Mechanical Constitutive Equations Research Articles

Thermo-hydro-mechanical modelling of the heterogeneous subsidence and swelling in the desiccation cracked clayey strata

Soil desiccation cracking as a consequence of severe environmental changes alters soil deformation mechanisms significantly. Therefore, this study aims to explore the effect of crack characteristics and environmental conditions on the heterogeneous deformation of desiccation-cracked soils using thermo-hydro-mechanical analyses. The model framework consists of balance equations, thermal, hydraulic, and mechanical constitutive equations, while the model scenarios were determined based on statistical analyses. The meteorological record of Qom city was used for three years, from 2015 to 2017, to capture long-term behaviour under wetting-drying cycles. Findings revealed that cracks extend the deformation range, potentially up to six times, with variation based on crack dimensions and spacing. Notably, narrower cracks experienced more pronounced deformation than wider ones. The cracked soil with a crack depth of 2.5 m showed 1.5 times higher swelling and subsidence than crack depth of 1 m. Furthermore, the wider cracks indicated a lower rate of increase in their dimensions compared to the initial state during drying. The investigation also highlights the mechanisms of soil surface shape due to swelling and shrinkage, resulting in concave and convex surfaces, respectively. The results provide new perspectives on the behaviour of fine-grained deposits in arid to semi-arid climates with deep groundwater levels.

Modeling the thermo-mechanical response and phase changes in metallic additive manufacturing (MAM) processes using a dissipative phase-field model

Additive manufacturing (AM) has emerged as a highly promising manufacturing technique, offering unprecedented possibilities for creating complex geometries and functional structures. However, harnessing the full potential of AM requires the development of a robust computational framework capable of capturing the intricate multi-scale and multi-physics nature of the process. The constitutive and structural responses encountered in AM are particularly challenging to reproduce due to the complex behavior of the material involved. This research aims to address these challenges by presenting a comprehensive computational approach that incorporates a material model capable of accurately representing the behavior of different phases occurring during AM. To achieve this, the finite element method, using the Lagrangian framework in the implicit time scheme, is employed through the widely adopted ABAQUS software. Computational implementation is facilitated using the FORTRAN programming language. By employing weakly coupled thermal and mechanical constitutive equations, the framework enables the analysis of thermal stresses, strains, and displacements during realistic solidification processes, which inherently involve highly nonlinear constitutive relations. Through a series of numerical examples, the capabilities of the proposed model are demonstrated across various computational scales, particularly during the rapid melting and solidification phases. These simulations reveal the formation of residual stresses, which can lead to part distortion and have detrimental effects on the mechanical properties of the manufactured components. This research contributes to the advancement of additive manufacturing by providing a reliable computational tool that integrates the complex interplay between thermal and mechanical phenomena. The developed framework enhances our understanding of the AM process, offering valuable insights into the factors influencing the structural integrity and performance of additively manufactured parts.

A new plastic flow theoretical model and verification for non-dense metals

The mechanical behavior and constitutive equations of isotropic non-dense metals, such as metal foams, porous metals, and lattice metals, have been extensively studied, but the subsequent yield surfaces depicted by different theoretical models are somewhat controversial and have not been fully validated in the whole permissible loading space. Based on two accepted assumptions for isotropic non-dense metals, we proposed a new plastic flow theoretical model. In order to verify its rationality, we established two mesoscopic models with different initial relative densities and different meso-structures. Then, the large amount of numerical simulation experimental data was established, which covers enough multiaxial loadings in the permissible principle-strain space. Our model solves some of the controversies in current models and adapts the equivalent stress, equivalent strain, and constitutive equations seamlessly to deformation from non-dense to dense state. Numerical results from two mesoscopic models show the relations between equivalent stress and plastic strain in our theoretical model have better consistency under all multiaxial loadings than those in some known models. We checked the topology of subsequent yield surfaces in the plastic principle-strain space and the results turn out the subsequent yield surfaces are not self-similar. The large amount of numerical test data not only well validates our theoretical model but also will be beneficial to the mechanical study of non-dense metals under multiaxial loadings.

Temperature and Strain Rate Dependence on the Tensile Mechanical Properties, Constitutive Equations, and Fracture Mechanisms of MarBN Steel.

The effect of strain rate and temperature on the thermomechanical behavior and microstructure of MarBN steel is studied with the strain rates of 5 × 10-3 and 5 × 10-5 s-1 from room temperature (RT) to 630 °C. At high strain rates of 5 × 10-3 s-1, the Holloman and Ludwigson equations can better predict tensile plastic properties. In contrast, under low strain rates of 5 × 10-5 s-1, coupling of the Voce and Ludwigson equations appears to predict the flow relationship at RT, 430, and 630 °C. However, the deformation microstructures have the same evolution behavior under strain rates and temperatures. Geometrically necessary dislocations appear along the grain boundaries and increase the dislocation density, which results in the formation of the low-angle grain boundaries and a decrease in the number of twinning. The strengthening sources of MarBN steel include grain boundary strengthening, dislocation interactions, and multiplication. The fitted R2 values of these models (JC, KHL, PB, VA, ZA) to plastic flow stress at 5 × 10-5 s-1 are greater than 5 × 10-3 s-1 for MarBN steel. Due to the flexibility and minimum fitting parameters, the phenomenological models of JC (RT and 430 °C) and KHL (630 °C) give the best prediction accuracy under both strain rates.

A deep learning approach for solving diffusion-induced stress in large-deformed thin film electrodes

Understanding the effects of large deformation on electrochemical performance of electrodes, such as silicon and tin, in lithium-ion battery has stimulated great interest in analyzing stress evolution during electrochemical cycling and in developing mechanochemical models for the stress analysis. As the complexity of mechanochemical models continues to increase, there is an urgent need to develop computational methods to reduce computational costs and increase computational efficiency. In this work, we propose a physics-inspired neural network (PINN) based on the DeepXDE deep learning library in analyzing diffusion-induced stresses (DIS) in a thin film electrode with large deformation, in which a loss function, which is associated with the loss functions of partial differential equations (PDEs) in the domain and the initial/boundary conditions for the mechanochemical problem, is introduced. Using the proposed physics-inspired neural network, the mechanical equations, including geometric, constitutive and equilibrium equations, in the framework of finite deformation theory as well as the mass transport equation are solved to obtain the stress evolution in the large-deformed thin-film electrode. The numerical results are in accord with the results obtained from finite element method. This work provides a unique approach for deep learning to solve the coupling mechanochemical problems in energy storage.

Mechanical behavior and constitutive equations of porcine brain tissue considering both solution environment effect and strain rate effect

Because brain tissue is immersed in cerebrospinal fluid, and studies such as TBI (traumatic brain injury) include impact mechanical properties of brain tissue, the constitutive relation of brain tissue needs to consider both solution environment effect and strain rate effect. In this article, we systematically study the effects of different external environments and different strain rates on the mechanical behavior of porcine brain tissue. Furthermore, a constitutive equation considering the above effects is proposed, which may well depict stress-strain relations of porcine brain tissue under these effects. Our work will have significant reference value for related research in biomedical engineering.

Size effect in piezoelectric semiconductor nanostructures

A gradient theory is applied to the mechanical constitutive equations for piezoelectric semiconductor nanostructures. This is achieved by considering the strain gradients in the constitutive equation with high-order stresses and electric displacements in advanced continuum model. The C1 continuous interpolations of displacements or a mixed formulation is required in the finite element method (FEM) due to the presence of the second-order derivative on the elastic displacements. A mixed FEM is then developed from the principle of virtual work. Numerical examples clearly show the significant effect of flexoelectricity on the induced electric potential and electric current in the piezoelectric semiconductor nanostructures.

A Review on Mechanical Models for Cellular Media: Investigation on Material Characterization and Numerical Simulation.

Cellular media materials are used for automobiles, aircrafts, energy-efficient buildings, transportation, and other fields due to their light weight, designability, and good impact resistance. To devise a buffer structure reasonably and avoid resource and economic loss, it is necessary to completely comprehend the constitutive relationship of the buffer structure. This paper introduces the progress on research of the mechanical properties characterization, constitutive equations, and numerical simulation of porous structures. Currently, various methods can be used to construct cellular media mechanical models including simplified phenomenological constitutive models, homogenization algorithm models, single cell models, and multi-cell models. This paper reviews current key mechanical models for cellular media, attempting to track their evolution from their inception to their latest development. These models are categorized in terms of their mechanical modeling methods. This paper focuses on the importance of constitutive relationships and microstructure models in studying mechanical properties and optimizing structural design. The key issues concerning this topic and future directions for research are also discussed.

Constitutive modeling of magneto-viscoelastic polymers, demagnetization correction, and field-induced Poynting effect

Magneto-active polymers are polymer-based composites consisting of magnetizable micro-particles embedded in an elastomeric matrix material. The presence of magnetic particles provides strong tunability to the stiffness and damping properties of the polymeric composites under the magnetic field. We propose here a novel free energy-based phenomenological modeling for magneto-active polymers. Coupled mechanical and magnetization constitutive equations are derived by considering magneto-viscoelasticity in a finite deformation framework. The model calibration takes into account the demagnetization effect, which depends on the specimen size, shape, and its own magnetization. The model prediction is verified with the available experimental data. Finally, a coupled simple shear problem is solved to demonstrate the influence of magnetic field on the Poynting effect. Classical nonlinear soft elastic material shows positive Poynting effect, i.e., shearing planes try to expand. We found the possibilities of field-induced reverse or negative Poynting effect in shear for magneto-active polymers. Such a negative Poynting effect could potentially affect a very diverse range of applications from the performances of miniature sensors and actuators to controlled drug delivery.

Prognostic evaluation of hip joint function following capsule repair based on a threedimensional finite element analysis model

To construct a three-dimensional (3D) finite element mechanical model of total hip arthroplasty for comparison of biomechanical differences of the hip joint following capsule repair and postoperative rehabilitation. Six frozen specimens of hip joint posterior capsule ligament complex were collected in a bone-capsule-bone manner, and the load-strain curve and other mechanical properties of the specimens were tested using a universal material testing machine. Thin-section CT data of the pelvis and lower limbs obtained from a volunteer were imported into Mimics software to construct a 3D model of the hip joint. Digital models of the cup, femoral prosthesis and joint capsule were created in CATIA software and imported into Mimics to simulate total hip arthroplasty; the assembled data were imported into ABAQUS software. The properties of the capsule were set according to results of the mechanical test, anatomical studies, and constitutive equations, and the biomechanics of the anatomically repaired and conventionally repaired capsules were compared during hip flexion. The results of testing on the 6 capsule specimens showed a mean ultimate tensile strain of (39.21±5.23)% and a mean of ultimate tensile strength of 1.65±0.38 MPa. The stress-strain curve of the finite element model was consistent with the results of mechanical test on the specimens and the biochemical characteristics of the capsule. The stress was distributed evenly in the anatomically repaired capsule during hip flexion but not in the capsule repaired through the conventional approach; the tensile stress in the lower part of the conventionally repaired capsule reached the ultimate tensile stress measured on the capsule specimens at a 90° flexion. The finite element model allows dynamic, quantitative and visual assessment of stress distribution in the hip joint capsule, and compared with the conventional approach, anatomical repair can achieve better biomechanical properties of the capsule.

Study on dynamic mechanical behavior of Q460JSC and HQ600 high strength steel

High-strength steel has been gradually applied in practical engineering and achieved good economic and social benefits. Due to the threat of terrorist attacks and accidental explosions, high-strength steel structures need to have high blast resistance. However, the studies on the mechanical properties and constitutive equations of high-strength steels under high strain rate are insufficient. The research on the anti-explosion properties of high-strength steel structures lacks material models and constitutive parameters. Therefore, this paper investigated the quasi-static and dynamic mechanical properties of Q460JSC and HQ600 high strength steels. A series of quasi-static tensile, high-speed tensile and SHTB tests were conducted to obtain the material properties under different strain rate in the range of 0.00025 s−1 to 5000 s−1. Combined with finite element simulation by LS-DYNA, the strain hardening relationship of specimens after necking was treated by inverse extrapolation method, and the true stress-strain curves at different strain rates were obtained. The effects of strain rate on materials strength and plasticity were analyzed. The quasi-static and dynamic constitutive parameters of these two high strength steels were fitted by Ludwik criterion and modified Johnson-Cook model, respectively. The results show that the yield strength, ultimate strength and fracture strain of these two kinds of high strength steels all increase with the increase of strain rate, but the uniform elongation is not sensitive to strain rate. The change of strength is not obvious in the strain rate range of 103. The inverse extrapolation method is effective in dealing with the true stress-strain curve after necking, and more accurate results can be obtained by continuously optimizing the strain hardening index. The improved Ludwik criterion and the new Johnson-Cook model correction formula can well describe the stress-strain relationship of these two high-strength steel materials in the range of 10−3 s−1 to 103 s−1, which provide basis for the investigation on anti-explosion properties of high-strength steel structures.

Particle Method Simulation for Formation and Flow of Cold Flakes in High-Pressure Die Casting

In high-pressure die casting (HPDC) processing of aluminum alloys, solidified layers generated in the sleeve of a die casting machine that flow into the mold cavity are known as “cold flakes.” The prediction and control of them are a crucially important issue for HPDC. This study developed a method to simulate their formation and flow using smoothed particle hydrodynamics. First, a solidified layer was modeled as a set of solid particles with behaviors defined by mechanical constitutive equations. Second, this study proposed an algorithm for ascertaining the phase of particles and for calculating liquid–solid particle interaction. Numerical results demonstrated that the method can predict the formations of solidified layers in the sleeve, their peeling and folding during the plunger movements, their inflow into the runner and the mold cavity, and flow disturbances caused by solidified layers trapped at the gate.

A physically-based model for strain-induced crystallization in natural rubber. Part I: Life cycle of a crystallite

Despite the numerous experimental investigations performed over the past century and more intensively in the last fifteen years, strain-induced crystallization in natural rubber still remains hardly understood in its precise mechanisms: a complete theoretical description for crystallization and melting of the involved crystallites is still needed to derive relevant physically-based mechanical constitutive equations. Therefore, the present Part I of our work proposes a coherent theory describing the full nucleation–growth–melting cycle of these crystallites, by using classical thermodynamics of phase transitions and by accounting for the topological constraints due to the network. A graphical representation of crystallite evolution involving strain, temperature, and crystallite size is then introduced, using a physical parameter to express the change of Gibbs free energy due to surface creation for a unit volume of crystalline phase. Finally, experimental results from literature exhibiting shape-memory effects in rubber are elucidated using this crystallite life cycle theory.

Effect of nonlinearities and objective function in optimization of an energy harvesting device

This work presents a study of the impact of the linearity assumption of the mechanical model in the overall performance of an energy harvesting piezoelectric beam. Also, a brief assessment of geometrical optimization solutions using different objective functions is presented. The mechanical model of the harvester is based on both linear and nonlinear variants of the electrical and mechanical constitutive equations for the piezoelectric material. The nonlinear elastic, damping and electromechanical coupling parameters are obtained via least squares identification using physical experimentation; the experimental tests are performed at different ground excitation accelerations. The computational optimization of the harvester is done using the genetic algorithm implemented in Matlab. Different objective functions are tested, i.e. broadband maximum peak power, maximum power at a particular frequency and broadband mean power; the influence of the selection of each of them in the total recovery of the power of the device is analyzed. The most suitable function to recover the vibratory energy from conventional transport vehicles is found.

Mechanical properties and constitutive equations of crumb rubber mortars

There will be a lot of pollution in the process of production and degradation of rubber. It is imperative to recycle rubber in order to build a green and saving environment. A new kind of mortar with rubber particles is an effective way to reuse the used rubber products. The crumb rubber mortar effectively resist external stress through its own deformation. Therefore, it is of practical significance to study the strength behavior of crumb rubber mortar. The stress-strain curves of crumb rubber mortars with 5 different rubber contents under a triaxial stress condition are obtained through laboratory testing. The internal variation law of the elastic modulus, Poisson’s ratio and damage variable are analyzed, and the necessity of incorporating the influence of the damage threshold into the damage evolution model is discussed. According to a rock damage model based on Lemaitre’s strain equivalence theory and the introduction of the damage threshold based on the traditional Weibull random distribution, the damage constitutive model of crumb rubber mortars under the influence of the damage threshold is established by considering two factors. The method of determining the model parameters is presented. The model can describe the entire process of strain softening and deformation of crumb rubber mortars for various levels of rubber content or confining pressure. The model can also fully reflect the linear elastic deformation characteristics of rock under small deformations and the nonlinear mechanical behavior after the peak strength is exceeded. The model is particularly suitable for describing the complex stress states in the field of underground engineering, and it provides a reference for the design of supporting structures. Finally, by comparing the measured results of three-axis compression tests of crumb rubber mortars with various rubber contents, the rationality of the model is verified.

Multiphase-field model of small strain elasto-plasticity according to the mechanical jump conditions

We introduce a small strain elasto-plastic multiphase-field model according to the mechanical jump conditions. A rate-independent $$J_2$$ -plasticity model with linear isotropic hardening and without kinematic hardening is applied exemplary. Generally, any physically nonlinear mechanical model is compatible with the subsequently presented procedure. In contrast to models with interpolated material parameters, the proposed model is able to apply different nonlinear mechanical constitutive equations for each phase separately. The Hadamard compatibility condition and the static force balance are employed as homogenization approaches to calculate the phase-inherent stresses and strains. Several verification cases are discussed. The applicability of the proposed model is demonstrated by simulations of the martensitic transformation and quantitative parameters.

Tensile mechanical properties, constitutive equations, and fracture mechanisms of a novel 9% chromium tempered martensitic steel at elevated temperatures

To explore the high-temperature tensile behavior and fracture mechanism of a novel 9% chromium tempered martensitic steel, G115, a series of tensile tests were conducted at 625, 650, and 675°C within the strain rate range of 5.2×10−5–5.2×10−3s−1. The results demonstrated that the tensile strength decreased with increasing temperature and decreasing strain rate. However, the elongation to failure increased with increasing temperature. The ductility at 675°C exhibited significant improvement. Furthermore, three typical constitutive equations were comparatively analyzed to accurately describe the tensile deformation behavior of G115 steel. A stress exponent of approximately 11.1 and an activation energy of approximately 639.865kJ/mol were obtained via the hyperbolic sine law. To explain the actual deformation mechanism, the threshold stress concept was introduced. The modified hyperbolic sine constitutive equation was proposed to determine the threshold stress σth, the true stress exponent nt, and the true activation energy Qt of G115 steel. The Qt value increased with increasing temperature and strain rate. Dislocation climb was the dominant deformation mechanism under the tested conditions. In addition, fracture surface investigations revealed a typical ductile fracture mode with a dense array of dimples at 625 and 650°C. However, transgranular facets with tear ridges were observed in the central region at 675°C. In addition, the microstructures of the fracture frontier and longitudinal section away from the fracture surface were studied to further understand the fracture mechanism.

Mechanical properties and constitutive equations of concrete containing a low volume of tire rubber particles

Uniaxial compressive tests were conducted for a low-volume tire rubber concrete (RC) with different rubber volume content levels and particle sizes (five of each). The influence of rubber content and particle size on the mechanical properties of RC was investigated, including axial compressive strength, elastic modulus, peak strain, ultimate strain, appearance of visible cracks and failure pattern of specimens. The mechanical analysis of test results was conducted based on uniaxial compressive stress–strain curves. Uniaxial compressive constitutive models of low-volume rubber concrete were established that included axial compressive strength, peak strain and constitutive parameters a and b. There was obvious regularity between constitutive parameters a and b, rubber content and rubber particle size. Moreover, the physical meanings of constitutive parameters a and b were given, and the established constitutive models were evaluated by tests. The test results demonstrate that the constitutive models have not only good predicting ability and high accuracy but also nice applicability within the limit of 50kg/m3 rubber content. The constitutive models were then improved by introducing the sand rate reduction factor k to reflect the influence of rubber additives. The improved constitutive models are able to predict the performance of concrete with low amounts of rubber.

Dynamic mechanical properties and constitutive equations of 2519A aluminum alloy

To analyze the effects of strain rate and temperature on the flow stress of 2519A aluminum alloy, the dynamic mechanical properties of 2519A aluminum alloy were measured by dynamic impact tests and quasi-static tensile tests. The effects of strain rate and temperature on the microstructure evolution were investigated by optical microscopy (OM) and transmission electron microscopy (TEM). The experimental results indicate that 2519A aluminum alloy exhibits strain-rate dependence and temperature susceptibility under dynamic impact. The constitutive constants for Johnson–Cook material model were determined by the quasi-static tests and Hopkinson bar experiments using the methods of variable separation and nonlinear fitting. The constitutive equation seems to be consistent with the experimental results, which provides reference for mechanical characteristics and numerical simulation of ballistic performance.

Continuum Modeling of Granular Media

This is a survey of the interesting phenomenology and the prominent regimes of granular flow, followed by a unified mathematical synthesis of continuum modeling. The unification is achieved by means of “parametric” viscoelasticity and hypoplasticity based on elastic and inelastic potentials. Fully nonlinear, anisotropic viscoelastoplastic models are achieved by expressing potentials as functions of the joint isotropic invariants of kinematic and structural tensors. These take on the role of evolutionary parameters or “internal variables,” whose evolution equations are derived from the internal balance of generalized forces. The resulting continuum models encompass most of the mechanical constitutive equations currently employed for granular media. Moreover, these models are readily modified to include Cosserat and other multipolar effects. Several outstanding questions are identified as to the contribution of parameter evolution to dissipation; the distinction between quasielastic and inelastic models of material instability; and the role of multipolar effects in material instability, dense rapid flow, and particle migration phenomena.