shape during drying. Using this assumption a relation between the strain tensor and volume fraction of water was developed. The viscoelastic stress–strain constitutive equation was used to calculate the magnitude of stresses at different spatial locations inside a corn kernel. In the companion paper (part 2), the experimental validation of the solution, simulation results for various drying conditions and the practical implications are discussed.

Revisiting the thermodynamics of hardening plasticity for unsaturated soils

A thermodynamically consistent extension of the constitutive equations of saturated soils to unsaturated conditions is often worked out through the use of a unique ‘effective’ interstitial pressure, accounting equivalently for the pressures of the saturating fluids acting separately on the internal solid walls of the pore network. The natural candidate for this effective interstitial pressure is the space averaged interstitial pressure. In contrast experimental observations have revealed that, at least, a pair of stress state variables was needed for a suitable framework to describe stress–strain–strength behaviour of unsaturated soils. The thermodynamics analysis presented here shows that the most general approach to the behaviour of unsaturated soils actually requires three stress state variables: the suction, which is required to describe the invasion of the soil by the liquid water phase through the retention curve; two effective stresses, which are required to describe the soil deformation at water saturation held constant. However a simple assumption related to the plastic flow rule leads to the final need of only a Bishop-like effective stress to formulate the stress–strain constitutive equation describing the soil deformation, while the retention properties still involve the suction and possibly the deformation. Commonly accepted models for unsaturated soils, that is the Barcelona Basic Model and any approach based on the use of an effective averaged interstitial pressure, appear as special extreme cases of the thermodynamic formulation proposed here.

Alternative stress-integration schemes for large-deformation problems of solid mechanics

In solving nonlinear problems of solid mechanics by the finite-element method, stresses at integration points are usually obtained by integrating nonlinear constitutive equations, given known incremental strains. In a large-deformation analysis, stress–strain relationships must be frame independent such that any rigid-body motion does not induce strain within the material. This principle is generally satisfied by introducing an objective stress rate, such as the Jaumann or Truesdell stress rates, into the constitutive equations. This paper investigates three alternative algorithms for integrating stress–strain relationships in a large-deformation analysis. It is shown that the effect of rigid-body motion is equivalent to a stress transformation and this transformation can be introduced before, after or during integration of the stress–strain constitutive equations. Although there is no theoretical advantage, in terms of accuracy, for selecting one of these strategies over the others, in terms of efficiency of algorithms one is more advantageous than the others. Performance of the proposed algorithms is studied and compared by means of numerical examples. The results of this study can be used in the development of fast and robust algorithms for stress integration of constitutive equations in nonlinear finite-element analysis.

Fractal ladder models and power law wave equations.

The ultrasonic attenuation coefficient in mammalian tissue is approximated by a frequency-dependent power law for frequencies less than 100 MHz. To describe this power law behavior in soft tissue, a hierarchical fractal network model is proposed. The viscoelastic and self-similar properties of tissue are captured by a constitutive equation based on a lumped parameter infinite-ladder topology involving alternating springs and dashpots. In the low-frequency limit, this ladder network yields a stress-strain constitutive equation with a time-fractional derivative. By combining this constitutive equation with linearized conservation principles and an adiabatic equation of state, a fractional partial differential equation that describes power law attenuation is derived. The resulting attenuation coefficient is a power law with exponent ranging between 1 and 2, while the phase velocity is in agreement with the Kramers-Kronig relations. The fractal ladder model is compared to published attenuation coefficient data, thus providing equivalent lumped parameters.

Friction in unconforming grain contacts as a mechanism for tensorial stress–strain hysteresis

Materials composed of consolidated grains and/or containing internal contacts are widespread in everyday life (e.g. rocks, geomaterials, concretes, slates, ceramics, composites, etc.). For any simulation of the elastic behavior of this class of solids, be it in seismology, in NDT, or in the modeling of building constructions, the stress–strain constitutive equations are indispensable. Since the most common loading patterns in nature considerably deviate from simple uniaxial compression, the problem of tensorial stress–strain representation arises. In simple loading cases it may be sufficient to use a phenomenological constitutive model. However, in a more general case, phenomenological approaches encounter serious difficulties due to the high number of unknown parameters and the complexity of the model itself. Simplification of the phenomenology can help only partly, since it may require artificial assumptions. For instance, is it enough just to link the volumetric stress to the volumetric strain, or do we have to include shear components as well, and if yes, in what form? We therefore propose a physical tensorial stress–strain model, based on the consideration of plane cracks with friction. To do this, we combine known relations for normal displacements of crack faces given by contact mechanics, the classical Amonton's law of dry friction for lateral displacements, and the equations of elasticity theory for a collection of non-interacting cracks with given orientation. The major advantages of this model consist in the full tensorial representation, the realistic stress–strain curves for uniaxial stress compression and quantitative comparison with experimental data, and a profound account for hysteretic memory effects.

Microcontact-based theory for acoustics in microdamaged materials

A universal theory describing the wide range of mechanical and acoustic phenomena in solids with internal contacts such as rocks, concrete, ceramics and composites is quite complex to develop. The goal of this paper is to demonstrate the potential to deduce the macroscopic stress–strain constitutive equation for a material as a whole starting from the microscopic hysteretic force–displacement relationship of individual asperities in contact. The material considered in the proposed model contains a large number of isotropic oriented penny-shaped cracks with rough internal surfaces. The stress–strain relationship we obtained for such a material is based on physical principles and laws. Even so, it displays close resemblance to the phenomenological Preisach–Mayergoyz model adopted for mechanical hysteresis and nonlinearity. This constitutive relationship is then used to simulate an experiment with standing acoustic waves in a resonant bar, and to compare model predictions to actual observations. We show that the most important experimentally measurable nonlinear features of these materials, such as the typical classical and nonclassical shifting behavior of the resonant frequency, the dependencies of the amplitudes of the generated harmonics, the softening due to intensive straining, and the subsequent relaxation effect (slow dynamics) can be attributed and explained in terms of the mechanics and the statistics of the internal contacts. The present model bridges the gap between three scales: macroscopic (material as a whole), mesoscopic (structure of intergranular contacts and cracks) and microscopic scale (contacts of individual asperities).

Sheet springback prediction based on non-linear combined hardening rule and Barlat89’s yielding function

A stress–strain constitutive equation of non-linear combined hardening rule has been proposed in this paper according to non-linear kinematic hardening theory of Lemaitre and Chaboche and Barlat89’s yielding function. The algorithm of elastic predicting, plastic correcting and radial returning is applied to calculate the stress increment. Taking a benchmark of NUMISHEET’93 2-D draw bending problem as an example, the effects of different hardening rules and yielding functions are investigated. Comparison with NUMISHEET’93 experiment data indicates that the isotropic hardening rule over-predicts springback since the predicted stress after forming is relatively larger, while the predicted springback by non-linear combined hardening rule coincides well with experimental data. In addition to that, the study indicates that Barlat89’s and Hill48’s yielding functions describe the practical material behavior better than Mise’s yielding function, and the predicted results by Barlat89’s and Hill48’s yielding criterion are very closer to experiment data.

Numerical analysis of dynamic thermoviscoelastic contact with damage of a rod

We describe and analyse a model for a problem of thermoviscoelastic dynamic contact which allows for the general evolution of the material damage. The effects on the mechanical properties of the material due to crack expansion are described by a damage field, which measures the decrease in the load-bearing capacity of the material. The damage process is assumed to be reversible and the microcracks which develop as a result of tension or compression may grow or disappear. The geometric setting is that of a 1D rod which may contact a deformable obstacle. The contact is modelled by the normal compliance condition and the stress–strain constitutive equation is of Kelvin–Voigt type. The model consists of a coupled system of energy–elasticity equations together with a non-linear parabolic inclusion for the damage field. The existence of a local weak solution is established using penalization, a finite element algorithm for the solution is constructed and analysed and the results of numerical simulations based on this algorithm are presented. The simulations illustrate how the size of the normal compliance coefficients, the damage rate coefficients and the applied force affect the character of the evolution of the damage. In particular, cycles of bonding and debonding can occur.

Stress–strain constitutive equations of concrete material at elevated temperatures

The paper presents a critical review of the currently available models for the mechanical behaviour of concrete at elevated temperatures. Based on these models and experimental data a stress–strain–temperature model is proposed which incorporates the effect of transient strain implicitly. This model can be easily incorporated into existing commercial finite element analysis software. A numerical example on a wall element heated on two opposite faces indicates that at very early stages of heating transient strain does not play an important part, but that as the exposure time increases the effect of ignoring transient strain progressively increases and produces unconservative estimates of load carrying capacity.