Stress Strain Space Research Articles

Nonlinear seismic response analysis of reef-coral sand site in the South China Sea

To investigate the nonlinear seismic response of the reef-coral sand site in the South China Sea, a visco-elasto-plastic dynamic constitutive model and corresponding subroutine for coral sand material are developed in this study. The Matasovic skeleton curve and the derived hysteretic curve are refined in this model by introducing a critical shear modulus. Hysteresis criteria under irregular cyclic loads are established based on the improved Masing criteria. Moreover, increments of the octahedral strain and stress are utilized to construct the mapping rules between the three-dimensional stress-strain relationship and the one-dimensional stress-strain space. On that basis, a nonlinear finite element model of the reef-coral sand site is established, and the distribution pattern of the ground motion field and key ground motion parameters, such as the seismic amplification ratio and response spectrum of the reef-coral sand site, are investigated. The findings suggest that the coral sand layer significantly increases the site seismic responses, especially near the edge of the reef flat, thus posing a threat to the seismic safety of the buildings and construction in this area. Furthermore, the deformation of the reef-coral sand site tends to be concentrated on one side, resulting in a certain residual displacement after the earthquake.

Experimental validation of the thermodynamic theory for predicting the strength of 6061 Al alloy with complex loadings

The safety and reliability of engineering structures face severe challenges all the time. Determining how to accurately predict the strength of material structures is a major issue for academia and industry. Recently, a general thermodynamic theory was proposed by Wang [Eng Fract Mech., 2021, 254: 107936] to predict the failure properties of material structures beyond classical strength criteria. In this study, the thermodynamics-based method is applied to predict the strength of 6061 Al alloy with complex loadings, accompanied by experimental validation. The plastic dissipation energy density function is established and its expansion coefficients are determined by uniaxial and equi-biaxial tensile tests. Then, the predicted failure strength of the 6061 Al alloy in stress space and strain space is obtained, which matches well with the experimental results. Moreover, a comparison between our prediction and the results of two classical strength theories, i.e., the maximum shear stress and maximum distortion energy theories, was performed. In summary, this study verifies the accuracy and reliability of the thermodynamic theory for predicting the strength of material structures with complex loadings, which lays the foundation for the engineering applications of the non-empirical thermodynamic method.

Characteristics of Elastoplastic Consolidation by Compaction and Its Effects on Coal Permeability

This paper presents a combined experimental–analytical investigation of coal strain development under fluctuating applied hydrostatic stress. The laboratory setup mimics the isotropic volumetric compaction of coal under burial–uplift cycles in the absence of tectonic stress. Special emphasis is placed on the corresponding permeability evolution of the coal strata. Our results show that the stress–strain path is exponential, approaching a linear relation in the logarithmic stress–strain space with the monotonic increase in stress. A similar behavior is found for the strain–permeability path in the logarithmic strain–permeability space. The permeability recovery undergoes hysteresis with respect to the stress in a stress loading–unloading cycle, but the hysteresis is not manifest with respect to the strain. A theoretical geomechanical consolidation analysis was performed using an elastoplastic modelling framework. The analysis suggests that plastic strain is the cause of the hysteresis of the strain recovery in a stress loading–unloading cycle. The closed hysteresis loops manifested in stress loading–unloading–reloading cycles are promoted by the plastic strain during stress unloading and the difference in evolution rates of the elastic core between loading and unloading. The results of this study are helpful for understanding the mechanism of permeability evolution and optimizing water and coal seam gas production.

Hodge duality between stress space and strain space in anisotropic media

AbstractDuality in a linear elasticity theory is studied based on a theory of differential form. A generalized expression of a Hodge star operator with an index is introduced. The index in the Hodge star operator means a superposition of ordinary Hodge star operators. By using the superposed Hodge star operator, a linear constitutive relation in anisotropic media can be expressed as a duality between a stress and strain spaces. Then, a basic elastodynamic equation is derived from the conservation law and the linear constitutive relation in the stress and strain spaces. Moreover, a geometric expression of equation of stress function and displacement function is derived by using the differential form and the dual relation of linear constitutive relation. These geometric results imply that the approach of differential forms is applicable to an analysis of deformation in anisotropic media. The superposed Hodge star operator discussed in this study has similar properties of a discrete Hodge operator, which represents the summation of differential forms on the decomposed regions. This means that the superposed Hodge star operator and the discrete Hodge operator express the constitutive relation in which physical quantities act on each other.

Model-free data-driven simulation of inelastic materials using structured data sets, tangent space information and transition rules

Model-free data-driven computational mechanics replaces phenomenological constitutive functions by numerical simulations based on data sets of representative samples in stress-strain space. The distance of strain and stress pairs from the data set is minimized, subject to equilibrium and compatibility constraints. Although this method operates well for non-linear elastic problems, there are challenges dealing with history-dependent materials, since one and the same point in stress-strain space might correspond to different material behaviour. In recent literature, this issue has been treated by including local histories into the data set. However, there is still the necessity to include models for the evolution of specific internal variables. Thus, a mixed formulation of classical and data-driven modeling is obtained. In the presented approach, the data set is augmented with directions in the tangent space of points in stress-strain space. Moreover, the data set is divided into subsets corresponding to different material behaviour. Based on this classification, transition rules map the modeling points to the various subsets. The approach will be applied to non-linear elasticity and elasto-plasticity with isotropic hardening.

On the use of elliptical bulge tests in material characterization through inverse methodologies

The hydraulic bulge test is an experimental technique that can be used to identify the properties of sheet metals up to large strain. The hydraulic bulge test, usually, is a free expansion of a sheet blank through a circular hole, driven by the pressure of a fluid, where the ends of the blank are blocked using a suitable die. The state of stress during the bulge test is mainly equi-biaxial, however, using elliptical holes with different aspect ratios, it is possible to produce heterogeneous stress-strain fields that can be used to identify the properties of the material through an inverse method; in this case, the non-linear Virtual Fields Method (VFM) was adopted. With respect to the traditional tests employed with the VFM in plasticity, i.e. double notched specimens or specimens with holes subjected to tensile tension, elliptical bulge tests allow to investigate a different zone in the stress-strain space, with a higher triaxiality. The capability of this type of test to calibrate the parameters of advanced anisotropic models is here assessed using simulated experiments.

Strength and Compressibility of Ammonia-Soda Residue from the Solvay Sodium Plant

This work presents the discussion of the results for an experimental study conducted to characterise the mechanical behaviour of ammonia-soda residue (ASR). The calcareous sludge is an alkaline waste formed during the production of soda ash and deposited at the area of the former Solvay Sodium Plant factory in Krakow, Poland. Isotropically consolidation drained (CID) triaxial tests and constant rate of strain (CRS) consolidation tests include the full saturation with water, completion of the consolidation, and the loading/strain rate choice. For this purpose, ASR undisturbed samples were collected from the ground and submitted to laboratory experiments. These samples show a distinct difference in the initial bulk density, the initial level of compaction, initial void ratio, and the natural water content. The CD triaxial tests were conducted under three different levels of confining pressure; in turn, CRS tests were run with two appropriate input strain rates. According to the physical state of ASR and the depth of sampling, two different evolutions of the critical state in the stress–strain space were observed. In the light of the assessed stress–strain–strength behaviour, key design engineering parameters of ASR were calculated.

New ductile fracture model for fracture prediction ranging from negative to high stress triaxiality

In this study, the ductile fracture of the alloy metal is analyzed. First, the key factors affecting the initiation of ductile fracture are investigated. Then, a new uncoupled ductile fracture model is established in the stress space by considering the volume change work and the distortion work simultaneously. The proposed ductile fracture model is not only dependent on stress triaxiality but also dependent on the Lode angle parameter. Subsequently, the proposed ductile fracture model is transformed to the mixed stress-strain space. Experimental tests are adopted to verify the validity of the fracture model. The comparative study with four typical ductile fracture criteria, viz., the pressure modified maximum shear stress criterion (PMMS), the extended Mohr-Coulomb criterion (EMC), the extended Hosford-Coulomb criterion (EHC), and the extended unified strength theory criterion (EUST), indicates that the proposed fracture model predicts the fracture strain with good stability and accuracy in a wide range of stress states, especially in the range of the high stress triaxiality. Finally, the asymmetry of the fracture locus and the realistic stress space are discussed.

Dynamic thermo-mechanical behaviors of SME TiNi alloys subjected to shock loading

Dynamic thermo-mechanical coupling behaviors of shape memory TiNi alloy in the strain rate ranging from 300 s−1 to 2000 s−1 are investigated by the split Hopkinson pressure bar (SHPB) device with an infrared (IR) detection system. In stress–strain space, dynamic response shows a strong strain hardening property, however, in stress–temperature space, transformation path is particularly sensitive to the strain rate. The corresponding temperature evolution measured synchronously shows that local temperature increased associated with the forward phase transition, and it would keep the loading maximum value unchanged or decreased for unloading, depending on the strain rate. Besides, local temperature evolution was consistent with the transformation stress and its values at different points are the same. Temperature evolution and transformation deformation mechanism is then analyzed by a simple one-dimensional model. The results show that latent heat and dissipated energy are responsible for temperature variation. Furthermore, the temperature evolution with strain rate reveals that with the increase of strain rate, the phase transformation deformation mechanism undergoes a transformation from phase transition fronts propagation during lower strain rates to combination of local nucleation and front propagation during middle strain rate and to uniform nucleations during higher strain rates. The results are helpful for a passive shape memory alloy (SMA) micro-valve design.

A kernel method for learning constitutive relation in data-driven computational elasticity

For numerical simulation of elastic structures, data-driven computational approaches attempt to use a data set of material responses, without resorting to conventional modeling of the material constitutive equation. In a material data set in the stress–strain space, the data points are considered to lie on or near a low-dimensional manifold, rather distribute ubiquitously in the space. This paper presents a kernel method for extracting this manifold. We formulate a regularized least-squares problem for learning a manifold, and show that its optimal solution corresponds to an eigenvector of a real symmetric matrix. Therefore, the method requires only simple computational task, and is easy to implement. We also give a description how to use the obtained solution in static equilibrium analysis of an elastic structure. Numerical experiments on two-dimensional continua are performed to demonstrate effectiveness and robustness of the proposed method.

Errata to “Stress inversion meets plasticity theory: A review of the theories of fault-slip analysis from the perspective of the deviatoric stress-strain space” [J. Struct. Geol. 125, 296–310

Errata to “Stress inversion meets plasticity theory: A review of the theories of fault-slip analysis from the perspective of the deviatoric stress-strain space” [J. Struct. Geol. 125, 296–310

Smart composites with tunable stress–strain curves

Smart composites with tunable stress–strain curves are explored in a numerical setting. The macroscopic response of the composite is endowed with tunable characteristics through microscopic constituents which respond to external stimuli by varying their elastic response in a continuous and controllable manner. This dynamic constitutive behavior enables the composite to display characteristics that cannot be attained by any combination of traditional materials. Microscopic adaptation is driven through a repetitive controller which naturally suits the class of applications sought for such composites where loading is cyclic. Performance demonstrations are presented for the overall numerical framework over complex paths in macroscopic stress–strain space. Finally, representative two- and three-dimensional tunable microstructures are addressed by integrating the control approach within a computational environment that is based on the finite element method, thereby demonstrating the viability of designing and analyzing smart composites for realistic applications.

On the sensitivity of the Mohr-Coulomb ductile fracture criterion in the mixed stress-strain space on varying hardening curve approximations

In recent years the use of the Mohr-Coulomb type fracture criteria has become popular in ductile fracture prediction. In ductile fracture prediction, fracture criteria are often transformed into the mixed stress strain space formulation and applied to numerical simulations in this formulation. This paper will investigate the shape of the Mohr Coulomb fracture criterion in the mixed stress strain space as a function of the used hardening curve formulation, in particular the hardening curve exponent n that determines the slope of the hardening curve approximation for large strains. It is found that the shape of the fracture criterion significantly depends on the slope of the hardening curve. This makes a correct extrapolation of the hardening curve for large strain values essential for a correct ductile fracture prediction using the Mohr-Coulomb fracture model in its mixed stress strain space formulation.

On the choice of stress–strain variables for unsaturated soils and its effect on plastic flow

The net stress plus suction and the average skeleton stress plus modified suction are two alternative sets of energetically consistent stress variables for modelling the hydro-mechanical behaviour of unsaturated soils. When used in conjunction with their work-conjugate strains, both sets of stress variables correctly calculate the first-order term of the hydro-mechanical work input into a soil element subjected to infinitesimal changes of deformation and water content. They therefore also correctly calculate the increment of internal energy along a given stress–strain path, that is the integral of the first-order term of the infinitesimal work input.This paper shows, however, that the above two sets of stress variables lead to different values of the second-order term of the hydro-mechanical work input. They are therefore no longer equivalent with respect to other aspects of material behaviour governed by the second-order work such as the flow rule of elasto-plastic models. The flow rule assumes the normality between plastic strains and equipotential surfaces defined in the conjugate stress–strain space. This normality is however lost when an elasto-plastic model originally formulated in terms of net stress plus suction is recast in terms of average skeleton stress plus modified suction (or vice versa) by means of standard mapping relationships between stress variables. To restore normality in both stress spaces, it is necessary to impose specific forms of elastic and plastic behaviour.

Comparative investigation on influences of concrete material constitutive models on structural behavior

This analysis was carried out based on the smeared crack model and plasticity damage model in general finite element program ABAQUS. Comparison and analysis were made on the key factors of concrete constitutive models that have effect on the static mechanical behavior of structural members. Then the mechanical behavior of reinforced concrete members and steel-concrete composite members under monotonic loading and cyclic loading were simulated using various concrete constitutive models. Both the smeared cracking model and the plasticity damage model could accurately simulate the macroscopic response of flexural concrete members or flexural composite members subjected to monotonic load, and the smeared cracking model was more accurate in simulating the cracking behavior of concrete, applicable for analyzing the stress situation of concrete during stress-strain space constantly changing after cracking. Besides, the plasticity damage model could be adopted to simulate the stress behavior of concrete structures or composite structures under the action of cyclic load. The two models could not accurately simulate the whole process of stress behavior of concrete in two-dimensional or three-dimensional stress state.

Data driven modeling of plastic deformation

In this paper the application of machine learning techniques for the development of constitutive material models is being investigated. A flow stress model, for strain rates ranging from 10−4 to 1012 (quasi-static to highly dynamic), and temperatures ranging from room temperature to over 1000 K, is obtained by beginning directly with experimental stress–strain data for Copper. An incrementally objective and fully implicit time integration scheme is employed to integrate the hypo-elastic constitutive model, which is then implemented into a finite element code for evaluation. Accuracy and performance of the flow stress models derived from symbolic regression are assessed by comparison to Taylor anvil impact data. The results obtained with the free-form constitutive material model are compared to well-established strength models such as the Preston–Tonks–Wallace (PTW) model and the Mechanical Threshold Stress (MTS) model. Preliminary results show candidate free-form models comparing well with data in regions of stress–strain space with sufficient experimental data, pointing to a potential means for both rapid prototyping in future model development, as well as the use of machine learning in capturing more data as a guide for more advanced model development.

Fatigue life prediction using multiaxial energy calculations with the mean stress effect to predict failure of linear and nonlinear elastic solids

An approach is presented that enables the calculation of elastic strain energy in linear and nonlinear elastic solids during arbitrary thermomechanical load cycles. The approach uses the simple fact that the variation of both strain and complementary energies always forms a rectangular shape in stress–strain space, hence integration is no longer required to calculate the energy. Furthermore, the approach considers the mean stress effect so that predictions of fatigue damage are more realistically representative of real-life experimental observations. By doing so, a parameter has been proposed to adjust the mean stress effect. This parameter α is based on the well-known Smith–Watson–Topper energy criterion, but allows consideration of other arbitrary mean stress effects, e.g. the Bergmann type criterion.The approach has then been incorporated into a numerical method which can be applied to uniaxial and multiaxial, proportional and non-proportional loadings to predict fatigue damage. The end result of the method is the cyclic evolution of accumulated damage. Numerical examples show how the method presented in this paper could be applied to a nonlinear elastic material.

THE TESTS AND FAILURE CRITERIA OF FULL LIGHTWEIGHT SHALE CERAMSITE CONCRETE UNDER TRUE TRIAXIAL COMPRESSION

The experiments of full lightweight shale ceramsite concrete LC30 were carried out under true a triaxial compressive state in proportional loading by the large-scale static and dynamic triaxial testing machine in Dalian University of Technology. The failure modes of samples were observed, and the ultimate triaxial strength, plastic strains, peak strains, total strains, and stress-strain curves were also recorded. The results show that the strength and deformation of full lightweight shale ceramsite concrete under triaxial compression are much larger than that of uniaxial ones, there is a very apparent ‘Platform of Plastic Flow (PoPF)’ on a stress-strain curve, and also present four failure modes as normal concrete, thusly the stress and the strain corresponding to PoPF should be design strength and strain during an engineering design for lightweight aggregate concrete. Also, the effects of intermediate principal stress on strength and deformation are analyzed under triaxial compression, the ultimate stresses were predicted by triaxial strength model, and the absolute values of relative errors are all less than 6%, which proves the test results are accurate and reliable. Finally, the failure criteria of stress and strain are established based on an octahedral stress-strain space respectively. The relevance of test points and curves are better to the former; the tension-compression meridians intersects the hydrostatic pressure axis in high-pressure stress area; and the failure surface is closed; although the latter is worse than the former, which can reflect effectively the outward expansion trend for the failure surface, the characteristics of meridians are smooth and convexity.

On the assessment of energy dissipated through hysteresis in finite element analysis

The accurate reproduction of the hysteretic behaviour exhibited by soils under cyclic loading is a crucial aspect of dynamic finite element analyses and is typically described using the concept of damping ratio. In this paper, a general algorithm is presented for assessing the damping ratio simulated by any constitutive model based on the registered behaviour in three-dimensional stress–strain space. A cyclic nonlinear elastic model capable of accurately reproducing a wide range of features of soil behaviour, including the variation of damping ratio with deformation level, is chosen to illustrate the capabilities of the proposed algorithm. The constitutive model is described and subsequently employed in two sets of finite element analyses, one involving the dynamic response of a sand deposit subjected to different types of motion and another focussing on the simulation of a footing subjected to cyclic vertical loading. The application of the presented algorithm provides insight into the processes through which energy is dissipated through hysteresis.

UNDRAINED SHEAR STRENGTH OF K0CONSOLIDATED SOFT CLAYS UNDER TRIAXIAL AND PLANE STRAIN CONDITIONS

Theoretical formulas for predicting the undrained shear strength of K0consolidated soft clays under the stress path related to triaxial and plane strain tests are presented within the framework of critical state soil mechanics. An inclined elliptical yield surface is adopted to take account of the initial anisotropic stress state. The undrained strength is determined by combining the undrained stress path in the volumetric stress–strain space and the initial yield surface in the deviator-mean stress space. The derived mathematical expressions are functions of the critical state frictional angle, the plastic volumetric strain ratio and the overconsolidation ratio, which can be simplified into the solutions for isotropically consolidated clays under triaxial tests or under plane strain tests. The results calculated by using the theoretical formulas obtained in this paper are in good agreement with the available collected test results. It indicates that these new formulas are applicable to triaxial and plane strain tests on normally and lightly to moderately overconsolidated soft clays.