Incremental Strain Energy Research Articles

An elastoplastic peridynamic beam model for large bending deformation and fracture analysis

The present study establishes an elastoplastic peridynamic (PD) beam model, which effectively incorporates the development of plasticity across both the cross-section and length of the beam, aiming to accurately capture the large bending deformation and fracture of metal frame structures. Firstly, the equivalent plastic modulus is derived to describe the plastic development of the beam cross-section, and then the expression of the incremental strain energy density of the beam in a large bending deformation elastoplastic process can be derived based on the classical continuum mechanics (CCM) theory. Secondly, the equivalent plastic micromoduli of the micro-beam bond are obtained from the incremental strain energy density of the beam in a large bending deformation elastoplastic process using the homogenization principle. Thirdly, the yield condition of the micro-beam bond is derived to determine its elastoplastic stage, and the elastoplastic PD-CCM coupling beam model is established via the morphing method to improve computational efficiency. Finally, the effectiveness and accuracy of the proposed elastoplastic PD-CCM beam model are verified by several numerical examples.

Shear behaviour of sand–rubber mixtures: interpretation from energy transformation and force chain evolution

This study investigates the effect of rubber particles on biaxial shear responses of sand–rubber mixtures (SRMs) using a multibody mesh-free method, which enables the flexible and robust modelling of visco-elastic deformation of rubber particles. From the energy evolutions in SRMs with different rubber contents (RCs), including the incremental elastic strain energy, friction slipping dissipation and damping dissipation, it is found that the slipping dissipation is much more severe than damping dissipation for specimens with lower RC, but less significant for specimens with greater RC during shearing. In addition, the fraction of incremental strain energy increases with RC. Further investigations on the evolutions of force chain networks of SRMs indicate that the incorporation of rubber particles facilitates the stability of strong force chains by mitigating particle rotations in the strong force chains and relative sliding in the weak contacts. It is observed that the energy transformation in sheared SRMs has a strong correlation with their microstructural evolution.

A two-dimensional ordinary state-based peridynamic model for plastic deformation based on Drucker-Prager criteria with non-associated flow rule

The plastic behaviors of geomaterials often obey non-associated flow rule. In this paper, a two-dimensional (2D) ordinary state-based peridynamic (OSB-PD) increment model with shear deformation based on Drucker-Prager (D-P) criterion with non-associated flow rule is proposed to study the plastic behavior of geomaterials. An incremental form of peridynamic (PD) strain energy density for plane elastic problems is first established. The material parameters are determined when the incremental PD strain energy density is equal to the incremental strain energy density in classical mechanics. The PD force density for the OSB-PD theory with shear deformation in plane elastic cases is derived accordingly. The relationship between the incremental second invariant of the stress tensor dJ2, the first invariant function of stress tensor dI1 in the classical mechanics and PD are established in this paper, and the Drucker-Prager yield surface is expressed in the framework of peridynamics. To avoid the excessive plastic dilation, the non-associated flow rule is proposed, in which isotropic and deviatoric plastic elongation are employed to describe the plastic behaviors of a bond. Also, the equivalent stress and equivalent plastic strain for the proposed PD model are given. An integration algorithm for the proposed increment model is developed. Numerical examples including circular opening in rocks are performed, and the PD results are compared with that obtained by the finite element method or analytical solution. Those numerical simulations have verified the validity of the proposed PD model to predict the elasto-plastic behavior of geomaterials.

Fatigue crack growth in the micro to large scale of 7075-T6 Al sheets at different R ratios

Transitionalized crack length (TCL) is derived from energy density based ΔS (incremental strain energy density factor) model that is inherently multiscaling, which is in contrast to the fictitious crack length (FCL) generated from da/dN–ΔK (incremental stress intensity factor) model. By employing transitional functions (TFs), the effects of material, loading and geometry (MLG) are incorporated and reflected in the multiscale fatigue crack growth process of 7075-T6 Al sheets. The constant change of MLG in metal fatigue is necessitated by the effects of Non-equilibrium and non-homogeneity (NENH). Variation of R ratios leads to the change of TFs accordingly. Particularly discussed are the effects of TFs on the fatigue crack growth of 7075-T6 Al sheets. Results turn out that TCLs have a relatively better agreement with test data than FCLs. The proposed model can possibly offer a predictive calibration for the ΔK model.

On statistical strain and stress energy bounds from homogenization and virtual testing

Computational homogenization for quasistatic stress problems is considered, whereby the macroscale stress is obtained via averaging on Statistical Volume Elements (SVE:s). The variational workhorse for the subscale problem is derived from the presumption of weak micro-periodicity, which was proposed by Larsson et al. (2011). Continuum (visco)plasticity is adopted for the mesoscale constituents, whereby a pseudo-elastic, incremental strain energy serves as the potential for the updated stress in a given time-increment. Strict bounds on the incremental strain energy are derived from imposing Dirichlet and Neumann boundary conditions, which are defined as suitable restrictions of the proposed variational format. For this purpose, both the standard situation of complete macroscale strain control and the (less standard) situation of macroscale stress control are considered. Numerical results are obtained from virtual testing of SVE:s in terms of mean values and a given confidence interval, and it is shown how these properties converge with respect to the SVE-size for different prescribed macroscale deformation modes and different statistical properties of the random microstructure. In addition, the upper and lower bounds for a sequence of increasing strain levels, for a fixed SVE-size, are used as data for the calibration of a macroscopic elastic-plastic constitutive model.

Effect of rate of deformation on electromagnetic radiation during quasi-static compression of sintered aluminium preforms

Abstract This paper presents some experimental results on the intermittent electromagnetic radiation (EMR) characteristics of sintered aluminium powder preforms under quasi-static compression. The stress level within the gross elastic limit and yielding of the partially compact preforms at which first EMR emission is observed, bears a distinct parabolic relation with the rate of compressive deformation. These observations can be developed into a new technique to detect the compactness of sintered metal powder preforms for industrial components. Further, each successive intermittent EMR emission at all rates of deformation requires increasing incremental strain energy during progressive plastic deformation. The electromagnetic energy release rate decreases sharply as the rate of deformation is increased and then attains a more or less constant value at higher rates of deformation. These results appear significant in understanding the mechanism of plastic deformation in metal powder preforms at the microscopic level, not yet reported in the literature.

The effects of nickel and carbon concentrations on the wear resistance of Fe–Ni–C austenitic alloys

The effects of nickel and carbon concentrations on the wear resistance of Fe– xNi– yC ( x = 14–20 wt.%, y = 0.6–1.0 wt.%) were investigated with respect to strain energy initiation of the martensitic transformation and hardness. The strain energy needed to initiate the martensitic transformation increased with increasing carbon and nickel concentrations, except in 1.0 wt.% C alloys. The wear resistance of the material decreased with increasing carbon concentration up to 0.9 wt.% C. This effect is most likely due to decrement of the martensite volume fraction with increasing carbon concentration induced by the incremental strain energy required to begin the martensitic transformation. In the case of 1.0 wt.% C, the improved wear resistance may be due to carbide precipitation.

Properties of mouse cutaneous rapidly adapting afferents: relationship to skin viscoelasticity.

When skin is stretched, stimuli experienced by a cutaneous mechanoreceptor neuron are transmitted to the nerve ending through the skin. In these experiments, we tested the hypothesis that the viscoelastic response of the skin influences the dynamic response of cutaneous rapidly adapting (RA) neurons. Cutaneous RA afferent neurons were recorded in 3 species of mice (Tsk, Pallid, and C57BL6) whose skin has different viscoelastic properties. Isolated samples of skin and nerve were stimulated mechanically with a dynamic stretch stimulus, which followed a pseudo Gaussian waveform with a bandwidth of 0-60 Hz. The mechanical response of the skin was measured as were responses of single RA cutaneous mechanoreceptor neurons. For each neuron, the strength of association between spike responses and the dynamic and static components of stimuli were determined with multiple logistic regression analysis. The viscoelastic material properties of each skin sample were determined indirectly, by creating a nonlinear (Wiener-Volterra) model of the stress-strain relationship, and using the model to predict the complex compliance (i.e., the viscoelastic material properties). The dynamic sensitivity of RA mechanoreceptor neurons in mouse hairy skin was weakly related to the viscoelastic properties of the skin. Loss modulus and phase angle were lower (indicating a decreased viscous component of response) in Tsk and Pallid than in C57BL6 mice. However, RA mechanoreceptor neurons in Tsk and Pallid skin did not differ from those in C57 skin with regard to their sensitivity to the rate of change of stress or to the rate of change of incremental strain energy. They did have a decreased sensitivity to the rate of change of tensile strain. Thus the skin samples with lower dynamic mechanical response contained neurons with a somewhat lower sensitivity to dynamic stimuli.

Damage integration in the strain space

The focus is on isotropic elastodamaging (softening) materials, where the damage parameter is expressed as a function of the total strain. By integrating the mechanical work in the strain space along a stepwise holonomic loading history, an incremental strain energy is obtained. A coaxiality condition for the incremental strain energy to be potential is identified, and its implications on the associativity of the damage evolution are discussed. Under some hypotheses, the increment of the mechanical work is shown to be minimum along strain radial paths. These results are used to construct a multifield variational framework supporting finite element (nonlocal) formulations.

EFFECT OF A CRACK ON THE DYNAMIC STABILITY OF A FREE–FREE BEAM SUBJECTED TO A FOLLOWER FORCE

In the present study, the effect of a crack on the dynamic stability of a free–free Timoshenko beam is investigated when it is subjected to a constant or a pulsating follower force. A crack, one of the structural damages, results in a change of transverse vibration frequencies and mode shapes. In addition, the crack may change the stability characteristics of the beam. A mathematical model for the crack is introduced in the form of the bending and shear compliance of equivalent incremental strain energy. To obtain numerical results, the finite element method and the method of multiple scales are applied in this work. As a conclusion, the effect of the crack location and depth on the stability characteristics of the system is illustrated.

RIGID‐PLASTIC IMPLICIT SCHEME FOR NON‐ AND THREE‐DIMENSIONAL ANALYSIS OF METAL FORMING BY FINITE ELEMENT METHOD

This paper is devoted to the analysis of metal forming with assumption of rigid‐plastic behaviour with strain hardening. As opposed to the classical rate problem formulation based on Markov's principle and the explicit scheme, a more satisfactory incremental approach is deduced from Moreau's catching up algorithm. This implicit scheme, although more complicated, gives better results concerning convergence and numerical stability. Using an internal variable representing the strain hardening, an incremental strain energy density is defined which leads to a principle of minimum of the total incremental strain energy. In the numerical approximation using finite elements, the non‐linear equilibrium equations are solved by classical Newton's method. An approximation of Coulomb's criterion is used in order to represent friction with a rigid foundation. The simple compression test is simulated and shows that the implicit scheme is faster than the explicit one.

Vibrational characteristics of cracked shafts

The vibrational characteristics of a cracked Timoshenko shaft are investigated in this paper. Analytical expressions and derived curves relating the crack depth and location on the shaft to changes in the first few natural frequencies of the shaft are presented. These expressions are obtained by modeling the crack as bending and shear compliances of equivalent incremental strain energy by using the J-integral concept from fracture mechanics. It is shown that knowledge of the changes in the first three natural frequencies relative to the uncracked shaft is sufficient to estimate the crack depth and crack location in the shaft. Each additional crack in the shaft requires the knowledge of the changes in two more higher natural frequencies.

Non-linear forced vibration of a non-prismatic beam due to combined flexure and stretching

Dynamic response of a non-prismatic beam with a fixed distance between its ends is analyzed by the variational method. A prismatic beam under similar conditions is used as a “reference structure” and the effect of non-uniform cross-section is accounted for by the minimization of the incremental strain energy in excess of the counterpart of the reference structure. Through the use of normal modes of the reference structure the problem is reduced to the solution of a system of non-linear, coupled, ordinary differential equations which are integrated numerically. Numerical results of a simply-supported haunched beam excited by a sinusoidal loading are presented.