A Phenomenological Model for the Hysteresis Behavior of Metal Sheets Subjected to Unloading/Reloading Cycles

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
P.-A. Eggertsen ◽  
K. Mattiasson ◽  
J. Hertzman
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Elastic Recovery ◽  
Yield Criterion ◽  
Experimental Investigations ◽  
Secant Modulus ◽  
New Model ◽  
Forming Simulation ◽  
Metal Sheets ◽  
Steel Grades

The springback phenomenon is defined as elastic recovery of the stresses produced during the forming of a material. An accurate prediction of the springback puts high demands on the material modeling during the forming simulation, as well as during the unloading simulation. In classic plasticity theory, the unloading of a material after plastic deformation is assumed to be linearly elastic with the stiffness equal to Young’s modulus. However, several experimental investigations have revealed that this is an incorrect assumption. The unloading and reloading stress–strain curves are in fact not even linear, but slightly curved, and the secant modulus of this nonlinear curve deviates from the initial Young’s modulus. More precisely, the secant modulus is degraded with increased plastic straining of the material. The main purpose of the present work has been to formulate a constitutive model that can accurately predict the unloading of a material. The new model is based on the classic elastic-plastic framework, and works together with any yield criterion and hardening evolution law. To determine the parameters of the new model, two different tests have been performed: unloading/reloading tests of uniaxially stretched specimens, and vibrometric tests of prestrained sheet strips. The performance of the model has been evaluated in simulations of the springback of simple U-bends and a drawbead example. Four different steel grades have been studied in the present investigation.

scholarly journals Hardness, Young’s Modulus and Elastic Recovery in Magnetron Sputtered Amorphous AlMgB14 Films

Crystals ◽  
2020 ◽  
Vol 10 (9) ◽  
pp. 823
Author(s):  
Alexander M. Grishin
Keyword(s):  
Young’S Modulus ◽  
Indentation Size Effect ◽  
Young's Modulus ◽  
Elastic Recovery ◽  
Rf Power ◽  
Magnesium Boride ◽  
Wide Range ◽  
Aluminum Magnesium Boride ◽  
Optical And Mechanical Properties ◽  
Diamond Probe

We report optical and mechanical properties of hard aluminum magnesium boride films magnetron sputtered from a stoichiometric AlMgB14 ceramic target onto Corning® 1737 Glass and Si (100) wafers. High target sputtering rf-power and sufficiently short target-to-substrate distance appeared to be critical processing conditions. Amorphous AlMgB14 films demonstrate very strong indentation size effect (ISE): exceptionally high nanohardness H = 88 GPa and elastic Young’s modulus E* = 517 GPa at 26 nm of the diamond probe penetration depth and almost constant values, respectively, of about 35 GPa and 275 GPa starting at depths of about 2–3% of films’ thickness. For comparative analysis of elastic strain to failure index  H/E*, resistance to plastic deformation ratio H3/E*2 and elastic recovery ratio We were obtained in nanoindentation tests performed in a wide range of loading forces from 0.5 to 40 mN. High authentic numerical values of H = 50 GPa and E* = 340 GPa correlate with as low as only 10% of total energy dissipating through the plastic deformations.

A New Method Developed for Brittle and Ductile Materials to Evaluate Mechanical Properties of a Lump Specimen in the Use of Indentation Test

2006 ◽  
Vol 129 (2) ◽  
pp. 284-292 ◽  
Author(s):  
Pal Jen Wei ◽  
Jen Fin Lin
Keyword(s):  
Yield Strength ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Elastic Recovery ◽  
Indentation Test ◽  
New Method ◽  
Ductile Materials ◽  
Conventional Material ◽  
Loading And Unloading ◽  
Logarithm Function

In this study, the load-depth (P‐h) relationships matching the experimental results of the nanoindentation tests exhibited at the subregions of small and large depths are obtained, respectively. The relationships associated with these two subregions are then linked by the hyperbolic logarithm function to attain a single expression that is applied in the evaluation of the specimen’s elastic recovery ability, as shown in the unloading process. A new method is developed in the present study to evaluate both Young’s modulus and the yield strength of either a ductile or brittle material through the uses of the appropriate P‐h relationships developed in the load and unloading processes. The results of the Young’s modulus and the yield strength achieved by the present method are compared to those obtained from the conventional material tests for a lump material. The scattering of the experimental data shown in the loading and unloading processes are also interpreted by different causes.

New expanding cavity model for indentation hardness including strain-hardening and indentation size effects

2006 ◽  
Vol 21 (5) ◽  
pp. 1317-1326 ◽  
Author(s):  
X.-L. Gao
Keyword(s):  
Strain Hardening ◽  
Young’S Modulus ◽  
Strain Gradient ◽  
Young's Modulus ◽  
Indentation Hardness ◽  
Cavity Model ◽  
Indentation Size ◽  
New Model ◽  
Plastic Materials ◽  
Classical Plasticity

An expanding cavity model (ECM) for determining indentation hardness of elastic–strain-hardening plastic materials is developed. The derivation is based on a strain gradient plasticity solution for an internally pressurized thick-walled spherical shell of an elastic linear-hardening material. Closed-form formulas are provided for both conical and spherical indentations. The formulas explicitly show that indentation hardness depends on Young's modulus, yield stress, strain-hardening index, and strain gradient coefficient of the indented material as well as on the geometry of the indenter. The newly formulated ECM can capture the indentation size effect, unlike classical plasticity based ECMs. The new model reduces to existing classical plasticity based ECMs (including Johnson's ECM for elastic-perfectly plastic materials) when the strain gradient effect is not considered. The presently developed ECM is validated by comparing with existing experimental hardness data. The numerical results obtained using the new model reveal that the hardness is indeed indentation size dependent when the indentation radius is very small: the smaller the indentation, the larger the hardness. Also, the indentation hardness is seen to increase with the Young's modulus and strain-hardening level of the indented material for both conical and spherical indentations. The strain-hardening effect on the hardness is observed to be significant for materials having strong strain-hardening characteristics. In addition, it is found that the indentation hardness increases with decreasing cone angle of the conical indenter or decreasing radius of the spherical indenter. These trends agree with existing experimental observations and model predictions.

Analysis of nanoindentation load-displacement loading curves

1996 ◽  
Vol 11 (8) ◽  
pp. 1987-1995 ◽  
Author(s):  
S. V. Hainsworth ◽  
H. W. Chandler ◽  
T. F. Page
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Elastic Recovery ◽  
Hard Coatings ◽  
Attractive Alternative ◽  
Indentation Hardness ◽  
Test Volume ◽  
Considerable Difficulty ◽  
Load Displacement ◽  
The Relationship

Nanoindentation load-displacement curves provide a “mechanical fingerprint” of a materials response to contact deformation. Over the last few years, much attention has been focused on understanding the factors controlling the detailed shape of unloading curves so that parameters such as true contact area, Young's modulus, and an indentation hardness number can be derived. When the unloading curve is well behaved (by which we mean approximating to linear behavior, or alternatively, fitting a power-law relationship), then this approach can be very successful. However, when the test volume displays considerable elastic recovery as the load is removed [e.g., for many stiff hard materials and many inhomogeneous systems (e.g., those employing thin hard coatings)], then the unloading curve fits no existing model particularly well. This results in considerable difficulty in obtaining valid mechanical property data for these types of materials. An alternative approach, described here, is to attempt to understand the shapes of nanoindentation loading curve and thus quantitatively model the relationship between Young's modulus, indentation hardness, indenter geometry, and the resultant maximum displacement for a given load. This paper describes the development and refinement of a previous approach by Loubet et al1 originally suggested for a Vickers indenter, but applied here to understand the factors that control the shape of the loading curve during nanoindentation experiments with a pointed, trigonal (Berkovich) indenter. For a range of materials, the relationship P = Kmδ2 was found to describe the indenter displacement, δ, in terms of the applied load P. For each material, Km can be predicted from the Young's modulus (E) and the hardness (H). The result is that if either E or H is known, then the other may be calculated from the experimental loading curve. This approach provides an attractive alternative to finite element modeling and is a tractable approach for those cases where analysis of unloading curves is infeasible.

EXPERIMENTAL STUDY ON COMPRESSIVE YOUNG’S MODULI AND TENSILE YOUNG’S MODULUS OF FLY ASH CONCRETE

2018 ◽  
Vol 5 (2) ◽  
Author(s):  
Yoichi Mimura ◽  
Vanissorn Vimonsatit ◽  
Yuki Watanabe ◽  
Itaru Horiguchi ◽  
Isamu Yoshitake
Keyword(s):  
Fly Ash ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Tangent Modulus ◽  
Early Age ◽  
Secant Modulus ◽  
Tensile Stresses ◽  
Fly Ash Concrete ◽  
Young’S Moduli ◽  
Initial Tangent Modulus

Initial cracks due to volume changes at an early age affect the durability of concrete structures, so numerical simulations are often conducted in order to predict cracks. Such prediction requires some mechanical properties of early age concrete. Tensile Young's modulus is directly dependent on the prediction of tensile stress and is one of the important input data for FEM analysis. However, direct tension test for tensile Young's modulus needs a unique apparatus and specimen, and such test is not suitable for evaluating Young's modulus at early ages of concrete. The present study compared tensile Young's modulus with compressive Young's moduli of Fly ash concrete. Compressive Young's moduli used in this study were secant modulus and initial tangent modulus. In addition, linear modulus taken from a regression line of a compressive stress-strain curve in the range of stresses less than the splitting tensile strength was also evaluated. It is found that the secant modulus, which is generally used as Young's modulus in Japan was clearly smaller than the tensile Young's modulus, which means that, tensile stresses evaluated using a secant modulus might be underestimated. On the other hand, linear modulus and initial tangent modulus were almost equal to the tensile Young's modulus. This result indicates that tensile stresses can be evaluated using Young's modulus obtained from a compression test with general apparatus and specimens.

Optimal Design of a Nonhomogeneous Annular Disk Under Pressure Loadings

1994 ◽  
Vol 116 (4) ◽  
pp. 989-996
Author(s):  
Chung-Yun Gau ◽  
Souran Manoochehri
Keyword(s):  
Optimal Design ◽  
Young’S Modulus ◽  
Variable Thickness ◽  
Volume Fraction ◽  
Young's Modulus ◽  
Yield Criterion ◽  
Constant Thickness ◽  
Annular Disk ◽  
Disk Thickness ◽  
Thickness Variations

A method for the design of nonhomogeneous, variable-thickness, annular disks under internal and external pressures satisfying Tresca yield criterion is presented in this paper. The effects of varying the disk thickness and stiffness properties to achieve a fully stressed design are investigated. Analytical solutions for distributions of Young’s modulus and disk thickness variations have been developed for the case of fully stressed designs. Examples are given for three different cases, namely, constant thickness with variable Young’s modulus, variable thickness with constant Young’s modulus, and variable thickness with variable Young’s modulus. In the last case, due to the existence of many alternative solutions, optimal design techniques have been utilized. Application of the developed methodology for optimal designs of short fiber composites with random fiber orientations is discussed. The optimization results of fiber volume fraction distributions and thickness variations for a disk made of nylon 66 matrix with E glass fiber are given under specified pressure loadings.

scholarly journals Experimental Study of Hardened Young’s Modulus for 3D Printed Mortar

Materials ◽  
2021 ◽  
Vol 14 (24) ◽  
pp. 7643
Author(s):  
Szymon Skibicki ◽  
Mateusz Techman ◽  
Karol Federowicz ◽  
Norbert Olczyk ◽  
Marcin Hoffmann
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Statistical Correlation ◽  
Test Methods ◽  
Cylindrical Sample ◽  
Experimental Investigations ◽  
Current Standard ◽  
Number Of Layers ◽  
3D Printed

Few studies have focused on determining the Young’s modulus of 3D printed structures. This study presents the results of experimental investigations of Young’s modulus of a 3D printed mortar. Specimens were prepared in four different ways to investigate possible application of different methods for 3D printed structures. Study determines the influence of the number of layers on mechanical properties of printed samples. Results have shown a strong statistical correlation between the number of layers and value of Young’s modulus. The compressive strength and Young’s modulus reduction compared to standard cylindrical sample were up to 43.1% and 19.8%, respectively. Results of the study shed light on the differences between the current standard specimen used for determination of Young’s modulus and the specimen prepared by 3D printing. The community should discuss the problem of standardization of test methods in view of visible differences between different types of specimens.

scholarly journals The elastic properties of steel at high temperatures

1905 ◽  
Vol 76 (512) ◽  
pp. 419-425 ◽  
Keyword(s):  
Young’S Modulus ◽  
Elastic Properties ◽  
High Temperatures ◽  
Young's Modulus ◽  
Elastic Recovery ◽  
Tensile Tests ◽  
Remarkable Change ◽  
Order Of Magnitude ◽  
Different Order ◽  
Slow Contraction

Hitherto, investigations into the elastic properties of metals have been confined to comparatively low temperatures. Gray, Dunlop, and Blyth have measured the modulus of rigidity and Young’s modulus for wires up to temperatures of 100° C., and found that both these quantities decrease as the temperature rises. Martens determined the influence of heat on the strength of iron up to temperatures of 600° C., but his experiments were the ordinary tensile tests carried to rupture, and though he also found a substantial diminution of Young’s modulus with rise of temperature, he did not go into the point fully, being mainly concerned with breaking stress and elongation. In the experiments here described the elastic properties of steel and iron have been investigated at higher temperatures, ranging up to 800° C., and for stresses greatly below that required to rupture the material. We have found that as the temperature rises the stress-strain relations undergo a remarkable change, which may best be expressed by saying that what is variously called the “time-effect,” or “elastische nachwirkung,” or “creeping,” increases greatly with the temperature. Steel, at high temperatures, behaves like indiarubber or glass; if it is stressed for a time, and the stress removed, it does not at once recover, but after the immediate elastic recovery there is a slow contraction perceptible for many minutes. Such “creeping” can be detected at ordinary temperatures, but at a red heat it attains a different order of magnitude, becoming (in its total amount) a substantial fraction of the whole deformation.

scholarly journals Tensile Mechanical Properties of Fly Ash Concrete at Early Age for Thermal Stress Analysis

2020 ◽  
Author(s):  
Yoichi Mimura ◽  
Vanissorn Vimonsatit ◽  
Itaru Horiguchi ◽  
Isamu Yoshitake
Keyword(s):  
Fly Ash ◽  
Young’S Modulus ◽  
Compression Test ◽  
Creep Test ◽  
Elastic Strain ◽  
Young's Modulus ◽  
Tensile Creep ◽  
Early Age ◽  
Secant Modulus ◽  
Fly Ash Concrete

Abstract The present study investigates tensile properties of concrete with and without fly ash at early age, such as tensile Young’s modulus, strength and creep. Some Young’s modulus of fly ash concrete for thermal crack analysis was compared with the tensile Young’s modulus, secant modulus, initial tangent modulus and linear modulus obtained from the direct tension test and compression test. The tensile creep test was also performed to obtain the specific creep behavior considering decrease in elastic strain due to stiffness development at early age during creep test. The results show that the Young's modulus for crack evaluation can be obtained from the compression test based on the stress range less than the splitting tensile strength, while the compressive secant modulus was smaller than the tensile Young's modulus. The decrease of the elastic strain at early age contributes more to the evaluation of the tensile creep than the use of the fly ash mixing.

Indentation at the Nanometer Scale on Ultra Thin Films of Diamond-Like Carbon

1994 ◽  
Vol 356 ◽  
Author(s):  
Sandrine Bec ◽  
André Tonck ◽  
Jean-Luc Loubet
Keyword(s):  
Thin Films ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Elastic Recovery ◽  
Contact Stiffness ◽  
Natural Diamond ◽  
Tangential Displacement ◽  
Diamond Like Carbon ◽  
New Instrument

AbstractUltra thin films (50 nm and 180 nm) of amorphous diamond-like carbon on a silicon substrate produced by laser ablation are tested by nanoindentation with a new instrument deriving from a Surface Force Apparatus. Quasi-static measurements of the load and dynamic measurements of the contact stiffness are continuously and simultaneously recorded versus the penetration depth. Scanning lines on the tested surface before and after indentation are made by means of tangential displacement of the diamond indenter on the surface.The tests are conducted with maximum loads from 50 μN to 2500 μN, which correspond to maximum indentation depths between 7 nm and 70 nm. The indentation curves show near elastic recovery but scanning lines and/or topographic images on the surfaces show detectable plastic prints. Despite the extremely small residual indentation depths for these ultra thin films, we show how the hardness value we calculate from the indentation curves with an elastoplastic theory is in good agreement with the hardness value we calculate from the indentation print profile. The determination of the Young's modulus, even at the smallest indentation depths, must take into account the mechanical properties of the substrate. The determination of both values, hardness and elastic modulus, also requires a calibration procedure for the geometry of the tip and knowledge of the piling-up effect.We find that the apparent hardness and the apparent Young's modulus of the tested diamondlike films are high. They are underestimated in comparison with the real values. A rough correction which overestimates the Young’s modulus gives higher values than those of natural diamond.

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