Inelastic Constitutive Equation Research Articles

Computer predictions of thermo-mechanical behavior and residual stresses in spray coating process

This paper presents a numerical method in order to forecast the thermo-mechanical behavior and the residual stresses in the thermal spray coating process by using high velocity oxygen fuel (HVOF). A set of coupled equations of the heat conduction and stress/strain and solidification based on the metallo-thermo-mechanical theory is introduced into the simulation of thermal spraying. Here, an inelastic constitutive equation with capacity to represent relation of stress/strain during rapid solidification is employed. The numerical modelling based on the finite element method is proposed to solve the heat conduction associated with solidification in the sprayed layer and residual stresses on the interface between multi-layer materials, especially. In this paper, the simulated results of the temperature field, solidified domain and residual stresses in the sprayed layer including interfacial combinations between substrate and spray layer are presented, and the validity of the calculated results is discussed in comparison with the measured results obtained by X-ray diffraction.

Rate-dependent inelastic constitutive equation: the extension of elastoplasticity

A rate-dependent inelastic constitutive equation is formulated by extending the elastoplastic constitutive equation so as to retain the latter's mathematical structure and thus reduce to the latter equation at an infinitesimal rate of deformation. That structure differs substantially from that of the over-stress model, the best-known rate-dependent inelastic constitutive model. The proposed constitutive model is a type of superposition model, which is premised on the additive decomposition of the inelastic strain rate into the plastic and creep strain rates. The plastic strain rate is formulated so as to become suppressed as the rate of deformation increases but is induced even at the infinite rate of deformation. This is the distinguishing features of this model from the existing superposition models. The present model can describe realistically the rate-dependent inelastic deformation for a wide range of strain rates. On the other hand, the over-stress model cannot predict appropriately the difference of mechanical response due to the rate of deformation, especially being inapplicable to the description of deformation at high rate of deformation as known from the unrealistic prediction of the infinite strength at an infinite rate of deformation. The proposed model is applied to various metals, and its adequacy is verified through comparisons with various test data under a wide variety of strain rates and temperatures.

Effect of the Shape of T–δ Cohesive Zone Curves on the Fracture Response

ABSTRACT Cohesive Zone Models (CZMs) are increasingly being used to simulate fracture and fragmentation processes in metallic, polymeric, ceramic materials, and composites thereof. A key feature of this approach is to represent the micromechanics of the fracture processes through a unique load-displacement relation. Most researchers consider magnitude of the energy, in addition to one of the two parameters (cohesive strength or critical displacement), to define the cohesive zone characteristics, ignoring the actual form (shape) of the relationship. Some of our recent work [1–3] and the work of others [17] has clearly shown that the energetics of the fracture process not only depends on the inelastic constitutive equation of the bounding material, but also on the choice of the cohesive zone model. CZM represents the embodiment of different inelastic micromechanisms active in the fracture process zone (FPZ). Since the micromechanisms are fundamental material characteristics, the choice of the CZM should depend on the specific material. The form (shape) of CZM represents the net effect of the processes and, hence, depends on the material system. In general, the shape of CZM is comprised of a rising, peak, and falling segment, and each segment exhibits a different influence on the energy dissipation, not only in the FPZ, but also (indirectly) on the bounding material. The commonly used exponential, bilinear, and trapezoidal models are analyzed to establish the relationships between the form (rise, peak value, and fall) characteristics of the T–δ curve and thermomechanical energy dissipation, plastic zone size, crack initiation load, and local stiffness behavior. By doing so, we provide specific guidelines to the CZM developers and users as to the criteria for the selection of appropriate CZM for a range of material system.

Thermal Fatigue Life Simulation for Sn-Ag-Cu Lead-Free Solder Joints

A key issue in using alternative lead-free solder is the evaluation of long-term reliability, such as thermal fatigue. In this paper, an estimation method for the thermal fatigue of Sn-Ag-Cu lead-free solder joints using an inelastic constitutive model was studied. First, the mechanical properties of Sn-Ag-Cu lead-free solders were investigated in order to model the inelastic constitutive equation for finite element analysis. Tensile tests and constant load creep tests were carried out. Next, an inelastic constitutive model was investigated. In the modeling, the inelastic strain rate was decomposed into the steady-state part depending on applied stress and the transient part driven by the difference between applied stress and back stress. Finally, in order to investigate the applicability of the proposed constitutive model for Sn-Ag-Cu lead-free solder, finite element analysis was carried out for the solder joints in a quad flat package. The estimation for the fatigue crack initiation was in good agreement with the actual thermal cycle tests for the quad flat package.

基礎的負荷条件に対するエポキシの非弾性変形と非弾性構成式の適用

Experiments on inelastic deformation of epoxy were performed under fundamental loading conditions systematically from the viewpoint of the formulation of an inelastic constitutive equation. In experiments under monotonic loading conditions, stress-strain relations depending on the loading rate and loading direction defined in stress space were elucidated. Moreover, creep curves and transient behaviors in tension with change in stress rate were obtained. In the case of loading conditions including unloading, on the other hand, strain recovery during unloading depending on the loading history was elucidated since the strain recovery is significant in epoxy. Then, the inelastic constitutive equation that has been developed for CFRP (epoxy reinforced by carbon fibers) so far was applied to the experimental results. In order to take into account the effect of hydrostatic pressure on the inelastic deformation of epoxy, the first invariant of stress tensor was introduced into the constitutive equation. The experimental results were simulated by the modified constitutive equation, and the describability of the inelastic deformation of epoxy by the modified constitutive equation was discussed.

Generalized Rate-Dependent Inelastic Constitutive Equation. Application to Metals.

The rate-dependent inelastic constitutive equation was formulated by extending the subloading model such that the plastic strain rate is suppressed with the increase of strain rate and incorporating the creep strain rate in the previous article. It retains the mathematical structure of the subloading surface model and thus reduces to the model itself at infinitesimal strain rate. It belongs to the superposition model premising on the additive decomposition of the inelastic strain rate into the plastic and the creep strain rates. It is applied to metals, and its adequacy is verified comparing with various test data at a wide variety of strain rates and temperatures.

Generalized Rate-Dependent Inelastic Constitutive Equation. The Extension of Elastoplasticity.

The rate-dependent inelastic constitutive equation is formulated by extending the elastoplastic constitutive equation so as to retain its mathematical structure and thus reduces to the latter equation at infinitesimal strain rate, whilst the over-stress model, the best-known inelastic constitutive model, has substantially different mathematical structure from the elastoplastic constitutive equation. It belongs to the superposition model pre on the additive decomposition of the inelastic strain rate into the plastic and the creep strain rates. It can describe precisely the rate-dependent inelastic deformation at a wide range of strain rate, whilst the over-stress model cannot predict appropriately the difference of mechanical response due to the rate of deformation, especially inapplicable to the description of deformation at high strain rate.

Inelastic analysis of new thermal ratchetting due to a moving temperature front

Abstract Results of the joint project by the Working Group of Inelastic Analysis, Committee on High Temperature Strength, Society of Materials Science, Japan, are summarized. The purpose of this project, consisting of two parts, is to find out an appropriate inelastic constitutive equation which can simulate the new thermal ratchetting of a cylinder subjected to a traveling axial temperature distribution. In the first part of this project, uniaxial strain and stress-controlled ratchetting behavior is simulated by plural constitutive equations in order to examine the ability to reproduce basic deformation behavior of 316FR stainless steel at room temperature and 650 °C. In the second part, inelastic analyses of the thermal ratchetting of a cylinder due to a traveling temperature distribution are performed. Experimental data of ten specimens made of SUS 316 and 316FR stainless steels are prepared for evaluating the following items: traveling distance, stress level and hold time at high temperature. A comparison between experiments and corresponding inelastic analyses using eight types of constitutive models is made, and the selection of the appropriate constitutive equation is discussed.

異方性および接線塑性に基づく非共軸性を持つ土の非弾性構成式の力学的応答

異方性および接線塑性に基づく非共軸性を持つ土の非弾性構成式の力学的応答

721 実用的な速度依存非弾性構成式の有限要素法インプレメンテーション

721 実用的な速度依存非弾性構成式の有限要素法インプレメンテーション

415 鉛フリーはんだ用非弾性構成式の有限要素法インプリメンテーション(OS3-3 構成式(1))(OS3 機械材料の変形とそのモデリング)

415 鉛フリーはんだ用非弾性構成式の有限要素法インプリメンテーション(OS3-3 構成式(1))(OS3 機械材料の変形とそのモデリング)

Inelastic Constitutive Equation for a Two-Phase System Considering Link with Structural Analysis. Micromechanical Approach.

This paper suggests a constitutive equation for a two-phase system in consideration of inelastic deformation, that is plastic and creep deformation, to which micromechanical model based on Eshelby's equivalent inclusion method and Mori-Tanaka's mean-field approximation is applied. This equation is expressed in piecewise linear incremental form, which can be easily put in structural analysis such as finite element method. In addition, the yield condition of this system is also developed. To verify this analysis, three numerical examples for SiC/Al unidirectional composites are shown. As the result of this calculation, it is said that this constitutive equation is appropriate.

高分子複合材料の非弾性変形における背応力の検討

In order to formulate an inelastic constitutive equation of polymer matrix composites, back stress in the kinematic hardening creep theory of Malinin and Khadjinsky extended to anisotropic materials is evaluated experimentally under various loading conditions. After brief discussion to evaluate the back stress from experimental results, we perform several tests for tubular specimens under various loading conditions ranging from monotonic loading with step-up change in stress rate to reversal loading in axial and torsional modes. Then the evolution of back stress with respect to the change of viscous strain is calculated for these tests. The stress-strain relations together with back stress-viscous strain relations are simulated also by using Armstrong-Frederick model of kinematic hardening. By comparing the results of Armstrong-Frederick model with those of the corresponding experiments, we discuss characteristics of the evolution of back stress in polymer matrix composites and further elaboration of kinematic hardening rule for the composites.

The tangential plasticity

Various constitutive models for describing the inelastic stretching due to the stress rate component tangential to the yield or loading surface, i.e. the non-coaxiality of stress and inelastic stretching have been proposed in the past. However, a pertinent model applicable to a general loading process for materials with an arbitrary yield surface has not been proposed up to the present. In this article, the inelastic constitutive equation extended so as to describe the non-coaxiality is formulated generalizing the Rudnicki and Rice’s [1] J2-deformation theory in rate form and incorporating it into the subloading surface model with a smooth elastic-plastic transition.

An Inelastic Constitutive Equation of Fiber Reinforced Plastic Laminates

A Constitutive model for describing the time dependent inelastic deformation of unidirectional and symmetric angle-ply CFRP (Carbon Fiber Reinforced Plastics) laminates is developed. The kinematic hardening creep law of Malinin and Khadjinsky and the evolution equation of Armstrong and Frederick are extended to describe the creep deformation of initially anisotropic materials. In particular, the evolution equations of the back stresses of the anisotropic material were formulated by introducing a transformed strain tensor, by which the expression of the equivalent strain rate of the an isotropic material has the identical form as that of the isotropic materials. The resulting model is applied to analyze the time dependent inelastic deformation of symmetric angle ply laminates. Comparison between the predictions and the experimental observations shows that the present model can describe well the time dependent inelastic behavior under different loadings.

A viscoplastic constitutive equation bounded between two generalized plasticity models

Abstract The present work addresses a new rate-dependent inelastic constitutive equation, whose response is always bounded between two distinct rate-independent generalized plasticity models. Main features of the proposed model are the following: (1) it approaches the two rate-independent generalized plasticity models for the case of fast and slow loading conditions, respectively; (2) it properly describes loading-unloading-reloading conditions for any loading rate; (3) it reproduces the experimentally observed difference between the dynamic and the static yielding conditions. These are all properties to be taken into account to obtain an accurate modeling for many different classes of materials, such as metals, polymers, geomaterials, shape-memory alloys. Both the time-continuous and the time-discrete version of the model are discussed. Moreover, a solution algorithmic within a return map setting as well as the form of the discrete consistent tangent tensor are addressed. Finally, some numerical simulations illustrate the performance of the proposed constitutive model.

Inelastic Constitutive Equation Based on a New Viscoelastic Material Model.

A new viscoelastic material model was developed to describe deformations under various loads at the high-temperature creep regime. In the model the inelastic strain rate is written in the form of a flow equation of the Norton-Bailey type and the material hardening during deformation under various loads is induced by an internal stress which is subdivided into back stress and friction stress. The back stress represents a conservative part of the creep resistance while the friction stress includes all the dissipative parts in the internal structure such as grain boundary sliding, diffusion and interaction between dislocation and precipitation. An inelastic constitutive equation is established based on the new viscoelastic material model. A case study was done for Hastelloy XR, a Ni-based superalloy. Deformation analyses show a good agreement between calculations and experimental results for creep, relaxation, tensile and stress dip tests.

損傷を考慮したＦＲＰの非弾性構成式 繰返し負荷を受けるＣＦＲＰ［±４５°］４への適用

This paper treats construction of the constitutive equation for inelastic behavior of CFRP, in which the damage effect is incorporated. At first, the authors propose an inelastic constitutive equation for CFRP subjected to cyclic loading. The constitutive equation is based on the constitutive model for cyclic plasticity proposed by the authors previously. Namely, the constitutive equation employs a damaged loading function that is similar to the loading function for metal alloys, the Ziegler type assumption to represent the movement of the damaged loading surface, and the Ramberg-Osgood stress-strain relation. To verify the applicability of the constitutive equation to the inelastic behavior of CFRP, a series of experiments, such as the cyclic tension-compression loading tests with several constant strain amplitudes, are carried out using laminated graphite/epoxy tubular specimens. The specimens have ±45 degree of fibers measured from the axial direction of the specimens. The predictions of the constitutive equation are compared with the experiments. As a result, it is found that the proposed constitutive equation can exactly predict the characteristic inelastic behavior of CFRP subjected to the cyclic loading, and that the damage variables are related to the strain amplitude of loading.

Implementation of the transformation field analysis for inelastic composite materials

The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (62); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.

High-Speed, Non-Isothermal Fiber Spinning

Abstract Spinning of PET fibers of low elasticity at high speed, and of polypropylene (of moderate elasticity) and polystyrene (of high elasticity) under non-isothermal conditions that contain their glass transition temperature, is analyzed by means of a unique BKZ-type constitutive equation that is augmented by the appropriate form of shift factors to apply in both, the rubbery and the glassy states of amorphous polymers. This approach eliminates the necessity of tracking the glassification point by trial and error, required in attempting to match different constitutive laws across it. Predictions of velocity and temperature distribution along the PET fiber agree with available experimental data, and the predicted glass transition point of polystyrene occurs indeed at about its glass transition temperature. Comparisons of predictions to those with an inelastic constitutive equation show that they apply to PET (of low elasticity) alone.