Finite element analysis of viscoelastic creep behaviors of deep-sea manned submersible viewport windows

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Highlights

  • A viscoelastic creep/damage coupled model is developed for creep analysis of PMMA viewport windows.

  • Numerical algorithm using ABAQUS-UMAT subroutine is developed for creep models.

  • Distributions of the creep strain, displacement and damage variable of viewport windows are discussed.

  • Creep lifetime of the PMMA main conical viewport window under constant 3000m water depth is evaluated.

Abstract

This paper studies the creep and damage behaviors of PMMA deep-sea manned submersible viewport windows under constant seawater pressure. First, the viscoelastic creep experiments of PMMA material specimens under four groups of tensile stresses (near 30 MPa) are performed. Second, we adopt the three-element spring/dashpot viscoelastic model to study the creep behaviors of the PMMA viewport windows without considering damage behaviors. Third, we develop a viscoelastic creep/damage coupled model to evaluate the creep behaviors of viewport windows by leveraging the Burgers viscoelastic model with four spring/dashpot elements. The main features are represented by three aspects: 1. the creep initiation, growth and acceleration stages are included, 2. the introduced damage variable into the creep model is featured by the void volume fraction, which represents the degradation of load-bearing ability of viewport windows and 3. the creep/damage coupling behaviors are reflected by the increased functional relationship between the damage variable and the equivalent strain. Fourth, the finite element algorithm using ABAQUS-UMAT (user material subroutine) is developed to implement the three-element and four-element spring/dashpot models, and the unknown viscoelastic parameters are identified by combining with the tensile creep experiments on PMMA material specimens. Finally, finite element analysis is performed to evaluate the distributions of the creep strain, displacement and damage variable of viewport windows by discussing the effects of the water depth (100–500 m and 3000 m). As an extreme case, the creep lifetime of the conical viewport window under 3000 m water depth is also evaluated from the conservative perspective.

Introduction

Now, China has successfully developed “Jiaolong” and “Fendouzhe” deep-sea manned submersibles for ocean exploration, which aim to dive into the maximum 7000–10000 m water depth. Among all system components, the pressure shell is one of the most important components because it guarantees the holistic safety and determines the pressure-bearing ability in deep sea. In order to guarantee structural strength and water tightness, the pressure shell is mainly composed of the titanium alloy with high strength and excellent corrosion resistance [1]. Particularly, the viewport window (as shown in Fig. 1) manufactured by transparent PMMA (polymethyl methacrylate) are specially set for underwater observation, which are connected with the titanium shell by the O-ring sealing equipments. One main viewport window and two lateral viewport windows are set in the “Jiaolong” submersible. Since mechanical properties between the seat and viewport window are distinctly different, the long-term in-serve performance of pressure shells requires favorable structural compatibility [2]. In order to perform safety evaluation, the design and analysis of PMMA viewport windows should resort to not only experimental approaches, but also analytical and numerical analysis [3]. As a long-term damage mechanism, the inherent creep issue has been generally considered for polymer viewport window. From experimental research, Du et al. [3] showed that the dwelling effect of viewport window under some deep-sea hydrostatic pressure results in lower lifetime than the fatigue issue of viewport window under variable seawater pressure. From material levels, Arnold [4], Spathis et al. [5], Zhou et al. [6] and Reza Adibeig et al. [7] performed creep analysis and lifetime prediction of PMMA material. Under different temperature, PMMA generally shows viscoelastic-viscoplastic mechanical properties, but viscoelastic features below the glassy transition temperature for seawater environments [8]. In addition, Khan et al. [9] performed experimental research on the non-monotonic creep behaviors of polymer. From the structural levels, Pranesh et al. [10,11] studied the stress levels of viewport windows with the contact area with the seat, and used the biological growth method to optimize the structural shape. Zhu et al. [12] performed finite element analysis (FEA) on the strength and stability behaviors of spherical pressure hulls with different viewport windows. Wang et al. [13,14] evaluated the creep lifetime of the titanium alloy submersible shells, and Liu et al. [15] and Tian et al. [16] evaluated the creep behaviors of PMMA viewport windows by experiments and analytical approaches. It should be emphasized that the macroscopic creep behaviors of PMMA viewport windows are involved by a series of microscopic failure mechanisms including void evolution, crazing and the rupture of molecular. However, the creep analysis above failed to consider these damage features.

In this work, we first attempt to adopt the three-element spring/dashpot viscoelastic model to study the creep behaviors of the PMMA main conical viewport window under 100–500 m and 3000 m water depth, without considering damage features above. Then, we further develop a viscoelastic creep/damage coupled model to evaluate the creep and damage behaviors of viewport window under 100–500 m and 3000 m water depth, based on the Burgers viscoelastic model with four spring/dashpot elements. The developed viscoelastic models are implemented by implicit FEA and the viscoelastic parameters are identified by combining with the experimental creep strain-time curves of the PMMA material specimens under near 30 MPa stress levels. The creep displacement of viewport window under 3000 m water depth by FEA is compared with the analytical results. Finally, the creep lifetime of the PMMA main conical viewport window under 3000 m water depth is evaluated. In this work, we do not attempt to study the creep behaviors of viewport window under 7000 m water depth with 71.6 MPa pressure since the tensile strength of PMMA material specimen is only about 60 MPa, which will be studied by performing multi-axial creep experiments of PMMA material specimens.

Section snippets

Viscoelastic creep/damage coupled model

Here, we first attempt to adopt the three-element spring/dashpot viscoelastic model in Fig. 2 to study the creep behaviors of PMMA viewport window, which is expressed as{σ=E1ε1,σ=E2ε2+η2ε˙1,ε=ε1+ε2where E1 is the instantaneous elastic modulus and E2 is the time-delay elastic modulus. ε1 is the elastic strain and ε2 is the time-delay elastic strain. η2 is the viscoelastic coefficient. ε˙1 is the viscoelastic strain rate. σ and ε are the total stress and strain.

By performing the Laplace

Numerical algorithm for the developed creep/damage coupled model

In the developed creep/damage coupled model, the damage variable f cannot be obtained independently since the damage and strain are coupled. First, the model parameters without considering damage in Eq. (12) can be rewritten asε(t)=σ0E1+σ0E2[1exp(E2η2t)]+M1+βσ0αtβ+1

One of the fundamental approach in the creep algorithm is to extend the uniaxial stress-strain relationship to the multi-axial equivalent one. The equivalent strain εc can be expressed asεc=a0σa1ta2where σ is the equivalent

Creep experiments of PMMA material specimens

The average tensile and compressive strengths of PMMA material are about 60 MPa and 100 MPa, respectively, which shows the multi-axial effect. The pressure at 100–500 m water depth is small (about 1–5 MPa) so that the creep time of PMMA specimens is very long. In addition, the pressure 71.6 MPa at 7000 m water depth exceeds the tensile strength 60 MPa of PMMA material. Therefore, the creep experiments of PMMA material specimens under near 30 MPa stress levels corresponding to 3000 m water depth

Creep analysis of PMMA viewport windows using the three-element viscoelastic model

In Section 2 Viscoelastic creep/damage coupled model, 3 Numerical algorithm for the developed creep/damage coupled model, the effects of elastic modulus on the creep stress and strain are explicitly described. However, the Poisson's ratio is also an important input parameter to construct the consistent tangent modulus in implicit FEA. Here, the Poisson's ratio 0.45 for PMMA is adopted.

Numerical analysis is performed on the PMMA viewport windows under 500 m and 3000 m water depth. By comparison,

Creep analysis of PMMA viewport windows using the developed four-element creep/damage coupled model

The critical damage variable fc=0.1 in Eq. (10) is used. Numerical analysis by FEA is performed on the PMMA conical viewport window under 100–500 m and 3000 m water depth.

Concluding remarks

This paper develops a viscoelastic creep/damage coupled model and numerical algorithm using FEA to predict the creep behaviors of PMMA deep-sea submersible viewport window under constant water pressure. By combining with the creep experiments of PMMA material specimens under near 30 MPa stress levels, the distributions of creep strain, displacement and damage variable with time for PMMA viewport windows by FEA are predicted. As extreme cases, this work provides an efficient method for

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work thanks funding support of the State Key Laboratory of Deep-Sea Manned Equipments affiliated to 702 institute of Chinese Shipbuilding Industry Corporation in Wu'xi city, China (2017.6–2020.6) and the National Natural Science Funding of China (No. 51875512).

References (17)

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