J-integral Evaluation from Displacement Fields around Crack Tip Measured by Digital Image Correlation
Hiroto YAMANE1, Shuichi ARIKAWA2, Satoru YONEYAMA3,Yasuaki WATANABE4, Tatsuhiko ASAI5, Kunio SHIOKAWA6 and Mitsuo YAMASHITA7 1, 3 Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara 252-5258, Japan
2Department of Mechanical Engineering Informatics, Meiji University, Kawasaki 214-5871, Japan 4, 5, 6 Fundamental Technology Research Center, Fuji Electric Co., Ltd., Tokyo 191-8502, Japan
7 New Industry Creation Hatchery Center, Tohoku University, Sendai 980-8579, Japan (Received 10 January 2017; received in revised form 12 April 2017; accepted 26 April 2017)
Abstract: In this paper, a method for evaluating J-integral for displacement fields obtained by digital image correlation (DIC) is proposed. First, the displacement gradient and strain are determined from the displacement using a least squares method on the domain of integration. Next, the stress components are determined from the strain using the Newton-Raphson method and material properties. Finally, the J-integral value is determined by the numerical integration on the domain of integration. The usefulness of this evaluation method is verified by applying this method to the displacement field obtained from the elastic-plastic finite element analysis, and by applying to the experimental displacement field obtained by DIC.
Keywords: J-integral, Digital image correlation, Elasto-plasticity, Fracture mechanics, Interface 1. Introduction
In the fields of aircraft industries and nuclear power plants, the soundness evaluations are performed using the fracture mechanics for the defects which occurred during production [1, 2]. Fracture mechanics parameters such as stress intensity factor K, J-integral and crack opening displacement δ are widely used to evaluate the behavior of materials having cracks and defects quantitatively [3
–
7]. Even if the geometry of an object, the loading condition and the dimension are unknown, the behavior of a crack can be known by evaluating these fracture parameters. Generally, the evaluation of the fracture mechanic parameter is performed referring the handbook [3, 4] or using finite element analysis [8–
10]. However, since actual products are complex structures and difficult to grasp actual boundary conditions, the reliability of the evaluation value is inaccurate. As a result, unexpected fractures often occur in actual products.Presently, displacement fields around cracks can be easily measured using optical methods such as digital image correlation (DIC) [11, 12]. Various studies have been performed for evaluating the fracture mechanics parameters from the measured displacement field [13
–
21]. Hence, it is expected that the fracture behavior can be evaluated through the measurement by optical methods even if the actual boundary condition cannot be known. Around a domain of a crack tip, fracture behavior of the material under small scale yielding can be evaluated by stress intensity factor K. However, since the material under a large scale yielding cannot be evaluated by stress intensity factor K, the fracture behavior under large scale yielding is evaluated using J-integral. As the past research, many researches have been carried out to study evaluating J-integral from measurements by optical methods [22–
33]. The J-integral values in these studies are calculated using the stress from the strain by modeling the stress/strain curve. Hence it is considered this approximation error affect to the J-integral value.In this study, J-integral values are calculated using the stress components from strain distributions without modeling the stress/strain curve. Accordingly, since the J-integral value is unaffected the approximation error for the stress/strain curve, it is considered J-integral evaluation from the displacement field around a crack tip can be evaluated more accurately. On the other hand, Yoneyama and coworkers [22] evaluate J-integral for the displacement field obtained by digital image correlation using both method of path integral and domain integral. From this result, the error of J-integral values caused by error of displacement field can be reduced by using the domain integral. Therefore, in this study, the evaluation is performed by domain integral. The effectiveness of the proposed method is demonstrated by applying to the displacement fields obtained from elasto-plastic finite element analysis, and the measured displacement fields of specimens with various crack lengths and interfacial strengths. Some of the experimental data used for the proposed evaluation are the same as that of the previous report [24]. Since J-integral can be evaluated easily and accurately from the measured displacement without the knowledge of boundary condition, it is expected that the proposed method can be applied to various experimental evaluation of structural components.
2. J-Integral Evaluation Method 2.1 J-integral
In fracture mechanics, J-integral is associated with strain energy release rate per unit crack area. The theoretical concept of the J-integral was developed by Rice [34, 35] who showed that it was independent of the path defined around the crack tip. The J-integral can be considered as both an energy parameter and a stress intensity parameter. It is defined as [5, 7, 34–36]
J
(WnxTiuix)ds(i, j = x, y) (1)
Fig. 1 Domain of integral
where nx is the x directional component of the unit normal of a contour path, and ui is the displacement components. W and Ti are the strain energy density and the traction along the contour Γ. W and Ti are expressed by the following equations W 0 ijdij ij
(2) j ij i n T (3)where σijand εijare the stress and strain tensors, respectively. Eq. (1) can be rewritten in domain integral in the form by the following equation [22, 37, 38].
dA x q x u x q W J A j i ij i
(i, j = x, y) (4)where A is the area of the domain, q is the arbitrary variable that satisfies q = 0 on the outer boundary and q = 1 on the inner boundary of the domain. The variable q can be defined as
2 in 2 out 2 0 2 0 2 out r r y y x x r q (5)where rout and rin express the radius of outer and inner boundaries of domain, x0 and y0represent the crack tip position as shown in Fig.1. In the case of the linear elasticity, this conversion is accurate. Furthermore, it is effective for the case of the non-linear elasticity without unloading. In the same as path integral, it is independent of the domain defined around the crack tip.
2.2 Evaluation procedure of J-integral
When the displacement fields around the crack tip are obtained by optical methods such as digital image correlation. The strains are obtained from the displacements using local least-squares method within the domain. Then, the stress components are calculated from the strain distribution. In this study, since the J-integral is evaluated from the measured displacements data at an instant, it is not considered the time dependency of the stress/strain relation. Therefore the total strain theory is employed for the constitutive equation in this study. The stress/strain relationship for the elastic-plastic material is expressed as [39]
Fig. 2 Specimen for simulation and experiment
ij ij G s e 2 1 (6)
where the G, sij and are elastic shear modulus, stress deviator and the parameter which depends on the work hardening, respectively. This is expressed by following equation. e ) p ( e 2 3 (7)
where σeand εe(p) are equivalent stress and equivalent plastic strain. Equivalent plastic strain is obtained from the value of the equivalent stress and the stress-strain relation. Kusayanagi and coworkers [23] have calculated stress components from the strain distributions in homogeneous materials by calculating using Ramberg-Osgood equation [40]. However, It is considered that this approximation error affect to the J-integral value. Therefore, in this study, stress components are calculated from strain distribution without modeling the stress/strain curve obtained by the experiment to evaluate J-integral with higher accuracy. The stress components are calculated from strain distribution by Newton-Raphson method. In this time, it needs to calculate the parameter of and derivative-type of about the stress. The parameter of is calculated from the stress/strain curve obtained by uniaxial tensile test because it is the function of σeand εe(p)(= εe– ε0). The derivative-type of about stress is calculated by numerical differentiation. For example the derivative-type of about σxis calculated as follow
x x x x x x / / 2 2 (8)Then, from Eq. (2), the strain energy density W is calculated as follows from stress and strain components.
1
J-integral Evaluation from Displacement Fields around Crack Tip Measured by Digital Image Correlation
Hiroto YAMANE1, Shuichi ARIKAWA2, Satoru YONEYAMA3,
Yasuaki WATANABE4, Tatsuhiko ASAI5, Kunio SHIOKAWA6 and Mitsuo YAMASHITA7
1, 3 Department of Mechanical Engineering, Aoyama Gakuin University, Sagamihara 252-5258, Japan
2Department of Mechanical Engineering Informatics, Meiji University, Kawasaki 214-5871, Japan
4, 5, 6 Fundamental Technology Research Center, Fuji Electric Co., Ltd., Tokyo 191-8502, Japan
7 New Industry Creation Hatchery Center, Tohoku University, Sendai 980-8579, Japan
(Received 10 January 2017; received in revised form 12 April 2017; accepted 26 April 2017)
Abstract: In this paper, a method for evaluating J-integral for displacement fields obtained by digital image correlation (DIC) is proposed. First, the displacement gradient and strain are determined from the displacement using a least squares method on the domain of integration. Next, the stress components are determined from the strain using the Newton-Raphson method and material properties. Finally, the J-integral value is determined by the numerical integration on the domain of integration. The usefulness of this evaluation method is verified by applying this method to the displacement field obtained from the elastic-plastic finite element analysis, and by applying to the experimental displacement field obtained by DIC.
Keywords: J-integral, Digital image correlation, Elasto-plasticity, Fracture mechanics, Interface 1. Introduction
In the fields of aircraft industries and nuclear power plants, the soundness evaluations are performed using the fracture mechanics for the defects which occurred during production [1, 2]. Fracture mechanics parameters such as stress intensity factor K, J-integral and crack opening displacement δ are widely used to evaluate the behavior of
materials having cracks and defects quantitatively [3
–
7].Even if the geometry of an object, the loading condition and the dimension are unknown, the behavior of a crack can be known by evaluating these fracture parameters. Generally, the evaluation of the fracture mechanic parameter is performed referring the handbook [3, 4] or using finite
element analysis [8
–
10]. However, since actual productsare complex structures and difficult to grasp actual boundary conditions, the reliability of the evaluation value is inaccurate. As a result, unexpected fractures often occur in actual products.
Presently, displacement fields around cracks can be easily measured using optical methods such as digital image correlation (DIC) [11, 12]. Various studies have been performed for evaluating the fracture mechanics parameters
from the measured displacement field [13
–
21]. Hence, it isexpected that the fracture behavior can be evaluated through the measurement by optical methods even if the actual boundary condition cannot be known. Around a domain of a crack tip, fracture behavior of the material under small scale yielding can be evaluated by stress intensity factor K. However, since the material under a large scale yielding cannot be evaluated by stress intensity factor K, the fracture behavior under large scale yielding is evaluated using J-integral. As the past research, many researches have been carried out to study evaluating J-integral from
measurements by optical methods [22
–
33]. The J-integralvalues in these studies are calculated using the stress from the strain by modeling the stress/strain curve. Hence it is considered this approximation error affect to the J-integral value.
In this study, J-integral values are calculated using the stress components from strain distributions without modeling the stress/strain curve. Accordingly, since the J-integral value is unaffected the approximation error for the stress/strain curve, it is considered J-integral evaluation from the displacement field around a crack tip can be evaluated more accurately. On the other hand, Yoneyama and coworkers [22] evaluate J-integral for the displacement field obtained by digital image correlation using both method of path integral and domain integral. From this result, the error of J-integral values caused by error of displacement field can be reduced by using the domain integral. Therefore, in this study, the evaluation is performed by domain integral. The effectiveness of the proposed method is demonstrated by applying to the displacement fields obtained from elasto-plastic finite element analysis, and the measured displacement fields of specimens with various crack lengths and interfacial strengths. Some of the experimental data used for the proposed evaluation are the same as that of the previous report [24]. Since J-integral can be evaluated easily and accurately from the measured displacement without the knowledge of boundary condition, it is expected that the proposed method can be applied to various experimental evaluation of structural components.
2. J-Integral Evaluation Method 2.1 J-integral
In fracture mechanics, J-integral is associated with strain energy release rate per unit crack area. The theoretical concept of the J-integral was developed by Rice [34, 35] who showed that it was independent of the path defined around the crack tip. The J-integral can be considered as both an energy parameter and a stress intensity parameter. It is defined as [5, 7, 34–36]
J
(WnxTiuxi)ds(i, j = x, y) (1)
2 Fig. 1 Domain of integral
where nx is the x directional component of the unit normal
of a contour path, and ui is the displacement components. W
and Ti are the strain energy density and the traction along
the contour Γ. W and Ti are expressed by the following
equations W 0 ijdij ij
(2) j ij i n T (3)where σijand εijare the stress and strain tensors, respectively.
Eq. (1) can be rewritten in domain integral in the form by the following equation [22, 37, 38].
dA x q x u x q W J A j i ij i
(i, j = x, y) (4)where A is the area of the domain, q is the arbitrary variable that satisfies q = 0 on the outer boundary and q = 1 on the inner boundary of the domain. The variable q can be defined as
2 in 2 out 2 0 2 0 2 out r r y y x x r q (5)where rout and rin express the radius of outer and inner
boundaries of domain, x0 and y0 represent the crack tip
position as shown in Fig.1. In the case of the linear elasticity, this conversion is accurate. Furthermore, it is effective for the case of the non-linear elasticity without unloading. In the same as path integral, it is independent of the domain defined around the crack tip.
2.2 Evaluation procedure of J-integral
When the displacement fields around the crack tip are obtained by optical methods such as digital image correlation. The strains are obtained from the displacements using local least-squares method within the domain. Then, the stress components are calculated from the strain distribution. In this study, since the J-integral is evaluated from the measured displacements data at an instant, it is not considered the time dependency of the stress/strain relation. Therefore the total strain theory is employed for the constitutive equation in this study. The stress/strain relationship for the elastic-plastic material is expressed as [39]
Fig. 2 Specimen for simulation and experiment
ij ij G s e 2 1 (6)
where the G, sij and are elastic shear modulus, stress
deviator and the parameter which depends on the work hardening, respectively. This is expressed by following equation. e ) p ( e 2 3 (7)
where σeand εe(p) are equivalent stress and equivalent plastic
strain. Equivalent plastic strain is obtained from the value of the equivalent stress and the stress-strain relation. Kusayanagi and coworkers [23] have calculated stress components from the strain distributions in homogeneous materials by calculating using Ramberg-Osgood equation [40]. However, It is considered that this approximation error affect to the J-integral value. Therefore, in this study, stress components are calculated from strain distribution without modeling the stress/strain curve obtained by the experiment to evaluate J-integral with higher accuracy. The stress components are calculated from strain distribution by Newton-Raphson method. In this time, it needs to calculate the parameter of and derivative-type of about the stress. The parameter of is calculated from the stress/strain curve obtained by uniaxial tensile test because it is the function of
σeand εe(p)(= εe– ε0). The derivative-type of about stress
is calculated by numerical differentiation. For example the
derivative-type of about σxis calculated as follow
x x x x x x / / 2 2 (8) Then, from Eq. (2), the strain energy density W is calculated as follows from stress and strain components.
e 0 e ) p ( e 2 e 2 2 e 3 2 ) 2 1 ( 2 1 d E W x y xy (9) W 0 ijdij ij
(2) j ij i n T (3) dA x q x u x q W J A j i ij i
(i, j = x, y) (4)
2 in 2 out 2 0 2 0 2 out r r y y x x r q (5) ij ij G s e 2 1 (6) e ) p ( e 2 3 (7)
x x x x x x / / 2 2 (8)
e 0 e ) p ( e 2 e 2 2 e 3 2 ) 2 1 ( 2 1 d E W x y xy (9) J
(WnxTiuxi)ds (i, j = x, y) (1)Advanced Experimental Mechanics, Vol.2 (2017)
Fig. 3 Stress/strain response of homogeneous material
(a) (b)
Fig. 4 Displacement distribution obtained by finite element
analysis (a = 25 mm): (a) ux; (b) uy (P = 1736 N)
In Eq. (9), the integration part is determined using linear interpolation from each stress values in the stress/strain curve. Finally, the J-integral value can be obtained by substituting the values of displacement gradients, strain and stress components into Eq. (4)
3. Applying to Displacement Field of Homogeneous Material
3.1 Verification by simulation
In order to verify the effectiveness of the proposed method, the method is applied to the displacement fields obtained by elasto-plastic finite element analysis. The analysis is performed by using MSC Marc Mentat 2010. The finite element model is shown in Fig. 2 and the crack length is a = 25 mm. The finite element mesh of this model involves 9060 elements and 9281 nodes, which consists of four-noded isoparametric elements. The material is assumed annealed pure aluminum (A1050-O). Its stress/strain curve is measured by tensile test and fitted curve are plotted in Fig. 3. The fixed boundary conditions to x-direction are applied to the upper and the lower load points, and the traction boundary condition is applied on the upper load point. The analysis is carried out under the plane stress condition. The
displacement distributions in the area of 12 × 12 mm2
around the crack tip are obtained at the step size of 0.2 mm. The displacement distribution around the crack tip under the load of P = 1736 N obtained by the analysis is shown in Fig. 4. The crack tip is located at the origin of the coordinate, and the crack surface is located in a left side of the crack tip. From the displacement field, J-integral is evaluated using the domain integral under various loads. As the domain of
integration, the inner radius is fixed in rin= 2.0 mm and the
outer radius rout is varied between 2.5 mm and 5.0 mm. In
this analysis, the J-integral value is evaluated both method of modeling the stress/strain curve by Ramberg-Osgood
Fig. 5 J values obtained from simulated displacement fields for various integration domain
(a)
(b)
Fig. 6 Compared J value obtained by finite element analysis: (a) compared VCEM with domain integral; (b) relative error model and without modeling stress/strain curve to verify the effectiveness of this method.
Figure 5 is the J-integral values obtained from the displacement field around the crack tip. The domain independence properties of J-integral are observed. Figure
Fig. 7 Images around a crack tip for A1050-O
(a) (b)
Fig. 8 Displacement distribution obtained by DIC (a = 25
mm): (a) ux, (b) uy (P = 1736 N)
Fig. 9 J values obtained from measured displacement fields for various integration domain
6(a) shows comparative J-integral values with domain integral (with and without modeling) and virtual crack extension method (VCEM) which is obtained by commercial finite element software. From this result, the J-integral values coincide between the domain J-integral and the virtual crack extension method. Additionally, Fig. 6(b) shows the relative error of both methods of modeling and without modeling using VCEM for the reference value. Compared the both method’s relative error, the proposed method can evaluate the J-integral value accurately in comparison with the method of modeling.
3.2 Application to measured displacement field The proposed method is applied to the displacement field obtained by DIC. The geometry and the material of the specimen is the same as the previous analysis. For the DIC
Fig. 10 J values compared with DIC and FEM
(a) (b)
Fig. 11 Tensile stress/strain responses of homogenous materials: (a) aluminum; (b) copper
measurement, a random pattern is applied to the surface of the specimen by black and white spray paints as shown in Fig. 7. The tensile test is performed and images of the specimen surface are recorded using a CCD camera (1024× 768 pixels) through a telephoto lens. The object size on the image is 35 m/pixel. The load is also measured using a load cell. The displacements are obtained using digital image correlation with the resolution of the displacement of
0.02 pixels (7 × 10-4 mm). A subset size of 25 × 25 pixels
with a step size of 5 pixels is used for the correlation between the images before and after the deformation. Figure 8 shows the displacement fields in the area of 12 ×
12 mm2 around the crack tip obtained by DIC. The data
includes rigid-body displacements. When the digital image correlation is used for the displacement measurement, it is difficult to obtain smooth and highly accurate measurement result near the crack because subset contains the upper and lower surface. In this case, since the subset size is 0.8 mm × 0.8 mm, the range for excluding data is 0.8 mm to the crack surface. The strain distribution around the crack surface is calculated using the extrapolation. The evaluation procedure is same as that in the simulation described in the previous section.
3 Fig. 3 Stress/strain response of homogeneous material
(a) (b)
Fig. 4 Displacement distribution obtained by finite element
analysis (a = 25 mm): (a) ux; (b) uy (P = 1736 N)
In Eq. (9), the integration part is determined using linear interpolation from each stress values in the stress/strain curve. Finally, the J-integral value can be obtained by substituting the values of displacement gradients, strain and stress components into Eq. (4)
3. Applying to Displacement Field of Homogeneous Material
3.1 Verification by simulation
In order to verify the effectiveness of the proposed method, the method is applied to the displacement fields obtained by elasto-plastic finite element analysis. The analysis is performed by using MSC Marc Mentat 2010. The finite element model is shown in Fig. 2 and the crack length is a = 25 mm. The finite element mesh of this model involves 9060 elements and 9281 nodes, which consists of four-noded isoparametric elements. The material is assumed annealed pure aluminum (A1050-O). Its stress/strain curve is measured by tensile test and fitted curve are plotted in Fig. 3. The fixed boundary conditions to x-direction are applied to the upper and the lower load points, and the traction boundary condition is applied on the upper load point. The analysis is carried out under the plane stress condition. The
displacement distributions in the area of 12 × 12 mm2
around the crack tip are obtained at the step size of 0.2 mm. The displacement distribution around the crack tip under the load of P = 1736 N obtained by the analysis is shown in Fig. 4. The crack tip is located at the origin of the coordinate, and the crack surface is located in a left side of the crack tip. From the displacement field, J-integral is evaluated using the domain integral under various loads. As the domain of
integration, the inner radius is fixed in rin= 2.0 mm and the
outer radius rout is varied between 2.5 mm and 5.0 mm. In
this analysis, the J-integral value is evaluated both method of modeling the stress/strain curve by Ramberg-Osgood
Fig. 5 J values obtained from simulated displacement fields for various integration domain
(a)
(b)
Fig. 6 Compared J value obtained by finite element analysis: (a) compared VCEM with domain integral; (b) relative error model and without modeling stress/strain curve to verify the effectiveness of this method.
Figure 5 is the J-integral values obtained from the displacement field around the crack tip. The domain independence properties of J-integral are observed. Figure
4 Fig. 7 Images around a crack tip for A1050-O
(a) (b)
Fig. 8 Displacement distribution obtained by DIC (a = 25
mm): (a) ux, (b) uy (P = 1736 N)
Fig. 9 J values obtained from measured displacement fields for various integration domain
6(a) shows comparative J-integral values with domain integral (with and without modeling) and virtual crack extension method (VCEM) which is obtained by commercial finite element software. From this result, the J-integral values coincide between the domain J-integral and the virtual crack extension method. Additionally, Fig. 6(b) shows the relative error of both methods of modeling and without modeling using VCEM for the reference value. Compared the both method’s relative error, the proposed method can evaluate the J-integral value accurately in comparison with the method of modeling.
3.2 Application to measured displacement field The proposed method is applied to the displacement field obtained by DIC. The geometry and the material of the specimen is the same as the previous analysis. For the DIC
Fig. 10 J values compared with DIC and FEM
(a) (b)
Fig. 11 Tensile stress/strain responses of homogenous materials: (a) aluminum; (b) copper
measurement, a random pattern is applied to the surface of the specimen by black and white spray paints as shown in Fig. 7. The tensile test is performed and images of the specimen surface are recorded using a CCD camera (1024× 768 pixels) through a telephoto lens. The object size on the image is 35 m/pixel. The load is also measured using a load cell. The displacements are obtained using digital image correlation with the resolution of the displacement of
0.02 pixels (7 × 10-4 mm). A subset size of 25 × 25 pixels
with a step size of 5 pixels is used for the correlation between the images before and after the deformation. Figure 8 shows the displacement fields in the area of 12 ×
12 mm2 around the crack tip obtained by DIC. The data
includes rigid-body displacements. When the digital image correlation is used for the displacement measurement, it is difficult to obtain smooth and highly accurate measurement result near the crack because subset contains the upper and lower surface. In this case, since the subset size is 0.8 mm × 0.8 mm, the range for excluding data is 0.8 mm to the crack surface. The strain distribution around the crack surface is calculated using the extrapolation. The evaluation procedure is same as that in the simulation described in the previous section.
Figure 9 shows the J-integral values obtained from the measured displacements in various integration regions. The result obtained by domain integral shows the domain independent property of J-integral. Additionally, Fig. 10 shows the J-integral values evaluated from the displacement field obtained by finite element analysis and DIC. Comparing both results, the J-integral values coincide closely. From the above results, J-integral can be evaluated from displacement field around the crack by proposed method.
Advanced Experimental Mechanics, Vol.2 (2017)
(a) (b)
Fig. 12 Displacement distribution obtained by DIC (a = 20
mm): (a) ux; (b) uy (P = 3004 N)
(a)
(b)
Fig. 13 J values obtained from measured displacement fields for various integration domain: (a) Set 1; (b) Set 2
4. Applying to Measured Displacement Field of Bi-Material
The geometry of the specimen is the same as the finite element model shown in Fig. 2. The bi-material specimen is composed of annealed pure aluminum (A1050-O) in the upper part and work hardened tough pitch copper (C1100-1/2H) in the lower part. The boundary is arranged on the crack tip extension line. Stress/strain curves of each material obtained by uniaxial tensile tests are shown in Fig. 11. Two rods made of these materials are jointed by a friction welding method. The boundary strength of the specimen cut from near the edge of the rod is different from that cut from the center of the rod, because the relative velocity of the two materials for welding is different for the edge and the center. The specimens cut from near the outer of the rod are called as the specimen group Set 1 whereas the specimens from the center are called as Set 2. In each Sets, the specimens which have the various crack length are created. The experimental procedure is the same as the case of homogeneous material.
The area imaged is 24 × 18 mm2. The object size on the
image is 24 m/pixel. The displacements are obtained using
(a)
(b)
Fig. 14 J values with various crack length: (a) Set 1; (b) Set 2 digital image correlation with the resolution of the
displacement of 0.02 pixels (4.8 × 10-4 mm). A subset size
of 31 × 31 pixels with a step size of 8 pixels is used for the correlation between the images before and after the deformation.
Figure 12 shows the displacement distributions in the
area of 12 × 12 mm2 around the interface crack tip for the
specimen under the load of P = 3004 N. Since the subset size is 0.8 mm × 0.8 mm, the range for excluding data is 0.8 mm to the crack surface. Figure 13 shows the J values obtained from the displacements measured using digital image correlation. From the result, the domain independent property is observed. Additionally, Fig. 14 represents the J-integral values with the load change for the various crack lengths for the specimen Sets 1 and 2. In each specimen, the topmost plots indicate the J-integral value when the specimen exhibits the fracture. Therefore, the topmost values can be considered as the critical J-integral for the fracture. The values of the critical J-integral coincide closely among various crack lengths in Fig. 14. From the results in Fig. 14, however, the difference of the crack length does not affect clearly the critical J value. On the other hand, the J values at break in Set 2 in Fig. 14(b) are lower than the values of Set 1. It is considered that the strength of the joint of the specimens in Set 2 is weaker than those in Set 1. As a result of a microscopic observation of the reaction layer [24], the reaction layer is thicker toward the outer region, and the poor jointing is observed around the center of the round bar. Moreover, exposed coppers are observed in the reaction layer. It is considered the reaction layer is not formed enough in Set 2 because the frictional heat is lower than Set 1. From the above, the critical J-integral values when the fractures occur can be evaluated even if crack
lengths are different. Therefore, the proposed method is effective for J-integral evaluations under various conditions.
5. Conclusion
The method for evaluating J-integral without modeling the elastic-plastic stress/strain relationship is proposed. Accordingly, the J-integral evaluation from the displacement around the crack tip can be evaluated more accurately. The effectiveness of the evaluation method is verified by applying to this method to the displacement fields obtained by the finite element analysis and digital image correlation. Therefore, it is expected that the proposed method can be applied to various experimental evaluation of structural components.
Nomenclature
x, y Cartesian coordinates
i, j Indices of coordinate axes
n Unit normal of a contour path
u Displacement components [m]
W Strain energy density [J/m3]
Γ Contour path
ds Length increment along contour path [m]
J J-integral [Pa･m]
T Traction along the contour Γ [Pa]
A Area of the domain [m2]
q Variable of the position between the inner and the
outer boundary
rin Radius of the inner boundary [m]
rout Radius of the outer boundary [m]
E Elastic modulus [Pa]
G Elastic shear modulus [Pa]
s Stress deviator [Pa]
σ Stress [Pa]
γ Shear stress [Pa]
ε Strain
φ Work hardening parameter [1/Pa]
σe Equivalent stress [Pa]
εe(p) Equivalent plastic strain References
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correlation and fracture: An advanced technique for estimating stress intensity factors of 2D and 3D cracks, J. Phys. D: Appl. Phys., 42 (2009), 214004(21pp).
[13] Yoneyama, S., Morimoto, Y. and Takashi, M., Automatic determination method of stress intensity factor utilizing digital image correlation and nonlinear least squares, Structural Health Monitoring and Intelligent Infrastructure, Wu, Z. and Abe, M., eds., Swets & Zeitlinger, (2003), 1357−1367.
[14] Yoneyama, S., Morimoto, Y. and Takashi, M.: Automatic evaluation of mixed-mode stress intensity factors utilizing digital image correlation, Strain, 42-1 (2006), 242-1–29.
[15] Yoneyama, S., Ogawa, T. and Kobayashi, Y.: Evaluating mixed-mode stress intensity factors from full-field displacement fields obtained by optical methods, Eng. Fract. Mech., 74-9 (2007), 1399–1412. [16] Mcneill, S.R., Peters, W.H. and Sutton, M.A.:
Estimation of stress intensity factor by digital image correlation, Eng. Fract. Mech., 28-1 (1987), 101–112. [17] Feng, Z., Rowlands, R.E. and Sanford, R.J.: Stress
intensity determination by an experimental-numerical hybrid technique, J. Strain Anal. Eng. Des., 26-4 (1991), 243–251.
[18] Yusof, F. and Withers, P.J.: Real-time acquisition of fatigue crack images for monitoring crack-tip stress intensity variations within fatigue cycles, J. Strain Anal. Eng. Des., 44-2 (2009), 149–158.
[19] Matvienko, Y.G., Pisarev, V.S. and Eleonsky, S.I.: Determination of fracture mechanics parameters on a base of local displacement measurements, Fract. Struct. Integrity, 25 (2013), 20–26.
[20] Barker, D.B., Sanford, R.J. and Chora, R.: Determine K and related stress-field parameters from displacement fields, Exp. Mech., 25-4 (1985), 309– 407.
[21] Hamam, R., Hild, F. and Roux, S.: Stress intensity factor gauging by digital image correlation: Application in cyclic fatigue, Strain, 43-3 (2007), 181–192.
[22] Yoneyama, S., Arikawa, S., Kusayanagi, S. and Hazumi, K.: Evaluating J-integral from displacement fields measured by digital image correlation, Strain,
5 (a) (b)
Fig. 12 Displacement distribution obtained by DIC (a = 20 mm): (a) ux; (b) uy (P = 3004 N)
(a)
(b)
Fig. 13 J values obtained from measured displacement fields for various integration domain: (a) Set 1; (b) Set 2
4. Applying to Measured Displacement Field of Bi-Material
The geometry of the specimen is the same as the finite element model shown in Fig. 2. The bi-material specimen is composed of annealed pure aluminum (A1050-O) in the upper part and work hardened tough pitch copper (C1100-1/2H) in the lower part. The boundary is arranged on the crack tip extension line. Stress/strain curves of each material obtained by uniaxial tensile tests are shown in Fig. 11. Two rods made of these materials are jointed by a friction welding method. The boundary strength of the specimen cut from near the edge of the rod is different from that cut from the center of the rod, because the relative velocity of the two materials for welding is different for the edge and the center. The specimens cut from near the outer of the rod are called as the specimen group Set 1 whereas the specimens from the center are called as Set 2. In each Sets, the specimens which have the various crack length are created. The experimental procedure is the same as the case of homogeneous material. The area imaged is 24 × 18 mm2. The object size on the image is 24 m/pixel. The displacements are obtained using
(a)
(b)
Fig. 14 J values with various crack length: (a) Set 1; (b) Set 2 digital image correlation with the resolution of the displacement of 0.02 pixels (4.8 × 10-4 mm). A subset size of 31 × 31 pixels with a step size of 8 pixels is used for the correlation between the images before and after the deformation.
Figure 12 shows the displacement distributions in the area of 12 × 12 mm2 around the interface crack tip for the specimen under the load of P = 3004 N. Since the subset size is 0.8 mm × 0.8 mm, the range for excluding data is 0.8 mm to the crack surface. Figure 13 shows the J values obtained from the displacements measured using digital image correlation. From the result, the domain independent property is observed. Additionally, Fig. 14 represents the J-integral values with the load change for the various crack lengths for the specimen Sets 1 and 2. In each specimen, the topmost plots indicate the J-integral value when the specimen exhibits the fracture. Therefore, the topmost values can be considered as the critical J-integral for the fracture. The values of the critical J-integral coincide closely among various crack lengths in Fig. 14. From the results in Fig. 14, however, the difference of the crack length does not affect clearly the critical J value. On the other hand, the J values at break in Set 2 in Fig. 14(b) are lower than the values of Set 1. It is considered that the strength of the joint of the specimens in Set 2 is weaker than those in Set 1. As a result of a microscopic observation of the reaction layer [24], the reaction layer is thicker toward the outer region, and the poor jointing is observed around the center of the round bar. Moreover, exposed coppers are observed in the reaction layer. It is considered the reaction layer is not formed enough in Set 2 because the frictional heat is lower than Set 1. From the above, the critical J-integral values when the fractures occur can be evaluated even if crack
6 lengths are different. Therefore, the proposed method is effective for J-integral evaluations under various conditions. 5. Conclusion
The method for evaluating J-integral without modeling the elastic-plastic stress/strain relationship is proposed. Accordingly, the J-integral evaluation from the displacement around the crack tip can be evaluated more accurately. The effectiveness of the evaluation method is verified by applying to this method to the displacement fields obtained by the finite element analysis and digital image correlation. Therefore, it is expected that the proposed method can be applied to various experimental evaluation of structural components.
Nomenclature
x, y Cartesian coordinates
i, j Indices of coordinate axes
n Unit normal of a contour path
u Displacement components [m]
W Strain energy density [J/m3] Γ Contour path
ds Length increment along contour path [m]
J J-integral [Pa･m]
T Traction along the contour Γ [Pa]
A Area of the domain [m2]
q Variable of the position between the inner and the outer boundary
rin Radius of the inner boundary [m]
rout Radius of the outer boundary [m]
E Elastic modulus [Pa]
G Elastic shear modulus [Pa]
s Stress deviator [Pa] σ Stress [Pa]
γ Shear stress [Pa] ε Strain
φ Work hardening parameter [1/Pa] σe Equivalent stress [Pa]
εe(p) Equivalent plastic strain References
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Stress-Strain Behavior of Ti-Nb Alloys under Compressions
along Linear Strain Paths and Bilinear Plane Strain Path
Ichiro SHIMIZU1, Yoshito TAKEMOTO2, Shinichi ISHIKAWA2 and Tomohiro KUMURA1 1 Department of Mechanical Engineering, Okayama University of Science, Okayama 700-0005, Japan 2 Graduate School of Natural Science and Technology, Okayama University, Okayama 700-8530, Japan
(Received 10 January 2017; received in revised form 20 March 2017; accepted 24 March 2017)
Abstract: Titanium alloys containing beta stabilizing elements exhibit several superior properties compared to other conventional metals. However, there are unknowns about the mechanical behavior of titanium alloys under biaxial compressive stress conditions. In this study, uniaxial compression and biaxial compressions along linear strain paths and bilinear strain path were performed on titanium-niobium alloys (Ti-Nb alloys) in order to investigate the influence of Nb contents on the compressive deformation behaviors. The results of compressions along linear strain paths revealed that the flow stress decreased while the ductility increased with the increase of the Nb content. It was found that the phenomenon was mainly due to changes of the microstructure and the primary plastic deformation mechanism. Those changes also induce anisotropic hardening. Based on those results, quantitative expression of the stress-strain relation with taking into account the Nb contents was discussed. The investigation on the effect of strain path change on the flow stress revealed the niobium dependency of the stress-strain response before and after the abrupt strain path change.
Keywords: Plasticity, Titanium alloy, Beta stabilizing element, Stress-strain relation, Biaxial compression 1. Introduction
Titanium alloys exhibit superior mechanical properties compared to other conventional metals, such as good corrosion resistance, high specific strength, and biocompatibility. In the last few decades, many studies have been reported on the development of the titanium alloys in order to widen the application fields in industries [1-3].
The common crystal structure of titanium alloys at room temperature is a hexagonal close-packed (HCP) structure, namely alpha phase. However, titanium alloy can be a body-centered cubic (bcc) structure, namely beta phase, by alloying with beta stabilizing elements, e.g. Mo, V, Nb, Cr, and so on. Since the beta phase improves the plastic formability, the influence of the beta stabilizer content on the microstructure have been investigated [4, 5]. Especially the metastable beta-type titanium alloys, which contain less amount of beta stabilizing elements than the stable beta alloys, obtained by solution treatment exhibits specific deformation characteristics. Those characteristics are closely related to the change of plastic deformation mechanisms. For instance,
332 113 twinning, which appears in some metastable beta-type titanium alloys [6, 7], is known to induce low yield strength and high work hardening. However, most of previous studies on the metastable beta titanium alloys utilize a uniaxial tensile test or a hardness testing, and there are still unknowns about the plastic behavior of titanium alloys under biaxial stress conditions.Under those circumstances, the authors have started to investigate uniaxial and biaxial compressive deformation behavior of titanium-niobium (Ti-Nb) alloys, which belong to typical metastable beta titanium alloys. In a previous work [8], we reported basic findings about the uniaxial and biaxial compressive deformation behaviors of Ti-Nb alloys containing different amount of niobium. In this study, we
performed the uniaxial and biaxial compressions along linear strain paths and bilinear strain path in order to investigate the influence of Nb contents on the compressive deformation behavior. Attention was paid especially on the change of primary deformation mechanism with the Nb contents under biaxial compressive conditions. Based on our previous work [8], this study aims to (1) clarify the compressive behavior under linear strain path with taking into account the plastic deformation mechanism, (2) its quantitative expression with regard to the stress-strain relation, and (3) clarify the influence of strain path change on the compressive behavior of Ti-Nb alloys.
2. Experimental Procedure 2.1 Materials
Materials used were Ti-Nb alloys with different niobium contents, from 20 to 45 mass% Nb. The materials were arc melted and hot rolled at 1173 K to the thickness reduction of 57%. The chemical compositions of Ti-Nb alloys were summarized in Table 1. Hereafter, the rolling direction, the width direction, and the thickness direction of plates were referred to as x-, y-, and z-direction, respectively. The cube shape specimens with the edge length of 5 mm were cut out from the plates by using an electric discharge machining. All the specimens were solution treated at 1223 K for 5 min in vacuum and then quenched into ice water. The hard
