Cowper-Symonds Model Research Articles

Effects of temperature and strain rate on mechanical behavior of low-silver lead-free solder under drop impact

This study uses finite element program to simulate the process of board-level drop for portable electronic devices, so as to create very-thin-profile fine-pitch BGA packages model. Based on the data from quasi-static tests and the experimental data measured by split Hopkinson pressure bar, the parameters of Cowper–Symonds model about solder joints were obtained, taking into consideration of strain rate effect and temperature effect. The results show that, the maximum stress in Cowper–Symonds material model considering strain rate effect is significantly lower than that in linear elastic material model, and the location of maximum stress is different. When the temperature is −45, 23 or 60 °C, low silver lead-free solder Sn0.3Ag0.7Cu shows temperature softening effect obviously. Under low temperature, stress is concentrated at the interface of solder, while under normal temperature or high temperature the phenomenon is different obviously. Additionally, compared with high silver solder, low silver solder has better resistance to drop at room temperature or under working conditions.

Experimental and numerical plastic response and failure of pre-notched transversely impacted beams

Abstract Experimental drop weight impact tests have been performed to examine the dynamic plastic response and failure of clamped pre-notched beams struck transversely by a mass with a spherical indenter. The laboratory results are compared with numerical simulations. The machined notches are 2.0 and 4.0 mm in depth, and the beams are impacted at the mid-span and at one-quarter from the support. The experiments are conducted using a fully instrumented impact testing machine. The obtained force–displacement responses show a good agreement with the simulations performed by the LS-DYNA finite element solver. The simulations aim at proposing techniques for defining the material characteristics and the boundary conditions of numerical models that analyse the large inelastic deformation and failure of ship structural components subjected to impact loading. The finite element model includes the experimental boundary condition so as to simulate small axial displacements of the specimen at the supports. The strain hardening of the material is defined using experimental data of the quasi-static tension tests and the strain rate sensitivity is evaluated using standard coefficients of the Cowper–Symonds constitutive model. The Cowper–Symonds model is also selected to estimate the dynamic critical failure strain by the use of the quasi-static failure strain predicted by numerical simulations of the tensile tests. Although the dynamic properties of the material should determine the response of the impacted specimens, the results show that the plastic response until failure is highly sensitive to the amount of restraint provided at the supports.

Identification of the plastic behavior of aluminum plates under free air explosions using inverse methods and full-field measurements

This article describes an inverse method for the identification of the plastic behavior of aluminum plates subjected to sudden blast loads. The method uses full-field optical measurements taken during the first milliseconds of a free air explosion and the finite element method for the numerical prediction of the blast response. The identification is based on a damped least-squares solution according to the Levenberg–Marquardt formulation. Three different rate-dependent plasticity models are examined. First, a combined model based on linear strain hardening and the strain rate term of the Cowper–Symonds model, secondly, the Johnson–Cook model and finally, a combined model based on a bi-exponential relation for the strain hardening term and the strain rate term of the Cowper–Symonds model. A validation of the method and its sensitivity to measurement uncertainties is first provided according to virtual measurements generated with the finite element method. Next, the plastic behavior of aluminum is identified using measurements from real free air explosions obtained from a controlled detonation of C4. The results show that inverse methods can be successfully applied for the identification of the plastic behavior of metals subjected to blast waves. In addition, the material parameters identified with inverse methods enable the numerical prediction of the material’s response with increased accuracy.

Establishment and comparison of four constitutive models of 5A02 aluminium alloy in high-velocity forming process

In high-velocity forming process, material constitutive relationships are remarkably changed, which has a significant effect on forming process and deformation behaviors. Based on the quasi-static tensile experiments and compression experiments at different temperatures, and dynamic tensile experiments and compression experiments at room temperature, the constitutive behaviors of 5A02 aluminium alloy and the mechanism of high strain rate influencing material dynamic responses are investigated in this paper. According to the above experimental results, Johnson–Cook model (J–C model), modified Cowper–Symonds model (C–S model), modified Rusinek–Klepaczko model (R–K model) and artificial neural network model (ANN model) have been proposed to describe the stress–strain relationship of 5A02 aluminium alloy in high-velocity forming process, and their predictability are compared by employing several statistical evaluation measures. The results show that the prediction precision of J–C model is limited, for the calculated coefficient of strain rate is too little to reflect the increasing trend of stress as strain rate is elevated; modified C–S model takes into consideration the effect of strain in the description of strain rate effect, thus it has a more precise correlation between predicted stress and measured stress; based on the influences of strain and strain rate on the dislocation evolution mechanism, the prediction precision of modified R–K model is higher when the strain rate is in the range of 10−3–3000s−1, while the precision decreases as the strain rate exceeds 3000s−1 due to the changed micromechanism of dislocation evolution; ANN model exhibits most powerful predictability due to its self-organizing ability, self-learning ability and high capacity of error tolerance.

The Forming Mechanism of Surface Defects on Machined Surface in High-Speed Cutting of SiCp/Al Composites

For investigating the machined surface defects in high-speed cutting of SiCp/Al composites. The simulation and experiment of high-speed cutting process is done. The simulation of high-speed cutting process using the Cowper-Symonds model is established to explore the forming mechanism of the machined surface defects. The results show that the machined surface defects include small pit, big pit, groove and the raised particle. The experiment which uses the same cutting parameters with the simulation of FEM (Finite Element Method) model is carried out to verify the results of FEM simulation. The results indicate that the forming mechanism of machined surface defects prove to be true.

Finite element modeling of ballistic impact on multi-layer Kevlar 49 fabrics

This paper presents a material model suitable for simulating the behavior of dry fabrics subjected to ballistic impact. The developed material model is implemented in a commercial explicit finite element (FE) software LS-DYNA through a user defined material subroutine (UMAT). The constitutive model is developed using data from uniaxial quasi-static and high strain rate tension tests, picture frame tests and friction tests. Different finite element modeling schemes using shell finite elements are used to study efficiency and accuracy issues. First, single FE layer (SL) and multiple FE layers (ML) were used to simulate the ballistic tests conducted at NASA Glenn Research Center (NASA-GRC). Second, in the multiple layer configuration, a new modeling approach called Spiral Modeling Scheme (SMS) was tried and compared to the existing Concentric Modeling Scheme (CMS). Regression analyses were used to fill missing experimental data – the shear properties of the fabric, damping coefficient and the parameters used in Cowper-Symonds (CS) model which account for strain rate effect on material properties, in order to achieve close match between FE simulations and experimental data. The difference in absorbed energy by the fabric after impact, displacement of fabric near point of impact, and extent of damage were used as metrics for evaluating the material model. In addition, the ballistic limits of the multi-layer fabrics for various configurations were also determined.

The compressive and tensile behavior of a 0/90 C fiber woven composite at high strain rates

An investigation into the compressive and tensile behavior of a carbon fiber reinforced resin matrix composite at high strain rates is carried out using a split Hopkinson bar. All the dynamic tests are performed under the condition of stress equilibrium and constant strain rate. The results of the compressive tests show that the failure strength and strain of the composite increase with the increase of strain rate. A plateau is observed in a typical stress–strain curve which prompts further study into the failure mechanism by monitoring the failure process with a high-speed camera. The three-phase failure mechanism of on-impact compression, crack-induced unloading, and crack deviation-caused further condensation, is found to have greatly increased the strength and toughness of the composite. In the tensile tests, an increase of strain rate produces a reduced fracture angle and extended crack path. In this process, more failure energy is absorbed, thus the failure strength and strain of the composite are improved. The Cowper–Symonds model of strain rate dependency indicates that the material has a higher tensile strength than compressive strength, and the strain rate sensitivity is more noticeable at high stain rates than quasi-static conditions.

Energy Absorption Capabilities of a Square Tube System

In the present paper the authors refer about a series of experimental tests, where an aluminium alloy square tube, filled with an aluminium foam, was crushed by a longitudinal load at a speed of 10 m/s. The test apparatus consisted of a sled installed on a very stiff frame moving on appropriate guides, as the specimen was set on a home-made fixture. Two arrangements of square tubes were considered as specimens: a “standard” one and an “optimized” one. Both crushing behaviours and energy absorption capabilities were analyzed experimentally and numerically simulated by means of the explicit FE code LS-DYNA®; the complete numerical model consisted of the striker, the assemblage of square tubes and the base. A high-speed video recording system was used to capture the images from the physical test. The results from the numerical analyses were compared to those obtained from the experiments: those results showed that the force–deflection response had been overestimated by the numerical model. The authors attempted to justify this inconsistency by considering the influence of the strain rate parameters of the considered Cowper-Symonds analytical model on the results. It was shown that the “optimized” energy absorber exhibited a more desirable force–deflection response than the standard one due to some easy design changes, which involved the insertion of aluminium foam dampers.

Finite Element Simulation on Failure of Elbow Impacted by Flat-Nosed Missile

Finite element model of the elbow interiorly impacted by flat-nosed missile was established using ANSYS/LS-DYNA. The Cowper-Symonds model was adopted. The rupture strain failure criterion was used to define the failure process. Numerical values were compared with experimental values obtained from the literature and the reliability of model was validated. The penetration failure mode of the elbow was analyzed. Factors of the critical rupture kinetic energy Er were acquired. It can be seen that the penetration failure mode is plugging induced by the extrusion and scraping dominated of axis stress. The effect of Do on Er can be neglected. Er increases with the increase of t/Do, Dm/t and R/Do when the missile mass m is invariable. The effect of m on Er should consider the factors of m and critical rupture velocity Vr.

MP-43 純銅の微小押込試験におけるひずみ速度の効果(ポスターセッション)

In order to elucidate the effect of strain rate on indentation load, relationship between micro-indentation test using sharp indenter (Berkovich type) and finite element method (FEM) analysis was investigated in pure copper (rolled and annealed copper). It was shown that the load-displacement curve obtained form the static FEM analysis was inconsistent with that obtained form the micro-indentation test in both the rolled and annealed coppers. On the other hand, in the dynamic FEM analysis (LS-DYNA), Cowper-Symonds model included the strain rate dependency of strength were employed. The calculated load-displacement curves by Cowper-Symonds model showed good agreement with the experimental result in both the coppers.

Temperature and Rate-Dependent Constitutive Relationship for Vanadium Alloy V-5Cr-5Ti and Failure Modes

In this work the static and dynamic properties of vanadium alloy V-5Cr-5Ti over a wide range of temperature from 20 to 1000 degree at strain rates ranged from 10-4/s~103/s were studied experimentally under uniaxial quasi-static tension with MTS universal testing machine, uniaxial dynamic compression and tension with split Hopkinson bar system with temperature control. The stress-strain curves of V-5Cr-5Ti at various temperatures and various strain rates were obtained. Experimental data show that V-5Cr-5Ti behaves strain-rate sensitive and temperature dependent, for instance the material parameters yield stress, tensile strength and failure strain. And fracture mode of the material is also dependent on strain-rate and temperature. Based on experimental data the temperature-rate-dependent constitutive relations were established in the form of Johnson-Cook and Cowper-Symonds models which are widely used in numerical simulation of dynamic processes of structures under impact loading. The material microstructures and failure modes were analyzed using optical microscope, TEM etc, and results shows that the yield stress and strength are increased with strain rate. The brittle-ductile transition strain-rate is from 101/s to102/s.

The Research on Thermal-Mechanical Coupling Simulation of Steam Turbine Vane Material’s Cutting Process

In order to explore the cutting rules of the 43-inch vane material --1Cr12Ni3Mo2VNbN on the turbine vane, and research the stress in cutting area, the strain as well as the temperature distribution and the variation, providing the reference data for the large-scale vane-root milling cutter's geometric parameter design and the cutting parameter optimization, the authors employed ANSYS / LS-DYNA finite element method to simulate and research the cutting process of the vane material. The authors adopted the two-dimensional plane element 2D SOLID l62 of the single-point integration Lagrange algorithm to establish the finite element cutting model, and established the Cowper-Symonds constitutive models based on the vane material, applied the 2D-r adaptive meshing to simulate the separation of the chip and workpiece. Obtain the distribution and rules of the stress, the strain as well as the temperature in the cutting area, and greatly shorten the period of the vane root milling cutter’s designing and trial-production.

Rate-dependent constitutive model of poly(ethylene terephthalate) for dynamic analysis

Uniaxial tensile testing at strain rates ranging from 10^(-3) to 10^(-1) s^(-1) was carried out to study the rate-dependent mechanical behavior for poly(ethylene terephthalate) (PET) used in the packaging industry. The experimental results show that a rate-dependent plastic behavior exists for PET material. The value of the yield strength was found to increase with the increasing strain rate. A new constitutive model based on the improved Cowper-Symonds rate-dependent constitutive model is proposed to describe the mechanical behavior of PET material in the strain rate ranging from 10^(-3) to 10^(-1) s^(-1), providing more accurate material data for the subsequent simulation analysis of drop test and dynamic buckling. The predictions obtained using the proposed model are compared with experimental results of the improved Cowper-Symonds model. The simulating results of the proposed model agree well with the experimental data. For a low strain rate, the predictions of this model are more precise than those obtained using the improved Cowper-Symonds model. This confirms that the new constitutive model is suitable for describing the mechanical behavior of PET material at a low strain rate and modeling impact problem.

Impact test simulations of stiffened plates using the micromechanical porous plasticity model

This paper first reviews the physical meanings and the expressions of two representative strain rate models: CSM (Cowper–Symonds model) and JCM (Johnson–Cook model). Since it is known that the CSM and JCM are suitable for low-intermediate and intermediate-high rate ranges, many studies regarding marine accidents such as ship-to-ship collisions, ship-to-rock groundings, and explosions in FPSO have employed the former in particular. A formula to predict the material constant of the CSM is introduced from a literature survey. The validity of the formula is proved by comparing with strain rate test results of high strength marine structural steels of DH-36. Numerical simulations with two different material constitutive equations, the classical metal plasticity model based on the von Mises yield function and the micromechanical porous plasticity model based on the Gurson yield function, have been carried out for stiffened plates under impact loading. It is concluded that the porous plasticity model with the porosity fracture criterion can quantitatively predict plastic deformation process and final fracture under impact loading if the material constants are properly chosen.

Dynamic Deformation Behavior Experiments of Alloy Galvanized Deep Drawing Steel Sheet for Automobile and Simulation

Quasi-static and dynamic tensile tests are carried out on alloyed galvanized deep drawing steel sheet for automobile(DC53D+ZF) to investigate its tensile mechanical behavior under different strain rates.Stress-strain relationships are obtained in strain rate range of 0～500 s–1.Johnson-Cook constitutive model of the steel is derived from the experiment results.Model without considering strain rate,Johnson-Cook,Cowper-Symonds model and experiment data are compared by simulation model of rectangular cross section beam collision.The results show that the J-C constitutive model derived is in good agreement with the test results,which proves the parameters obtained can well describe the dynamic deformation behavior of DC53D+ZF under high strain rate.

Effect of Constitutive Equations on Dynamic Bend Buckling Analysis of Thin Steel Sheet Hollow Columns

Selecting a suitable constitutive equation which takes account of the precise strain rate sensitivity of the material is, in general, very important for dynamic-crash analyses. Tanimura-Mimura (T-M) model is one of a few constitutive equations which are able to describe the specific characteristics of many ferritic steels that the strain rate sensitivity in flow stress decreases with increasing plastic strain, however, it has not been examined well how the crash analyses are improved when this model is employed. In this paper, both experimental and numerical analyses of dynamic three-points bend test of thin steel sheet hollow column were conducted at the impact velocity of 4m/s in order to evaluate the effectiveness of T-M model. In the numerical analyses, Cowper-Symonds (C-S) model and strain rate insensitive (I-S) model as well as T-M model were employed, and their calculated results were compared with the experimental result. The calculated load-stroke relations by T-M model showed good agreement with the experimental ones over the whole stroke, on the contrary C-S model was in agreement only in the small stroke region. Shape profiles of the buckling portions of columns in the experiment were compared with the simulated results based on each model. The width of the buckling zone was getting smaller in the order of T-M model, I-S model and C-S model, and T-M model was found to give the best agreement with the experiment since its low strain hardening profiles in stress-strain relations at high strain rates promote the localization in the buckling zone. From these results, T-M model can be considered to improve accuracy of numerical simulation, especially in the case that significant strain localization such as buckling occurs.

미시역학 소성모델을 이용한 충격하중을 받는 보강판의 파단 예측

This paper first reviews the physical meanings and the expressions of two representative strain rate models: CSM (Cowper-Symonds Model) and JCM (Johnson-Cook Model). Since it is known that the CSM and the JCM are suitable for low-intermediate and intermediate-high rate ranges, many studies regarding marine accidents such as ship collision/grounding and explosion in FPSO have employed the CSM. A formula to predict the material constant of the CSM is introduced from literature survey. Numerical simulations with two different material constitutive equations, classical metal plasticity model based on von Mises yield function and micromechanical porous plasticity model based on Gurson yield function, have been carried out for the stiffened plates under impact loading. Simulation results coincide with experimental results better when using the porous plasticity model.

Adiabatic shear failure of high reinforcement content aluminum matrix composites

Dynamic failure behaviors of high reinforcement content TiB2/Al composites were experimentally investigated using split Hopkinson pressure bar (SHPB). The TiB2/Al composites showed high flow stresses and good plastic deformation ability at high strain rates. Adiabatic temperature rise decreased the flow stresses of TiB2/Al composites, which was verified by the prediction of Johnson–Cook model. While the predictions by Cowper–Symonds model exhibited obvious strain hardening characteristic, the values of which were much higher than those of the Johnson–Cook model and the experimental. The composites were failed macroscopically in brittle fracture and some phase transformation bands were found on the shearing surfaces. The dynamic failure behavior of TiB2/Al composites was predominated by the formation of adiabatic shear bands.

Subsurface deformation and damage accumulation in aluminum–silicon alloys subjected to sliding contact

The study of plastic deformation and damage accumulation below the contact surfaces is important in order to understand the dry sliding wear behaviour of aluminum alloys. Experimental evidence exists for the nucleation of voids and microcracks around second phase particles in the material layers adjacent to the contact surface. Propagation of these cracks at a certain depth below the surface may lead to the creation of long, thin plate-like wear debris particles. This work studied the deformation processes during sliding wear by means of metallographic observations of subsurface layers in an Al–7% Si (A356 Al) alloy and by finite element analyses. Specifically, the accumulation of subsurface stresses and strains was investigated, using a coupled structural-thermal finite element model based on the Voce-type exponential stress–strain relationship obtained from the sliding wear tests. Additionally, temperature and strain rate effects were taken into account using a constitutive equation based on Johnson–Cook and Cowper–Symonds models. Accordingly during sliding, the flow stress in subsurface layers increased rapidly and reached a saturation stress after a finite number of sliding contacts. The variation of hydrostatic pressure for different loading conditions was also determined as a function of sliding passes: as the sliding process progressed from the first to the seventh contacts, the hydrostatic pressure at the surface increased from 1150 to 1300 MPa. A total temperature increase of 45 K occurred at the surface after the seventh sliding contact. A debris formation model was proposed in which the presence of a maximum damage gradient at critical depth was considered. The model showed that, with a sliding velocity of 10 m/s, and a normal load of 150 N per unit thickness in mm, the material location where the maximum rate of damage occurred corresponded to a normalized depth (depth/counterface diameter) of 0.060. Increasing the load to 250 N/mm caused an increase in the critical depth of damage (a normalized depth of 0.085). Comparisons with the experimental subsurface crack observations indicate that the proposed damage rate calculations provide a good estimation of the subsurface crack propagation depth.