Design of a Practical Metal-Made Cold Isostatic Pressing (CIP) Chamber Using Finite Element Analysis

The fast development of deep-ocean engineering equipment requires more deep-ocean pressure chambers (DOPCs) with a large inner diameter and ultra-high-pressure (UHP). Using the pre-stressed wire-wound (PSWW) concept, cold isostatic pressing (CIP) chambers have become a new concept of DOPCs, which can provide 100% performance of materials in theory. This paper aims to provide a comprehensive design process for a practical metal-made CIP chamber. First, the generalized design equations are derived by considering the fact that the cylinder and wire have different Young’s moduli and Poisson’s ratios. Second, to verify the theory and the reliability of the CIP chamber, the authors proposed a series of FEA models based on ANSYS Mechanical, including a two-dimensional (2D) model with the thermal strain method (TSM) and a three-dimensional (3D) model with the direct method (DM). The relative errors of the pre-stress coefficient range from 0.17% to 5%. Finally, the crack growth path is predicted by using ANSYS’s Separating Morphing and Adaptive Remeshing Technology (SMART) algorithm, and the fatigue life is evaluated by using the unified fatigue life prediction (UFLP) method developed by the authors’ group. This paper provides a more valuable basis to the design of DOPCs as well as to the similar pressure vessels than the previous work.


Introduction
Pressure testing with a deep-ocean pressure chamber (DOPC) is a fast and effective method to validate the safety of deep-ocean engineering equipment [1]. To develop fullocean-depth (FOD) submersibles, large biomimetic robotic fish, new buoyancy materials, and so forth require a DOPC with a large inner diameter and ultra-high-pressure (UHP) [2]. However, such a monobloc chamber must have a large wall thickness. For instance, with a 3 m inner diameter and 90 MPa maximum working pressure (MWP), the 930 Pressure Chamber of China Ship Scientific Research Center (CSSRC) has a 530 mm wall-thickness, which may reach the ceiling of manufacture [2]. To conduct a FOD level's pressure testing or tests for buoyancy materials, a DOPC should operate at more than 180 MPa (1.5 times of FOD pressure) or even above 200 MPa. Thus, the structural form of monobloc chambers cannot meet these design requirements of such a DOPC with a large inner diameter [2]. In 2009, the Deep Ocean Exploration and Research (DOER) company developed three innovative DOPCs with the introduced technologies of cold isostatic presses; thus, the DOER company called them cold isostatic pressing (CIP) chambers [3,4]. Using these CIP chambers, DOER conducted a full range of FOD testing for their Deepsearch submersible components. The results show that CIP chambers can become a new design concept of DOPCs by replacing the monobloc chambers [2,3]. Figure 1 shows the nomenclature for a CIP chamber, which consists of a cylinder pre-stressed by a wire-wound layer. The wire-wound layer is formed by a wire helically wounded edge-to-edge in pre-tension many turns around the outside of the cylinder. To deduce the generalized design equations of CIP chambers, the following basic assumptions are introduced. drawn. Figure 1 shows the nomenclature for a CIP chamber, which consists of a cylinder pre-stressed by a wire-wound layer. The wire-wound layer is formed by a wire helically wounded edge-to-edge in pre-tension many turns around the outside of the cylinder. To deduce the generalized design equations of CIP chambers, the following basic assumptions are introduced.

Assumption 1 (A1). The cylinder and the wire are made of two different materials with different Young's moduli and Poisson's ratios. Let the Young's moduli of the cylinder and the wire be E1
and E2, Poisson's ratios for them are μ1 and μ2, and the allowable stresses for them are [σ]1 and [σ]2, respectively.

Assumption 2 (A2). The variation of stresses is under linear-elastic conditions, and yielding is not permitted.
Assumption 3 (A3). The wire is under an ideal winding condition; that is, the wire-wound layer is closely contacted with the cylinder without slipping, and so it is the same with every two adjacent wire layers. Consequently, the displacement continuity condition is satisfied on the interface and the contact surfaces between every two adjacent wire layers. Figure 1. Nomenclature for a CIP chamber: the contact surface between the cylinder and the wirewound layer is defined as the interface for which its radius is represented as RIF; RI and RO are the inner radius of the cylinder and the outer radius of the CIP chamber, respectively. Nomenclature for a CIP chamber: the contact surface between the cylinder and the wirewound layer is defined as the interface for which its radius is represented as R IF ; R I and R O are the inner radius of the cylinder and the outer radius of the CIP chamber, respectively. Assumption 1 (A1). The cylinder and the wire are made of two different materials with different Young's moduli and Poisson's ratios. Let the Young's moduli of the cylinder and the wire be E 1 and E 2 , Poisson's ratios for them are µ 1 and µ 2 , and the allowable stresses for them are [σ] 1 and [σ] 2 , respectively. Assumption 2 (A2). The variation of stresses is under linear-elastic conditions, and yielding is not permitted.

Assumption 3 (A3).
The wire is under an ideal winding condition; that is, the wire-wound layer is closely contacted with the cylinder without slipping, and so it is the same with every two adjacent wire layers. Consequently, the displacement continuity condition is satisfied on the interface and the contact surfaces between every two adjacent wire layers.
According to the above assumptions, the wire-wound layer can be simplified as a cylinder. Thus, the stress distributions of the cylinder and the wire-wound layer can be determined by Lamé formulas, respectively. Lamé formulas represent the stress distribution in a cylinder submitted to uniform internal pressure P I and external pressure P O , which takes the following form [18]: where σ t and σ r are the tangential stress and the radial stress, respectively. It is easy to understand that the stress state of CIP chambers can be categorized into the pre-stressed state (non-working state, without internal pressure) and the working state [2,19]. In the pre-stressed state, the cylinder will only be under the action of the prestress generated by the wire-wound layer, so the stress in this state is known as pre-stress (marked with a superscript "P"). If we ignore pre-stress in the entire CIP chamber, the stress generated by the internal pressure can be called the Lamé stress (marked with a superscript "L") because it is determined by Lamé formulas. In the working state, the cylinder will be under the action of both the pre-stress and the internal pressure, so the stress in this state is known as composite stress [2,19].
When designing a CIP chamber, the primary consideration is to what extent we expect to offset the tensile stress in the cylinder. Therefore, we introduce the pre-stress coefficient η here [2,19]: where σ P tI and σ L tI are the tangential pre-stress and the tangential Lamé stress on the innersurface of the cylinder, respectively. Obviously, tensile stress will be eliminated when η ≥ 1.
To deduce the generalized design equations of CIP chambers, we let the pre-stress generated by the wire-wound layer to be equivalent to radial stress. Therefore, the cylinder can be regarded as a one that is submitted to uniform pressure on its inner surface and outer surface. In this paper, this equivalent stress is known as interface pressure P IF .

Equilibrium Equation
The stress distribution of the wire-wound layer in 2D polar coordinates is depicted in Figure 2, and the equilibrium equation can be expressed as follows.
Then, we have the following.
The strength criterion of the wire-wound layer has the following form [18,20]: where τ max is the maximum shear stress of the wire-wound layer. The boundary conditions of the wire-wound layer can be described as follows. Thus, we can obtain the stress distribution of the wire-wound layer in the working state.
When r = R IF , the interface pressure in the working state can be obtained as follows. Then, we have the following.
The strength criterion of the wire-wound layer has the following form [18,20]: where τmax is the maximum shear stress of the wire-wound layer. The boundary conditions of the wire-wound layer can be described as follows.
Thus, we can obtain the stress distribution of the wire-wound layer in the working state.
When r = RIF, the interface pressure in the working state can be obtained as follows.

Displacement Continuity Condition
According to basic assumptions, the radial displacement of the outer surface of the cylinder caused by internal pressure u1 should always be equal to the radial displacement of the inner surface of the wire-wound layer caused by internal pressure u2. For the

Displacement Continuity Condition
According to basic assumptions, the radial displacement of the outer surface of the cylinder caused by internal pressure u 1 should always be equal to the radial displacement of the inner surface of the wire-wound layer caused by internal pressure u 2 . For the cylinder, it is under the action of the internal pressure and the interface pressure. Therefore, according to Lamé formulas, the stress distribution on the interface has the following form: where σ tIF and σ rIF are the tangential stress and radial stress on the interface respectively. Thus, according to the elasticity theory [18], we have the following.
Similarly, for the wire-wound layer, it is only under the action of the interface pressure. Therefore, we have the following.
Thus, we can obtain the following.
Materials 2022, 15, 3621 6 of 23 As u 1 = u 2 , the interface pressure caused by the internal pressure has the following form.

Wall-Thickness Equation
According to Lamé formulas, we have the following.
Thus, the strength criterion of the cylinder has the following form.
If Equations (16) and (17) are combined, the generalized design equations of CIP chambers take the following form.
Thus, interface radius R IF can be determined by solving Equation (18) with numerical methods, and the outer radius R O can be determined by substituting R IF into Equation (16) or Equation (17). Therefore, the wall thickness of the cylinder δ 1 and the thickness of the wire-wound layer δ 2 can be determined, where δ 1 = R IF − R I and δ 2 = R O − R IF .

Winding Stress Formula
The winding stress of the wire σ 0 is the essential process parameter of CIP chambers, which must be provided in the design. The winding stress is the tangential stress σ P t generated in the wire due to pre-tension in the winding process. When the winding process is finished, one wire layer (except the outmost layer) will be under the reaction of the tangential stress σ P t * of its adjacent outer layer. Then, we have the following.
Moreover, we can obtain the following.
According to Lamé formulas and the displacement continuity condition, we can obtain the reaction tangential stress of the adjacent outer layer as Equation (21). To avoid repetition, the following equations describe the conditions.
Thus, the winding stress formula takes the following form.

Stress Distribution
Based on the above derivation, the generalized stress distribution of CIP chambers has the following form: where σ P t and σ P r are the tangential pre-stress and radial pre-stress, respectively. According to the elasticity theory [18], the radial displacements of the inner surface of the cylinder in the pre-stressed state and in the working state has the following form.
where u P I and u I represent the two displacements, respectively. When E 1 = E 2 and µ 1 = µ 2 , we can prove that the derived generalized design equations in this paper are equivalent to the equations in the ASME code, which have the following form [7]: where χ is the radius coordinates, and χ 1 and χ 2 are the radius coordinates of the cylinder and wire-wound layer, respectively.

Pre-Stress Coefficient
The pre-stress coefficient can facilitate the design of CIP chambers, but determining a proper pre-stress coefficient is still a key problem. Because DOPCs are under the action of low-cycle loads, the pre-stress degree of CIP chambers should be essentially dependent upon the requirement of fatigue resistance. As we know, many fatigue tests show that the fatigue resistance of a material increases with increased pre-stress. Moreover, it is further known that fatigue almost never occurs if the stress pulsations are always below the yield stress of a material and stress always remains compressive [5,16]. From the perspective of the fracture mechanics, if the residue stress of the cylinder is always compressive, the stress intensity factor (SIF) must be negative, which can make result in cracks always tending to be closed and restrains the nucleation and propagation of cracks [21]. From the perspective of the elasticity theory, although the tensile stress in the cylinder is eliminated when η = 1.0, tensile strain will still exist. Therefore, a proper pre-stress coefficient should eliminate not only the tensile stress but also the tensile strain in the cylinder to reduce the possibility of crack nucleation [2].
According to the elasticity theory [18], we have the following: where ε P tI is the tangential strain of the inner-surface of the cylinder in the pre-stressed state, and ε L tI is the tangential strain of the inner surface of the cylinder generated only by the internal working pressure.
When ε P tI ≥ ε L tI , there is no tensile strain in the cylinder. Thus, we have the following. Replacing Equation (15) with Equation (29), the pre-stress coefficient has the following form.

Case Study
To illustrate how to use the generalized design equations, a case study is presented here. In this case, the design problem is about a CIP chamber with 200 MPa MWP and 500 mm inner diameter intended to be built in our laboratory. According to the ASME code, the design internal pressure P I should be 1.25 times MWP [7]. The materials of the CIP chamber are all high-strength low alloy steels. Here, the cylinder is forged from the steel SA-723 Class 2a, and the wire comprised drawn and cold rolled steel wire SA-905 Class 2 with a rectangular cross-section of 4.06 mm × 1.02 mm (simplified as 4 mm × 1 mm in design). The materials' specifications are shown in Table 1 [22]. According to Equation (30), the pre-stress coefficient is 1.074. Therefore, the dimensions of the CIP chamber can be obtained by generalized design equations, which are shown in Table 2. It is known that the cylinder should be wound with 80 layers of wire.   1 The design values must ensure that the maximum stress in the CIP chamber does not exceed the allowable stress. Figure 3 presents the variation curves of the tangential stress in the CIP chamber, which can provide a better understanding of the design principles and the stress characteristics of CIP chambers. The results are shown as follows: (1) The tangential stress in the cylinder is compressive when the pre-stress coefficient is greater than 1.0. In the pre-stress state, the tangential stress in the cylinder gradually increases with a decreased radius, and it reaches the maximum on the inner-surface, which is close to the allowable stress of the cylinder. In the working state, the tangential stress caused by the internal pressure is greatly offset and reduced by the residual stress in the cylinder generated due to the pre-stressed wire-wound. (2) The tangential stress in the wire is always tensile stress. In the pre-stress state, tangential stress gradually increases with an increased radius of the wire. In the working state, the tangential stress further increases due to internal pressure, and reaches the maximum on the outermost layer, which is close to the allowable stress of the wire. (3) The tangential stress on the inner surface of the cylinder shall not exceed its allowable stress, and this principle can guarantee full use of the cylinder's material. Likewise, the tangential stress in the outermost wire layer shall not exceed its allowable stress, and this principle can guarantee full use of the wire's material.     The winding stress of the wire is a function of radius r, which can be determined by Equation (22). However, the variable tension control in the winding process is very difficult to achieve. In order to reduce the difficulty and the cost of the winding process, the eighty layers of the wire are simplified as eight isotension stages, and each wire layer in one isotension stage is wound with equal tension instead of variable tension [16]. The actual values of the winding stress in each isotension stage should be appropriately increased based on the theoretical value of the central layer of the stage to compensate for the loss of pre-stress coefficient caused by isotension winding, which are shown in Table 3 and     (22) (22) at the central position of each isotension stage in the radial direction.

Finite Element Model
To verify the theory and reliability of the CIP chamber in the above case, the authors proposed a series of FEA models based on ANSYS Mechanical. The key point of FEA modeling is how to impose the winding stress as the boundary condition to simulate the effect of PSWW. Therefore, TSM is applied to the 2D model, and the DM proposed by the authors is used in the 3D model.
The TSM originated from a proposed methodology of wire winding simulation conducted by Alegre et al. [23]. The basic idea of this methodology is to convert the elastic strains of each wire layer of a CIP chamber into the corresponding thermal strains, which take the following form [23]: where ε T is the equivalent thermal strain, and δ w is the thickness of the wire. According to the theory of ANSYS [24], the thermal strain can be expressed as follows: where α T is the material's isotropic secant coefficient of thermal expansion (in 1/ • C), T 0 is the initial temperature (in • C), and T is the thermal load (in • C). By combining Equation (31) with (32), the equivalent thermal loads can be converted from the winding stresses of the wire, which are shown in Table 4. In the FEA 2D model, the wire-wound layer of CIP chambers is simplified into eight iso-tension stages. According to the stress characteristics of CIP chambers, the finite element types are set to be plane strain and axisymmetric, respectively, shown in Figure 5. The element size is optimized by using a mesh sensitivity analysis and the quadratic elements with 5 mm are used to improve the precision of the results. The frictional type of contacts is used, and the frictional coefficient is 0.1. In this setting, the two contacting geometries can carry shear stresses up to a certain magnitude across their contact face before they start sliding relative to each other [24]. The Augmented Lagrange formulation is used for contact pairs to provide a better performance, which takes the following form [24]: where F n is the normal contact force, K n is the contact normal stiffness, u n is the contact gap size, and λ i+1 is the Lagrange multiplier force at iteration i + 1. The boundary conditions of the wire are imposed by the thermal loads in Table 4. Then, the related results can be obtained by using the post-processing function of AN-SYS Mechanical, which will be provided in the next subsection. The main parameters of the ANSYS Mechanical are listed in Table 5.
Using TSM, the proposed 2D model can reflect the stress changes of the cylinder, but it cannot present the stress distribution of the wire-wound layer itself. Obviously, the wire winding of CIP chambers is a typical 3D problem. Therefore, a 3D model is also proposed by the authors. To impose the winding stress of the wire, the wire-wound layer is divided into two symmetric parts with a notch of 10°. Thus, the 3D model can directly impose the winding stress of the wire as the boundary condition, and this is the meaning of the direct method, shown in Figure 6. The element size is also optimized by using a mesh sensitivity analysis and the quadratic elements with 10 mm are used to improve the precision of the results. The main parameters of the ANSYS Mechanical are very similar to those of the 2D model (also see Table 5), and the related results will also be provided in the next subsection.

Boundary Conditions
Step 1: only thermal loads Step 2: thermal loads and internal pressure Step 1: only winding stresses Step 2: winding stresses and internal pressure The boundary conditions of the wire are imposed by the thermal loads in Table 4. Then, the related results can be obtained by using the post-processing function of ANSYS Mechanical, which will be provided in the next subsection. The main parameters of the ANSYS Mechanical are listed in Table 5.

Boundary Conditions
Step 1: only thermal loads Step 2: thermal loads and internal pressure Step 1: only winding stresses Step 2: winding stresses and internal pressure Using TSM, the proposed 2D model can reflect the stress changes of the cylinder, but it cannot present the stress distribution of the wire-wound layer itself. Obviously, the wire winding of CIP chambers is a typical 3D problem. Therefore, a 3D model is also proposed by the authors. To impose the winding stress of the wire, the wire-wound layer is divided into two symmetric parts with a notch of 10 • . Thus, the 3D model can directly impose the winding stress of the wire as the boundary condition, and this is the meaning of the direct method, shown in Figure 6. The element size is also optimized by using a mesh sensitivity analysis and the quadratic elements with 10 mm are used to improve the precision of the results. The main parameters of the ANSYS Mechanical are very similar to those of the 2D model (also see Table 5), and the related results will also be provided in the next subsection. Table 6 provides the main results obtained by the above proposed FEA models, and Figure 7 presents the deformation distributions of the CIP chamber. The results are shown as follows:

Results
(1) The deformation distributions of the CIP chamber obtained by ANSYS all present in a rainbow image, and the maximum displacement of the CIP chamber appears on the outermost wire layer, which is in accordance with the theory.   Table 6 provides the main results obtained by the above proposed FEA models, and Figure 7 presents the deformation distributions of the CIP chamber. The results are shown as follows:

Results
(1) The deformation distributions of the CIP chamber obtained by ANSYS all present in a rainbow image, and the maximum displacement of the CIP chamber appears on the outermost wire layer, which is in accordance with the theory. In the FEA 2D model, the displacement of the inner surface of the cylinder in the pre-stressed state is about 0.68 mm, while in the working state, it decreased to about 0.048 mm. In the FEA 3D model, the displacement of the inner surface of the cylinder in the prestressed state is about 0.73 mm, while in the working state it decreased to about 0.049 mm. The deformation results obtained by the FEA models are very close to the theory values, and the maximum relative errors are about 4.3% in the 2D model and 2.9% in the 3D model. (2) The FEA 2D model can only present the stress distribution of the cylinder. In the pre-stressed state, the maximum stress of the cylinder is the tangential stress on the inner-surface of the cylinder, which is about −585 MPa with a relative error of 5%.
In the working state, the tangential stress on the inner-surface of the cylinder decreased to about −41 MPa with a maximum relative error of 8.6%. The pre-stress coefficient is about 1.13 with a relative error of 5%. (3) The FEA 3D model can present stress distributions of both the cylinder and the wire-wound layer. In the pre-stressed state, the maximum stress of the wire-wound layer is the tangential stress in the outermost wire layer, which is about 678.64 MPa with a relative error of only 0.78%. In the working state, this maximum stress increased to about 832 MPa with a relative error of 12.7%. The pre-stress coefficient is about 1.076 with a relative error of only 0.17%. (4) Two FEA models can also provide interface pressure. In the 2D model, the maximum relative errors range from 0.3% to 3.2%, while the maximum relative errors range from 3.2% to 8.6% in the 3D model.  (2) The FEA 2D model can only present the stress distribution of the cylinder. In the pre-stressed state, the maximum stress of the cylinder is the tangential stress on the inner-surface of the cylinder, which is about −585 MPa with a relative error of 5%. In the working state, the tangential stress on the inner-surface of the cylinder decreased to about −41 MPa with a maximum relative error of 8.6%. The pre-stress coefficient is about 1.13 with a relative error of 5%. (3) The FEA 3D model can present stress distributions of both the cylinder and the wirewound layer. In the pre-stressed state, the maximum stress of the wire-wound layer is the tangential stress in the outermost wire layer, which is about 678.64 MPa with a relative error of only 0.78%. In the working state, this maximum stress increased to about 832 MPa with a relative error of 12.7%. The pre-stress coefficient is about 1.076 with a relative error of only 0.17%. (4) Two FEA models can also provide interface pressure. In the 2D model, the maximum relative errors range from 0.3% to 3.2%, while the maximum relative errors range from 3.2% to 8.6% in the 3D model. PSWW, and these results confirm the correctness of the generalized design equation rived in this paper. Overall, the FEA 2D model using TSM has a fast calculation s (several minutes) with slightly larger errors, while the FEA 3D model using DM higher accuracy with a very slow calculation speed (tens of hours). However, the 3D model can directly impose the winding stress as the boundary condition and present the stress distributions not only in the cylinder but also in the wire-wound which is a more realistic simulation model. Therefore, the proposed FEA models of the CIP chamber can reflect the effect of PSWW, and these results confirm the correctness of the generalized design equations derived in this paper. Overall, the FEA 2D model using TSM has a fast calculation speed (several minutes) with slightly larger errors, while the FEA 3D model using DM has higher accuracy with a very slow calculation speed (tens of hours). However, the FEA 3D model can directly impose the winding stress as the boundary condition and can present the stress distributions not only in the cylinder but also in the wire-wound layer, which is a more realistic simulation model.

Fatigue Crack Propagation
As the above-mentioned discussion, the inner cylinder of the CIP chamber can be considered to have infinite fatigue life when the pre-stress coefficient is greater than 1.0. Thus, the fatigue strength of the CIP chamber mainly depends on the steel wire, which is always under the action of low-cyclic tensile stress. The stress ratio R of the steel wire can be defined by σ min /σ max , where σ min and σ max are the minimum stress level and the maximum stress level, respectively. In the above case, the stress ratio of the steel wire is about 0.7.
To illustrate how to predict the crack growth path of the steel wire and to evaluate the fatigue life of the CIP chamber, a FEA model of SMART crack growth of a piece of the steel wire used in the CIP chamber is proposed based on ANSYS Mechanical, which is shown in Figure 8. In Case 1, the initial crack is set to be a semi-elliptical crack with the initial depth of 0.2 mm and the aspect ratio of 1:3 on the side of the steel wire [17]. In Case 2, the initial crack is set to be a V-notch pre-meshed crack with the initial depth of 0.2 mm and the aspect ratio of 1:3 at the bottom of the steel wire [17]. In ANSYS's SMART algorithm, the fatigue life is evaluated by using the Paris equation, which takes the following form [15,21]: where a is the crack length, N is the load cycles, da/dN represents the crack growth rate, ∆K is the stress intensity factor (SIF) range, C and m are experimentally determined constants.
Here, C = 2.29 × 10 −10 m/cycle (MPa·m 1/2 ) −m and m = 2 [15].  The stress level of the outermost layer in the CIP chamber (here is 953 MPa) is imposed in the FEA model. The predicted trajectories of the crack growth can be obtained by increasing the number of substeps in ANSYS Mechanical (see Figure 9). The results are shown as follows: (1) The semi-elliptical crack on the side of the steel wire will gradually be opened and extended under the cyclic tensile load. The steel wire will be fractured along the direction of its thickness, and bending failure will eventually occur. (2) The V-notch at the bottom of the steel wire will also gradually be opened and extended under the cyclic load. The steel wire will be fractured along the direction of its height, and shear failure will eventually occur.

Unified Fatigue Life Prediction Method
To provide an accurate prediction of the fatigue life of the CIP chamber, the UFLP method is applied here. The general constitutive relation in the UFLP method takes the following form [9]: where K max and K min are the maximum SIF level and the minimum SIF level, respectively, K C is the actual fracture toughness, ∆K effth is the threshold effective SIF range, f op is the crack opening function, A is a material and environmentally sensitive constant of dimensions in the crack growth rate model, m is a material constant, and n is an index indicating the unstable fracture. r e is an empirical material constant of the inherent flaw length, σ v is the virtual strength representing the material strength at the condition of r e = 0, and Y(a) is the geometrical factor.
The actual fracture toughness K C can be determined by the following Equation [9]: where K IC is the plane strain fracture toughness, and λ is the crack tip plastic zone confident. The value of λ can be calculated by the follwing Equation [9]: where p is the strain hardening exponent of the material. The value of K IC can be estimated by the following Equation [9]: Where ∆K th0 is the threshold SIF range under zero load ratio, and ε f is the fracture strain of material. When 0.5 ≤ R < 1, the value of ∆K th0 can be determined by the following Equation [9]: where ∆K th is the threshold SIF range. It should be observed that K max , K C , and f op are all the functions of crack length a.
Based on the results of FEA, the fatigue life of the CIP chamber can be predicted more accurately by using the UFLP method. The related parameters are estimated according to Ref. [9] and the ASME code [7,8,22,25], which are shown in Table 7. The comparison curves of the crack growth rate obtained by ANSYS and the UFLP method are shown in Figure 10, and the predicted fatigue life of the CIP chamber is provided in Table 8. The results show that the predicted fatigue life can be significantly different by using different methods. Since ANSYS's SMART algorithm is only based on the Paris equation, the direct results of ANSYS Mechanical are more conservative. However, we can still use the values of SIF obtained in ANSYS Mechanical to provide a more convincing result by applying the UFLP method. Anyhow, the shear failure of the steel wire is always the most dangerous situation. shown in Figure 10, and the predicted fatigue life of the CIP chamber is provided in Table 8. The results show that the predicted fatigue life can be significantly different by using different methods. Since ANSYS's SMART algorithm is only based on the Paris equation, the direct results of ANSYS Mechanical are more conservative. However, we can still use the values of SIF obtained in ANSYS Mechanical to provide a more convincing result by applying the UFLP method. Anyhow, the shear failure of the steel wire is always the most dangerous situation.

Discussion
The fast development of deep-ocean technologies needs more DOPCs with a large inner diameter and UHP. As the working pressure and the inner diameter increases, the wall-thickness of monobloc chambers will be increased dramatically. The fundamental defect of monobloc chambers is that the outer material of chambers cannot be fully utilized. With the PSWW concept, CIP chambers are widely regarded to be the most reliable and durable pressure chambers ever designed [2]. Moreover, when a fatigue crack nucleates in CIP chambers, the consequences of fracture would not be as sudden and violent as it would be in the case for monobloc chambers. This is because the fatigue cracks in the cylinder will not continue to propagate through the wire-wound layer, and correspondingly, a fatigue crack at one point of the wire will not immediately initiate new cracks [5,6].
The stress distribution equations of CIP chambers in the ASME code limit the scope of application. Therefore, the generalized design equations were derived considering that the cylinder and the wire have different Young's moduli and Poisson's ratios. To reduce the difficulty and the cost of the winding process, numerous wire layers should be simplified as a few isotension stages, and the isotension winding is used in each wire

Discussion
The fast development of deep-ocean technologies needs more DOPCs with a large inner diameter and UHP. As the working pressure and the inner diameter increases, the wall-thickness of monobloc chambers will be increased dramatically. The fundamental defect of monobloc chambers is that the outer material of chambers cannot be fully utilized. With the PSWW concept, CIP chambers are widely regarded to be the most reliable and durable pressure chambers ever designed [2]. Moreover, when a fatigue crack nucleates in CIP chambers, the consequences of fracture would not be as sudden and violent as it would be in the case for monobloc chambers. This is because the fatigue cracks in the cylinder will not continue to propagate through the wire-wound layer, and correspondingly, a fatigue crack at one point of the wire will not immediately initiate new cracks [5,6].
The stress distribution equations of CIP chambers in the ASME code limit the scope of application. Therefore, the generalized design equations were derived considering that the cylinder and the wire have different Young's moduli and Poisson's ratios. To reduce the difficulty and the cost of the winding process, numerous wire layers should be simplified as a few isotension stages, and the isotension winding is used in each wire layer instead of variable tension winding [16]. However, the winding stress should be appropriately increased to compensate the loss of the pre-stress coefficient caused by isotension winding.
The critical stress of a CIP chamber should be analyzed by FEA in the design stage. The fatigue life of CIP chambers mainly depends on the steel wire if the design pre-stress coefficient is greater than 1.0. The UFLP method proposed by Cui et al. can explain most of the observed macroscopic fatigue phenomena satisfactorily, and it can provide an effective method for evaluating the fatigue life of CIP chambers. As a matter of fact, more than 90% of over 1900 deployed CIP chambers since the 1960s can still operate well today [2]. Thus, the fatigue life of CIP chambers must not be estimated conservatively due to their ingenious structure. In other words, a CIP chamber behaves as though it was manufactured from a stronger material than it actually was. In addition, the shear failure of the steel wire should always be paid more attention to.

Conclusions
A comprehensive design process of a practical metal-made CIP chamber intended to be constructed in our laboratory was presented in this paper, which can be illustrated in Figure 11. The generalized design equations of CIP chambers were derived by the authors. To illustrate how to use the design equations, a case study of the practical CIP chamber was given, and the stress distribution of the CIP chamber was investigated. To verify the theory and the reliability of the CIP chamber, the authors proposed a series of FEA models based on ANSYS Mechanical, including a 2D model using TSM and a 3D model using DM. Furthermore, ANSYS's SMART algorithm was used to investigate two different types of cracks, including a semi-elliptical crack and a V-notch pre-meshed crack. To predict the fatigue life of the CIP chamber accurately, the UFLP method is applied instead of the very simplified Paris equation. This paper can provide more valuable basis to the design of DOPCs in marine engineering as well as to similar pressure vessels than the previous work.
The fatigue life of CIP chambers mainly depends on the steel wire if the design pre-stress coefficient is greater than 1.0. The UFLP method proposed by Cui et al. can explain most of the observed macroscopic fatigue phenomena satisfactorily, and it can provide an effective method for evaluating the fatigue life of CIP chambers. As a matter of fact, more than 90% of over 1900 deployed CIP chambers since the 1960s can still operate well today [2]. Thus, the fatigue life of CIP chambers must not be estimated conservatively due to their ingenious structure. In other words, a CIP chamber behaves as though it was manufactured from a stronger material than it actually was. In addition, the shear failure of the steel wire should always be paid more attention to.

Conclusions
A comprehensive design process of a practical metal-made CIP chamber intended to be constructed in our laboratory was presented in this paper, which can be illustrated in Figure 11. The generalized design equations of CIP chambers were derived by the authors. To illustrate how to use the design equations, a case study of the practical CIP chamber was given, and the stress distribution of the CIP chamber was investigated. To verify the theory and the reliability of the CIP chamber, the authors proposed a series of FEA models based on ANSYS Mechanical, including a 2D model using TSM and a 3D model using DM. Furthermore, ANSYS's SMART algorithm was used to investigate two different types of cracks, including a semi-elliptical crack and a V-notch pre-meshed crack. To predict the fatigue life of the CIP chamber accurately, the UFLP method is applied instead of the very simplified Paris equation. This paper can provide more valuable basis to the design of DOPCs in marine engineering as well as to similar pressure vessels than the previous work. The main conclusions can be summarized as follows: (1) The generalized design equations of CIP chambers are derived by considering the fact that the cylinder and wire have different Young's moduli and Poisson's ratios, which expands the scope of application. The main conclusions can be summarized as follows: (1) The generalized design equations of CIP chambers are derived by considering the fact that the cylinder and wire have different Young's moduli and Poisson's ratios, which expands the scope of application. (2) To increase the fatigue resistance of CIP chambers, the design pre-stress coefficient should be slightly greater than 1.0, which can guarantee that the residual stress of the cylinder is always compressive. The tangential stress on the inner surface of the cylinder shall not exceed its allowable stress, which can guarantee full use of the cylinder's material. Likewise, the tangential stress in the outermost wire layer shall not exceed its allowable stress, which can guarantee full use of the wire's material. (3) The proposed FEA models of the CIP chamber can reflect the effect of PSWW, including a 2D model with TSM and a 3D model with DM. The 2D model has a fast calculation speed with slightly larger errors, while the 3D model has higher accuracy with a very slow calculation speed. However, the 3D model can present stress distributions not only in the cylinder but also in the wire-wound layer, which is a more realistic simulation model. An empirical material constant of the inherent flaw length (m) u 1 The radial displacement of the outer-surface of the cylinder caused by the internal pressure (mm) u 2 The radial displacement of the inner-surface of the wire-wound layer caused by the internal pressure (mm) u I The radial displacements of the inner-surface of the cylinder in the working state (mm) u P I The radial displacements of the inner-surface of the cylinder in the prestess state (mm) u n Contact gap size (mm) ∆K Stress intensity factor range (MPa·m 1/2 ) ∆K effth Threshold effective stress intensity factor range (MPa·m 1/2 ) ∆K th Threshold stress intensity factor range (MPa·m 1/2 ) ∆K th0 The Threshold stress intensity factor range under zero load ratio (MPa·m 1/2 ) The radius coordinates in the ASME equations The radius coordinates of the cylinder in the ASME equations The radius coordinates of the wire-wound layer in the ASME equations