Anisotropic Elastoplastic Damage Mechanics Method to Predict Fatigue Life of the Structure

New damagemechanicsmethod is proposed to predict the low-cycle fatigue life ofmetallic structures undermultiaxial loading.The microstructure mechanical model is proposed to simulate anisotropic elastoplastic damage evolution. As the micromodel depends on fewmaterial parameters, the present method is very concise and suitable for engineering application.Thematerial parameters in damage evolution equation are determined by fatigue experimental data of standard specimens. By employing further development on the ANSYS platform, the anisotropic elastoplastic damage mechanics-finite element method is developed. The fatigue crack propagation life of satellite structure is predicted using the present method and the computational results comply with the experimental data very well.


Introduction
For multiaxial fatigue problem, many methods [1][2][3][4] have been developed over the past decades.A series of fatigue failure approaches are classified as follows: the linear rule method [5], the equivalent stress method [6], the critical plane method [7,8], the stress invariant method [9], the damage mechanics method [10][11][12][13][14][15][16], and so on.However, there is no existing method accepted extensively in the engineering field at present.Hence, further work still must be done.Recently, there is a trend in favor of both the critical plane approach and the damage mechanics approach.
The critical plane method [17,18] can predict the fatigue crack orientation.For the damage mechanics method [15,19], a great deal of attention has been paid.The damage mechanics method [13,14,[20][21][22][23] deals with the mechanical behavior of the deteriorated materials.Damage models can be classified as follows: isotropic damage and anisotropic damage.As the isotropic damage model is characterized with simple constitutive relationship, it has been widely applied in the engineering field.For isotropic damage model, damage mechanicsfinite element method [19] is proposed to predict the fatigue life of the engineering structure.However, most of the fatigue problem is anisotropic.Thus anisotropic damage model will be more valuable.Many models [24][25][26] have been proposed to describe anisotropic damage properties of materials.An elastic microstructure model in [27] is proposed to evaluate the anisotropic elastic fatigue problem.Some models [11,12] for dealing with elastoplastic fatigue problems are developed.However, in engineering applications we do not find extensive applications of them because isotropic schemes can be put more easily into available computer codes, although such approaches are less realistic.Thus, for the low-cycle fatigue problem, it will be very valuable to propose an anisotropic elastoplastic model for the engineering application.
In this paper, damage mechanics method is proposed to predict the low-cycle fatigue life of metallic structures.Referring to the elastic micromodel [27] and the critical plane method [17,18], a micro elastoplastic mechanical model is established.Considering the hysteresis energy [28], the damage evolution equation is advanced.The material parameters in the damage evolution equation are determined by the low-cycle fatigue experimental data of standard specimens.Based on the further development on the ANSYS platform, anisotropic elastoplastic damage mechanics-finite element method is developed.In this paper, the low-cycle fatigue crack initiation and propagation life of the structure are predicted using the present method.For a real component of

The Anisotropic Elastoplastic
Damage Theory As we all know, for the metallic materials, the iterative appearance of the shear slip will result in the crack initiation.Boom-panel structure in the micromodel simulates shearing slip band.The present micromodel is built at three orthogonal planes with maximum shear stress.Then the relationship between the micromodel and each element in the structure is built and shown in Figure 2. Global coordinate system belongs to the element in the structure and local coordinate system belongs to the micromodel.

Stress-Strain
Relationships for the Micromodel.The minimum cell of the micromodel is shown in Figure 3. Based on the minimum cell in Figure 3, parameters involved in the micromodel will be discussed in detail in this section.
In the local coordinate system, the points for the micromodel are as follows: ,   denote the stress tensor components and the strain tensor components of the micromodel, respectively,  1 ,  1 denote the stress tensor components and the strain tensor components of elastic block, respectively,  2 ,  2 denote the stress tensor components and the strain tensor components of boom-panel structure, respectively,  1 is the length of elastic block, and  2 is the length of boom-panel structure.

Elastic Analysis of the Micromodel.
For elastic block, the linear elastic constitutive model is expressed as For boom-panel structure, an elastic analysis is written as for the boom for the panel (3) in which  1 is the elastic shear modulus of panel and   is the elastic modulus of boom.

The Analysis of the Elastic Constitutive Relationship.
In the local coordinate system, the elastic part of constitutive model of the micromodel is expressed as One element in the structure The micromodel The global coordinate system The local coordinate system The relationship between one element in the structure and the micromodel.

Elastic block without simulating damage
Boom-panel structure with simulating damage in which  1 is Young's modulus of elastic block,  1 is the shear modulus of elastic block, and  1 is Poisson's ratio of elastic block.
In the global coordinate system, the elastic part of constitutive model can be expressed as in which  0  ,  0  stand for the stress tensor components and the strain tensor components of one element in the structure, respectively,  is Young's modulus,  is the shear modulus, and  is Poisson's ratio.

Elastic Material Parameters of the Micromodel.
From ( 4) and ( 5), we have The shear modulus  1 of elastic block is also written as From ( 6) and ( 7), then in which ,  are known and  =  2 / 1 is a constant.Then   ,  1 ,  1 ,  1 ,  1 ,  1 , and  2 are known. constant is a material parameter that can vary in the range defined by (26)..For the micromodel in Figure 3, block is elastic and boom-panel structure is elastoplastic.For boom-panel structure, panel is assumed as bilinear plastic and boom is elastic.For the flow rule of the plastic behavior, the kinematic hardening law is adopted in this article.In the local coordinate system, the elastoplastic constitutive relation of the micromodel is expressed as

The Elastoplastic Constitutive Equation
where in which  2 is the shear yield stress of boom-panel structure,  2 indicates the shear yield strain of boom-panel structure, and  2 signifies the plastic shear modulus of panel.

Plastic Material Parameters of the Micromodel.
In order to determine  2 , a simple stress state is considered.Schematic of a plate subject to tension is shown in Figure 4.The structure is in plane-stress condition.For this case, all of the micromodels are the same and built at the position with  = 45 ∘ .
In the global coordinate system, the strain  0 of the structure is defined as in which Δ 0  is the displacement increment of the micromodel along  0 direction in the global coordinate system.By the coordinate system transformation, the displacement increment Δ 0  is expressed as in which in which Δ  , Δ  are the displacement increment of the micromodel along  or  direction in the local coordinate system, respectively.The stress values   ,   , and   in the local coordinate system are From ( 15)- (20), then the strain  0 of the structure is expressed as In the global coordinate system, the strain  0 of the structure is expressed as From ( 21) and ( 22), the following requirement must be satisfied: Then we have From ( 11), (13), and (24), then the inverse of the plastic shear modulus  2 is For the plastic shear modulus  2 , the condition 0 ≤  2 <  1 must be satisfied.From ( 11) and ( 25), the range of the constant  is Substituting ( 8)-( 14), (16), and ( 25) into (15), we obtain in which   is a variable and   =   /2.

Damage Constitutive Law for the Micromodel.
For this micromodel, elastic block does not undergo damage and boom-panel structure can simulate the damage failure.In this section, a simple case in Figure 4 is discussed first.For this case, the micromodel is at the position with  = 45 ∘ or 135 ∘ .For boom-panel structure, the constitutive law coupling with damage is displayed in Figure 5.
For a plate subject to uniaxial loading, constitutive equations including damage can be written as follows:   (29) in which   ,   denote the damage variables of boom and   denotes the damage variable of panel.

Boom Panel
If the plate is subject to multiaxial loading, constitutive equations including damage become in which   ,   ,   denote the damage variables of boom and   ,   ,   denote the damage variables of panel.

Relationship between Boom Damage and Panel Damage.
For a plate subject to uniaxial loading (see Figure 4), the following requirements need to be satisfied: For the case in Figure 4, the isotropic property is still satisfied in  plane when  2 ≤  2 ⋅ (1 −   ).So the requirement is as follows: From ( 28), we have Substituting ( 10)-( 13), (A.1), (32), and (34) into (33), then From ( 32) and ( 35), we have Similarly, For the plate in the  plane, while   = 0 and For the plate in the  plane, while   = 0 and   =   = 0 In order to satisfy (36)-(38), then the following conclusions are obtained: Thus three damage variables   ,   ,   are independent.

Damage Evolution Equation.
Based on hysteresis energy, the damage evolution model depends on three variables,   ,   ,   .Let us now consider the damage equation of the first variable   .For the micromodel, the damage failure of the materials is simulated by boom-panel structure.The hysteresis loop is considered by the shear stress of panel  2 .The hysteresis loop for the panel of the micromodel is shown in Figure 6.For this method, the same behavior of the hysteresis loop remains in time when fatigue increases.
For each loop, the hysteresis energy of panel is The total hysteresis energy of panel is assumed as Then the damage evolution equation is defined as in which , , , and   are material parameters.
Similarly, the damage evolution equation for the damage variables   ,   can also be obtained.Then the damage evolution equations of the micromodel for three damage variables   ,   ,   are ( = , , ) . (43)

The Damage Mechanics-Finite Element Method
The present damage mechanics model is implemented in the commercial finite element software ANSYS.Computations proceed as follows: (1) Stress distribution of the structure is computed first in order to find the critical element.(2) The increment of damage extent Δ of critical element is given, Δ = constant, and the magnitude of damage extent increment will be checked by the convergence verification.Then the corresponding fatigue life increments Δ  (), Δ  (), and Δ  () of critical element are (4) From the damage evolution equation ( 43), the damage variable increment of the elements can be obtained when Δ min () is known.The damage extent increments of critical element are Then damage extent increments of other elements are (5) Modify the local coordinate system and material properties of damaged elements according to (A.1).
The new stress field acting in the structure is determined via FE analysis.At the same time, the level of damage is computed for each element:  8 is discussed while   > 1.The low-cycle fatigue life of the crack initiation and propagation is predicted using the present damage mechanics method.The mean fatigue crack life for the structure of 35Cr2Ni4MoA is presented in Figure 9.The crack propagation life curve for notched structure is shown in Figure 10 with constant strain Δ/2 = 2.8 − 3.

Real Satellite Structure.
In this section, a real satellite structure of 5A06 in Figure 11 is investigated while  = −1.

The low-cycle fatigue experimental curve
The results derived from the present method   The material parameters in damage evolution equation are listed in Table 2.
The finite element model of real satellite structure is shown in Figure 11.In this paper, the fatigue crack propagation life of real satellite structure is predicted using the present method and shown in Figure 12.
For the satellite structure, the experimental low-cycle fatigue lifetime is as follows: Log  = 2.7604 when the crack length  = 4.875 mm.
For the satellite structure, the computational low-cycle fatigue lifetime is as follows: Log  = 2.7419 when the crack length  = 4.875 mm.
The relative error between the calculated results and the experimental data is 0.67%.Hence, the fatigue life prediction

Advances in Materials Science and Engineering
The present method

Conclusions
In this article, a new damage mechanics method is proposed to predict the low-cycle fatigue life of metallic structures under multiaxial loading: (1) A microstructure mechanical model is proposed to simulate the anisotropic damage failure.As the micromodel depends on few material parameters, the present method is very concise and suitable for engineering application.
(2) Considering the hysteresis energy, the damage evolution equation is constructed.The material parameters are obtained by the low-cycle fatigue experimental results of standard specimens.
(3) Based on the further development on the ANSYS platform, anisotropic elastoplastic damage mechanicsfinite element method is developed.
(4) The fatigue crack initiation and propagation life for notched structure of 35Cr2Ni4MoA are predicted using the present method.
(5) The fatigue crack growth life of a satellite structure is predicted and the computational results fit well with the experimental data.

Figure 3 :
Figure 3: The minimum cell of the micromodel.

Figure 4 :
Figure 4: Schematic of a plate subject to tension.

Figure 5 :
Figure 5: The constitutive law with damage for boom-panel structure.

Figure 6 :
Figure 6: The hysteresis loop for the panel in  plane in the local coordinate system.

Figure 7 :
Figure 7: Comparison between FE results and experimental data for material 35Cr2Ni4MoA.

Figure 9 :Figure 10 :
Figure 9: The mean fatigue crack initiation life curve for notched structure.

Figure 11 :
Figure 11: The finite element model of satellite structure.
(  ) + Δ  (  ) , ( = , , )   ( +1 ) =   (  ) + Δ  (  ) , From (39), the damage variables   ,   ,   of boom can be obtained when   ,   ,   are known.The process from steps (2) to (5) will be repeated until one of the boom damage extents   ,   ,   of critical element is equal to 1.In this step, the failure of critical element means crack initiation in the structure.
(7)Repeat the process from (2) to(7), until the crack length of the engineering structure is equal to .The corresponding fatigue life   with crack length  is   = ∑ Δ min () .

Table 1 :
Material properties in damage evolution equation for material 35Cr2Ni4MoA with   = 1.
for satellite structure is acceptable in the engineering application.

Table 2 :
Material properties in damage evolution equation for material 5A06 with   = 1.