1、NPL Manual Manual for the Calculation of Elastic-Plastic Materials Models Parameters This Electronic Guide was produced as part of the Measurements for Materials System Programme on Design for Fatigue and Creep in Joined Systems Introduction Testing Calculation of Parameters Typical Predictions Fail
2、ure Criteria Full contents page is available Crown copyright 2007 Reproduced with the permission of the Controller of HMSO and Queens Printer for Scotland Contents Introduction Overview of elastic-plastic models Von Mises parameters Von Mises information Typical von Mises parameters Drucker-Prager m
3、odels Linear Drucker-Prager parameter calculation Associated and non-associated flow Exponent Drucker-Prager parameter calculation Data analysis Equivalent data points Effect of equivalent data point selection on parameters Typical linear Drucker-Prager parameters Typical exponent Drucker-Prager par
4、ameters Cavitation model parameters Cavitation model information Typical cavitation model parameters Calculation of parameters Typical Predictions Scarf joint Lap joint Tpeel joint Butt joint Failure Criteria Tensile Data input Tensile Testing Typical tensile data Tensile data for elastic-plastic mo
5、dels Elastic and plastic behaviour Hardening curves Calculation of parameters Shear Data input Shear Testing Typical shear data Shear data parameters Calculation of shear parameters Compression Data input Compression Testing Typical compression data Compression data calculations HOME INTRODUCTION Fi
6、nite element analysis is a computational tool that can be used for calculating forces, deformations, stresses and strains throughout a bonded structure. These predictions can be made at any point in the structure including within the adhesive layer. Furthermore, the element mesh can accurately descr
7、ibe the geometry of the bond line so the influence of geometrical features, such as the shape of fillets and boundaries with adherends, on joint performance can be accounted for in the analysis. This is particularly important in the design of adhesive joints because these features are usually associ
8、ated with regions of stress and strain concentration likely to initiate joint failure. Since adhesives are generally tough materials, they can sustain large strains before failure and, under these conditions, relationships between stress and strain are highly non-linear. The origin of non-linearity
9、in a stress-strain curve for a polymeric adhesive can be explained in terms of enhanced creep brought about by an increase in molecular mobility caused by the application of stress. At moderate stresses this non-linear deformation is recoverable. At higher stresses, the molecular mobility is high en
10、ough for yielding to occur by plastic flow, which is not recoverable. Elastic-plastic materials are employed to describe deformation under large strain. There are several models available for modelling the adhesive based on different criteria for plastic deformation. Predictions of joint performance
11、 at large strains close to joint failure depend on the model used. For the prediction of failure, stress and strain distributions in the adhesive need to be accurately calculated and a failure criterion for the adhesive needs to be established. In its current form, this manual demonstrates the calcu
12、lation of parameters for use with four different elastic-plastic material models. These four elastic plastic models are: von Mises, linear Drucker-Prager, exponent Drucker-Prager and Cavitation model. Material Models Data requirements Von Mises tensile only Linear Drucker-Prager tensile data and she
13、ar data at the same strain rate Exponent Drucker-Prager tensile data and shear data at the same strain rate Cavitation model tensile data, shear data and compressive data at the same strain rate Click here for an overview of the models BACK HOME NEXT Material models overview Von Mises criterion is t
14、he most simple yield criterion. It interprets yielding as a purely shear deformation process which occurs when the effective shear stress reaches a critical value. Simple equations relate the tensile yield stress, shear yield stress and compressive yield stress to a material property. Tests on adhes
15、ives under additional stress states reveal that in many cases yielding is sensitive to the hydrostatic component of stress in addition to the shear component. Therefore the von Mises criterion is not realistic for many adhesives. The linear Drucker-Prager model is a simple modification of the von Mi
16、ses criterion that includes some hydrostatic stress sensitivity. A parameter () is introduced which characterises the sensitivity of yielding to hydrostatic stress. Two different stress states e.g. tension and shear are required to determine this parameter. The linear Drucker-Prager model is not cap
17、able of accurately describing the non-linear behaviour of an important class of tough adhesives - the rubber toughened materials. The exponent Drucker-Prager is a more complex elastic-plastic model, which is better able to describe behaviour under stress states in which there is a high component of
18、hydrostatic tension. Two different stress states e.g. tension and shear are required to determine the exponent Drucker-Prager hydrostatic stress sensitivity parameter. In rubber-toughened epoxy, nucleation of cavities occurs in the rubber phase. The Cavitation model includes the influence of void nu
19、cleation on ductility through adaptation of the linear Drucker-Prager model. This model has been developed at NPL and is currently being used to evaluate the accuracy of predictions of deformation in various joint geometries. BACK HOME TENSILE DATA INPUT Tensile data are required for all four elasti
20、c-plastic materials models considered here. The data obtained from tensile test measurements of bulk adhesive are nominal (or engineering) values, where stresses and strains have been calculated using the initial specimen dimensions. Measurements of the following data are required: Nominal Stress (M
21、Pa) Nominal Axial Strain Nominal Transverse Strain For more information on tensile testing please click here Typical data for a rubber-toughened epoxy BACK HOME NEXT Tensile Testing Tensile tests 1 for the determination of Youngs modulus (E) and Poissons ratio () are carried out on standard specimen
22、s 2, 3 under constant deformation rate in a tensile test machine at relatively low strain rates e.g. 10 mm/min. For best accuracy, contacting extensometers should be used for the measurement of axial and transverse strain, T and t . Two extensometers mounted on opposite faces of the specimen should
23、preferably be used for the axial strain measurement to eliminate small non-uniformity in the strain through the thickness of the specimen caused by bending. The transverse strain measurement should be made close to the axial gauge section and, if possible, between the contact points of the extensome
24、ters. The contact pressure used to attach the extensometers to the specimen should be large enough to prevent slippage but insufficient to indent the specimen surface. Strain gauges are not recommended as they locally stiffen the specimen 1. Values for Youngs modulus and Poissons ratio are calculate
25、d from the regression slopes in the linear region of the T - T and t - T curves. Use of regression slopes is preferable to single point values owing to the potential scatter in the data points (particularly the t - T data) that is mainly due to uncertainties in the small extensions measured. Whilst
26、elastic values can be determined over any strain range where the data appear linear, the slight curvature due to viscoelastic effects will tend to reduce the value of E as the strain range widens. The measurement of tensile hardening curves involves use of the same tests out to larger strains. Conta
27、cting extensometers, unless they have been modified, typically have an upper strain limit of around 0.05. They may also initiate premature failure in the specimen at a point of contact. For these tests, the use of a video extensometer is therefore preferable for the measurement of axial strain. The
28、videoextensometer gauge markings are visible on the tensile test specimen shown above. These instruments are generally unsatisfactory for the measurement of small displacements and so a contacting device is best used to measure the lateral strain for the determination of true stresses and the plasti
29、c component of Poissons ratio. 1. G.D. Dean and B.C. Duncan, Preparation and Testing of Bulk Specimens of Adhesives. NPL Measurement Good Practice Guide No 17, July 1998. 2. ISO 3167:1993, Plastics - Multipurpose test specimens. 3. ISO 527-2:1993, Plastics - Determination of tensile properties - Par
30、t 2: Test conditions for moulding and extrusion plastics. BACK HOME Tensile Data Nominal Strain Nominal Stress (MPa) 0.0004 0.0015 0.0035 0.0053 0.0070 0.0086 0.0103 0.0120 0.0135 0.0153 0.0172 0.0189 0.0211 0.0230 0.0251 0.0274 0.0298 0.0322 0.0348 0.0373 0.0399 0.0423 0.0452 0.0476 0.0504 0.0533 0
31、.0560 0.0588 0.0615 0.0643 0.0670 0.0699 0.0728 0.0758 0.0786 0.0812 0.0842 1.44 4.80 10.34 15.73 20.71 24.28 28.02 31.59 34.93 37.89 40.46 43.13 45.03 46.81 48.57 49.68 50.65 51.46 52.25 52.90 53.36 53.63 54.07 54.46 54.80 54.98 55.33 55.84 55.97 56.19 56.32 56.44 56.79 56.86 57.09 57.17 57.26 Here
32、 are typical tensile data for a rubber-toughened epoxy. The data are shown in tabular and graphical form. BACK HOME Tensile Data for Elastic-Plastic Models Elastic-plastic models in FE systems require data in the form of elastic constants to describe elastic behaviour and parameters that describe th
33、e yield, hardening and flow behaviour to describe plastic (non-linear) behaviour. Further information on elastic and plastic behaviour is available. Data from tensile tests are required in the form of true stresses and strains for the calculation of the elastic-plastic model parameters. The change i
34、n cross-section of the specimen during tensile tests needs to be included to calculate true stress and strains. Under tensile loading, the specimen cross-section reduces. True values can be calculated from the nominal (engineering) values, which are based on the original specimen dimensions. The fol
35、lowing data can be calculated: True stress, T True strain, T True transverse stress, t Nominal Poissons ratio, Youngs Modulus, E True Elastic modulus, E True Poissons ratio, P True plastic strain, T True transverse plastic strain, P t True plastic Poissons ratio, p Hardening curve For more informati
36、on on the calculation of these parameters please click here Of the above data, the Youngs modulus, Poissons ratio, true stress, true plastic strain, and true plastic Poissons ratio are required for the elastic-plastic models parameter calculations. The hardening curve is also required. This is a plo
37、t of true stress versus true plastic strain, which is sampled to reduce the number of data points and then used in tabular form within elastic-plastic materials models. More information on the hardening curve can be obtained by clicking on the link above. BACK HOME NEXT Elastic-Plastic Behaviour Wit
38、h elastic-plastic models, calculations of stress and strain distributions at low strains are based on linear elasticity. The onset of non-linearity is attributed to plastic deformation and occurs at a stress level regarded as the first yield stress. The subsequent increase in stress with strain is a
39、ssociated with the effects of strain hardening, and increases to a maximum corresponding to the flow region. In this non- linear region, the total strain is considered to be the sum of a recoverable elastic component and a plastic component, which is non- recoverable. Stress analysis calculations th
40、en involve the use of multiaxial yield criteria and a flow law. The yield criterion relates components of applied stress field to material parameters after the onset of yielding. The material parameters will depend upon the plastic strain for a strain hardening material. The calculation of plastic s
41、train components is achieved in plasticity theory using a flow rule, which relates increments of plastic strain to a plastic flow potential. If the flow behaviour for a particular material is such that the flow potential can be identified with the yield function then this is termed associated flow.
42、In general, this will be an approximation and extra information is needed to characterise non-associated flow. In order to calculate some of the parameters in elastic-plastic models, it is necessary to select stress values from different tests under the same state of yielding. This requires the defi
43、nition of an effective plastic strain, and equivalent stresses are then a set of stresses that characterise stress states having the same effective plastic strain. BACK HOME Hardening Curve Strain hardening data can be given in either single or multiple rate form. If a single strain rate hardening c
44、urve is used, this strain rate is assumed to be the average strain rate in the adhesive layer. The use of multiple rate hardening curves allows for regions of varying strain rate within the adhesive layer. Generally four strain hardening curves separated by a factor of 10 in strain rate are used. On
45、e is chosen with a strain rate approximately the same as the average strain rate in the joint, one higher rate curve and one lower rate curve. An extremely low rate curve is also required and is designated the zero rate curve. Analysis difficulties can occur if more hardening curves are used or if t
46、oo many data points are used to characterise each curve. The plastic strain value must always increase otherwise the analysis will not run. The best approach is to sample the hardening data until a representative curve is achieved. The sampling density of the hardening curves shown in the figure has
47、 been used successfully. The hardening curve data are required in the tabular form of yield stress with plastic strain where the first pair of numbers must correspond to the initial yield stress at zero plastic strain. If strains in the analysis exceed the maximum effective strain supplied in the ha
48、rdening curve then the analysis will assume that additional extension occurs with no hardening. One means of getting around this is to extrapolate a point on the hardening curve at a significantly higher strain. BACK HOME Tensile Data Calculations Data from tensile tests are required in the form of
49、true stresses and strains for the calculation of the elastic-plastic model parameters. These are related to nominal (engineering) values and based on the original specimen dimensions by the equations shown below. Click here for a definition of the symbols True Stress () 2 T T T -1 = True Strain )(1ln TT += Tru
