Elastic Constitutive Relation Research Articles

Biomechanics of soft biological tissues and organs, mechanobiology, homeostasis and modelling.

The human body consists of many different soft biological tissues that exhibit diverse microstructures and functions and experience diverse loading conditions. Yet, under many conditions, the mechanical behaviour of these tissues can be described well with similar nonlinearly elastic or inelastic constitutive relations, both in health and some diseases. Such constitutive relations are essential for performing nonlinear stress analyses, which in turn are critical for understanding physiology, pathophysiology and even clinical interventions, including surgery. Indeed, most cells within load-bearing soft tissues are highly sensitive to their local mechanical environment, which can typically be quantified using methods of continuum mechanics only after the constitutive relations are determined from appropriate data, often multi-axial. In this review, we discuss some of the many experimental findings of the structure and the mechanical response, as well as constitutive formulations for 10 representative soft tissues or organs, and present basic concepts of mechanobiology to support continuum biomechanical studies. We conclude by encouraging similar research along these lines, but also the need for models that can describe and predict evolving tissue properties under many conditions, ranging from normal development to disease progression and wound healing. An important foundation for biomechanics and mechanobiology now exists and methods for collecting detailed multi-scale data continue to progress. There is, thus, considerable opportunity for continued advancement of mechanobiology and biomechanics.

Nonlinear behavior simulation of ceramic-matrix composites using constituent-volume homogenization method

The nonlinear behavior of ceramic-matrix composites is affected by interface debonding and fiber pull-out. In this paper, we propose the constituent-volume homogenization method (CVHM) to replace the conventional bundle homogenization method, in order to model the behavior of trans-element debonding in woven structures using finite element method. Based on the CVHM, we establish the elastic relation of the relative motion between the fiber and the matrix, and the elastic constitutive relations for the constituents. Methods have also been developed to calculate the microscopic local stress of bundle under pull-out and multiaxial loads. Based on the above studies, we develop a procedure to analyze the stress-strain relation of braided composites using the CVHM. To validate the proposed method, we estimate the nonlinear behavior of the 2D plain-weave ceramic-matrix composites and compare the results with experimental data. The influence of interface shear strength (ISS) and the fiber ultimate tensile strength (UTS) on the nonlinear behavior is investigated.

Modelling of non-linear elastic constitutive relationship and numerical simulation of rocks based on the Preisach–Mayergoyz space model

SUMMARY The existence of pores, cracks and cleavage in rocks results in significant non-linear elastic phenomena. One important non-linear elastic characteristic is the deviation of the stress–strain curve from the linear path predicted by Hooke's law. To provide a more accurate description of the non-linear elastic characteristics of rocks and to characterize the propagation of non-linear elastic waves, we introduce the Preisach–Mayergoyz space model. This model effectively captures the non-linear mesoscopic elasticity of rocks, allowing us to observe the stress–strain and modulus–stress relationships under different stress protocols. Additionally, we analyse the discrete memory characteristics of rocks subjected to cyclic loading. Based on the Preisach–Mayergoyz space model, we develop a new non-linear elastic constitutive relationship in the form of an exponential function. The new constitutive relationship is validated through copropagating acousto-elastic testing, and the experimental result is highly consistent with the data predicted by the theoretical non-linear elastic constitutive relationship. By combining the new non-linear elastic constitutive relationship with the strain–displacement formula and the differential equation of motion, we derive the non-linear elastic wave equation. We numerically solve the non-linear elastic wave equation with the finite difference method and observe two important deformations during the propagation of non-linear elastic waves: amplitude attenuation and dispersion. We also observe wave front discontinuities and uneven energy distribution in the 2D wavefield snapshot, which are different from those of linear elastic waves. We qualitatively explain these special manifestations of non-linear elastic wave propagation.

Nonlinear hydroelastic vibration of foamed concrete beams via peridynamic differential operator

Evaluating the hydroelastic responses of underwater cementitious structural elements is critical for ensuring the sustainability and durability of energy-saving marine infrastructures. Existing work on the hydroelastic analysis of porous structures has been mostly developed using the general elastic constitutive relation; however, it fails to capture the influence of saturation. To fill this knowledge gap, we for the first time propose a novel fluid-porous structure interactive model that incorporates the combined effects of hydrodynamic pressure and saturation-induced pore pressure. One more pioneering effort is to solve this nonlinear hydroelastic problem by introducing peridynamic differential operator (PDDO). It is worth noting that the introduction of PDDO removes the inherent drawback employing the local-theory based techniques, namely being prone to singularities arising from the presence of discontinuity. The accuracy and reliability of the proposed numerical framework are validated by comparing the results with the degraded model in the reported literature. Moreover, our results highlight that the angular frequencies are underestimated when ignoring the effect of saturation in foamed concrete beams. The presented method provides a profound understanding of the underwater structural dynamic monitoring that benefits the design of marine infrastructures.

Semi-destructive methods for evaluating the micro-scale residual stresses of carbon fiber reinforced polymers

Micro-scale residual stress in carbon fiber reinforced polymers have a significant impact on their mechanical performance. The residual stresses in the carbon fibers were released by micro-slotting and micro-ring-core methods using focused ion beam (FIB). The released deformation fields were captured by cross-gratings and mapped by geometric phase analysis (GPA) and digital image correlation (DIC) methods. The in-plane residual stresses in the carbon fibers were experimentally determined to be −40.5 MPa using elastic constitutive relations, which is consistent with the composites cylinder model. The axial compressive residual stresses in the fibers were found to be greater than −100 MPa. Moreover, the maximum value of residual stress in the polymer matrix was observed at the interface, potentially serving as a crack initiation site.

Mechanical properties of poplar (NL-3412) branches

ABSTRACT The mechanical properties of poplar (NL-3412) branches are the basis for designing and optimizing the working parameters of branch pruning and crushing machinery. In this study, the tensile, compression, bending and shear properties of poplar branches were investigated by utilizing a precision microcomputer-controlled electronic universal testing machine. The test results suggested that the elastic constitutive relationship of the stress–strain curves of poplar branches under longitudinal compression, transverse compression, and bending conformed to Hooke's law, and longitudinal tensile and shear exhibited plastic damage; the anti-failure strength, anti-failure elastic modulus, and external work of poplar branches under longitudinal compression were greater than those under transverse compression. The analysis results indicated that the correlation function between the anti-failure strength and the external force work of poplar branches is positively correlated, and the external force work required to destroy poplar branches can be estimated by the established regression mathematical model. The results of the study can provide important data support for the optimal design of poplar branch pruning and shredding machinery and related simulation experimental research. HIGHLIGHTS The basic parameters of mechanical properties of poplar branches were determined. The variation law of stress–strain curve of poplar branch mechanical test was proved. The mathematical model between the strength and external force work was established. Provide reference for the optimal design of branch pruning and crushing machinery. Provide data support for the simulation cutting test of poplar branch.

A modified approach for a scaled boundary shell formulation in structural isogeometric analysis

In this contribution, an efficient modification of the scaled boundary finite element method for shells is presented to compute accurate results for h-refinement. While previous works on the scaled boundary shell formulation showed inaccurate and unstable results for fine meshes, this modification overcomes such problems by the assumption of a linear force distribution over the thickness. Furthermore, the shell formulation is derived in the framework of isogeometric analysis by discretizing the bottom surface of the shell as a reference surface. This leads to a solution for in-plane directions in a weak sense, while the scaling direction is solved analytically by a modified Padé expansion. Therefore, the normal vector on the reference surface is constructed and the shell is scaled along that leading to an extensible shell formulation incorporating a three-dimensional linear elastic constitutive relation. Since the isogeometric analysis is utilized, an exact derivation of the normal vector and its gradient is possible. Initial curvature locking and Poisson’s thickness locking are eliminated inherently while shear locking and volumetric locking are alleviated by mesh refinement. The power of the modified approach is presented by standard benchmark problems and compared to the results of the unmodified approach and standard shell formulations.

Implementation and validation of a bonded particle model to predict rheological properties of viscoelastic materials

This work focuses on implementing a particle-based method able to characterize viscoelastic materials whose rheological properties, such as storage modulus G’ and loss modulus G”, are known. It is based on the Bonded Particle Model, with the elastic constitutive relation here substituted with a viscoelastic one to capture time-scale effects. The Burgers model, vastly used in literature to model viscoelastic systems, is discretized and implemented. The test case used for calibration comprises of a cubic lattice, sheared with a periodic motion, to mimic the effect of a shear rheometer. After appropriate filtering of the stress response, the rheological properties are obtained, highlighting the effect of the lattice geometry, as well as the particle size, on the accuracy of the model. Moreover, the Burgers parameters are calibrated by analytically fitting the experimental dataset, showing the limitation of the Burgers model. The micro-contact parameters are obtained from the macro parameters through appropriate scaling. After completing a frequency sweep, the simulated G’ and G” show a relatively large error, around 25% for G’ for example. For this reason, a more robust model, namely the Generalized Maxwell Model, has been implemented. The calibration procedure is performed in the same fashion as for the Burgers model. Moreover, the tangential micro-contact parameters are scaled w.r.t. the normal ones. This scaling parameter, called α, is calibrated by minimizing the Root Mean Square Error between simulation and experimental data, giving errors below 10% in both G’ and G” for a large dataset. Additionally, a full ring plate-plate rheometer setup is simulated, and the simulation is compared with the given experimental dataset, again finding a good agreement.

A computational approach to obtain nonlinearly elastic constitutive relations of strips modeled as a special Cosserat rod

This study presents a computational approach to obtain nonlinearly elastic constitutive relations of strip/ribbon-like structures modeled as a special Cosserat rod. Starting with the description of strips as a general Cosserat plate, the strip is first subjected to a strain field which is uniform along its length. The Helical Cauchy–Born rule is used to impose this uniform strain field which deforms the strip into a six-parameter family of helical configurations — the six parameters here correspond to the six strain measures of rod theory. Two vector variables are introduced to model the position of the deformed centerline of the strip’s cross-section and to model orientation of thickness lines along the strip’s width. The minimization of the strip’s plate energy under the constraint of the aforementioned uniformity in strain field reduces the partial differential equations of plate theory from the entire mid-plane of the strip to just a system of nonlinear ordinary differential equations along the strip’s width line for the above mentioned two vector variables. A nonlinear finite element formulation is further presented to solve this set of ordinary differential equations. This, in turn, yields the strip’s stored energy per unit length as well as the induced internal force, moment and stiffnesses of the strip for every prescribed set of six strain measures of rod theory. The proposed scheme is used to study uniform bending, twisting and shearing of a strip. For the case of uniform twisting and shearing, the strip is also seen to buckle along its width into a more complex configuration which are accurately captured by the presented scheme. We demonstrate that the presented scheme is more general and accurate than the existing schemes available in the literature.

Load introduction to composite columns revisited—Significance of force allocation and shear connection stiffness

The AISC 360-16 Specification recommends that the design shear force between parts of a composite column in the load introduction area shall be calculated based on the force allocation at ultimate limit state. Applicability of this straightforward method to the load levels that usually arise in slender composite columns is questionable, as this capacity-based force allocation is only true when the axial force is equal to the plastic resistance of the composite cross-section. Next, the number of required shear connectors is calculated as a quotient of the design shear force and the strength of a single shear connector. We demonstrate that: first, for the lower load levels, the stiffness-based force allocation gives a more accurate estimate of the shear force; second, the number of shear connectors satisfying the strength requirement can lead to insufficient force transfer between parts of the composite cross-section. To investigate the shear transfer mechanism in composite columns, we derive an analytical model with linear elastic constitutive relations both for steel and concrete and three types of shear force slip laws: elastic, elastic plastic, and rigid plastic. The case studies carried out for different shear transfer scenarios demonstrate the importance of the shear connection stiffness on the effectiveness of the load introduction. The remaining portion of the shear force is transferred outside the load introduction area, which hampers the column’s ability to withstand shearing from varying bending moments or incipient buckling. To control the shear force transfer efficiency by enhancing the shear connection stiffness, we propose an original Stiffness Method and provide design charts as an aid in the design process.

A thermodynamics-based micro-macro elastoplastic micropolar continuum model for granular materials

A micro–macro elastoplastic micropolar continuum model for granular materials in the dry and pendular states is proposed on the basis of thermodynamics. The independent rotational degrees of freedom are considered besides the translational degrees of freedom for particles among the discrete particle system, which is described by the equivalent micropolar continuum. In the framework of the micropolar continuum, the elastic part and yield locus at macroscale are firstly rigorously deduced from the general elastic constitutive relation and Coulomb-type yield criterion at inter-particle contacts, respectively. Furthermore, the micro–macro elastoplastic micropolar continuum model is proposed, in which the model parameters have clear physical meanings. Subsequently, a return mapping algorithm for the integration of the micropolar elastoplastic model is presented. Finally, the proposed model, coded into ABAQUS software, is validated by comparisons of test and discrete element method results in terms of the stress–strain relation and strain localization phenomenon. The stress–strain relation and shear band modes are well reproduced, and the effects of the confining pressure and the partially saturated state on granular materials can also be correctly predicted by the proposed model.

Orthotropic Elastic Model and Its Parameters in Anchored Layered Rock Mass under Varying Incident Angles: Development and Validation

Rock bolting is one of the most effective and economical methods for jointed rock mass reinforcement. Therefore, the elastic constitutive relationship in an anchored rock mass, which could well reflect the rules of strength and deformation, should be studied. An orthotropic elastic model, which considered varying anchorage angles (α) and the corresponding mechanical parameters, was investigated based on the theory of mechanics of composite materials. To verify the suitability of the model and the mechanical parameters, a WAW-3,000 kN servo universal hydraulic testing machine was used to undertake a series of uniaxial compression tests on the anchored layered rock specimens. The mechanical parameters of composite specimens, such as elastic modulus (E), Poisson’s ratio (ν), peak strength, and its variation with α were obtained. The results showed that the anchored effect of a bolt on the layered rock mass was very significant and anisotropic, for instance, increments in the elastic parameters in three orthogonal directions were quite different with the increase in α from 0° to 90°, which mainly depended on the relative volume content of the component materials in a certain direction. Comparing the calculated E with the experimental values, the variation rules for the mechanical parameters maintained good consistency, and this proved that the calculated equations in this work were reasonable and had acceptable precision. The spillway slope of a reservoir was simplified and simulated and the optimal α was obtained, which further demonstrated the reasonability and applicability of the proposed model from the viewpoint of engineering practice.

Study of finite fracture of the micro contact on the friction interface induced by stress waves

Frictional interface commonly exist in nature and kinds of materials. The dynamic behavior of the frictional interface has a profound impact on the macro strength and other properties of materials. This paper mainly studies the influence of finite fracture of the micro contact the interface under stress wave loading theoretically and simulatively. Based on the micro contact fracture mechanism of friction, a two-dimensional interface friction model including a triangular micro bulge is established with linear elastic constitutive relationship and D-P failure criterion. The results show that the interaction of the incoming stress disturbance and the micro bulge will induce the fracture of the bulge, and with the decrease of the inclination angle of the micro-contact, the fracture position of the triangular micro contact gradually moves upward from the root, and a crack propagating towards the matrix occurs in the top corner of the triangle. Moreover, in the micro process of fracture, a three wave profile centered on the fracture surface: longitudinal wave, transverse wave and interface wave, will be generated A new interesting phenomenon is that at the moment of loading, a micro stress disturbance is produced synchronously from the interface and propagates to the substrate in the form of longitudinal wave. More comparative examples and analysis show that the mechanism of this disturbance is related to the overall gravity micro adjustment acting on the interface. This work connects fracture and wave, and reveals the finite fracture of micro contact of the friction interface and wave effect, which is expected to provide an effective way for earthquake prediction, so as to advance the earthquake prediction time.

Recent Advancements in the Tribological Modelling of Rough Interfaces

This paper analyses some effective strategies proposed in the last few years to tackle contact mechanics problems involving rough interfaces. In particular, we present Boundary Element Methods capable of solving the contact with great accuracy and, at the same time, with a marked computational efficiency. Particular attention is paid to non-linearly elastic constitutive relations and, specifically, to a linearly viscoelastic rheology. Possible implications deal with all the tribological mechanical systems, where contact interactions are present, including, e.g., seals, bearings and dampers.

Elastic Constitutive Relationship of Metallic Materials Containing Grain Shape

The grain shape and orientation distribution of metal sheets at mesoscales are usually irregular, which has an impact on the elastic properties of metal materials. A grain shape function (GSF) is constructed to represent the shape of grains. The expansion coefficient of GSF on the basis of the Wigner D function is called the shape coefficient. In this paper, we study the influence of average grain shape on the elastic constitutive relation of orthogonal polycrystalline materials, and obtain a new expression of the elastic constitutive relation of polycrystalline materials containing grain shape effects. The seven string method is proposed to fit the shape of irregular grains. Experiments show that the GSF can better describe the shape of irregular grains. Using the microscopic images of the grains, we carried out the experimental measurement of micro and macrostrain at grain scale. The experimental results show that the grain shape parameter (slenderness ratio) is consistent with the theoretical results of the material macroscopic mechanical properties.

Dynamic simulation of aircraft electro-impulse de-icing using bond-based peridynamics

Electro-impulse de-icing is a lightweight type of mechanical de-icing system that consumes little energy, provides high reliability, and offers wide application prospects. In this study, the bond-based peridynamics theory was used to simulate the dynamic damage process and evolution law of aircraft electro-impulse de-icing. The ice layer was simplified as a brittle material that satisfies the linear elastic constitutive relation. A numerical analysis model of the ice layer, aircraft skin substrate, and their interface was established to simulate the dynamic responses under a high strain rate. Considering the anisotropy of ice adhesion on the substrate, the interfacial bonds were described as shear bonds and tensile bonds, and the critical elongations of the two were derived. The tensile and shear capacities of the ice on the substrate were then simulated using these critical elongations, and the peeling rates of single ice particles and interfacial ice layers were taken as indexes describing the efficacy of electro-impulse de-icing. The process of electro-impulse de-icing was then analyzed for an aluminum substrate with two adjacent clamped edges. Finally, the results of the peridynamic simulation were compared with those of existing experiments and finite element models to verify the effectiveness of the peridynamic approach.

Nonlinear elastic constitutive relations of residually stressed composites with stiff curved fibres

Nonlinear elastic constitutive relations of residually stressed composites with stiff curved fibres

The theory of compression–shear coupled composite wave propagation in rock

AbstractThe wave velocity analysis of rock medium is the main method used to explore the internal compositions in the crust and research seismic. In this paper, a compression–shear coupled nonlinear elastic constitutive relation is established, which is consistent with the mechanical properties of rock and mineral medium under high pressure. On this basis, numerical solutions of the wave equation and plane wave analytical solutions for the primary and secondary wave velocities are obtained. As is indicated by the comparison with the linear elastic constitutive theory, the results reflect the compression–shear coupling characteristics of the rock, including the stress path effect and the compression–shear coupling wave effect. With different parameter values, the velocity of the secondary wave changes from lower than that of the elastic shear wave, to higher than that of the elastic shear wave. The research results are expected to provide meaningful explanations for the physical mechanisms of the supershear wave and sub‐Rayleigh wave, and guidance for the detection of rock and soil composition and the observation of seismic waves.

Studying the Effects of Negative Skin Friction on Single Piles in Basrah Governorate

The finite element method capable of simulating the behavior of deep foundations subjected to negative skin friction in Basrah soil is investigated. Single piles under drag forces are analyzed using the PLAXIS program with an axisymmetric model. Linear elastic, Soft Soil and Mohr-Coulomb constitutive relations are adopted, where higher order triangular element is chosen for pile and soil clusters. Both pile and soil are modeled using (15)-node triangular elements. Three sites in Basrah province (Umm Qasr Port, Khor Al-Zubair, and Shatt AlArab Hotel) were selected to perform this study. The soil profile and layer characteristics are obtained from the soil investigation reports. Where the negative skin friction is evaluated due to filling loads. It is Conclusion thatSmall relative displacements are necessary to activate the negative skin friction. The elastic shorting for pile effect negative skin friction, due to increase relative displacement. The elastic shorting of the driven pile is more than that of the bored pile due to the less cross-sectional area of the driven pile. The results revealed proportional relation between the developed drag forces and pile section dimensions, interface friction factor, and fill height, with a maximum effect on the section dimension and minimum effect on the interface factor. The locations of neutral points are not sensitive to the above-mentioned factors.

Constitutive Relations of Anisotropic Polycrystals: Self-Consistent Estimates.

In this paper, the elastic constitutive relation of polycrystals contains the effect of the mesostucture coefficients. We consider a general case and derive the average elastic constitutive relation pertaining to polycrystals of cubic crystals with any symmetry of crystalline orientation in their statistical distribution. Following Budiansky and Wu, we used self-consistent estimates of eigenstrain to obtain the effective elastic constitutive relation of polycrystals in an explicit form. For the Voigt assumption and the Reuss assumption, the effective elastic constitutive relation of polycrystals on cubic crystals contains the the mesostructure coefficients up to linear terms. In general, the linear term expression works well for materials such as aluminum, the single crystal of which has weak anisotropy. However the same expression (which allows the anisotropic part of the effective elastic constitutive relation to depend only linearly on the mesostructure coefficients) does not suffice for materials such as copper, in which the single crystal is strongly anisotropic. Per the Taylor theorem, we expand the expression based on the self-consistent estimates with respect to the mesostructure coefficients up to quadratic terms for anisotropic polycrystals of cubic crystals. While our numerical data are very close to those of Morris, our expression is much simpler.