Analysis of longitudinal cracked two-dimensional functionally graded beams exhibiting material non-linearity

An analytical study of longitudinal fracture in two-dimensional functionally graded cantilever beam configurations is carried-out with taking into account the non-linear behavior of material. A longitudinal crack is located arbitrary along the beam cross-section height. The material is functionally graded along the width as well as along the height of beam. The external loading consists of a bending moment applied at the free end of lower crack arm. Fracture is studied in terms of the strain energy release rate by considering the beam complementary strain energy. The solution derived is verified by analyzing the longitudinal crack with the help of the J-integral. The distribution of J-integral value along the crack front is studied. The effects of crack location, material gradients and non-linear behavior of material on the fracture are elucidated. The analysis reveals that the material non-linearity has to be taken into account in fracture mechanics based safety design of structural members and components made of two-dimensional functionally graded materials.


INTRODUCTION
he functionally graded materials are new inhomogeneous materials produced by mixing of two constituent materials in different proportions in order to satisfy the requirements of material properties in different parts of a structural member [1][2][3][4][5][6][7][8][9][10][11][12].Since the composition of constituent materials varies continuously along one or more spatial coordinates during manufacturing without sudden changes of material properties, the interfaces and stress concentrations are largely avoided in contrast to laminated composite materials [13][14][15].Besides, the performance of functionally graded structural members and components to the external load can be optimized by tailoring the variation of material properties in a desired way during manufacturing.Therefore, the functionally graded materials have been widely applied as advanced structural materials in aeronautics, electronics, nuclear energy, optics, engineering and biomedicine for T the last three decades [16].Fracture behavior is crucial for structural integrity and functionality of members and components made of functionally graded materials.Therefore, considerable attention has been paid to fracture in these novel materials by the international research community [17][18][19][20][21][22].The present paper is concerned with a longitudinal fracture analysis of a two-dimensional functionally graded cantilever beam configuration with taking into account the non-linear behavior of material.Fracture is analyzed in terms of the strain energy release rate.The material is functionally graded along the height as well as along width of the beam.The mechanical behavior of material is described by a power-law stress-strain relation.An additional fracture analysis is carried-out by applying the J-integral approach and the results obtained are compared with the strain energy release rate.The effects of material gradients, crack location and material non-linearity on the fracture behavior are elucidated.

ANALYSIS OF THE STRAIN ENERGY RELEASE RATE
he present paper analyses the longitudinal fracture in the two-dimensional functionally graded cantilever beam configuration shown schematically in Fig. 1.A longitudinal crack of length, a, is located arbitrary along the beam cross-section height (it should be mentioned that the present paper is motivated also by the fact that functionally graded materials can be built up layer by layer [16] which is a premise for appearance of longitudinal cracks between layers).The lower and upper crack arms have different thicknesses denoted by 1 h and 2 h , respectively.The beam is loaded by one bending moment, y M , applied at the free end of lower crack arm (Fig. 1).Thus, the upper crack arm is free of stresses.The beam has a rectangular cross-section of width, b, and height, 2h.The beam is clamped in section, D. The beam non-linear mechanical behavior is described by the following power-law stress-strain relation [23]: where σ is the longitudinal normal stress, ε is the longitudinal strain, B and m are material properties.
Zircona-titanium is one of the functionally graded materials which exhibit non-linear mechanical behavior.Data for mechanical properties of zircona-titanium can be found in [24].The material is functionally graded along the width as well as along the height of the beam (Fig. 1).The distribution of material property, B, in the beam cross-section is expressed as a function of 3 y and 3 z by the following quadratic law: In ( 2), 0 B is the value of B in the centre of the beam cross-section, 1 B and 2 B are material properties which govern the material gradients along axes, 3 y and 3 z , respectively (Fig. 1).It should be mentioned that Eq. ( 2) is suitable for two-dimensional functionally graded beam configurations which have high mechanical properties at the lower, upper and lateral surfaces of the beam as shown in Fig. 2. The fracture behavior is studied in terms of the strain energy release rate.For this purpose, an elementary increase, da , of the crack length is assumed (the external load is kept constant).The strain energy release rate, G, is expressed via the changes of external work, ext dW , and strain energy, dU , as where the increase of crack area is The change of external work is written as where * dU is the change of complementary strain energy.By combining of (4), ( 5) and (6), the strain energy release rate can be expressed as It should be noted that the present analysis holds for non-linear elastic behavior of material.The analysis is applicable also for elastic-plastic behavior of material if the beam considered undergoes active deformation, i.e. if the external loading increases only [25].It should also be mention that the present study is based on the small strains assumption.The beam complementary strain energy, * U , is written as , respectively.Axes, 1 y and 1 z , are shown in Fig. 3, 2 y and 2 z are the centroidal axes of the cross- section of the un-cracked beam portion.
In principle, the complementary strain energy density is equal to the area OQR that supplements the area OPQ, enclosed by the stress-strain curve, to a rectangle (Fig. 4).Therefore, the complementary strain energy density in the lower crack arm is written as where the strain energy density, L u 0 , is equal to the area OPQ (10) From ( 1), ( 9) and ( 10), the complementary strain energy density can be obtained as The strain distribution is analyzed by applying the Bernoulli's hypothesis for plane sections since the span to height ratio of the beam under consideration is large.Concerning the application of the Bernoulli's hypothesis in the present study, it should also be noted that since the beam is loaded in pure bending (Fig. 1) the only non-zero strain is the longitudinal strain, ε.Therefore, according to the small strain compatibility equations, ε is distributed linearly along the height of the beam cross-section.Thus, ε in the lower crack arm is written as where 1 1 n z is the coordinate of the neutral axis, 1 1 n n  , (the neutral axis shifts from the centroid since the material is functionally graded (Fig. 3)), 1  is the curvature of the lower crack arm.The following equilibrium equations of the lower crack arm cross-section (Fig. 3) are used in order to determine 1 1 n z and 1  : where 1 N and 1 y M are the axial force and the bending moment, respectively.It is obvious that By using (2), the distribution of material property, B, in the lower crack arm cross-section is written as where By substituting of ( 1), ( 12) and ( 16) in ( 13) and ( 14), one derives and / f q m  ( f and q are positive integers).In (19), 1 N and 1 y M are determined by (15).It is obvious that at m=1 the non-linear stress-strain relation (1) transforms into the Hooke's law, assuming that B=E (here E is the modulus of elasticity).This means that at m=1 Eq. ( 19) should transform into the formula for curvature of linear-elastic beam.Indeed, at m=1, 0 B E  and 1 2 0 B B   Eq. ( 19) yields which is exact match of the formula for curvature of a linear-elastic homogeneous beam of width, b, and height, 1 h .Eqs. (18), and ( 19) should be solved with respect to 1 1 n z and 1  by using the MatLab computer program.
By substituting of ( 12) and ( 16) in (11) the distribution of complementary strain energy density in the lower crack arm is written as Formula ( 21) can also be applied to obtain the distribution of complimentary strain energy density, * 0U u , in the un-cracked beam portion.For this purpose, r , 1 y , 1 z , 1 1 n z and 1  have to be replaced with 0 , 2 y , 2 z , 2 2 n z and 2  , respectively ( 2 2 n z is the neutral axis coordinate of the cross-section of un-cracked beam portion, 2  is the curvature of un-cracked beam portion).The quantities, 2 2 n z and 2  , can be determined from Eqs. ( 18) and (19).For this purpose, r , 1 1 n z , 1 h and 1  have to be replaced with 0 , 2 2 n z , 2h and 2  , respectively.
Finally, by substituting of * 0L u , * 0U u and ( 8) in (7), one derives Formula (22) calculates the strain energy release rate in the beam configuration shown in Fig. 1 when the beam mechanical behavior and the material gradient are described by formulae (1) and ( 2), respectively.It should be noted that at m=1, which is exact match of the expression for the strain energy release rate when the beam considered is linear-elastic and homogeneous [26].
In order to verify (22), the fracture is analyzed also by using the J-integral written as [27] 0 cos where Γ is a contour of integration going from the lower crack face to the upper crack face in the counter clockwise direction, 0 u is the strain energy density, α is the angle between the outwards normal vector to the contour of integration and the crack direction, x p and y p are the components of stress vector, u and v are the components of displacement vector with respect to the crack tip coordinate system xy (x is directed along the crack), ds is a differential element along the contour.The J-integral is solved by using an integration contour, Γ, that coincides with the beam contour (Fig. 1).It is obvious that the J-integral value is non-zero only in segments 1  and 2  of the integration contour.Therefore, the J-integral solution is written as where 1 J  and 2 J  are the J-integral values in segments 1  and 2  , respectively ( 1  and 2  coincide with the free end of lower crack arm and the clamping, respectively).The components of J-integral in segment, 1  , of the integration contour are written as (Fig. 3) where 1 z varies in the interval 24) is found as where 1 1 n z and 1  are obtained from Eqs. ( 18) and (19).The strain energy density is calculated by substituting of (1) in ( 10) By substituting of ( 1), ( 12), ( 16), ( 26), ( 27), ( 28) and ( 29) in (24) and integrating in boundaries from The J-integral solution in segment, 2  , of the integration contour (Fig. 1) can be determined by (30).For this purpose, r ,  and 1  have to be replaced with 0 , 2 2 n z , 1u  , 2u  and 2  , respectively.Besides, the sign of (30) must be set to "minus" because the integration contour is directed upwards in segment, 2  .
The J-integral final solution is found by substituting of 1 J  and 2 J  in ( 25) Formula (31) describes the distribution of the J-integral value along the crack front.The average value of the J-integral along the crack front is written as It should be noted that the J-integral solution derived by substituting of (31) in (32) is exact match of the strain energy release rate (22).This fact verifies the fracture analysis developed in the present paper.

RESULTS
he distribution of J-integral value along the crack front is analyzed.For this purpose, calculations are performed by using formula (31).It is assumed that b=0.020 m, h=0.004 m, m=0.7, f=7, q=10,  The effects of the material gradient and the crack location on the strain energy release rate are evaluated too.For this purpose, calculations of the strain energy release rate are carried-out by using formula (22).The strain energy release rate is presented in non-dimensional form by using the formula, Figure 6: The strain energy release rate in non-dimensional form plotted against 1 0 / B B ratio at three 1 / 2 h h ratios.
The crack location along the height of the beam cross-section is characterized by 1 / 2 h h ratio.In the calculations, 0 B is kept constant.Therefore, 1 B is varied in order to generate various 1 0 / B B ratios.The strain energy release rate in nondimensional form is plotted against 1 0 / B B ratio at three 1 / 2 h h ratios and 2 0 / 2.0 B B  in Fig. 6.The curves in Fig. 6 indicate that the strain energy release rate decreases with increasing of 1 / 2 h h ratio.This finding is attributed to the increase of the lower crack arm stiffness.It can be observed also in Fig. 6 that the strain energy release rate decreases with increasing of 1 0 / B B ratio.The influence of non-linear behavior of the material on the fracture is also analyzed.For this purpose, the strain energy release rate in non-dimensional form is presented as a function of 2 0 / B B ratio at 1 / 2 0.5 h h and 1 0 / 0.5 B B  in Fig. 7.It can be observed that the strain energy release rate decreases with increasing of 2 0 / B B ratio (Fig. 7).The strain energy release rate obtained assuming linear-elastic behavior of the two-dimensional functionally graded material is also presented in Fig. 7 for comparison with the non-linear solution (the linear-elastic solution is derived by substituting of m=1 in formula ( 22)).The curves in Fig. 7 indicate that material non-linearity leads to increase of the strain energy release rate.

CONCLUSIONS
he longitudinal fracture behavior of two-dimensional functionally graded cantilever beam that exhibits material non-linearity is investigated analytically.The beam is loaded by a bending moment applied at the free end of the lower crack arm.The fracture is studied in terms of the strain energy release rate.The beam mechanical behavior is described by using a power-law stress-strain relation.The material property, B, varies continuously in the beam crosssection according to a quadratic law.The solution derived is applicable for longitudinal crack located arbitrary along the height of the beam cross-section.In order to verify the solution, the fracture is analyzed also by applying the J-integral approach.The distribution of the Jintegral value along the crack front is investigated.The physical meanings of the dependence of the J-integral value on material gradient are discussed.The effects of material gradients and crack location along the beam height on the fracture behavior are also analyzed.The analysis revealed that the strain energy release rate decreases with increasing of the lower crack arm thickness.It is found also that the strain energy release rate decreases with increasing of 1 0 / B B and 2 0 / B B ratios.This finding is attributed to the increase of the beam stiffness.The influence of material non-linearity on the fracture is studied too.It is found that the material non-linearity leads to increase of the strain energy release rate.Therefore, non-linear behavior of the material has to be taken into account in fracture mechanics based safety design of structural members and components made of two-dimensional functionally graded materials.The results obtained in the T present study indicate that the longitudinal fracture in two-dimensional functionally graded beams with material nonlinearity can be efficiently regulated in their design stage by employing appropriate material gradients.Besides for the beam shown in Fig. 1, the analysis developed in the present paper can be applied to determine the strain energy release rate for longitudinal cracks in two-dimensional functionally graded non-linear elastic beam configurations loaded in pure bending (for instance, the double cantilever beam loaded with uneven bending moments, the four-point bending beam when the crack tip is located in middle beam portion which is loaded in pure bending).It should be noted that the approach developed in the present paper is applicable also for beam configurations loaded by a vertical load when the shear stresses can be neglected, i.e. when the span to height ratio of the beam is large.

Figure 1 :
Figure 1: The geometry and loading of the two-dimensional functionally graded cantilever beam.

Figure 2 :
Figure 2: Distribution of the material property, B, in the beam cross-section.

Figure 3 :
Figure 3: The free end of the lower crack arm (. 1 1 n n . is the neutral axis).

u
are the complementary strain energy densities in the lower crack arm and the un-cracked beam portion

Figure 4 :
Figure 4: Schematic of a non-linear stress-strain curve ( 0 u and * 0 u are the strain energy density and the complementary strain energy density, respectively).
.The J-integral value is presented in non-dimensional form by using the formula, .The material gradient along the beam width is characterized by 1 0 / B B ratio.Fig.5shows the distribution of the J-integral value in non-T dimensional form along the crack front for two 1 0 / B B ratios at 1 / 2 0.75 h h and 2 0 / 2 B B  .Only the right-hand half of the crack front is shown in Fig.5since the distribution is symmetrical with respect to the crack front centre.The horizontal axis is defined such that is in the right-hand lateral surface of the beam.One can observe in Fig.5that for 1 0 / 1.2 B B  , the J-integral value is maximum in the crack front centre and decreases towards the right-hand lateral surface (this is due to the fact that for 1 0 / 1.2 B B  the material property, B, is minimum in the crack front centre and increases towards the beam lateral surfaces).Fig.5shows that for  , the J-integral value increases from its minimum in the crack front centre towards the right-hand lateral surface of the beam.

Figure 5 :
Figure 5: The distribution of the J-integral value in non-dimensional form along the crack front (curve 1 -at 1 0 / 1.2 B B  , curve 2 -at 1 0 / 0.4 B B   ).

Figure 7 :
Figure 7: The strain energy release rate in non-dimensional form presented as a function of 2 0 / B B ratio (curve 1 -at non-linear elastic behavior of material, curve 2 -at linear-elastic behavior of material).