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.

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.

Further analysis of the size effect in indentation hardness tests of some metals

1995 ◽  
Vol 10 (11) ◽  
pp. 2908-2915 ◽  
Author(s):  
M. Atkinson
Keyword(s):  
Size Effect ◽  
Strain Hardening ◽  
Indentation Size Effect ◽  
Plastic Hinge ◽  
Small Scale ◽  
Indentation Hardness ◽  
Vickers Indentation ◽  
Indentation Size ◽  
Hardness Tests ◽  
Indentation Tests

The variation of apparent hardness observed in previously reported Vickers indentation tests of metals is reexamined. Common deseriptions of the effect are shown to be inaccurate: the variation of apparent hardness is monotonic but not simple. The effect is consistent with varying size of a previously postulated “plastic hinge” at the perimeter of the indent. This complexity confers uncertainty on the estimation of characteristic macrohardness from small scale tests. Association of the indentation size effect with friction and with strain hardening is confirmed.

A systematic study of the validation of Oliver and Pharr’s method

2007 ◽  
Vol 22 (12) ◽  
pp. 3385-3396 ◽  
Author(s):  
Siqi Shu ◽  
Jian Lu ◽  
Dongfeng Li
Keyword(s):  
Mechanical Properties ◽  
Strain Hardening ◽  
Young’S Modulus ◽  
Basic Assumption ◽  
Young's Modulus ◽  
Strain Hardening Exponent ◽  
The Finite Element Method ◽  
Solid Materials ◽  
Simulation Results ◽  
Depth Curves

Oliver and Pharr’s method (O&P’s method) is an efficient and popular way of measuring the hardness and Young’s modulus of many classes of solid materials. However, there exists a range of errors between the real values and the calculated values when O&P’s method is applied to materials not included in the basic assumption proposed initially. In this article, the dimensional analysis theorem and the finite element method are applied to evaluate errors for high elastic (E/σY → 5) to full plastic (E/σY→ 1000) materials with different strain-hardening exponents from 0 to 0.5. A new method is proposed to correct errors obtained using O&P’s method. The numerical simulation results show that the errors obtained using O&P’s method, given in the form of charts, are mainly dependent on the ratio of the reduced Young’s modulus to the yield stress (i.e., Er/σY) and the strain-hardening exponent, n, for an indenter with a fixed included angle. The two mechanical properties, which can be extracted from the load–depth curves of two indenters with different included angles, are used to correct the errors in the hardness and Young’s modulus of the indented materials produced by O&P’s method.

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.

Change of modulus and yielding properties of syndiotactic polypropylene with the glass transition

e-Polymers ◽  
2002 ◽  
Vol 2 (1) ◽  
Author(s):  
Yongfeng Men ◽  
Gert Strobl ◽  
Patrick Wette
Keyword(s):  
Glass Transition ◽  
Strain Hardening ◽  
Young’S Modulus ◽  
Temperature Variation ◽  
Yield Point ◽  
Critical Points ◽  
Tensile Deformation ◽  
Young's Modulus ◽  
Syndiotactic Polypropylene ◽  
Crystalline Polymers

AbstractSemi-crystalline polymers like syndiotactic polypropylene pass during tensile deformation different stages with cross-overs at four critical points. Temperature variation above ambient changes the stresses at the critical points, but leaves the strains constant. Cooling down to the glass transition leads to the following changes: Fracture occurs before the onset of strain hardening. Yield point strain shifts to lower values. Young’s modulus increases by a factor on the order of the inverse of the crystallinity, as can be expected when the amorphous regions also contribute to the transmission of forces.

Identifying the limitation of Oliver and Pharr method in characterizing the viscoelastic-plastic materials with respect to indenter geometry

2008 ◽  
Vol 1137 ◽  
Author(s):  
Keerthika Balasundaram ◽  
Yanping Cao ◽  
Dierk Raabe
Keyword(s):  
Mechanical Properties ◽  
Experimental Study ◽  
Young’S Modulus ◽  
Young's Modulus ◽  
Polymeric Materials ◽  
Analysis Method ◽  
Correct Evaluation ◽  
Nano Scale ◽  
Plastic Materials ◽  
Indenter Geometry

AbstractNanoindentation tests are widely used in recent years to characterize the mechanical properties of viscoelastic-plastic materials like polymers and biomaterials at the micro or nano-scale using the analysis method proposed by Oliver & Pharr (OP). However, recent studies revealed that the mechanical properties of viscoelastic-plastic (polymeric) materials determined using the OP method does not lead to a correct evaluation of Young's modulus. A systematic experimental study is performed with different indenter geometries like spherical and Berkovich geometries using various polymers in order to identify the limitations of the OP method.

On the Relation between Indentation Hardness and Young's Modulus

1958 ◽  
Vol 31 (4) ◽  
pp. 896-906 ◽  
Author(s):  
A. N. Gent
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Empirical Relation ◽  
Rigid Sphere ◽  
Indentation Hardness ◽  
Shore Hardness ◽  
Theoretical Relation ◽  
Classical Elasticity ◽  
British Standard ◽  
Classical Elasticity Theory

Abstract A relation between British Standard and International rubber hardness and Young's modulus is derived from classical elasticity theory, and compared with the empirical relation given in B.S.903:1950. An experimental examination of the load-indentation relationship for a rigid sphere pressed into a flat rubber pad is described ; it indicates that the theoretical relation is more appropriate than the empirical one for small indentations, corresponding to hardnesses exceeding about 60° B.S. & I.R.H., and equally valid for hardnesses between about 40° and 60°. Moreover, the numerical constants are not subject to experimental uncertainty. If reduced major loads are stipulated for determining the hardness of rubbers of less than 35° to 40° B.S. & I.R.H., the theoretical relation should apply over the entire useful range. An approximate relation between Shore hardness and Young's modulus is derived similarly. The approximate equivalence of the British Standard and International rubber hardness and Shore hardness scales over the major part of the hardness range is confirmed.

Relationship between Indenter Calibration and Measured Mechanical Properties of Elastic-Plastic Materials

2015 ◽  
Vol 662 ◽  
pp. 27-30 ◽  
Author(s):  
Jaroslav Čech ◽  
Petr Haušild ◽  
Jiri Nohava
Keyword(s):  
Young’S Modulus ◽  
Young's Modulus ◽  
Plastic Behavior ◽  
Calibration Function ◽  
Simple Method ◽  
Area Function ◽  
Elastic Plastic ◽  
Plastic Materials ◽  
Pile Up ◽  
Elastic Plastic Materials

Calibration of Berkovich indenter area function was performed on materials with different elastic-plastic behavior resulting in pile-up and sink-in, respectively. Experimentally obtained results were compared with the results obtained by the application of theoretical area function. The values of Young’s modulus and hardness were significantly affected by the calibration function used. Since the effects of pile-up and sink-in are already included in the used area function, this simple method can lead to more accurate results of Young’s modulus and hardness measurements.

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