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

Authors: Hualiang Wan; Qizhi Wang; Zheng Zhang
Journal: Advances in Materials Science and Engineering
 (2016)

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## Abstract

New damage mechanics method is proposed to predict the low-cycle fatigue life of metallic structures under multiaxial loading. The microstructure mechanical model is proposed to simulate anisotropic elastoplastic damage evolution. As the micromodel depends on few material parameters, the present method is very concise and suitable for engineering application. The material 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.


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## Body

## 1. Introduction

For multiaxial fatigue problem, many methods [1–4] 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 [10–16], 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, 20–23] 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 mechanics-finite 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 [24–26] 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 satellite structure, the fatigue life is predicted by the present method.


## 2. The Anisotropic Elastoplastic Damage Theory

### 2.1. The Microstructure Mechanical Model

In this section, a microstructure mechanical model is proposed to simulate the anisotropic elastoplastic damage failure. The microstructure mechanical model, as shown in Figure1, is constituted by elastic block and boom-panel structure unit. Boom-structure is used to simulate the damage failure and elastic block does not undergo damage. For boom-panel structure model, boom is used to transfer the tensile and compressive loading and panel is used to transfer the shear loading.Figure 1
The microstructure mechanical model.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 Figure2. Global coordinate system belongs to the element in the structure and local coordinate system belongs to the micromodel.Figure 2
The relationship between one element in the structure and the micromodel.


### 2.2. Stress-Strain Relationships for the Micromodel

The minimum cell of the micromodel is shown in Figure3. Based on the minimum cell in Figure 3, parameters involved in the micromodel will be discussed in detail in this section.Figure 3
The minimum cell of the micromodel.In the local coordinate system, the points for the micromodel are as follows:(1)εij=l1l1+l2·εij1+l2l1+l2·εij2σij=l1l1+l22·σij1=l2l1+l22·σij2.σij, εij denote the stress tensor components and the strain tensor components of the micromodel, respectively, σij1, εij1 denote the stress tensor components and the strain tensor components of elastic block, respectively, σij2, εij2 denote the stress tensor components and the strain tensor components of boom-panel structure, respectively, l1 is the length of elastic block, and l2 is the length of boom-panel structure.


### 2.3. Elastic Analysis of the Micromodel

For elastic block, the linear elastic constitutive model is expressed as(2)εij1=1E11+μ1·σij1-δij·μ1E1·σkk1.For boom-panel structure, an elastic analysis is written as(3)εx2=1KF·σx2εy2=1KF·σy2εz2=1KF·σz2for the boomεxy2=1KS1·τxy2εyz2=1KS1·τyz2εxz2=1KS1·τxz2for the panelin which KS1 is the elastic shear modulus of panel and KF is the elastic modulus of boom.


#### 2.3.1. The Analysis of the Elastic Constitutive Relationship

In the local coordinate system, the elastic part of constitutive model of the micromodel is expressed as(4)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·l1+l2l2·σzεxy=12G1·l1+l2l1+1KS1·l1+l2l2·τxyεxz=12G1·l1+l2l1+1KS1·l1+l2l2·τxzεyz=12G1·l1+l2l1+1KS1·l1+l2l2·τyzin which E1 is Young’s modulus of elastic block, G1 is the shear modulus of elastic block, and μ1 is Poisson’s ratio of elastic block.In the global coordinate system, the elastic part of constitutive model can be expressed as(5)εx0=1E·σx0-μ·σy0+σz0εy0=1E·σy0-μ·σx0+σz0εz0=1E·σz0-μ·σx0+σy0εxy0=12G·τxy0εxz0=12G·τxz0εyz0=12G·τyz0in which σij0, εij0 stand for the stress tensor components and the strain tensor components of one element in the structure, respectively, E is Young’s modulus, G is the shear modulus, and μ is Poisson’s ratio.


#### 2.3.2. Elastic Material Parameters of the Micromodel

From (4) and (5), we have(6)1E=1E1·l1+l2l1+1KF·l1+l2l2μ=μ1/E1·l11/E1·l1+1/KF·l2=μ11+E1/KF·l1/l212G=1+μE=12G1·l1+l2l1+1KS1·l1+l2l2.The shear modulus G1 of elastic block is also written as(7)12·G1=1+μ1E1.From (6) and (7), then(8)KF=E·1+μ·μ1μ1-μ2(9)E1=E·1+μ1+μ1·μ1μ2(10)12G1=1E·1+μ·1+μ1·μμ12(11)1KS1=1E·1+μ·μ1-μμ12(12)l1+l2l1=1+c=1+μ1+μ1·μ1μ(13)l1+l2l2=1+1c=μ1·1+μμ1-μ(14)μ1=μ·1+c1-μ·c,in which E, μ are known and c=l2/l1 is a constant. Then KF, E1, G1, KS1, μ1, l1, and l2 are known.c constant is a material parameter that can vary in the range defined by (26).


### 2.4. Plastic Analysis of the Micromodel

#### 2.4.1. The Elastoplastic Constitutive Equation

For the micromodel in Figure3, 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(15)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·l1+l2l2·σzεxy=τxy2G1·l1+l2l1+ε2S+τxy2-τ2SKS2·l2l1+l2εxz=τxz2G1·l1+l2l1+ε2S+τxz2-τ2SKS2·l2l1+l2εyz=τyz2G1·l1+l2l1+ε2S+τyz2-τ2SKS2·l2l1+l2,where(16)τ2S=σs2·l1+l2l22τxy2=τxy·l1+l2l22ε2S=τ2SKS1·l1+l2l22in which τ2S is the shear yield stress of boom-panel structure, ε2S indicates the shear yield strain of boom-panel structure, and KS2 signifies the plastic shear modulus of panel.


#### 2.4.2. Plastic Material Parameters of the Micromodel

In order to determineKS2, 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 α=45°.Figure 4
Schematic of a plate subject to tension.In the global coordinate system, the strainε0 of the structure is defined as(17)ε0=Δuy02·l1+l2in which Δuy0 is the displacement increment of the micromodel along Y0 direction in the global coordinate system.By the coordinate system transformation, the displacement incrementΔuy0 is expressed as(18)Δuy0=22·Δux+Δuyin which(19)Δux=εx·l1+l2+εxy·l1+l2Δuy=εy·l1+l2+εxy·l1+l2in which Δux, Δuy are the displacement increment of the micromodel along X or Y direction in the local coordinate system, respectively.The stress valuesσx, σy, and τxy in the local coordinate system are(20)σx=σy=σ02,τxy=σ02.From (15)–(20), then the strain ε0 of the structure is expressed as(21)ε0=εx+εy2+εxy=1E+l1+l22·l2·1KS2-1KS1·σ0-l1+l22·l2·1KS2-1KS1·σs.In the global coordinate system, the strain ε0 of the structure is expressed as(22)ε0=εs+σ0-σsK2=σsE+σ0-σsK2.From (21) and (22), the following requirement must be satisfied:(23)εx+εy2+εxy=εs+σ0-σsK2.Then we have (24)1E+l1+l22·l2·1KS2-1KS1=1K2.From (11), (13), and (24), then the inverse of the plastic shear modulus KS2 is(25)1KS2=1K2-1E·2·μ1-μμ1·1+μ+1E·1+μ·μ1-μμ12.For the plastic shear modulus KS2, the condition 0≤KS2<KS1 must be satisfied. From (11) and (25), the range of the constant c is(26)c≥2·K2-EK2·E+2·E-K2.Substituting (8)–(14), (16), and (25) into (15), we obtain(27)εx=1E·σx-μ·σy+σzεy=1E·σy-μ·σx+σzεz=1E·σz-μ·σx+σyεxy=12G+2·1K2-1E·τxy-2·1K2-1E·τSεyz=12G+2·1K2-1E·τyz-2·1K2-1E·τSεxz=12G+2·1K2-1E·τxz-2·1K2-1E·τSin which τS is a variable and τS=σS/2.


### 2.5. 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 Figure4 is discussed first. For this case, the micromodel is at the position with α=45° or 135°. For boom-panel structure, the constitutive law coupling with damage is displayed in Figure 5.Figure 5
The constitutive law with damage for boom-panel structure.For a plate subject to uniaxial loading, constitutive equations including damage can be written as follows:(i)
Whenτxy2≤τ2S·1-Dxy, (28)εxD=1E1·l1+l2l1·σx-μ1·σy+1KF·1-Dx·l1+l2l2·σx.εyD=1E1·l1+l2l1·σy-μ1·σx+1KF·1-Dy·l1+l2l2·σyεxyD=τxy2G1·l1+l2l1+τxyKS1·1-Dxy·l1+l2l2=τxy2·G1D.(ii)
Whenτxy2>τ2S·1-Dxy,(29)εxD=1E1·l1+l2l1·σx-μ1·σy+1KF·1-Dx·l1+l2l2·σxεyD=1E1·l1+l2l1·σy-μ1·σx+1KF·1-Dy·l1+l2l2·σyεxyD=τxy2G1·l1+l2l1+ε2S+τxy2-τ2S·1-DxyKS2·1-Dxy·l2l1+l2in which Dx, Dy denote the damage variables of boom and Dxy denotes the damage variable of panel.If the plate is subject to multiaxial loading, constitutive equations including damage become(30)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·1-Dx·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·1-Dy·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·1-Dz·l1+l2l2·σzεxy=τxy2G1·l1+l2l1+ε2S+τxy2-τS·1-DxyKS2·1-Dxy·l2l1+l2εxz=τxz2G1·l1+l2l1+ε2S+τxz2-τS·1-DxzKS2·1-Dxz·l2l1+l2εyz=τyz2G1·l1+l2l1+ε2S+τyz2-τS·1-DyzKS2·1-Dyz·l2l1+l2in which Dx, Dy, Dz denote the damage variables of boom and Dxy, Dxz, Dyz denote the damage variables of panel.


### 2.6. Relationship between Boom Damage and Panel Damage

For a plate subject to uniaxial loading (see Figure4), the following requirements need to be satisfied:(31)Dz=0Dxz=Dyz=0(32)Dx=Dy.For the case in Figure 4, the isotropic property is still satisfied in XY plane when τxy2≤τ2S·1-Dxy. So the requirement is as follows:(33)1+μ12DE1D=12·G1D.From (28), we have(34)12·G1D=12G1·l1+l2l1+1KS1·1-Dxy·l1+l2l2.Substituting (10)–(13), (A.1), (32), and (34) into (33), then(35)1E·μ+μμ1+11-Dx·1-μμ1=1E·μ·1+μ1μ1+11-Dxy·μ1-μμ1.From (32) and (35), we have(36)Dxy=Dx=Dy.Similarly, For the plate in the XZ plane, while Dx=0 and Dxy=Dxz=0(37)Dx=Dz=Dxz.For the plate in the YZ plane, while Dx=0 and Dxy=Dxz=0(38)Dy=Dz=Dyz.In order to satisfy (36)–(38), then the following conclusions are obtained:(39)Dx=Dxy+DxzDy=Dxy+DyzDz=Dxz+Dyz.Thus three damage variables Dxy, Dyz, Dxz are independent.


### 2.7. Damage Evolution Equation

Based on hysteresis energy, the damage evolution model depends on three variables,Dxy, Dxz, Dyz. Let us now consider the damage equation of the first variable Dxy. 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 τxy2. 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.Figure 6
The hysteresis loop for the panel inXY plane in the local coordinate system.For each loop, the hysteresis energy of panel is(40)W0=4·τxy-σs2·1-Dxy·1KS2-1KS1·σs2·l1+l2l24.The total hysteresis energy of panel is assumed as (41)Wcr=ωcr·1KS2-1KS1·σs2·l1+l2l24.Then the damage evolution equation is defined as(42)dDxydN=α·W0/Wcrm1-Dxyβ=ατxy-σs/2·1-Dxymωcrm·1-Dxyβ=τxy-σs/2·1-Dxymωcr∗·1-Dxyβ,ωcr∗=αωcrmin which α, β, m, and ωcr are material parameters.Similarly, the damage evolution equation for the damage variablesDxz, Dyz can also be obtained. Then the damage evolution equations of the micromodel for three damage variables Dxy, Dxz, Dyz are(43)dDijdN=α·W0/Wcrm1-Dijβ=ατij-σs/2·1-Dijmωcrm·1-Dijβ=τij-σs/2·1-Dijmωcr∗·1-Dijβ,ij=xy,xz,yz.


## 2.1. The Microstructure Mechanical Model

In this section, a microstructure mechanical model is proposed to simulate the anisotropic elastoplastic damage failure. The microstructure mechanical model, as shown in Figure1, is constituted by elastic block and boom-panel structure unit. Boom-structure is used to simulate the damage failure and elastic block does not undergo damage. For boom-panel structure model, boom is used to transfer the tensile and compressive loading and panel is used to transfer the shear loading.Figure 1
The microstructure mechanical model.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 Figure2. Global coordinate system belongs to the element in the structure and local coordinate system belongs to the micromodel.Figure 2
The relationship between one element in the structure and the micromodel.


## 2.2. Stress-Strain Relationships for the Micromodel

The minimum cell of the micromodel is shown in Figure3. Based on the minimum cell in Figure 3, parameters involved in the micromodel will be discussed in detail in this section.Figure 3
The minimum cell of the micromodel.In the local coordinate system, the points for the micromodel are as follows:(1)εij=l1l1+l2·εij1+l2l1+l2·εij2σij=l1l1+l22·σij1=l2l1+l22·σij2.σij, εij denote the stress tensor components and the strain tensor components of the micromodel, respectively, σij1, εij1 denote the stress tensor components and the strain tensor components of elastic block, respectively, σij2, εij2 denote the stress tensor components and the strain tensor components of boom-panel structure, respectively, l1 is the length of elastic block, and l2 is the length of boom-panel structure.


## 2.3. Elastic Analysis of the Micromodel

For elastic block, the linear elastic constitutive model is expressed as(2)εij1=1E11+μ1·σij1-δij·μ1E1·σkk1.For boom-panel structure, an elastic analysis is written as(3)εx2=1KF·σx2εy2=1KF·σy2εz2=1KF·σz2for the boomεxy2=1KS1·τxy2εyz2=1KS1·τyz2εxz2=1KS1·τxz2for the panelin which KS1 is the elastic shear modulus of panel and KF is the elastic modulus of boom.


### 2.3.1. The Analysis of the Elastic Constitutive Relationship

In the local coordinate system, the elastic part of constitutive model of the micromodel is expressed as(4)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·l1+l2l2·σzεxy=12G1·l1+l2l1+1KS1·l1+l2l2·τxyεxz=12G1·l1+l2l1+1KS1·l1+l2l2·τxzεyz=12G1·l1+l2l1+1KS1·l1+l2l2·τyzin which E1 is Young’s modulus of elastic block, G1 is the shear modulus of elastic block, and μ1 is Poisson’s ratio of elastic block.In the global coordinate system, the elastic part of constitutive model can be expressed as(5)εx0=1E·σx0-μ·σy0+σz0εy0=1E·σy0-μ·σx0+σz0εz0=1E·σz0-μ·σx0+σy0εxy0=12G·τxy0εxz0=12G·τxz0εyz0=12G·τyz0in which σij0, εij0 stand for the stress tensor components and the strain tensor components of one element in the structure, respectively, E is Young’s modulus, G is the shear modulus, and μ is Poisson’s ratio.


### 2.3.2. Elastic Material Parameters of the Micromodel

From (4) and (5), we have(6)1E=1E1·l1+l2l1+1KF·l1+l2l2μ=μ1/E1·l11/E1·l1+1/KF·l2=μ11+E1/KF·l1/l212G=1+μE=12G1·l1+l2l1+1KS1·l1+l2l2.The shear modulus G1 of elastic block is also written as(7)12·G1=1+μ1E1.From (6) and (7), then(8)KF=E·1+μ·μ1μ1-μ2(9)E1=E·1+μ1+μ1·μ1μ2(10)12G1=1E·1+μ·1+μ1·μμ12(11)1KS1=1E·1+μ·μ1-μμ12(12)l1+l2l1=1+c=1+μ1+μ1·μ1μ(13)l1+l2l2=1+1c=μ1·1+μμ1-μ(14)μ1=μ·1+c1-μ·c,in which E, μ are known and c=l2/l1 is a constant. Then KF, E1, G1, KS1, μ1, l1, and l2 are known.c constant is a material parameter that can vary in the range defined by (26).


## 2.3.1. The Analysis of the Elastic Constitutive Relationship

In the local coordinate system, the elastic part of constitutive model of the micromodel is expressed as(4)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·l1+l2l2·σzεxy=12G1·l1+l2l1+1KS1·l1+l2l2·τxyεxz=12G1·l1+l2l1+1KS1·l1+l2l2·τxzεyz=12G1·l1+l2l1+1KS1·l1+l2l2·τyzin which E1 is Young’s modulus of elastic block, G1 is the shear modulus of elastic block, and μ1 is Poisson’s ratio of elastic block.In the global coordinate system, the elastic part of constitutive model can be expressed as(5)εx0=1E·σx0-μ·σy0+σz0εy0=1E·σy0-μ·σx0+σz0εz0=1E·σz0-μ·σx0+σy0εxy0=12G·τxy0εxz0=12G·τxz0εyz0=12G·τyz0in which σij0, εij0 stand for the stress tensor components and the strain tensor components of one element in the structure, respectively, E is Young’s modulus, G is the shear modulus, and μ is Poisson’s ratio.


## 2.3.2. Elastic Material Parameters of the Micromodel

From (4) and (5), we have(6)1E=1E1·l1+l2l1+1KF·l1+l2l2μ=μ1/E1·l11/E1·l1+1/KF·l2=μ11+E1/KF·l1/l212G=1+μE=12G1·l1+l2l1+1KS1·l1+l2l2.The shear modulus G1 of elastic block is also written as(7)12·G1=1+μ1E1.From (6) and (7), then(8)KF=E·1+μ·μ1μ1-μ2(9)E1=E·1+μ1+μ1·μ1μ2(10)12G1=1E·1+μ·1+μ1·μμ12(11)1KS1=1E·1+μ·μ1-μμ12(12)l1+l2l1=1+c=1+μ1+μ1·μ1μ(13)l1+l2l2=1+1c=μ1·1+μμ1-μ(14)μ1=μ·1+c1-μ·c,in which E, μ are known and c=l2/l1 is a constant. Then KF, E1, G1, KS1, μ1, l1, and l2 are known.c constant is a material parameter that can vary in the range defined by (26).


## 2.4. Plastic Analysis of the Micromodel

### 2.4.1. The Elastoplastic Constitutive Equation

For the micromodel in Figure3, 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(15)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·l1+l2l2·σzεxy=τxy2G1·l1+l2l1+ε2S+τxy2-τ2SKS2·l2l1+l2εxz=τxz2G1·l1+l2l1+ε2S+τxz2-τ2SKS2·l2l1+l2εyz=τyz2G1·l1+l2l1+ε2S+τyz2-τ2SKS2·l2l1+l2,where(16)τ2S=σs2·l1+l2l22τxy2=τxy·l1+l2l22ε2S=τ2SKS1·l1+l2l22in which τ2S is the shear yield stress of boom-panel structure, ε2S indicates the shear yield strain of boom-panel structure, and KS2 signifies the plastic shear modulus of panel.


### 2.4.2. Plastic Material Parameters of the Micromodel

In order to determineKS2, 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 α=45°.Figure 4
Schematic of a plate subject to tension.In the global coordinate system, the strainε0 of the structure is defined as(17)ε0=Δuy02·l1+l2in which Δuy0 is the displacement increment of the micromodel along Y0 direction in the global coordinate system.By the coordinate system transformation, the displacement incrementΔuy0 is expressed as(18)Δuy0=22·Δux+Δuyin which(19)Δux=εx·l1+l2+εxy·l1+l2Δuy=εy·l1+l2+εxy·l1+l2in which Δux, Δuy are the displacement increment of the micromodel along X or Y direction in the local coordinate system, respectively.The stress valuesσx, σy, and τxy in the local coordinate system are(20)σx=σy=σ02,τxy=σ02.From (15)–(20), then the strain ε0 of the structure is expressed as(21)ε0=εx+εy2+εxy=1E+l1+l22·l2·1KS2-1KS1·σ0-l1+l22·l2·1KS2-1KS1·σs.In the global coordinate system, the strain ε0 of the structure is expressed as(22)ε0=εs+σ0-σsK2=σsE+σ0-σsK2.From (21) and (22), the following requirement must be satisfied:(23)εx+εy2+εxy=εs+σ0-σsK2.Then we have (24)1E+l1+l22·l2·1KS2-1KS1=1K2.From (11), (13), and (24), then the inverse of the plastic shear modulus KS2 is(25)1KS2=1K2-1E·2·μ1-μμ1·1+μ+1E·1+μ·μ1-μμ12.For the plastic shear modulus KS2, the condition 0≤KS2<KS1 must be satisfied. From (11) and (25), the range of the constant c is(26)c≥2·K2-EK2·E+2·E-K2.Substituting (8)–(14), (16), and (25) into (15), we obtain(27)εx=1E·σx-μ·σy+σzεy=1E·σy-μ·σx+σzεz=1E·σz-μ·σx+σyεxy=12G+2·1K2-1E·τxy-2·1K2-1E·τSεyz=12G+2·1K2-1E·τyz-2·1K2-1E·τSεxz=12G+2·1K2-1E·τxz-2·1K2-1E·τSin which τS is a variable and τS=σS/2.


## 2.4.1. The Elastoplastic Constitutive Equation

For the micromodel in Figure3, 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(15)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·l1+l2l2·σzεxy=τxy2G1·l1+l2l1+ε2S+τxy2-τ2SKS2·l2l1+l2εxz=τxz2G1·l1+l2l1+ε2S+τxz2-τ2SKS2·l2l1+l2εyz=τyz2G1·l1+l2l1+ε2S+τyz2-τ2SKS2·l2l1+l2,where(16)τ2S=σs2·l1+l2l22τxy2=τxy·l1+l2l22ε2S=τ2SKS1·l1+l2l22in which τ2S is the shear yield stress of boom-panel structure, ε2S indicates the shear yield strain of boom-panel structure, and KS2 signifies the plastic shear modulus of panel.


## 2.4.2. Plastic Material Parameters of the Micromodel

In order to determineKS2, 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 α=45°.Figure 4
Schematic of a plate subject to tension.In the global coordinate system, the strainε0 of the structure is defined as(17)ε0=Δuy02·l1+l2in which Δuy0 is the displacement increment of the micromodel along Y0 direction in the global coordinate system.By the coordinate system transformation, the displacement incrementΔuy0 is expressed as(18)Δuy0=22·Δux+Δuyin which(19)Δux=εx·l1+l2+εxy·l1+l2Δuy=εy·l1+l2+εxy·l1+l2in which Δux, Δuy are the displacement increment of the micromodel along X or Y direction in the local coordinate system, respectively.The stress valuesσx, σy, and τxy in the local coordinate system are(20)σx=σy=σ02,τxy=σ02.From (15)–(20), then the strain ε0 of the structure is expressed as(21)ε0=εx+εy2+εxy=1E+l1+l22·l2·1KS2-1KS1·σ0-l1+l22·l2·1KS2-1KS1·σs.In the global coordinate system, the strain ε0 of the structure is expressed as(22)ε0=εs+σ0-σsK2=σsE+σ0-σsK2.From (21) and (22), the following requirement must be satisfied:(23)εx+εy2+εxy=εs+σ0-σsK2.Then we have (24)1E+l1+l22·l2·1KS2-1KS1=1K2.From (11), (13), and (24), then the inverse of the plastic shear modulus KS2 is(25)1KS2=1K2-1E·2·μ1-μμ1·1+μ+1E·1+μ·μ1-μμ12.For the plastic shear modulus KS2, the condition 0≤KS2<KS1 must be satisfied. From (11) and (25), the range of the constant c is(26)c≥2·K2-EK2·E+2·E-K2.Substituting (8)–(14), (16), and (25) into (15), we obtain(27)εx=1E·σx-μ·σy+σzεy=1E·σy-μ·σx+σzεz=1E·σz-μ·σx+σyεxy=12G+2·1K2-1E·τxy-2·1K2-1E·τSεyz=12G+2·1K2-1E·τyz-2·1K2-1E·τSεxz=12G+2·1K2-1E·τxz-2·1K2-1E·τSin which τS is a variable and τS=σS/2.


## 2.5. 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 Figure4 is discussed first. For this case, the micromodel is at the position with α=45° or 135°. For boom-panel structure, the constitutive law coupling with damage is displayed in Figure 5.Figure 5
The constitutive law with damage for boom-panel structure.For a plate subject to uniaxial loading, constitutive equations including damage can be written as follows:(i)
Whenτxy2≤τ2S·1-Dxy, (28)εxD=1E1·l1+l2l1·σx-μ1·σy+1KF·1-Dx·l1+l2l2·σx.εyD=1E1·l1+l2l1·σy-μ1·σx+1KF·1-Dy·l1+l2l2·σyεxyD=τxy2G1·l1+l2l1+τxyKS1·1-Dxy·l1+l2l2=τxy2·G1D.(ii)
Whenτxy2>τ2S·1-Dxy,(29)εxD=1E1·l1+l2l1·σx-μ1·σy+1KF·1-Dx·l1+l2l2·σxεyD=1E1·l1+l2l1·σy-μ1·σx+1KF·1-Dy·l1+l2l2·σyεxyD=τxy2G1·l1+l2l1+ε2S+τxy2-τ2S·1-DxyKS2·1-Dxy·l2l1+l2in which Dx, Dy denote the damage variables of boom and Dxy denotes the damage variable of panel.If the plate is subject to multiaxial loading, constitutive equations including damage become(30)εx=1E1·l1+l2l1·σx-μ1·σy+σz+1KF·1-Dx·l1+l2l2·σxεy=1E1·l1+l2l1·σy-μ1·σx+σz+1KF·1-Dy·l1+l2l2·σyεz=1E1·l1+l2l1·σz-μ1·σx+σy+1KF·1-Dz·l1+l2l2·σzεxy=τxy2G1·l1+l2l1+ε2S+τxy2-τS·1-DxyKS2·1-Dxy·l2l1+l2εxz=τxz2G1·l1+l2l1+ε2S+τxz2-τS·1-DxzKS2·1-Dxz·l2l1+l2εyz=τyz2G1·l1+l2l1+ε2S+τyz2-τS·1-DyzKS2·1-Dyz·l2l1+l2in which Dx, Dy, Dz denote the damage variables of boom and Dxy, Dxz, Dyz denote the damage variables of panel.


## 2.6. Relationship between Boom Damage and Panel Damage

For a plate subject to uniaxial loading (see Figure4), the following requirements need to be satisfied:(31)Dz=0Dxz=Dyz=0(32)Dx=Dy.For the case in Figure 4, the isotropic property is still satisfied in XY plane when τxy2≤τ2S·1-Dxy. So the requirement is as follows:(33)1+μ12DE1D=12·G1D.From (28), we have(34)12·G1D=12G1·l1+l2l1+1KS1·1-Dxy·l1+l2l2.Substituting (10)–(13), (A.1), (32), and (34) into (33), then(35)1E·μ+μμ1+11-Dx·1-μμ1=1E·μ·1+μ1μ1+11-Dxy·μ1-μμ1.From (32) and (35), we have(36)Dxy=Dx=Dy.Similarly, For the plate in the XZ plane, while Dx=0 and Dxy=Dxz=0(37)Dx=Dz=Dxz.For the plate in the YZ plane, while Dx=0 and Dxy=Dxz=0(38)Dy=Dz=Dyz.In order to satisfy (36)–(38), then the following conclusions are obtained:(39)Dx=Dxy+DxzDy=Dxy+DyzDz=Dxz+Dyz.Thus three damage variables Dxy, Dyz, Dxz are independent.


## 2.7. Damage Evolution Equation

Based on hysteresis energy, the damage evolution model depends on three variables,Dxy, Dxz, Dyz. Let us now consider the damage equation of the first variable Dxy. 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 τxy2. 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.Figure 6
The hysteresis loop for the panel inXY plane in the local coordinate system.For each loop, the hysteresis energy of panel is(40)W0=4·τxy-σs2·1-Dxy·1KS2-1KS1·σs2·l1+l2l24.The total hysteresis energy of panel is assumed as (41)Wcr=ωcr·1KS2-1KS1·σs2·l1+l2l24.Then the damage evolution equation is defined as(42)dDxydN=α·W0/Wcrm1-Dxyβ=ατxy-σs/2·1-Dxymωcrm·1-Dxyβ=τxy-σs/2·1-Dxymωcr∗·1-Dxyβ,ωcr∗=αωcrmin which α, β, m, and ωcr are material parameters.Similarly, the damage evolution equation for the damage variablesDxz, Dyz can also be obtained. Then the damage evolution equations of the micromodel for three damage variables Dxy, Dxz, Dyz are(43)dDijdN=α·W0/Wcrm1-Dijβ=ατij-σs/2·1-Dijmωcrm·1-Dijβ=τij-σs/2·1-Dijmωcr∗·1-Dijβ,ij=xy,xz,yz.


## 3. 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 extentΔD of critical element is given, ΔD= constant, and the magnitude of damage extent increment will be checked by the convergence verification. Then the corresponding fatigue life increments ΔNxy(e), ΔNyz(e), and ΔNxz(e) of critical element are (44)ΔNxye=ωcr∗·1-Dxyβτxy-σs/2·1-Dxym·ΔDΔNyze=ωcr∗·1-Dyzβτyz-σs/2·1-Dyzm·ΔDΔNxze=ωcr∗·1-Dxzβτxz-σs/2·1-Dxzm·ΔD,where the symbol in right parentheses(e) means critical element and (x) denotes other elements. Subscript without parentheses xy means XY plane in the local coordinate system.(3)
Then the minimum valueΔNmin(e) is found and defined as(45)ΔNmine=min⁡ΔNxye,ΔNyze,ΔNxze.(4)
From the damage evolution equation (43), the damage variable increment of the elements can be obtained when ΔNmin(e) is known. The damage extent increments of critical element are(46)ΔDxye=τxy-σs/2·1-Dimωcr∗·1-Dβ·ΔNmine.ΔDyze=τyz-σs/2·1-Dimωcr∗·1-Dβ·ΔNmineΔDxze=τxz-σs/2·1-Dimωcr∗·1-Dβ·ΔNmine.Then damage extent increments of other elements are(47)ΔDxyx=τxy-σs/2·1-Dimωcr∗·1-Dβ·ΔNmineΔDyzx=τyz-σs/2·1-Dimωcr∗·1-Dβ·ΔNmineΔDxzx=τxz-σs/2·1-Dimωcr∗·1-Dβ·ΔNmine.(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: (48)Dijek+1=Dijek+ΔDijek,ij=xy,yz,xzDijxk+1=Dijxk+ΔDijxk,ij=xy,yz,xzin whichDij(e) is the damage extent of critical element and Dij(x) indicates the damage extent of other elements.(6)
From (39), the damage variables Dx, Dy, Dz of boom can be obtained when Dxy, Dyz, Dxz are known. The process from steps (2) to (5) will be repeated until one of the boom damage extents Dx, Dy, Dz of critical element is equal to 1. In this step, the failure of critical element means crack initiation in the structure.(7)
Modify local coordinates and material properties of the damaged element and recompute stress distribution by performing a new FE analysis. Determine the next critical element.(8)
Repeat the process from (2) to (7), until the crack length of the engineering structure is equal toa. The corresponding fatigue life Na with crack length a is(49)Na=∑ΔNmine.


## 4. Prediction of Fatigue Life

### 4.1. Smooth Specimen Made of 35Cr2Ni4MoA

The material parameters in damage evolution equation are determined with low-cycle fatigue experimental data of standard specimens in [29]. The low-cycle fatigue curve Δε/2-2Nf of the material 35Cr2Ni4MoA with stress ratio R=-1 is considered in this article. The material parameters of 35Cr2Ni4MoA are expressed as follows: E=206.5 GPa, μ=0.377, and G=75 GPa. The material parameters in damage evolution equation are listed in Table 1. Numerical results and experimental data are compared in Figure 7. The results derived from the present model agree with the experimental observations very well.Table 1
Material properties in damage evolution equation for material 35Cr2Ni4MoA withKT=1.
m ω c r ∗ /Pam β 1.05 4.10848 E 11 1.05Figure 7
Comparison between FE results and experimental data for material 35Cr2Ni4MoA.


### 4.2. Notched Structure Made of 35Cr2Ni4MoA

In this section, a notched structure of 35Cr2Ni4MoA in Figure8 is discussed while KT>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 Δε/2=2.8E-3.Figure 8
Notched structure of 35Cr2Ni4MoA.Figure 9
The mean fatigue crack initiation life curve for notched structure.Figure 10
The crack propagation curve for notched structure.


### 4.3. Real Satellite Structure

In this section, a real satellite structure of 5A06 in Figure11 is investigated while R=-1. The material parameters in damage evolution equation are listed in Table 2.Table 2
Material properties in damage evolution equation for material 5A06 withKT=1.
m ω c r ∗/Pam β 1.0 2.4051 E 10 1.0Figure 11
The finite element model of satellite structure.The finite element model of real satellite structure is shown in Figure11. In this paper, the fatigue crack propagation life of real satellite structure is predicted using the present method and shown in Figure 12.Figure 12
The crack propagation curve of satellite structure.For the satellite structure, the experimental low-cycle fatigue lifetime is as follows:LogNm=2.7604 when the crack length a=4.875 mm.For the satellite structure, the computational low-cycle fatigue lifetime is as follows:LogNm=2.7419 when the crack length a=4.875 mm.The relative error between the calculated results and the experimental data is 0.67%. Hence, the fatigue life prediction for satellite structure is acceptable in the engineering application.


## 4.1. Smooth Specimen Made of 35Cr2Ni4MoA

The material parameters in damage evolution equation are determined with low-cycle fatigue experimental data of standard specimens in [29]. The low-cycle fatigue curve Δε/2-2Nf of the material 35Cr2Ni4MoA with stress ratio R=-1 is considered in this article. The material parameters of 35Cr2Ni4MoA are expressed as follows: E=206.5 GPa, μ=0.377, and G=75 GPa. The material parameters in damage evolution equation are listed in Table 1. Numerical results and experimental data are compared in Figure 7. The results derived from the present model agree with the experimental observations very well.Table 1
Material properties in damage evolution equation for material 35Cr2Ni4MoA withKT=1.
m ω c r ∗ /Pam β 1.05 4.10848 E 11 1.05Figure 7
Comparison between FE results and experimental data for material 35Cr2Ni4MoA.


## 4.2. Notched Structure Made of 35Cr2Ni4MoA

In this section, a notched structure of 35Cr2Ni4MoA in Figure8 is discussed while KT>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 Δε/2=2.8E-3.Figure 8
Notched structure of 35Cr2Ni4MoA.Figure 9
The mean fatigue crack initiation life curve for notched structure.Figure 10
The crack propagation curve for notched structure.


## 4.3. Real Satellite Structure

In this section, a real satellite structure of 5A06 in Figure11 is investigated while R=-1. The material parameters in damage evolution equation are listed in Table 2.Table 2
Material properties in damage evolution equation for material 5A06 withKT=1.
m ω c r ∗/Pam β 1.0 2.4051 E 10 1.0Figure 11
The finite element model of satellite structure.The finite element model of real satellite structure is shown in Figure11. In this paper, the fatigue crack propagation life of real satellite structure is predicted using the present method and shown in Figure 12.Figure 12
The crack propagation curve of satellite structure.For the satellite structure, the experimental low-cycle fatigue lifetime is as follows:LogNm=2.7604 when the crack length a=4.875 mm.For the satellite structure, the computational low-cycle fatigue lifetime is as follows:LogNm=2.7419 when the crack length a=4.875 mm.The relative error between the calculated results and the experimental data is 0.67%. Hence, the fatigue life prediction for satellite structure is acceptable in the engineering application.


## 5. 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 mechanics-finite 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.



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*Source: 7128347-2016-10-31.xml*