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УДК 539.421
Оценка разрушения образцов из бейнитных функционально-градиентных сталей с надрезом при нагружении смешанного типа (I + II)
H. Salavati1, Y. Alizadeh2, F. Berto3
1 Университет Шахида Бахонара, Керман, Иран 2 Технологический университет Амир-Кабир, Тегеран, 15875-4413, Иран 3 Падуанский университет, Виченца, 36100, Италия
Среднее значение плотности энергии деформации по объему служит эффективным критерием оценки статической прочности образцов с U- и V-образным надрезом. В статье впервые исследовано влияние параметров надреза (радиус, глубина и угол раскрытия надреза) на разрушение образцов из бейнитных функционально-градиентных сталей с тупым V-образным надрезом при нагружении смешанного типа (I + II). С помощью численного метода определена граница контрольного объема, вычислены среднее значение плотности энергии деформации, а также критическая разрушающая нагрузка. Рассмотрены разные значения радиуса надреза (0.5, 1.0, 1.5 и 2.0 мм), глубины надреза (5.5, 6.0 и 6.5 мм), угла раскрытия надреза (30°, 60° и 90°) и расстояния от места приложения нагрузки до биссектрисы надреза (5 и 10 мм). Для функционально-градиентных сталей показана возможность применения грубой расчетной сетки для вычисления среднего значения плотности энергии деформации в контрольном объеме.
Ключевые слова: функционально-градиентная сталь, плотность энергии деформации, критическая разрушающая нагрузка, радиус надреза, глубина надреза, угол раскрытия надреза
Fracture assessment of notched bainitic functionally graded steels under mixed mode (I + II) loading
H. Salavati1, Y. Alizadeh2, and F. Berto3
1 Shahid Bahonar University of Kerman, Kerman, Iran 2 Amirkabir University of Technology, 15875-4413, Tehran, Iran 3 University of Padova, Vicenza, 36100, Italy
The averaged value of the strain-energy density over a well-defined volume is one of the powerful criteria to assess the static strength of U- and V-notched specimens. This contribution is the first to investigate the effect of notch parameters (notch radius, notch depth and notch opening angle) for fracture assessment of specimens weakened by blunt V-notches made of bainitic functionally graded steels under mixed mode loading (I + II). A numerical method has been used to evaluate the boundary of the control volume, the mean value of the strain-energy density and the critical fracture load. Different values of the notch radius (0.5, 1.0, 1.5 and 2.0 mm), notch depth (5.5, 6.0 and 6.5 mm), notch opening angle (30°, 60° and 90°) and distance of the applied load from the notch bisector line (5 and 10 mm) have been considered. Moreover, this contribution shows that the mean value of the strain-energy density over the control volume can also be accurately determined from a coarse mesh for functionally graded steels.
Keywords: functionally graded steel, strain energy density, critical fracture load, notch radius, notch depth, notch opening angle
1. Introduction
Several criteria have been proposed by many researchers for the failure assessment of notched components. The failure criterion proposed by Novozhilov [1] and developed by Seweryn [2] called as theory of critical distances suggests that failure occurs when the average normal stress along the characteristic length scale denoted by d0 equals a material dependent stress at failure without the presence of a notch. The successful application of theory of critical
© Salavati H., Alizadeh Y., Berto F., 2015
distances on the notched components is widely investigated in [3-5]. Leguillon [6, 7] proposed a criterion for the failure initiation at a sharp V-notch based on a combination of the Griffith energy criterion for a crack, and the strength criterion for a straight edge.
Neuber [8] first suggested the idea of linking the stress averaging to the fictitious notch rounding approach and other researchers investigated the influence of plane stress and plane strain conditions on the application of this approach
and in particular on the calculation of the multiaxiality factors [9-12].
Marsavina et al. [13] investigated the dynamic and static fracture toughness of polyurethane rigid foams and in another work [14], four fracture criteria (maximum circumferential tensile stress, minimum strain energy density, maximum energy release rate, equivalent stress intensity factor) were applied to evaluate the mixed mode fracture of polyurethane foams using asymmetric semi-circular specimens. In particular, the authors showed that the equivalent stress intensity factor criterion predicted well the mixed mode fracture more precisely. In another work [15], the theory of critical distances was applied to investigate the fracture properties and notch effect of polyurethane materials. In that work, a wide range of mixed modes from pure mode I to pure mode II was considered by changing only the position of one support.
The other worth mentioning approach is based on the cohesive zone model. The major advantages of cohesive zone model over the conventional methods in fracture mechanics like those including linear elastic fracture mechanics, crack tip open displacement, etc. is that it is able to adequately predict the behaviour of uncracked structures, including those with blunt notches. Moreover, the size of the nonlinear zone has not to be negligible in comparison with other dimensions of the cracked geometry in cohesive zone model. This approach was first proposed for concrete and later successfully extended to brittle or quasi-brittle failure of a large bulk of materials [16-21] and in particular polymethylmethacrylate specimens tested at room and low temperature [16, 20]. In those works both sharp and blunt U- and V-notches were considered.
A review on the coupled modes in the V-notched plates of finite thickness and describing the contribution of these modes into the overall stress state in the close vicinity of the notch tip has been done by Berto [22]. Berto et al. [23] investigated the effect of free-fixed boundary conditions along the notch edges in three dimensional plates weakened by pointed V-notches and quantified the intensity of the out-of-plane singularity occurring under this constraint configuration. They also showed the capability of the strain energy density approach to capture all the combined effects due to the out-of-plane mode make.
A recent approach, based on the strain energy density, reminding the well-known strain-energy density criterion by Sih and Macdonald [24] has been proposed to assess the static and fatigue strength of notched specimens [25, 26]. This approach is based on the averaged value of the strain-
energy density over a well-defined volume and has been applied to assess the fracture behavior of different materials under mode I, mixed mode and torsion loading [2734]. The strain-energy density criterion can be applied to assess the fatigue strength of welded joints and notched components as shown in Refs. [35-39]. Moreover, the strain-energy density is able to take into account 3D effects through the plate thickness and different boundary conditions [40]. An important advantage of the approach with respect to the stress based criteria is the possibility to use coarse meshes for the direct evaluation of the strain-energy density in a control volume [41].
A complete review of the strain-energy density approach applied to V-notches and welded structures have been done in [42, 43].
Functionally graded materials may be characterized by the variation in composition and structure gradually over volume, resulting in corresponding changes in the properties of the material. The materials can be designed for specific function and applications. Various approaches based on the bulk (particulate processing), preform processing, layer processing and melt processing are used to fabricate the functionally graded materials. In this regard, well-known metal-ceramic functionally graded materials are usually used to enhance the properties of thermal-barrier systems. In fact, functionally graded materials may show unexpected properties, which differ from those of single ingredient materials and also from those of traditional composite materials having the same mean composition [44].
Functionally graded steels are one of the main group of functionally graded materials with elastic-plastic behavior which have recently been produced from austenitic stainless steel and carbon steel using electroslag refining method [45, 46]. As well discussed in [40], by selecting the appropriate arrangement and thickness of the primary ferritic and austenitic steel electrodes, it is possible to obtain composites with several layers consisting of ferrite, austenite, bainite and martensite. The tensile behavior of functionally graded steels with different configurations has been modeled by using the modified rule of mixture and was also investigated experimentally [46]. In other works, the mechanism-based strain gradient plasticity theory was applied to investigate the tensile strength as well as the Vickers micro-hardness profile of functionally graded steels [47, 48].
Other important aspects are related to the characterization of functionally graded steels under hot working conditions. The flow stress assessment of functionally graded steels under hot compression loading is well discussed in
Table 1
Chemical composition of original ferritic and austenitic steels
C, % Ni, % Cr, % Mo, % Cu, % Si, % Mn, % S, % P, %
AISI 1020 (y) 0.01 9.58 16.69 1.89 0.43 0.53 1.5 0.04 0.04
AISI 316L (a) 0.11 0.07 0.12 0.02 0.29 0.19 0.63 0.08 0.01
Table 2
Mechanical properties of single phase steels present in the considered functionally graded steels [46, 58]
Single phase Yield strength, MPa Ultimate strength, MPa KIC, MPa-m05 Poisson's ratio v Elasticity module, GPa
Ferritic 245 425 45.72 0.33 207
Austenitic 207 0.33 107.77 480 200
Bainitic 207 0.33 82.08 1125 1025
Refs. [49, 50] by applying the constitutive equations in combination with the rule of mixture. In another work [51], the mechanism-based strain gradient plasticity theory is applied to model the hot deformation behavior of functionally graded steels.
The Charpy impact energy of functionally graded steel in the crack divider configuration has been investigated and is well highlighted in Refs. [52-54] while for crack arrester configuration one can refer to [55, 56].
The brittle or quasi-brittle static failure of martensitic functionally graded steels was studied by Barati et al. [57] and the bainitic one was studied by Salavati et al. [58]. In these works, the strain-energy density approach over the control volume which has been obtained by a numerical method was applied to predict the critical facture load. In both these studies, the Young's modulus and the Poisson's ratio have been assumed to be constant, while the ultimate tensile strength Gut and the fracture toughness KIC have been varied exponentially through the specimen width. In a recent work, Salavati et al. [59] trained an artificial neural network to obtain a new simple model to predict the critical fracture load of functionally graded steel. The output of the this model sounds a good agreement with the experimental and finite element data.
In the present paper, a bainitic functionally graded steel made of initial ferritic and austenitic electrodes has been studied. The flow (yield/ultimate) strength of the graded regions has been obtained by using the mechanism-based theory of strain gradient plasticity while the fracture toughness has been found to vary exponentially along the notch depth. The main aim of the present work is to study the effect of different notch parameters (notch root radius, notch depth and notch opening angle) as well as different mode mixity on the mean value of stain energy density and critical fracture load.
2. Experimental procedures
The functionally graded steels considered in the present investigation were obtained from ingots made of ferritic AISI 1020 and austenitic AISI 316L steels. The diameter of the ingots was equal to 45 mm. The chemical composition of the two materials is summarised in Table 1. A single bar of austenitic steel, with an initial length of 210 mm, and a single bar of ferritic steel with an initial length of 150 mm spot-welded together were used to produce the two-piece
electrode for the fabrication of bainitic specimens. The mechanical properties of single phase steels present in the considered functionally graded steels are summarized in Table 2.
During the fabrication process, the electrodes were welded to a steel bar and successively inserted vertically in the electroslag refining furnace. The furnace contained a 70x70 mm2 squared copper mould.
As first step of the process, some slag was inserted in the starter and melt by activating the electrical power. The liquid, generated by the starter during the melting process, filled the plate cavity. Afterwards, other slag was continuously added to the mould with the aim to increase the electrical resistance. The slag was made of 70% fluoride-calcium and 30% aluminium oxide for a total weight of 1.5 kg. The final ingots were 60 mm height. By means of a hot hydraulic press (at 980°C), the height of the ingots was reduced to 22 mm. The final height of 18 mm was reached by grinding.
For metallographic examinations, the plates were sliced, ground, polished, and etched in a "Kalling" solution and 1 pct "Nital".
Vickers hardness tests were carried out using 50 kgf weight. Figure 1 shows the Vickers microhardness profile of the aPy composite.
The aPy specimens drawn from the ingots were characterized by 90 mm in length, 18 mm in width and 9 mm in the thickness direction. The geometry is in agreement with the main standard in force (ASTM E1820) [60] for the crack arrester configuration.
To measure the critical fracture load, three-point bending load was used. The load was applied normally to the in-
Fig. 1. Vickers microhardness profile versus depth in aPy composite
F\
Bainitic layer
Austenitic region \
Hardness profile
\
1
Ferritic region
7T
7T
J
Fig. 2. Geometry of the specimen under three-point bending in the crack arrester configuration (S = 72 mm, W = 18 mm)
terface layers at a distance b from the notch bisector line in order to obtain a mixed mode loading condition (Fig. 2). The tests were performed by a Zwick 1494 testing machine under load displacement control with constant displacement-rate of 1 mm/min. The load-displacement curves were recorded and used to obtain the critical fracture load.
3. Mechanical properties
Regarding to the Vickers microhardness profile of the aPY composite (Fig. 1), the thickness of the bainitic layer is approximately equal to 2 mm. Moreover, the thickness of a and Y graded regions are 2.5 and 3.5 mm, respectively. In addition, the thickness of the original ferritic a and the original austenitic layers Y are 4.5 mm and 5.5 mm, respectively.
In order to model flow (yield/ultimate) strength as well as the fracture toughness KIC of the considered aPY composite, the graded regions (a and y) have been divided into different layers. The number of layers in a region is ma and the number of layers in y region is mY. The flow (yield/ ultimate) strength of the constituent elements in a region changes from the original a0 steel on one side, to the bainitic layer on the other side. Similarly, the flow (yield/ ultimate) strength of the constituent elements in y region varies from the bainitic layer on one side to the original y 0 steel on the other side. In addition, the mechanism-based theory of strain gradient plasticity was applied to obtain the flow (yield/ultimate) strength of each element in both a and Y regions.
According to mechanism-based theory of strain gradient plasticity, the flow (yield/ultimate) strength at each point of the graded regions (a and y regions) is related to the density of the statistically stored dislocations of that point according to the following equation [48]:
(CTfiow )i =V3aG&V (pfow)i, (1)
where (^flow)i indicates the flow (yield/ultimate) strength of a generic point i inside the material, G is the elastic shear modulus (equal to 80 GPa for steel specimens), a is an empirical coefficient ranging between 0.2 and 0.5 and considered here equal to 0.3, b is the Burgers vector which is equal to a0V2/2 for fcc crystals (such as an austenitic steel) and a0 V3/2 for bcc crystals (such as a ferritic steel). The parameter a0 is the lattice parameter and is equal to 4-K/V2 for
fcc crystals and 4_r/V3 for bcc crystals whereas R is the atomic radius which is equal to 0.127 nm for iron. The dislocation density around the point i is indicated as (pfow ). The hardening mechanisms which contribute to the gradual variation of the hardness profile in a certain layer, such as the solid solution hardening and the hardening effect due to an increase of the density of the statistically stored dislocations, may be considered as a pseudodislocation density p* [48]. Therefore, instead of analyzing a functionally graded composite in which the strengthening effect of each element has been obtained by means of the solid solution mechanism, a functionally graded composite with gradual density of dislocation having the same strengthening effect has been analyzed. Therefore, the flow strength of each layer is determined by expression (1). In agreement with Ref. [48], psflow has been considered to vary exponentially in the graded regions. Therefore,
*flow
( * )(a) = (if™)
Aa)
PSfl0W( x )( y) = (Psfl0W)p ^0 -
(p;flow)p \p:flOW)a0
(P*f,Ow)p
(2)
where xi is the position of each element in the graded regions and xa0, xam, xym and xy0 are the position of the boundary layers. The pseudodislocation density for the boundary layers could be expressed as:
t \ l2 (aflow)a0
(p; )a0 =
(p;flOw)p =
(p:flow)Y0=
SaGb
(aflow)P
SaGb
(aflow)y0
V3aGb
(3)
where (°flow)a0,(CTflow)p and (^flow)y0 are the flow
strength of the boundary layers as shown in Table 2.
Similarity to pseudodislocation density, the fracture toughness KIC of each element has also been assumed to obey the exponential law reported below:
-ln-
( Kic)p
KIC (xi )( a) = (KIC)a0 e
Xi —Xym
(Kic)(x-U) = (Kick) ^0"x'm
am xa 0 ( K IC)a0
lniKIC)(y 0)
(K IC)(P )
(4)
where (KIC)a0, (KIC)(p) and (Kic)y0 are the fracture toughness corresponding to a0, P and Y0, respectively as shown in Table 2.
4. Strain energy density over the control volume under mixed mode loading
4.1. Control volume for functionally graded steel
The averaged strain-energy density criterion as reported in [25] states that brittle or quasi-brittle failure occurs when the mean value of the strain energy density over a control
\ ° * /
JPA
Crack initiation angle
Fig. 3. Control volume for homogenous materials, mode I loading (a) and mixed mode loading (b)
volume is equal to the critical energy for the unnotched material, Wc. The strain-energy density approach is based both on a precise definition of the control volume and the fact that the critical energy does not depend on the notch sharpness. Such a method was applied first to sharp, zero radius, V-notches and later extended to blunt U- and V-notches under mode I loading [26].
The control volume in blunt V-notched specimens under mode I loading conditions is centered in relation to the notch bisector line (Fig. 3, a). Under mixed mode loading the critical volume is no longer centered on the notch tip, but rather on the point where the principal stress reaches its maximum value along the edge of the notch (Fig. 3, b). It is assumed that the crescent shape volume rotates rigidly under mixed mode, with no change in shape and size [27].
Under plane strain conditions, Yosibash et al. [61] derived the expression of the control radius Rc as a function of fracture toughness KIC, ultimate tensile strength aut, and Poisson's ratio v of the material:
& =
(1 + v)(5 -8 v)
4 n
K
IC
(5)
In homogeneous materials, Rc is constant all over the specimen. However, in nonhomogeneous materials, Rc varies from point by point because of the material gradient. In a nonhomogeneous medium with a smooth unidirectional variation in mechanical properties in the x-direction (along the notch depth), the outer boundary of the control volume assumes elliptic shape.
In the present work, a numerical approach has been used to obtain the boundary of the control volume. In the numerical approach, firstly X0, the coordinate of the crack initiation point as shown in Fig. 4, has to be determined by solving the following equation numerically:
Xo -(a-p) = (+p)cos
(6)
RCn (control radius at the crack initiation point) can be calculated as:
(1 + v)(5-8v) 2
£
4n
K IC ( Xo)
°ut( Xo)
(7)
The intersection of control volume boundary and of the notch boundary (the angles 0* and shown in Fig. 4) has been obtained by some close-form solutions for homog-
Fig. 4. Notch depth and X-Y Cartesian coordinate system, X0 the coordinate of the crack initiation point
Fig. 5. Boundary discretization of the control volume
Table 3
Crack initiation angle 9° of the specimens
a, mm b, mm p, mm 2a 9 a, mm b, mm p, mm 2a 9
5.5 5 0.5 30° 3.99° 5.5 10 0.5 60° 6.41°
5.5 5 1.0 30° 7.92° 5.5 10 1.0 60° 9.17°
5.5 5 1.5 30° 7.89° 5.5 10 1.5 60° 9.47°
5.5 5 2.0 30° 7.68° 5.5 10 2.0 60° 9.37°
6.0 5 0.5 30° 7.93° 6.0 10 0.5 60° 6.41°
6.0 5 1.0 30° 7.92° 6.0 10 1.0 60° 9.58°
6.0 5 1.5 30° 7.91° 6.0 10 1.5 60° 9.45°
6.0 5 2.0 30° 3.85° 6.0 10 2.0 60° 9.37°
6.5 5 0.5 30° 7.93° 6.5 10 0.5 60° 6.41°
6.5 5 1.0 30° 7.92° 6.5 10 1.0 60° 9.58°
6.5 5 1.5 30° 8.00° 6.5 10 1.5 60° 9.49°
6.5 5 2.0 30° 7.68° 6.5 10 2.0 60° 11.40°
5.5 10 0.5 30° 7.93° 5.5 5 0.5 90° 4.74°
5.5 10 1.0 30° 7.92° 5.5 5 1.0 90° 4.73°
5.5 10 1.5 30° 8.01° 5.5 5 1.5 90° 4.72°
5.5 10 2.0 30° 11.51° 5.5 5 2.0 90° 6.85°
6.0 10 0.5 30° 7.93° 6.0 5 0.5 90° 4.74°
6.0 10 1.0 30° 7.92° 6.0 5 1.0 90° 4.73°
6.0 10 1.5 30° 12.09° 6.0 5 1.5 90° 7.11°
6.0 10 2.0 30° 11.51° 6.0 5 2.0 90° 6.85°
6.5 10 0.5 30° 7.93° 6.5 5 0.5 90° 4.74°
6.5 10 1.0 30° 11.86° 6.5 5 1.0 90° 4.73°
6.5 10 1.5 30° 11.83° 6.5 5 1.5 90° 7.11°
6.5 10 2.0 30° 11.51° 6.5 5 2.0 90° 6.85°
5.5 5 0.5 60° 6.41° 5.5 10 0.5 90° 4.74°
5.5 5 1.0 60° 6.40° 5.5 10 1.0 90° 7.12°
5.5 5 1.5 60° 6.39° 5.5 10 1.5 90° 7.11°
5.5 5 2.0 60° 6.26° 5.5 10 2.0 90° 9.15°
6.0 5 0.5 60° 6.41° 6.0 10 0.5 90° 4.74°
6.0 5 1.0 60° 6.40° 6.0 10 1.0 90° 7.12°
6.0 5 1.5 60° 6.21° 6.0 10 1.5 90° 9.50°
6.0 5 2.0 60° 6.26° 6.0 10 2.0 90° 9.15°
6.5 5 0.5 60° 6.41° 6.5 10 0.5 90° 4.74°
6.5 5 1.0 60° 6.20° 6.5 10 1.0 90° 7.12°
6.5 5 1.5 60° 6.37° 6.5 10 1.5 90° 9.50°
6.5 5 2.0 60° 6.26° 6.5 10 2.0 90° 9.15°
enous steel [57] but there is not any analytical equation to obtain the angles 6* and 62 for nonhomogeneous steels. Therefore, in the current numerical approach, the angle 6 increases from zero until the control volume boundary and
the notch boundary reach their intersection. In addition, in the present work, the outer boundary of the control volume is discretized in n1 +1 points where n1 is the integer part of the angle 6*, n2 +1 points where n2 is the integer
part of the angle 9 and n3 +1 points where n3 is the integerpart of the angle (02 -9) (Fig. 5).
For example, X and Y coordinate of point i (9 < 0i <0* ) can be calculated as follows:
Xi =a -p + (Rc (Xi ) + ro)cos(0 + 9) +
+ (p- ^o)cos 9, Y = ( Rc( Xt ) + ro)sin(9 + i) + (p- ^o) sin 9,
where Rc( Xi ) is calculated as follow: (1 + v)(5 -8 v)
(8)
Rc( X- ) =-
4n
3 K ic ( X )42
MY )
(9)
Subscript i is related to the ith layer in the graded region. It is clear that the coordinates of point i in x—y coordinate system are:
x, = X - (a - re), yi = Y. (10)
Finally, the values of R and ^ can be calculated as follows:
R = Vxf + yf >
^ = tan
3 „ 4
xi i
(11)
(12)
The above procedure should be repeated for all points to obtain R value as a function of T.
4.2. Crack initiation angle and notch stress intensity factor
In this work, three different values of the notch depth (5.5, 6.0, 6.5 mm), four values of the notch radius (0.5, 1.0,
Fig. 6. Maximum principal stress contour lines
1.5 and 2.0 mm) and three values of the notch opening angle (30°, 60° and 90°) have been considered. To achieve different mixed mode loading, two values of the length b (5 and 10 mm) have been selected. For each geometrical, finite element code ABAQUS 6.13 has been applied in order to determine the point where the maximum principal stress was located. All the analyses have been carried out by means of triangular elements under plane strain conditions and linear elastic hypotheses. Figure 6 shows the maximum principal stress for the configuration with p = 1.5 mm, a = 6 mm, 2 a = 30° and b = 10 mm. The values of the crack initiation angle 9 are summarized in Table 3. Moreover, for the experimental tests, the crack initiation angles have been measured by using a high resolution camera and a dedicate software and the results are summarized in Table 4. As can be seen in the table, there is a sound agreement between the experimental values of the crack initiation angles and the finite-element mechanics results.
Table 4
Comparison between the finite-element mechanics and experimental values of the crack initiation angle
a, mm b, mm p, mm 2a 9exp 9FEM
5.5 5 1.0 60° 5.50° 6.40°
5.5 5 1.0 30° 11.22° 7.92°
6.5 5 1.0 30° 8.49° 7.92°
6.5 5 1.0 60° 6.77° 6.20°
6.5 5 1.5 60° 8.98° 6.37°
6.5 5 1.5 30° 8.97° 8.00°
5.5 5 1.5 30° 6.04° 7.89°
5.5 5 1.5 60° 10.74° 6.39°
5.5 10 1.0 60° 9.90° 9.17°
5.5 10 1.0 30° 8.15° 7.92°
6.5 10 1.0 30° 9.58° 11.86°
6.5 10 1.0 60° 11.90° 9.58°
6.5 10 1.5 60° 9.69° 9.49°
6.5 10 1.5 30° 9.65° 11.83°
5.5 10 1.5 30° 11.85° 8.01°
5.5 10 1.5 60° 8.79° 9.47°
Table 5
Notch stress intensity factors and mode mixity of the specimens
a, mm b, mm p, mm 2a Kvpl, MPa • mm1-1 Kp n, MPa • mm1-12 Kp, „/Kpv,„ mm^-12
5.5 5 0.5 30° 4.2760 -0.560 0.1309
5.5 5 1.0 30° 4.3695 -0.604 0.1382
5.5 5 1.5 30° 4.4643 -0.694 0.1555
5.5 5 2.0 30° 4.5556 -0.780 0.1713
6.0 5 0.5 30° 4.5578 -0.609 0.1337
6.0 5 1.0 30° 4.6588 -0.697 0.1496
6.0 5 1.5 30° 4.7683 -0.766 0.1606
6.0 5 2.0 30° 4.8491 -0.842 0.1736
6.5 5 0.5 30° 4.8912 -0.664 0.1358
6.5 5 1.0 30° 4.9572 -0.732 0.1476
6.5 5 1.5 30° 5.0805 -0.822 0.1617
6.5 5 2.0 30° 5.1838 -0.918 0.1771
5.5 10 0.5 30° 3.6071 -0.663 0.1839
5.5 10 1.0 30° 3.6759 -0.723 0.1968
5.5 10 1.5 30° 3.7538 -0.811 0.2161
5.5 10 2.0 30° 3.8281 -0.898 0.2347
6.0 10 0.5 30° 3.8381 -0.717 0.1868
6.0 10 1.0 30° 3.9163 -0.800 0.2044
6.0 10 1.5 30° 4.0041 -0.875 0.2186
6.0 10 2.0 30° 4.0664 -0.957 0.2352
6.5 10 0.5 30° 4.1134 -0.766 0.1861
6.5 10 1.0 30° 4.1578 -0.843 0.2028
6.5 10 1.5 30° 4.2585 -0.926 0.2174
6.5 10 2.0 30° 4.3389 -1.019 0.2348
5.5 5 0.5 60° 4.3557 -0.722 0.1658
5.5 5 1.0 60° 4.4220 -0.718 0.1625
5.5 5 1.5 60° 4.5119 -0.754 0.1672
5.5 5 2.0 60° 4.5810 -0.824 0.1798
6.0 5 0.5 60° 4.6336 -0.624 0.1348
6.0 5 1.0 60° 4.7209 -0.796 0.1686
6.0 5 1.5 60° 4.8119 -0.826 0.1716
6.0 5 2.0 60° 4.8813 -0.882 0.1807
6.5 5 0.5 60° 4.9574 -0.836 0.1686
6.5 5 1.0 60° 5.0392 -0.865 0.1717
6.5 5 1.5 60° 5.1285 -0.901 0.1757
6.5 5 2.0 60° 5.2198 -0.972 0.1862
5.5 10 0.5 60° 3.6729 -0.834 0.2270
5.5 10 1.0 60° 3.7214 -0.841 0.2259
5.5 10 1.5 60° 3.7940 -0.878 0.2313
5.5 10 2.0 60° 3.8491 -0.948 0.2463
Continued Table 5
a, mm b, mm p, mm 2a KpVI, MPa • mm1-xi KpV;II, MPa • mm1-x2 KpV, II / KpV, i. mmXl -X2
6.0 10 0.5 60° 3.8979 -0.753 0.1931
6.0 10 1.0 60° 3.9690 -0.914 0.2302
6.0 10 1.5 60° 4.0392 -0.947 0.2344
6.0 10 2.0 60° 4.0932 -1.004 0.2452
6.5 10 0.5 60° 4.1662 -0.947 0.2272
6.5 10 1.0 60° 4.2280 -0.976 0.2308
6.5 10 1.5 60° 4.2984 -1.011 0.2353
6.5 10 2.0 60° 4.3718 -1.074 0.2456
5.5 5 0.5 90° 4.6301 -0.573 0.1237
5.5 5 1.0 90° 4.6882 -0.599 0.1279
5.5 5 1.5 90° 4.7374 -0.684 0.1444
5.5 5 2.0 90° 4.7850 -0.702 0.1468
6.0 5 0.5 90° 4.9657 -0.739 0.1488
6.0 5 1.0 90° 4.9978 -0.686 0.1372
6.0 5 1.5 90° 5.0464 -0.726 0.1438
6.0 5 2.0 90° 5.1000 -0.754 0.1479
6.5 5 0.5 90° 5.2677 -0.713 0.1353
6.5 5 1.0 90° 5.3283 -0.713 0.1338
6.5 5 1.5 90° 5.4006 -0.791 0.1464
6.5 5 2.0 90° 5.4544 -0.822 0.1507
5.5 10 0.5 90° 3.8931 -0.727 0.1868
5.5 10 1.0 90° 3.9431 -0.712 0.1807
5.5 10 1.5 90° 3.9805 -0.786 0.1975
5.5 10 2.0 90° 4.0169 -0.804 0.2002
6.0 10 0.5 90° 4.1724 -0.837 0.2006
6.0 10 1.0 90° 4.1946 -0.788 0.1880
6.0 10 1.5 90° 4.2312 -0.824 0.1948
6.0 10 2.0 90° 4.2746 -0.85 0.1988
6.5 10 0.5 90° 4.4168 -0.811 0.1836
6.5 10 1.0 90° 4.4640 -0.811 0.1816
6.5 10 1.5 90° 4.5214 -0.875 0.1936
6.5 10 2.0 90° 4.5629 -0.898 0.1969
In order to quantify the mode mixity in the considered specimens, some finite element analysis has been carried out. In agreement with Ref. [62], the definition of generalized notch stress intensity factors for mode I and mode II is as follows:
-x, (ae )e=c
K^ = V2rcr1'
1 + ' -x2 (Tr6 )e=o
(13)
where ae and Tre are the stresses at a distance r from the local frame origin. Equations (13) is not expected to give a
constant value for notch stress intensity factors but slight variations are possible. The slightly oscillating trend ahead of the notch tip is widely investigated in [63, 64]. In order to eliminate the weak dependence on the notch tip distance, the following expressions have been defined to calculate the mean values of the generalized notch stress intensity factors [62]:
_ 1 ro+np
KV, =— J (K^r, np r0
_ 1 ro+np
KPii =— J K>n)dr,
p' np r„ p
Notch depth, mm ' ' Notch depth,'mm
Fig. 7. Critical fracture load versus notch depth for the homogenous material corresponding to the layer including the notch tip in the functionally graded steels (a) and the homogenous material corresponding to the layer including the notch tip in the functionally graded steels with the notch depth a = 5.5 mm (b). p = 0.5 (•), 1.0 (■), 1.5 (*), 2.0 mm (a)
where n is set equal to 0.2 in the present paper. The values of Kp,i and ^p,n as we^ as the mode mixity KP,n
/K pV,I
of the considered specimens are listed in Table 5.
4.3. Critical fracture load in functionally graded steel with blunt V-notch under mixed model loading
To evaluate the critical fracture load, one needs to know the local strain energy density averaged over a control volume and use the following expression:
(15)
where Fap is the applied load, Wap is the averaged strain-energy density over the control volume related to Fap, Fcr is the critical fracture load, and Wcr is the critical strain-energy density as follows:
Wc c 2 E
(16)
where Gut is the ultimate tensile strength of each layer, E is the Young's modulus. We must emphasize that the Young's modulus (E = 207 GPa) and the Poisson's ratio (v = 0.33) have been assumed to be constant along the specimen width.
The average value of the strain energy density and the critical fracture load Fcr was evaluated by the numerical simulations with an accurate definition of the control volume over where the strain energy density was averaged. All the analyses have been carried out by means of triangular elements under plane strain conditions and linear elastic hypotheses both for functionally graded steel and the specimen made of the same homogeneous material corresponding to the layer including the notch tip in the functionally graded steels. The material corresponding to the layer in-
Table 6
Experimental results of critical fracture load and comparison with the strain-energy density approach
a, mm b, mm p, mm 2a Fcerxp, kN FcfM, kN
5.5 5 1.0 60° 14.74 13.03
5.5 5 1.0 30° 12.48 11.84
6.5 5 1.0 30° 15.01 14.01
6.5 5 1.0 60° 16.27 15.78
6.5 5 1.5 60° 18.32 15.96
6.5 5 1.5 30° 15.74 14.09
5.5 5 1.5 30° 14.00 11.91
5.5 5 1.5 60° 15.48 13.18
5.5 10 1.0 60° 15.94 15.14
5.5 10 1.0 30° 13.83 13.77
6.5 10 1.0 30° 18.05 16.33
6.5 10 1.0 60° 19.97 18.35
6.5 10 1.5 60° 20.93 18.59
6.5 10 1.5 30° 18.45 16.48
5.5 10 1.5 30° 14.28 13.88
5.5 10 1.5 60° 16.68 15.35
cluding the notch tip is called the homogenous material in the following of the paper.
The variation of the critical fracture load with respect to the notch depth for the homogenous material is shown Fig. 7, a. It can be observed that the critical fracture load increases with increasing the notch depth. This is due to the fact that when the notch depth increases in the a region, the ultimate strength and the fracture toughness of the material ahead of the notch tip increase. The variation of the critical fracture load with respect to the notch depth for the homogenous steel with the mechanical properties of the layer with the notch depth a = 5.5 mm is shown in Fig. 7, b. It can be observed that the critical fracture load decreases with increasing the notch depth. This is due to the fact that all the notch depths have the same ultimate strength and fracture toughness.
4.3.1. Experimental results
The experimental values of the critical fracture load and the strain energy density output based on the finite element analysis are reported in Table 6. It may be observed from the table that the agreement is remarkable. In addition, it can be observed that the strain energy density and the critical fracture load increase with increasing the notch depth. This is due to the fact that when the notch depth increases in the
a region, the ultimate strength and the fracture toughness ahead of the notch tip increase.
4.3.2. Comparison of the strain energy density and critical fracture load between functionally graded steel and homogeneous materials
It may be interesting to compare the strain energy density and the critical fracture load between functionally graded steel and homogeneous materials. The mechanical properties of the functionally graded steel at the notch tip were considered as those of the homogeneous material.
The variation of the strain energy density with respect to the notch radius for both homogeneous and nonhomo-geneous materials is shown in Fig. 8. All the results have been obtained under the same applied load (F = 1 kN). It can be observed that the strain energy density decreases with increasing the notch radius keeping constant the notch depth and notch opening angle both for the functionally graded steel and the homogeneous material. Moreover, the strain energy density of the functionally graded steel is lower than the homogenous material.
The variation of the critical fracture load with respect to the notch radius for both homogeneous and nonhomo-geneous materials is shown in Fig. 9. All the results have
Fig. 8. Variation of the strain energy density versus the notch radius for different notch depths: 2a = 30o (a), 60o (b) and 90o (c), b = 5 mm, functionally graded steel (solid line), homogeneous materials (dotted line), a = 5.5 ( ), 6.0 ( ), 6.5 mm ( )
Fig. 9. Variation of the critical fracture load versus the notch radius for different notch depths: 2a = 30o (a), 60o (b) and 90o (c), b = = 5 mm, functionally graded steel (solid line), homogeneous materials (dotted line), a = 5.5 ( ), 6.0 ( ), 6.5 mm ( )
Table 7
Comparison of critical fracture load of functionally graded steel and the homogeneous material (b = 5 mm)
a, mm p, mm 2a Homogeneous material Fcr,kN Functionally graded steel Percentage of Fcr increasing, %
WFEM, MJ/m3 Fcr,kN
5.5 0.5 30° 11.28 0.0069 11.75 4.2
5.5 1.0 30° 11.37 0.0068 11.84 4.1
5.5 1.5 30° 11.44 0.0067 11.91 4.1
5.5 2.0 30° 11.51 0.0066 11.99 4.1
6.0 0.5 30° 11.85 0.0085 12.85 8.5
6.0 1.0 30° 11.92 0.0084 12.93 8.5
6.0 1.5 30° 12.02 0.0083 13.03 8.4
6.0 2.0 30° 12.09 0.0082 13.10 8.4
6.5 0.5 30° 12.45 0.0107 13.93 11.9
6.5 1.0 30° 12.52 0.0106 14.01 11.9
6.5 1.5 30° 12.62 0.0104 14.09 11.7
6.5 2.0 30° 12.69 0.0104 14.14 11.4
5.5 0.5 60° 11.76 0.0057 12.87 9.5
5.5 1.0 60° 11.87 0.0056 13.03 9.8
5.5 1.5 60° 11.99 0.0055 13.18 10.0
5.5 2.0 60° 12.08 0.0054 13.29 10.1
6.0 0.5 60° 12.25 0.0069 14.22 16.1
6.0 1.0 60° 12.38 0.0068 14.40 16.3
6.0 1.5 60° 12.51 0.0066 14.57 16.5
6.0 2.0 60° 12.62 0.0065 14.73 16.8
6.5 0.5 60° 12.78 0.0085 15.57 21.9
6.5 1.0 60° 12.90 0.0083 15.78 22.3
6.5 1.5 60° 13.05 0.0081 15.96 22.3
6.5 2.0 60° 13.17 0.0080 16.09 22.2
5.5 0.5 90° 12.24 0.0049 13.93 13.8
5.5 1.0 90° 12.41 0.0047 14.15 14.0
5.5 1.5 90° 12.56 0.0046 14.35 14.3
5.5 2.0 90° 12.66 0.0045 14.50 14.5
6.0 0.5 90° 12.72 0.0058 15.58 22.5
6.0 1.0 90° 12.90 0.0056 15.78 22.3
6.0 1.5 90° 13.09 0.0055 16.03 22.5
6.0 2.0 90° 13.22 0.0053 16.25 23.0
6.5 0.5 90° 13.18 0.0070 17.20 30.5
6.5 1.0 90° 13.38 0.0068 17.50 30.8
6.5 1.5 90° 13.62 0.0065 17.78 30.6
6.5 2.0 90° 13.78 0.0064 18.01 30.8
Table 8
Comparison of critical fracture load of functionally graded steel and the homogeneous material (b = 10 mm)
a, mm p, mm 2a Homogeneous material Fcr,kN Functionally graded steel Percentage of Fcr increasing, %
WFEM, MJ/m3 Fcr,kN
5.5 0.5 30° 13.17 0.0051 13.63 3.52
5.5 1.0 30° 13.29 0.0050 13.77 3.55
5.5 1.5 30° 13.41 0.0049 13.88 3.55
5.5 2.0 30° 13.51 0.0049 13.99 3.56
6.0 0.5 30° 13.86 0.0063 14.92 7.68
6.0 1.0 30° 13.97 0.0062 15.05 7.75
6.0 1.5 30° 14.11 0.0061 15.19 7.69
6.0 2.0 30° 14.22 0.0060 15.31 7.64
6.5 0.5 30° 14.60 0.0079 16.20 10.93
6.5 1.0 30° 14.71 0.0078 16.33 11.06
6.5 1.5 30° 14.83 0.0076 16.48 11.12
6.5 2.0 30° 14.97 0.0075 16.58 10.71
5.5 0.5 60° 13.69 0.0043 14.92 8.99
5.5 1.0 60° 13.86 0.0041 15.14 9.24
5.5 1.5 60° 14.02 0.0040 15.35 9.48
5.5 2.0 60° 14.16 0.0039 15.51 9.58
6.0 0.5 60° 14.29 0.0052 16.49 15.35
6.0 1.0 60° 14.47 0.0050 16.72 15.59
6.0 1.5 60° 14.64 0.0049 16.96 15.80
6.0 2.0 60° 14.81 0.0047 17.19 16.12
6.5 0.5 60° 14.90 0.0064 18.06 21.23
6.5 1.0 60° 15.11 0.0061 18.35 21.43
6.5 1.5 60° 15.30 0.0060 18.59 21.50
6.5 2.0 60° 15.48 0.0059 18.79 21.39
5.5 0.5 90° 14.25 0.0036 16.15 13.30
5.5 1.0 90° 14.50 0.0035 16.45 13.45
5.5 1.5 90° 14.69 0.0034 16.72 13.78
5.5 2.0 90° 14.83 0.0033 16.91 14.01
6.0 0.5 90° 14.80 0.0043 17.99 21.51
6.0 1.0 90° 15.06 0.0042 18.33 21.72
6.0 1.5 90° 15.33 0.0040 18.67 21.80
6.0 2.0 90° 15.50 0.0039 18.97 22.37
6.5 0.5 90° 15.36 0.0052 19.92 29.67
6.5 1.0 90° 15.64 0.0050 20.35 30.10
6.5 1.5 90° 15.94 0.0048 20.72 29.96
6.5 2.0 90° 16.17 0.0047 21.04 30.06
been obtained under the same applied load (F = 1 kN). It can be observed that the critical fracture load increases with increasing the notch radius keeping constant the notch depth and notch opening angle both for the functionally graded steel and the homogeneous material. Moreover, the critical fracture load of functionally graded steel is higher than the homogenous material because of higher material fracture toughness along the specimen width near the notch tip.
The detailed values of the critical fracture load are summarized in Tables 7 and 8. As these tables show, the critical fracture load of the functionally graded steel is larger that of the homogenous material for all cases. In addition, the percentage difference of the critical fracture load between the functionally graded steel and homogeneous material increases by increasing the notch depth. By increasing the notch depth, the ultimate strength and the material fracture toughness increase and also the control volume becomes larger along the specimen width, resulting in decreasing the value of the strain energy density and increasing the critical fracture load.
5. Effect of mesh size on the mean values of strain energy density and critical fracture load
Lazzarin and Berto [41] showed that when the material behaviour is ideally linear elastic or obeys a power hardening law, the mean value of the strain energy density over the control volume as well as the critical fracture load can be precisely determined from a coarse mesh. This contribution shows that this is true for both the functionally graded
steel and the homogenous material. It means that the mean value of the strain energy density over the control volume as well as the critical fracture load can be precisely determined from a coarse mesh for functionally graded materials. This result is of interest in the practical application of the strain-energy density approach to real components. Figure 10 shows two different kind of mesh (fine and coarse mesh) over the control volume for the homogenous material and functionally graded steel, respectively. The results of strain energy density as well as the critical fracture load for the fine and coarse mesh are summarized in Table 9 for homogenous material. These values are summarized in Table 10 for functionally graded steel. As these table show, the average deviation between the fine and coarse mesh in terms of the strain-energy density have been found to be 1.006-10-7% and 7.35 • 10-9% for the functionally graded steel and the homogenous material, respectively. In addition, the average deviation between the fine and coarse mesh in terms of the critical fracture load have been found to be 0.00082% and 0.000025% for the functionally graded steel and the homogenous material, respectively.
6. Conclusions
In the present work, the average value of strain energy density over a well-defined control volume ahead of the notch tip was used to obtain the critical fracture load of V-notched specimens made of functionally graded steels under mixed mode loading. The main findings of the present work can be summarised as follows.
Fig. 10. Fine (a, c) and coarse (b, d ) meshes for the homogeneous material (a, b) and the functionally graded steel (c, d )
Table 9
Comparison of strain energy density and critical fracture load in two cases of fine and coarse mesh for the homogenous material
Notch geometry Strain energy density, MJ/m3 Fcr,kN
a, mm b, mm p, mm 2a Fine mesh Coarse mesh Fine mesh Coarse mesh
5.5 5 0.5 30° 0.00747 0.00747 11.28 11.28
5.5 5 1.0 30° 0.00735 0.00736 11.37 11.36
5.5 5 1.5 30° 0.00727 0.00726 11.44 11.44
5.5 5 2.0 30° 0.00717 0.00717 11.51 11.51
6.0 5 0.5 30° 0.00999 0.01000 11.85 11.84
6.0 5 1.0 30° 0.00988 0.00988 11.92 11.92
6.0 5 1.5 30° 0.00971 0.00973 12.02 12.01
6.0 5 2.0 30° 0.00960 0.00960 12.09 12.09
6.5 5 0.5 30° 0.01337 0.01338 12.45 12.44
6.5 5 1.0 30° 0.01321 0.01322 12.52 12.51
6.5 5 1.5 30° 0.01301 0.01303 12.62 12.61
6.5 5 2.0 30° 0.01285 0.01287 12.69 12.69
5.5 5 0.5 60° 0.00688 0.00689 11.76 11.75
5.5 5 1.0 60° 0.00674 0.00675 11.87 11.87
5.5 5 1.5 60° 0.00661 0.00662 11.99 11.98
5.5 5 2.0 60° 0.00652 0.00652 12.08 12.07
6.0 5 0.5 60° 0.00935 0.00936 12.25 12.25
6.0 5 1.0 60° 0.00915 0.00916 12.38 12.38
6.0 5 1.5 60° 0.00897 0.00897 12.51 12.51
6.0 5 2.0 60° 0.00881 0.00882 12.62 12.61
6.5 5 0.5 60° 0.01269 0.01272 12.78 12.76
6.5 5 1.0 60° 0.01244 0.01245 12.90 12.90
6.5 5 1.5 60° 0.01215 0.01217 13.05 13.05
6.5 5 2.0 60° 0.01194 0.01194 13.17 13.17
5.5 5 0.5 90° 0.00634 0.00635 12.24 12.23
5.5 5 1.0 90° 0.00617 0.00618 12.41 12.40
5.5 5 1.5 90° 0.00603 0.00603 12.56 12.55
5.5 5 2.0 90° 0.00593 0.00593 12.66 12.66
6.0 5 0.5 90° 0.00866 0.00868 12.72 12.71
6.0 5 1.0 90° 0.00843 0.00844 12.90 12.90
6.0 5 1.5 90° 0.00819 0.00820 13.09 13.08
6.0 5 2.0 90° 0.00803 0.00803 13.22 13.22
6.5 5 0.5 90° 0.01193 0.01196 13.18 13.16
6.5 5 1.0 90° 0.01156 0.01157 13.38 13.38
6.5 5 1.5 90° 0.01117 0.01119 13.62 13.61
6.5 5 2.0 90° 0.01091 0.01091 13.78 13.78
Average deviation, % 7.35 • 10-9% 25 • 10-5%
Table 10
Comparison of strain energy density and critical fracture load in two cases of fine and coarse mesh for the functionally graded steel
Notch geometry Strain energy density, MJ/m3
a, mm b, mm p, mm 2a Fine mesh Coarse mesh Fine mesh Coarse mesh
5.5 5 0.5 30° 0.00688 0.00685 11.75 11.78
5.5 5 1.0 30° 0.00678 0.00675 11.84 11.86
5.5 5 1.5 30° 0.00670 0.00667 11.91 11.94
5.5 5 2.0 30° 0.00662 0.00663 11.99 11.97
6.0 5 0.5 30° 0.00849 0.00843 12.85 12.90
6.0 5 1.0 30° 0.00839 0.00834 12.93 12.97
6.0 5 1.5 30° 0.00827 0.00824 13.03 13.05
6.0 5 2.0 30° 0.00818 0.00819 13.10 13.09
6.5 5 0.5 30° 0.01067 0.01058 13.93 13.99
6.5 5 1.0 30° 0.01056 0.01040 14.01 14.11
6.5 5 1.5 30° 0.01042 0.01036 14.09 14.14
6.5 5 2.0 30° 0.01036 0.01036 14.14 14.14
5.5 5 0.5 60° 0.00573 0.00570 12.87 12.91
5.5 5 1.0 60° 0.00560 0.00559 13.03 13.04
5.5 5 1.5 60° 0.00547 0.00547 13.18 13.18
5.5 5 2.0 60° 0.00538 0.00539 13.29 13.28
6.0 5 0.5 60° 0.00694 0.00691 14.22 14.25
6.0 5 1.0 60° 0.00677 0.00673 14.40 14.44
6.0 5 1.5 60° 0.00661 0.00661 14.57 14.57
6.0 5 2.0 60° 0.00646 0.00650 14.73 14.70
6.5 5 0.5 60° 0.00854 0.00846 15.57 15.64
6.5 5 1.0 60° 0.00831 0.00827 15.78 15.83
6.5 5 1.5 60° 0.00813 0.00814 15.96 15.95
6.5 5 2.0 60° 0.00800 0.00801 16.09 16.08
5.5 5 0.5 90° 0.00490 0.00489 13.93 13.95
5.5 5 1.0 90° 0.00475 0.00474 14.15 14.17
5.5 5 1.5 90° 0.00461 0.00460 14.35 14.38
5.5 5 2.0 90° 0.00452 0.00454 14.50 14.47
6.0 5 0.5 90° 0.00578 0.00579 15.58 15.57
6.0 5 1.0 90° 0.00563 0.00558 15.78 15.85
6.0 5 1.5 90° 0.00546 0.00543 16.03 16.07
6.0 5 2.0 90° 0.00531 0.00536 16.25 16.18
6.5 5 0.5 90° 0.00700 0.00695 17.20 17.26
6.5 5 1.0 90° 0.00676 0.00672 17.50 17.56
6.5 5 1.5 90° 0.00655 0.00652 17.78 17.82
6.5 5 2.0 90° 0.00638 0.00641 18.01 17.97
Average deviation, % 1.006 •10-7% 8.2 • 10-4%
The ultimate yield stress of each layer of the considered aßy functionally graded steel has been obtained by using the mechanism-based theory of strain gradient plasticity while the fracture toughness has been considered to vary exponentially along the specimen width.
The approach based on the strain energy density evaluated in a control volume previously used for homogeneous materials has been successfully extended to functionally graded steels. The outer contour of the volume has been determined numerically.
The average deviation between the theoretical and the experimental values in terms of the critical fracture loads has been found to be limited (8.4%).
In the same condition of notch parameters and applied load, the strain energy density of the functionally graded steel is lower than that measured in the homogeneous material. On the contrary the critical fracture load is higher for the functionally graded steel than for the homogeneous material.
The average deviation between the fine and coarse mesh in terms of the strain energy density has been found to be 1.006 • 10-7% for the functionally graded steel. In addition, this value has been found to be 0.082% in term of critical fracture load.
Further studies are necessary to investigate more in detail the influence on the critical fracture loads of the geometrical parameters as well as for extending the present method to different functionally graded steels.
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Поступила в редакцию 22.07.2015 г.
Сведения об авторах
Hadi Salavati, Assist. Prof., Shahid Bahonar University of Kerman, Iran, hadi_salavati@uk.ac.ir Yoness Alizadeh, Assoc. Prof., Amirkabir University of Technology, Tehran, Iran, alizadeh@aut.ac.ir Filippo Berto, Prof., University of Padova, Italy, berto@gest.unipd.it
