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.

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 Figure 1, 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 Figure 2. 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 Figure 3. 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: , denote the stress tensor components and the strain tensor components of the micromodel, respectively, , denote the stress tensor components and the strain tensor components of elastic block, respectively, , denote the stress tensor components and the strain tensor components of boom-panel structure, respectively, is the length of elastic block, and 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 For boom-panel structure, an elastic analysis is written asin which is the elastic shear modulus of panel and 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 asin which is Young’s modulus of elastic block, is the shear modulus of elastic block, and is Poisson’s ratio of elastic block.

In the global coordinate system, the elastic part of constitutive model can be expressed as in which , 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.

2.3.2. Elastic Material Parameters of the Micromodel

From (4) and (5), we haveThe shear modulus of elastic block is also written asFrom (6) and (7), thenin which , are known and is a constant. Then , , , , , , and are known.

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 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 aswherein which is the shear yield stress of boom-panel structure, indicates the shear yield strain of boom-panel structure, and signifies the plastic shear modulus of panel.

2.4.2. Plastic Material Parameters of the Micromodel

In order to determine , 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 .

Figure 4 

Schematic of a plate subject to tension.

In the global coordinate system, the strain of the structure is defined asin which is the displacement increment of the micromodel along direction in the global coordinate system.

By the coordinate system transformation, the displacement increment is expressed asin whichin which , 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 areFrom (15)–(20), then the strain of the structure is expressed asIn the global coordinate system, the strain of the structure is expressed asFrom (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 isFor the plastic shear modulus , the condition must be satisfied. From (11) and (25), the range of the constant isSubstituting (8)–(14), (16), and (25) into (15), we obtainin which is a variable and .

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 Figure 4 is discussed first. For this case, the micromodel is at the position with or . 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)When , (ii)When ,in which , denote the damage variables of boom and denotes the damage variable of 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.

2.6. 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 . So the requirement is as follows:From (28), we haveSubstituting (10)–(13), (A.1), (32), and (34) into (33), thenFrom (32) and (35), we haveSimilarly, For the plate in the plane, while and For the plate in the plane, while and In order to satisfy (36)–(38), then the following conclusions are obtained:Thus three damage variables , , are independent.

2.7. 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 . 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 in plane in the local coordinate system.

For each loop, the hysteresis energy of panel isThe total hysteresis energy of panel is assumed as Then the damage evolution equation is defined asin 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