Dynamic Constitutive Relations Research Articles

The Development of a New Shock Absorbing Uniaxial Graded Auxetic Damper (UGAD).

Auxetic structures are efficient cellular materials that can absorb blast/impact energy through plastic deformation, thus protecting the structure. They are developing sacrificial solutions with light weight, high specific strength, high specific toughness and excellent energy dissipating properties, due to its negative Poison’s ratio nature. The use of auxetic and non-auxetic panels in blast resistant structures had been relatively perceived by researchers. Nonetheless, implementation of those energy dissipaters, explicitly as a uni-axial passive damper is restrained to limited studies, which highlight the potential need for further explorations. The aim of this paper is the design of a new uniaxial graded auxetic damper (UGAD) that can be used as a blast/impact/shock absorber in different scales for different structural applications. First, the geometry, material, numerical model and loading are introduced. Then, a detailed parametric study is conducted to achieve the most efficient graded auxetic system. Moreover, the designed auxetic damper is numerically tested and its static and dynamic constitutive relations are derived and validated analytically. The selection of optimum parameters was based on the ratio of the reaction force to the applied load (RFd/P) and plastic dissipation energy (PDE). The final designed UGAD contains three auxetic cores that have the same geometry, material grade (6063-T4), size and number of layers equal to eight. The cell-wall thickness t of the three auxetic cores is 1.4 mm, 1.8 mm and 2.2 mm, respectively; composing a graded auxetic system. The performance of the three auxetic cores together have led to a wide plateau region (80% of total crushing strain) and variant strength range (1–10 MPa), which in return, can justify the superior performance of the UGAD under different blast levels. Finally, the 3D printed prototype of the UGAD is presented and the possible applications are covered.

Dynamic Deformation Behaviors and Constitutive Relations of High-Strength Weldox700E Steel

High-strength Weldox700E steel has been increasingly applied to dynamic environment because of its high strength and good toughness. The plastic deformation behaviors of Weldox700E steel at strain rates ranging from $$10^{-4}$$ to $$6200~ \hbox {s}^{-1}$$ are investigated by the quasi-static and dynamic uniaxial compression tests. The Weldox700E steel exhibits rate-related plastic behavior, work hardening and thermal softening behaviors. Due to the nonlinear strain rate effect of the material and the adiabatic temperature rise caused by high-speed impact compression, the properties of material are greatly affected. By improving the strain rate enhancement term and temperature term in the Johnson–Cook constitutive model, a new constitutive model was proposed to describe the dynamic mechanical behavior of Weldox700E steel. This constitutive equation was implemented into the finite element software, ABAQUS, via an explicit user material subroutine utilizing the stress update algorithm. The simulation results at different strain rates were in good agreement with the experimental data, verifying the validity of the constitutive model.

Dynamic deformation behaviors and constitutive relations of an AlCoCr1.5Fe1.5NiTi0.5 high-entropy alloy

A new high-entropy alloy, AlCoCr1.5Fe1.5NiTi0.5, with dual-phase body-centered cubic structure, exhibits excellent compression strength and large plasticity upon quasi-static and dynamic loadings. Positive strain-rate sensitivity on the yielding strength is available, and the work-hardening behavior weakens with increasing the strain rates due to the adiabatic heating effect upon dynamic loadings. Considering the adiabatic temperature rise converted by plastic deformation work at high strain rates, the flow behavior is characterized by employing the modified Johnson-Cook model.

Stochastic micro-vibration response characteristics of a sandwich plate with MR visco-elastomer core and mass

The magneto-rheological visco-elastomer (MRVE) is used as a smart core to control the stochastic micro-vibration of a sandwich plate with supported mass. The micro-vibration response of the sandwich plate with MRVE core and supported mass under stochastic support motion excitations is studied and compared to evaluate the vibration suppression capability. The effects of the supported mass and localized magnetic field on the stochastic micro-vibration response of the MRVE sandwich plate are taken into account. The dynamic characteristics of the MRVE core in micro-vibration are described by a non-homogeneous complex modulus dependent on vibration frequency and controllable by applied magnetic fields. The partial differential equations for the coupled transverse and longitudinal motions of the MRVE sandwich plate with supported mass are derived from the dynamic equilibrium, constitutive and geometric relations. The simplified ordinary differential equations are obtained for the transverse vibration of the MRVE sandwich plate under localized magnetic fields. A frequency-domain solution method for the stochastic micro-vibration response of sandwich plates with supported mass is developed based on the Galerkin method and random vibration theory. The expressions of frequency-response functions, response power spectral densities and root-mean-square velocity responses of the plate in terms of the one-third octave frequency band are obtained for micro-vibration evaluation. Finally, numerical results are given to illustrate the large response reduction capacity of the MRVE sandwich plate with supported mass under stochastic support motion excitations, and the influences of MRVE parameters, supported mass and localized magnetic field placement on the micro-vibration response.

Test on dynamic characteristics of subgrade of heavy-haul railway in cold regions

Dynamic characteristics of heavy-haul railway subgrade under vibratory loading in cold regions are investigated via low-temperature dynamic triaxial tests with multi-stage cyclic loading process. The relationship between dynamic shear stress and dynamic shear strain of frozen soil of subgrade under train loading and the influence of freezing temperatures on dynamic constitutive relation, dynamic shear modulus and damping ratio are observed in this study. Test results show that the dynamic constitutive relations of the frozen soils with different freezing temperatures comply with the hyperbolic model, in which model parameters a and b decrease with increasing freezing temperature. The dynamic shear modulus of the frozen soils decreases with increasing dynamic shear strains initially, followed by a relatively smooth attenuation tendency, whereas increases with decreasing freezing temperatures. The damping ratios decrease with decreasing freezing temperatures. Two linear functions are defined to express the linear relationships between dynamic shear modulus(damping ratio) and freezing temperature, respectively, in which corresponding linear coefficients are obtained through multiple regression analysis of test data.

Stochastic micro-vibration suppression of a sandwich plate using a magneto-rheological visco-elastomer core

Magneto-rheological visco-elastomer (MRVE) is used to construct sandwich plates for micro-vibration control. The micro-vibration response of a sandwich plate with an MRVE core under stochastic support motion excitation is studied to evaluate the vibration suppression capability. The dynamic behavior of MRVE in micro-vibration is characterized by a non-homogeneous complex modulus dependent on the vibration frequency and controllable by an applied magnetic field, in which the effect of the localized magnetic field distribution is considered. The partial differential equations for the coupled transverse and longitudinal motions of the sandwich plate are derived from the dynamic equilibrium, constitutive and geometric relations. A frequency-domain solution method for the stochastic micro-vibration response of sandwich plates is developed based on the Galerkin method and random vibration theory. The partial differential equations are first converted into ordinary differential equations according to the Galerkin method. Then the expressions for the frequency-response function, response power spectral density and root-mean-square velocity spectrum in terms of the one-third octave frequency band for micro-vibration are obtained. Finally, numerical results are given to illustrate the high response reduction capacity of the MRVE sandwich plate under stochastic support motion excitation, and the influence of the MRVE parameters and localized magnetic field placement on the micro-vibration response.

Stochastic Dynamic Reliability Analysis of Seismic Response of Soft Ground Including Silt

Due to the uncertainties and randomness of earthquake loads, it is more logical that reliability method should be employed in dynamic geotechnical engineering analysis. A methodology of 2-D effective stress FEM analysis based on dynamic reliability of seismic response of soft ground including silt is proposed in this paper, Duncan-Chang and Hardin-Drnevich hyperbolic models are used for static and dynamic constitutive relations of soils, respectively. Furthermore, three stochastic dynamic reliability analysis approaches based on the cumulative damage, cyclic shear stress and pore water pressure ratio are respectively suggested. The proposed methodology was applied to the stochastic dynamic reliability analysis of ground seismic response of two typical soft soil profiles in Shanghai in terms of seismic wave input depths of 50m and 280m. Basically consistent conclusions can be drawn in the three aforementioned perspectives. In addition, the filtering effect of frequency components by deeper soil layers plays perhaps an important role in dynamic reliability analysis in view of the fact that remarkable differences of dynamic reliability analysis results exist between 50m and 280m seismic wave input depths.

Dynamic constitutive model for soils considering asymmetry of skeleton curve

Based on the asymmetric characteristic of skeleton curve obtained from dynamic tests on soils, a function with double asymptotes is proposed for describing the dynamic constitutive relations of soils. The hysteresis loops observed during unloading and reloading show the same form as the skeleton curve and are constructed by taking the ultimate stress as the corresponding asymptote. The coefficient of initial unloading modulus is used to ensure that the constructed hysteresis loop fits well with the experimental data. Then, a new dynamic constitutive model considering the asymmetry of skeleton curve is elaborated. The verification tests on saturated Nanjing fine sand are performed using a hollow cylinder apparatus to verify the applicability of the UD model. It is found that the predicted curves by the UD model agree well with the test data.

Effective Dynamic Constitutive Relations for 3-D Periodic Elastic Composites

ABSTRACTCentral to the idea of metamaterials is the concept of dynamic homogenization which seeks to define frequency dependent effective properties for Bloch wave propagation. Recent advances in the theory of dynamic homogenization have established the coupled form of the constitutive relation (Willis constitutive relation). This coupled form of the constitutive relation naturally emerges from ensemble averaging of the dynamic fields and automatically satisfies the dispersion relation in the case of periodic composites. Its importance is also notable due to its invariance under transformational acoustics. Here we discuss the explicit form of the effective dynamic constitutive equations. We elaborate upon the existence and emergence of coupling in the dynamic constitutive relation and further symmetries of the effective tensors.

Dynamic compressive behavior and constitutive relations of lanthanum metal

The dynamic compressive behavior and constitutive relations of Lanthanum (La) metal was determined by using the first compression in split Hopkinson pressure bar (SHPB) tests at different strain rates and temperatures. The constitutive relation of La metal determined in a certain range of strains was employed and adjusted in numerically simulating large deformations of La metal specimens generated by multi-compression in SHPB tests and recorded by a high-speed camera. The dynamic compressive behavior and constitutive relations of La metal under multiple SHPB tests loading was also revealed. The results of scanning electron microscope (SEM) investigation of the recovered La metal specimens for typical tests showed that there was a variety of deformation microstructures depending on strain rate, temperature and stress state.

Overall dynamic constitutive relations of layered elastic composites

A method for the homogenization of a layered elastic composite is presented. It allows direct, consistent, and accurate evaluation of the averaged overall frequency-dependent dynamic material constitutive relations without the need for a point-wise solution of the field equations. When the spatial variation of the field variables is restricted by Bloch-form (Floquet-form) periodicity, then these relations together with the overall conservation and kinematical equations accurately yield the displacement or stress mode-shapes and, necessarily, the dispersion relations. The method can also give the point-wise solution of the elastodynamic field equations (to any desired degree of accuracy), which, however, is not required for the calculation of the average overall properties. The resulting overall dynamic constitutive relations are general and need not be restricted by the Bloch-form periodicity. The formulation is based on micromechanical modeling of a representative unit cell of the composite. For waves in periodic layered composites, the overall effective mass-density and compliance (stiffness) are always real-valued whether or not the corresponding unit cell (representative volume element used as a unit cell) is geometrically and/or materially symmetric. The average strain and linear momentum are coupled and the coupling constitutive parameters are always each others' complex conjugates. We separate the overall constitutive relations, which depend only on the composition and structure of the unit cell, from the overall field equations which hold for any elastic composite; i.e., we use only the local field equations and material properties to deduce the overall constitutive relations. Finally, we present solved numerical examples to further clarify the structure of the averaged constitutive relations and to bring out the correspondence of the current method with recently published results.

Micro-vibration response of a stochastically excited sandwich beam with a magnetorheologicalelastomer core and mass

Magnetorheological (MR) elastomers are used to construct a smart sandwich beam formicro-vibration control. The micro-vibration response of a clamped–free sandwich beamwith an MR elastomer core and a supplemental mass under stochastic supportmicro-motion excitation is studied. The dynamic behavior of MR elastomer as a smartviscoelastic material is described by a complex modulus which is controllable by externalmagnetic field. The sixth-order partial differential equation of motion of the sandwich beamis derived from the dynamic equilibrium, constitutive and geometric relations. Afrequency-domain solution method for the stochastic micro-vibration responseof the sandwich beam is developed by using the frequency-response function,power spectral density function and spatial eigensolution. The root-mean-squarevelocity response in terms of the one-third octave frequency band is calculated, andthen the response reduction capacity through optimizing the complex modulusof the core is analyzed. Numerical results illustrate the influences of the MRelastomer core parameters on the root-mean-square velocity response and the highresponse reduction capacity of the sandwich beam. The developed analysis method isapplicable to sandwich beams with arbitrary cores described by complex shearmoduli under arbitrary stochastic excitations described by power spectral densityfunctions.

Analytical–numerical study of interfacial stresses in plated beams subjected to pulse loading

In strengthening an existing beam using externally bonded soffit plates, interfacial stresses play an important limiting role on the attainment of full flexural capacity of the composite system. This paper deals with spatial and temporal distributions of shear and normal interfacial stresses in a plated beam subjected to pulse loading. Interfacial stresses are related to displacement coordinates using a set of assumptions and the coupled partial differential equations in the space of these displacement coordinates have been derived by application of dynamic equilibrium, compatibility and constitutive relations. The equations have been solved numerically for the dynamic response. The results for three pulse loading cases have been compared to finite element analyses and reasonable correlation is observed. Compared to dynamic or quasi-static ranges the correlation is weaker when loading is in the impulsive range. Spatial stress distribution suggests that unlike static loading, in the case of pulse loading interfacial stresses are not necessarily concentrated at the edges.

Study on dynamic constitutive relations for concrete with finite deformation

The differences between finite deformation and infinitesimal deformation are discussed. They are exercised on elasto-viscoplastic constitutive relations of concrete. Then, a rate-dependent mechanics model was presented on the basis of Ottosen's four-parameter yield criterion, where different loading surface transferring laws were taken into account, when material was in hardening stage or in softening stage, respectively. The model is well established, so that it can be applied to simulate the response of concrete subject to impact loading. Green-Naghdi stress rate was introduced as objective stress rate. Appropriate hypothesis was postulated in accordance with many experimental results, which could reflect the mechanical behaviour of concrete with large deformation. Available thoughts as well as effective methods are also provided for the research on related engineering problems.

Dynamic constitutive relations for polarization and magnetization.

In this paper we develop constitutive relations for materials where the magnetization and polarization may depend on both the electric and magnetic fields. The approach is general, and is based on a previously developed statistical-mechanical theory. We include the quadrupole-moment density as well as the dipole-moment density in the microscopic displacement field. This yields an electric gradient term in the constitutive equations. This leads to origin invariance in the multipole moments from which Maxwell's equations are defined. We present generalizations of Debye and Landau-Lifshitz equations of motion which are valid for nonequilibrium and contain memory. The reversible and relaxation terms in the polarization and magnetization evolution equations include the possibility of magnetoelectric coupling. Using constitutive relationship, we derive evolution equations for the displacement and induction fields from a Hamiltonian approach.

Determination of Dynamic Constitutive Equations Based on an Information Criterion by the Elasto-Plastic Waves Propagation Analysis.

Previously, we presented the determination method of a dynamic constitutive equation with strain- rate sensitivity from the results of an uniaxial longitudinal impact compression test. The proposed method could derive an appropriate strain-rate sensitive dynamic constitutive equation as a polynomial without consideration based on underlying material physics. However, the method needs a lot of test results over a wide strain-rate range. This paper presents the determination method of a dynamic constitutive equation from a few impact test results. The proposed method determines unknown coefficients involved in a polynomial that appears dynamic constitutive relations of a material by nonlinear least squares method. Furthermore, the proposed method determines the optimum degree of a polynomial constitutive equation by AIC (Akaike information criterion). This paper, further, presents the split Hopkinson pressure bar method using a stepped striker bar. The proposed method can obtain the relations between the dynamic stresses and strain-rates in a wide range from a single impact test.

Influences of Strain Rate and Temperature Variation on Material Intrinsic Length of Plasticity Strain Gradient Theory

Cowper-Symonds and Johnson-Cook dynamic constitutive relations are used to study the influence of both strain rate effect and temperature variation on the material intrinsic length scale in strain gradient plasticity. The material intrinsic length scale decreases with increasing strain rates, and this length scale increases with temperature.

Attenuation of stress wave in sandstone

This article has processed the experimental study of stress wave propagation using Hopkinson Pressure Bar, and measured with groups of PVDF stress sensors, obtained the waveform characteristics and the attenuation constant, and got the dynamic constitutive relations using Lagrangian analysis method.

Dynamic Equilibrium Constitutive Relations of Copper Powder Within Dies Subjected to Mutliple Shock Compactions

An approximate dynamic equilibrium pressure p-specific volume V relation exists for porous materials of a simple type undergoing mutliple shock compaction processes. A copper powder medium in dies is assumed to be of such a type, and the relation is constructed when the medium is compacted by a punch. It is given by an expression in the form of p=(V−Vi)/[b{Vi(1−a)−V}], where Vi, a and b are the material constants. These constants are estimated by matching the computational and experimental results obtained for the mean green density of the medium. Similarly, a dynamic equilibrium lateral pressure p1-specific volume V relation is also estimated for the medium after being given by p1=αp2+βp for ρ<3430 kg/m3, where α and β are the material constants and ρ is the density, while p1=0.5(Vsolid/V)νp+c for ρ≧3430 kg/m3, where Vsolid is the specific volume of solid copper, ν the material constant, and the contant c continuously connects the lateral pressures of the above two equations at ρ=3430 kg/m3. The compaction processes analyzed using the estimated relations agree favorably with the powder particle movement and shock wave front paths from experiments, suggesting the validity of the simple type assumption and that of the estimated relationships.

Three dimensional dynamic electromechanical constitutive relations for ferroelectric materials

We propose, in this paper, three dimensional constitutive relations suitable for describing the dynamic electromechanical responses of ferroelectric materials. Our approach is phenomenological and is based on the assumptions that the macroscopic electrical properties can be represented by electric dipoles whose magnitude and direction can be altered by external stimuli. The constitutive relations for the stress and the electric displacement take into account the transient and instantaneous responses of the dipole moments and the transient response of domain switching. A specific respesentation of the theory is also derived; this representation is suitable for studying the responses of ferroelectic materials to dynamic mechanical loading and rapidly applied electric field.