Static progressive damage finite element analysis of a certain type of composite pressure vessel

2.1. Theoretical fundamentals of composite material mechanics

The carbon fiber winding layer is composed of carbon fibers and a resin matrix, exhibiting significant anisotropy. The constitutive equation is used to represent the stress–strain relationship of the material under external loads, thereby enabling the prediction of stress distribution, deformation, and failure mechanisms. In the stress calculation and strength verification of the winding layer, the overall stiffness distribution must first be determined. The principal material parameters are converted into off-axis parameters using the windangle function provided by the WCM plugin [2] (as shown in Fig. 1, where $X$ and $Y$ denote the off-axis directions and $\theta$ represents the angle between the $X$-axis and direction 1). This conversion is then used to compute the constitutive equation, thus providing a more accurate description of the mechanical characteristics of the winding layer [3].

Fig. 1Schematic diagram of the transformation between off-axis and principal axes

The transformation relationship for principal stress and strain is:

The transformation relationship for off-axis stress and strain [3] is given by ：

where:$m=\mathrm{c}\mathrm{o}\mathrm{s}\left(\theta \right)$, $n=\mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right)$, $\left[Q\right]$ is the material’s principal stiffness matrix.

2.2. Damage initiation criterion

The carbon fiber composite layer is the primary load-bearing component of the pressure vessel, and its damage model can be divided into intra-laminar damage (such as failure of fibers and matrix under tension or compression) and interlaminar damage. In the analysis, composite materials can be simplified as orthotropic materials to focus on overall structural failure while neglecting internal damage. In 1980, Japanese scholars proposed the Hashin criterion as an empirical method, treating composite materials as a multi-layer assembly and considering multiple failure modes, including fiber breakage, matrix failure, and shear failure [5].

Fiber Tension $({\sigma }_{11}\ge 0)$:

Fiber Compression $({\sigma }_{11}<0)$:

Matrix Tension $({\sigma }_{22}+{\sigma }_{33}\ge 0)$:

Matrix Compression $({\sigma }_{22}+{\sigma }_{33}<0)$:

2.3. Damage evolution criterion

After damage initiation, a progressive stiffness degradation model based on the studies by Lapczyk [5] and Fang [6] is employed in conjunction with Murakami’s damage theory [7] to analyze the evolution of damage. In this model, independent damage variables are defined for each failure mode, and the material degradation process is characterized by a continuous reduction in stiffness. For each failure mode, the damage variable can be expressed as:

where ${\delta }_{1,eq}^f$ is the equivalent displacement at final failure, ${\delta }_{1,eq}^0$ is the equivalent displacement at damage initiation, and ${\delta }_{1,eq}$ current is the equivalent displacement at the current analysis step.

2.4. Construction of the static damage identification and progressive evolution model based on the three-dimensional Hashin criterion and equivalent displacement method

The static damage identification and progressive evolution model developed in this study is implemented in three stages. First, a mapping relationship between the finite element and the windangle data is established, and the material parameters in the principal direction are converted into off-axis parameters to compute the constitutive equation. Next, the stress is calculated using the constitutive equation, and damage indices such as fiber tension are computed based on the three-dimensional Hashin criterion. When any one of these indices reaches the critical value of 1, the material is considered to have entered the initial stage of damage evolution. Finally, during the evolution phase, the damage coefficient is calculated using the equivalent displacement method, and a damage matrix $D$ is constructed to achieve a gradual degradation of the winding layer’s stiffness, thereby fully describing the entire process of the composite material transitioning from an undamaged to a damaged state:

4.1. Damage process analysis

A stepwise loading approach with increments of 1 MPa was used to simulate over-designed burst conditions: initial damage occurs when the internal pressure reaches 62 MPa, with the damage coefficient increasing nonlinearly from zero as the pressure rises; when the pressure increases to 73.4 MPa, the damage coefficient reaches 1, indicating complete structural failure. Fig. 6 illustrates the trend of the damage variable as a function of internal pressure.

Fig. 4Flowchart of the computational process

Fig. 5Damage evolution process of the pressure vessel under high-intensity loading

Fig. 6Damage variable trend graph

4.2. Ultimate burst pressure prediction

Numerical simulations indicate that the vessel experiences complete failure at an internal pressure of 73.4 MPa, with the failure primarily concentrated in the head region. The simulation results are in excellent agreement with experimental data, thereby verifying the accuracy of the model.

Fig. 7Failure location prediction