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УДК 539.421.4
Особенности /-интеграла в условиях упругопластической деформации для материалов, описываемых законом Рамберга-Осгуда
P. Gallo, F. Berto
Падуанский университет, Виченца, 36100, Италия
При расчете прочности конструкций /-интеграл успешно применяется как параметр разрушения в условиях упругопластической деформации. В литературе, посвященной трещинообразованию, /-интеграл принято оценивать как сумму вкладов упругих и пластических деформаций. Однако использование данного метода в модели Рамберга-Осгуда имеет ряд ограничений, в особенности при полномасштабной текучести.
В работе выполнена оценка /-интеграла для деталей с трещиной в условиях упругопластических деформаций. Рассмотрены два варианта нелинейного поведения материала: закон Рамберга-Осгуда и степенной закон. Проведено сравнение результатов численного исследования и конечно-элементного анализа, полученных в рамках различных подходов. Показано, что наиболее адекватная оценка /-интеграла для материала, описываемого законом Рамберга-Осгуда, может быть получена при независимом исследовании методом конечных элементов вкладов упругих деформаций (расчет на основе линейно-упругой зависимости) и пластических деформаций (нелинейный расчет с учетом степенного закона).
Ключевые слова: /-интеграл, трещина, упругопластическое поведение, закон Рамберга-Осгуда, степенной закон
Some considerations on the /-integral under elastic-plastic conditions for materials obeying a Ramberg-Osgood law
P. Gallo and F. Berto
University of Padua, Vicenza, 36100, Italy
Among the approaches available for structural analysis, the /-integral has received an excellent feedback as a fracture parameter under elastic-plastic conditions. In the literature and dealing with the crack case, it is proposed to evaluate the /-integral as a sum of elastic and plastic contributions. However, some uncertainties arise when applying this method to a Ramberg-Osgood law, especially under large scale yielding conditions.
The aim of the present paper is to discuss how the /-integral evaluation can be performed for elastic-plastic cracked components. Two different non-linear behaviours have been considered for the material: the Ramberg-Osgood law and power law. Numerical and finite element results from different approaches have been accurately compared, proving that the most appropriate way to evaluate the /-integral for a material obeying Ramberg-Osgood law is to perform two finite element analyses evaluating separately the elastic contribution (through a linear elastic analysis) and the plastic contribution (through a nonlinear analysis considering power law behaviour).
Keywords: /-integral, crack, elastic-plastic behaviour, Ramberg-Osgood law, power law
1. Introduction
Design of mechanical components requires the application of an acceptable procedure based on engineering principles. The procedure is applied to verify the integrity and/or functionality of components. Sometimes, for simple problems, the use of handbooks and simplified formulas could be sufficient. Unfortunately, more often the analysis relates to complex components and applications, and the use of more accurate methodology becomes essential.
Nowadays, it is very common for an engineer to design a component that, in operating conditions, is subjected to
stress and strain well beyond the elastic limit of the material and has the ability to exhibit a nonlinear stress-strain response (material nonlinearity). Components subjected to plasticity, crushing, and cracking are good examples, but time-dependent effects such as visco-elasticity or visco-plas-ticity (creep) can be also mentioned. Due to advanced industrial applications in critical environments (e.g. very high temperature applications), the design procedure under linear elastic conditions is sometimes not applicable. This situation is not so rare, for example, considering the aerospace and automotive industries.
© Gallo P., Berto F., 2015
Among the local approaches [1-12] available for nonlinear analysis, the contour ./-integral has received an excellent feedback as a fracture parameter characterizing nonlinear materials under different loading conditions, considering also thermomechanical fatigue and creep [13-15].
In 1968, Rice [ 16] provided the basis for extending fracture mechanics methodology well beyond the validity limits. It was originally proposed for near tip stresses characterization of notches and cracks under two-dimensional deformation field and consists in a path independent integral surrounding cracks or notches. The path independence is maintained only if the notch opening angle is equal to zero. Hutchinson [17] and Rice and Rosengren [18] also showed that / can characterize crack-tip stresses and strain in nonlinear materials, underlining the double form of the /-integral: energy parameter and/or stress intensity parameter.
Under linear elastic conditions /-integral can be easily correlated to mode I, II and III stress intensity factors as well as to the strain energy density. For opening angles different from zero, because of the loss of the path independence, a direct correlation between /-integral and nonlinear stress intensity factors is far from easy. However, different attempts have been made recently to obtain closed form relationships for the linear elastic strain energy density averaged over the control volume [7-11, 19, 20] and the /-integral at the tip of notches under different loading conditions [1]. Moreover other attempts have been made in the past years for sharp and blunt notches [21-26].
Considering the crack case under linear elastic conditions, analytical formulas are available to evaluate the /-integral as a function of the stress intensity factor, Young's modulus and Poisson's ratio. Under elastic-plastic conditions, instead, different approaches are available. In [2729], extensive tables for calculating / for various geometries, crack lengths and for different nonlinear stress-strain curves are provided. However, sometimes, different parameters are not available in the handbooks or some geometries are not considered at all. In these cases the finite element analysis becomes fundamental for the evaluation of the /-integral.
The aim of the paper is to present a brief overview on /-integral evaluation for cracked components obeying an elastic-plastic rule. The influence of the chosen material law on the results in terms of stress field is investigated and the nonlinear finite element analysis is provided to support the conclusions. In particular, the Ramberg-Osgood law and power law have been considered for modeling the material behavior. The objectives of this paper are to precisely define all parameters necessary for the calculation of /, to summarize the analytical equations available to evaluate these parameters and to present some practical guidelines for conducting the /-integral evaluation for a cracked plate obeying a Ramberg-Osgood law by means of the finite element analysis. Three methods will be presented, underlin-
ing the main advantages and drawbacks. This paper wants to give useful guidelines for an easy evaluation of the /-integral in practical applications, when facing the Ram-berg-Osgood material law. The manuscript is also aimed to clarify the use of the available handbook formulas and the finite element analysis to quantify elasto-plastic /-integral for the Ramberg-Osgood material law.
In details, a plate with a central crack will be considered, and the following objectives will be addressed step by step:
- to clarify the definition of all parameters necessary for the calculation of /-integral;
- to make a short review of the different analytical equations available in the literature;
- to propose some simple guidelines that can suggest a correct way for the estimation of / for an elastic-plastic Ramberg-Osgood cracked material by the finite element analysis.
Regarding the last point, three methods will be presented, underling the main advantages and drawbacks. This paper is aimed to be a useful tool giving some practical guidelines for an easy evaluation of the /-integral for engineers in their practical applications, when the Ramberg-Osgood material law is employed. The present investigation creates more clarity both in the case of using handbook formulas and the finite element analysis to quantify elastic-plastic /-integral from the Ramberg-Osgood material law.
2. Material models
2.1. Geometry and mechanical properties
The geometry investigated in the present paper is shown in Fig. 1. It consists in a plate with a central crack. The crack length 2a is equal to 20 mm, while the plate is 200 mm width and 200 mm height. The mechanical properties instead are reported in Table 1: the Young's modulus E is equal to 206 000 MPa, the strength coefficient K to 950 MPa and the hardening exponent n is 8.33. These properties characterized the AISI 1045 steel.
TTTTTTTTTTTTTT
Fig. 1. Geometrical parameters
Table 1
Mechanical properties of AISI 1045
K, MPa
950
;.33
E, MPa
206000
öy, MPa
450
2.2. Ramberg-Osgood and power law
In 1943, Ramberg and Osgood [30] proposed the following simple formula for describing the nonlinear relationship between stress and strain:
a
£ = —+ E
'a ^
vKy
(1)
The parameters K and n are constants and describe the hardening behavior of the material. The curve represented by the Ramberg-Osgood model depicts a continuous transition from the elastic to plastic behavior of the material. The adequacy of Eq. (1) was tested by Ramberg and Osgood by plotting on log-log graph the stress-deviation curves for the different materials taken from the literature [31], showing that the formula is adequate for aluminum alloy, carbon steel and chromium-nickel steel.
The resulting Ramberg-Osgood curve, considering the mechanical properties of Table 1, is depicted in Fig. 2.
Neglecting the elastic contribution of the Ramberg-Osgood law, one obtains the simple power law form reported below:
e =
/ a ^
K
(2)
Assuming also in this case the mechanical properties previously proposed (Table 1), one obtains the power-law curve shown in Fig. 3. The elastic length of the Ramberg-Osgood law is clearly visible from the figure. For higher loads the two curves collapse into a single one.
3. /-integral evaluation
3.1. The J contour integral
When dealing with the Ramberg-Osgood law, in the literature it is suggested an approximation of the elastic-plas-
tic J that consists in an interpolation formula combining the linear elastic and the fully plastic conditions [27]:
J = Je(ae) + Jp(a, n), (3)
where Je (ae ) is the elastic contribution based on an adjusted crack length ae = a + §ry, i.e. Irwin's effective crack length modified to account for strain hardening,
ßn
n -1 n +1
K
\2
1
1 + (P/P))2'
Jp (a, n) is the plastic contribution based on the material hardening exponent n and crack length, for plane strain p=6.
Under small-scale yielding, the plastic contribution is small compared with the elastic contribution and Eq. (3) reduces to the well-known elastic solution reported in [28]. On the other side, in the fully plastic range, the elastic contribution is negligible.
Equation (3) expresses the total J as a sum of the elastic and plastic contribution. Regarding the elastic contribution, the evaluation of J is very straightforward since it can be easily linked to various fracture mechanics parameters such as the stress intensity factor K that, in turn, can be evaluated through the finite element analysis [28]:
Je = K 2 i ] • (4)
E
p
Considering the plastic contribution Jp (a, n), Kumar et al. [27] provide the following expression:
ba
Jp = aea
0U0
W
-h
' P Nn+1
(5)
Expressed for the following power law constitutive model:
= a
n
(6)
where a0 is the reference stress value that is usually equal to the yield strength, e0 = a0 /E or in general f (g0 ), a is the dimensionless constant, n is the hardening exponent, h1 = f (a/W, n) and it is provided by Anderson [28] for fully plastic J for a middle tension specimen in plane strain and plane stress, P0 = (2/V3 )2ba 0 is the reference load
Fig. 2. Ramberg-Osgood curve
Fig. 3. Power law curve
(only for plane strain conditions), P is the applied load referred to the gross area, b, a and Ware geometrical parameters, while B is the plate thickness.
3.2. Parameters a, P0 and hj
When dealing with the estimation of /p, different parameters of Eq. (5) need to be quantified by tables and/or figures available in the literature [27, 28, 32]. Two main difficulties can be pointed out: first, the considered geometries are limited; secondly, the parameters are defined only for the power law form expressed by Eq. (6), and so in terms of ct0 and e0 while nowadays a power law expressed in terms of K and n, as shown in Eq. (2), is usually employed. For these reasons, some considerations deserve to be spent about the definition of parameters P0, hj and a when dealing with Eq. (2).
In the configuration under investigation, since the material constitutive model and its related ct-e curve are fully available, the dimensionless constant a can be easy determined through a best fitting procedure, once defined the value of ct0 and e0. As suggested by Anderson's handbook [28] and other authors [27], ct0 can be assumed as the yield strength of the material, here called cty. This assumption made possible to write e0 as a function of the yield strength:
e0 = -
(7)
= _y
E E '
An alternative procedure, avoiding the best fitting pro cess, can be obtained by few simple considerations. Rear ranging Eq. (6) and taking into account the relation expressed by Eq. (7), it is possible to express a as follows:
a = -
eE ( a.
eE
(a Y
(8)
Since Eqs. (2) and (6) represent the same material curve, with the same mechanical properties, the deformations can be defined also by the latter power law form. For this reason, substituting Eq. (2) into Eq. (8) and rearranging the equation, it leads to the following formula of the dimen-sionless coefficient a as a function of K and n:
a = l K I En?-1
(9)
Substituting the mechanical properties considered here, Eq. (9) returns a = 0.9145.
Considerations deserve to be spent also on the definition of the limit load P0. It is usually defined by a limit load solution for the geometry of interest, and normally corresponds to the load at which the net cross section yields. In general, it is defined as:
P0 = CTy,nom Aet, (10)
where CTy,nom is the stress at the limit condition. Considering now the von Mises equivalent stress for fully plastic and plane strain conditions and remembering the relations within stress components, one obtains:
aeq =
4
ay,nom + ay,nom °'5ay,nom
= ay
V3
y,nom 2
From the latter equation, it is possible to express CTy,nom as a function of CTe
eq •
(12)
"y.nom ^ eq'
Considering the definition of P0, once reached the yield condition, CTeq must equal ct0:
r 2
CTy,nom = CTy,nom (ct0) = CT0- (13)
Substituting Eq. (13) into Eq. (10), the final expression for P0 can be obtained:
P = CT0 Anet. (14)
Parameter hj is a function of a/W and n. It is catalogued for several values of a/W and n under plain strain and plain stress conditions [28]. However, in order to obtain the correct hj for a not listed half-crack length plate width ratio and hardening exponent, an extrapolation/interpolation process based on the available curves is necessary: first, by interpolation, the value of hj for a n = 8.33 is determined; subsequently, fixed the value of n = 8.33, the value of hj for the desired value of a/W is determined by extrapolation process. The equations obtained through interpolation/extrapolation techniques are those reported in Table 2 of the present paper. The resulted value of the dimensionless parameter hj for the considered geometry and mechanical properties is 4.456.
4. Numerical analysis
The geometry shown in Fig. 1 is modeled in ANSYS APDL. A multilinear isotropic hardening model is chosen. This model requires (like other nonlinear models) the definition of the stress-strain curve point by point. As the model suggests, the resulting curve will be composed of many linear segments connecting the points defined. The multilinear isotropic hardening option can contain up to 20 different temperature curves, with up to 100 different stress-strain points allowed per curve. For this reason, the points have been distributed in order to have a better trend definition of the transition zone, near the yielding stress.
Only one quarter of the plate has been modeled, imposing symmetry BC structural constraints. Element Plane 183 (ANSYS V.14) with plane strain option is used. The applied nominal load is 360 MPa. As general rule, the generalized yielding condition is verified at least for a load equal to 0.8ay, which is equal to 360 MPa for the considered material. A very refined mesh is modeled near the crack tip; with the smallest element size of 10-4 mm.
The finite element analysis can be conducted following three methods that have been investigated in the present paper:
Table 2
Solution y = bx6 + cx5 + dx4 + kx3 + lx2 + mx + n
b c d k l m n
y = hi, x = n *
a/W = 0.125
-1.8136-10-6 1.2447-10-4 -3.4256-10-3 4.9069-10-2 -3.9982-10-1 1.7118 1.4433
a/W = 0.25
- 1.27735-10-5 -7.53207-10-4 1.70327-10-2 -1.85014-10-1 8.78505-10-1 1.84269
a/W = 0.375
-1.21519-10-6 8.51464-10-5 -2.37417-10-3 3.35170-10-2 -2.49108-10-1 8.05512-10-1 1.75703
ajW = 0.5
-1.39740-10-6 9.12734-10-5 -2.33391 -10-3 2.95003-10-2 -1.87929-10-1 4.50323-10-1 1.92481
ajW = 0.625
-5.43849-10-7 3.38873-10-5 -8.09214-10-4 9.13681 -10-3 -4.32795-10-2 -9.46892-10-2 2.25234
a/W = 0.75
-2.70960-10-8 -2.32908-10-6 2.05096-10-4 -5.38089-10-3 6.84170-10-2 -5.12603-10-1 2.52013
ajW = 0.875
-2.70960-10-8 -2.32908-10-6 2.05096-10-4 -5.38089-10-3 6.84170-10-2 -5.12603-10-1 2.52013
y = h1, x = a/W **
a/W = 0.1
- - - - 7.02819 -11.5034 5.63628
* It must be underlined that the previous equations are valid only for interpolation process and not for extrapolation; the valid n-range is 1-20.
** Fixed the desired value of n = 8.33, this equation is used to determine the desired value of hx.
(i) Under large scale yielding, it could be assumed that the elastic contribution of the Ramberg-Osgood can be neglected. This assumption leads to the conclusion that the J-integral of the Ramberg-Osgood law has to be close to that obtained considering the simple power law.
(ii) The J-integral of Ramberg-Osgood law can be obtained by simply modelling the complete curve (elastic and
plastic contribution) and carrying out a single nonlinear analysis.
(iii) Superposition principle. The total J of the Ramberg-Osgood law can be obtained performing two finite element models: a linear elastic analysis, in order to determine the elastic J, and a nonlinear analysis considering only the power law in order to determine the plastic contribution of J.
Fig. 5. Comparison within elastic portion of the Ramberg-Osgood Fig. 4. Power law fictitious linear portion (1) and power law (2)
2000
200-1—.........—.........—.........—.........—........
0.0001 0.001 0.01 0.1 1 10 Tip distance x, mm
Fig. 6. Influence of the fictitious elastic portion on the stress distribution. CTmax = 1209 (o), 1066 (o), 927 (a), 862 (□), 861 MPa (+), end of the elastic portion 600 (o), 500 (o), 400 (a), 300 (□), 200 MPa (+)
4.1. Power law convergence problem
The power law constitutive model has an infinity slope in the first portion of the curve, as shown in Fig. 3. The high slope may result in ANSYS code solution convergence problems. In order to avoid this issue, a fictitious linear trend is imposed. This fictitious trend is shown in Fig. 4 and compared with the Ramberg-Osgood linear portion in Fig. 5. It is clear that the linear trend is very small and does not affect the final results. On the other hand, the improvement of the solution convergence is appreciated. To better support these last comments, Fig. 6 shows the influence of the length of the fictitious power low elastic portion on the stress distribution, for an applied constant load. Every curve in the picture depicts the stress distribution ahead the crack for different length of the fictitious linear trend. The first curve for example has the end of the linear elastic portion at 600 MPa. This means that the fictitious elastic portion starts from the value of zero and ends at 600 MPa, that is in turn where the plastic portion starts. If the final point of the elastic portion is under the value of 400 MPa, no differences emerge from the stress analysis. While for higher loads, the stress distribution is strongly influenced. In the present paper, the point of the elastic portion is settled at 200 MPa. In general it seems that, as a good rule, the ratio within the value of the end point of the fictitious linear portion and the yielding stress should be lower than 0.6.
5. Results
5.1. Assuming negligible elastic contribution of the Ramberg-Osgood law
As stated in the previous section, under fully plastic condition, assured by the applied load of 360 MPa, it could be assumed that the elastic contribution has a negligible value if compared with the plastic contribution. As general rule, in fact, the generalized yielding condition is verified at least for a load equal to 0.8ay, which is equal to 360 MPa for the considered material. This assumption permits to approximate the total Ramberg-Osgood J with that obtained considering a simple power law. With the aim to verify this simplification, a finite element analysis has been conducted assuming the power law material model. The results are compared with the analytical ones obtained through Eq. (3).
Table 3 reports the obtained results and shows an evident difference between the analytical and finite element results; Jth is the theoretical contour J-integral obtained through Eq. (3) while Jj?EM is the plastic J obtained by a finite element analysis considering a pure power law: also with a very high applied load, the elastic contribution cannot be neglected.
5.2. Unified assessment of elastic and plastic contribution
The J-integral of Ramberg-Osgood law can be obtained by simply modeling the complete curve (elastic and plastic contribution) and carrying out a single nonlinear analysis. In this case, the elastic contribution is no longer neglected, and no simplifications are introduced.
Table 4 reports the obtained results and shows a percentage difference between theoretical Jth and numerical value (considering the Ramberg-Osgood material law) jfem,ro equal to 10%.
5.3. Evaluation of J through superposition principle
The total J of the Ramberg-Osgood law can be obtained performing two finite element calculations: a linear elastic analysis, in order to determine the elastic J, and a nonlinear analysis considering only a pure power law in order to determine the plastic contribution of J. In other words, Dowling's approximation [33] is respected also through the finite element analysis, taking advantage of a superposition effect.
Table 5 reports the obtained results and shows a very good agreement between numerical and analytical values,
Table 3
Comparison between J-integral values, applied load 360 MPa
kJ/m2 JFEM, kJ/m2
22.46 4.56
Table 4
Comparison between J-integral values, applied load 360 MPa
Jh, kJ/m2 JFEM,RO ' kJ/m2 Jh,RO
22.46 24.68 10%
Table 5
Comparison between /-integral values (kJ/m2), applied load 360 MPa
J e J FEM JFEM JFEM Jth ^Jth,FEM
18.38 4.56 22.94 22.46 2%
where Jem is the / obtained by finite element linear elastic analysis; JFEM is the plastic J obtained through a nonlinear finite element analysis and considering a pure power law material model; JFEM is the arithmetic sum of the elastic and plastic JFEM and /FEM> Jth is the expected total J contour integral obtained through Eq. (3).
This procedure returns a very low value of percentage difference between the numerical and theoretical /-integral. The best solution is to calculate the J-integral for an elastic-plastic material implementing two models: one for the plastic contribution, considering a power law, and a second one for the elastic contribution considering a pure elastic material, taking advantage of superposition principle.
6. Conclusions
The J-integral has received an excellent feedback as a fracture parameter under elastic-plastic condition. Considering a crack, the literature proposes to evaluate the J-inte-gral as a sum of elastic and plastic contributions. However, some uncertainties arise when applying this method to the Ramberg-Osgood law, especially under large scale yielding.
The aim of the present paper is to discuss how the J-integral evaluation can be carried out for elastic-plastic cracked components. Two different nonlinear behaviors have been considered: the Ramberg-Osgood law and power law. Theoretical and numerical results of different approaches have been compared. The following main results can be identified.
In terms of/-integral, the constitutive models return very different results. It seems that the elastic contribution of the Ramberg-Osgood law cannot be neglected.
The results support Dowling's approximation: / = = Je + Jp.
The Ramberg-Osgood material law could be considered as a sum of plastic and elastic contribution. Following this methodology, the calculation of the theoretical J-inte-gral for an elastic-plastic materials can be conducted by considering independently the elastic and plastic contribution, as it is shown by Eq. (3) and by the given references.
Also under fully plasticity, the elastic contribution cannot be neglected by the analysis. The best procedure to determine the correct value of the total J-integral for an elastic-plastic material is to implement two nonlinear models: one for the plastic contribution, considering a power law, and a second one for the elastic contribution considering a
pure elastic material. The sum of these values corresponds perfectly to the expected value of /-integral.
For the theoretical evaluation for J-integral, interpolation/extrapolation procedure is proposed for the estimation of the dimensionless parameter hv
The parameter a is given as a function of the mechanical properties K, n and yielding stress, allowing and easy and fast evaluation based on the common mechanical properties of the Ramberg-Osgood and power law material.
More extensive analysis has been conducted to support the conclusions. Different loading condition and mechanical properties have been considered but for the sake of brevity are here omitted. All the detailed results will be presented in a future work, which the authors are currently drawing up.
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nocTynnna b pe^aKUHro 06.07.2015 r.
CeedeHun 06 aemopax
Pasquale Gallo, Dr. Eng., University of Padua, Italy, pgallo@gest.unipd.it Filippo Berto, Prof., University of Padua, Italy, berto@gest.unipd.it
