A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model
Abstract
:1. Introduction
2. The Governing Equations of the SCLS1 Model
2.1. Notations and Model Description
- The subscript indicate layer i, and the interface between layers with , respectively. By extension, the superscripts refer to the lower face and the upper face , respectively.
- In each layer are, respectively, the bottom, the top and the mid-plane z coordinate of the layer, and is the thickness. Thus, we have for all , and we set . See Figure 1.
- Greek subscripts indicate the in-plane components.
- Latin subscripts indicate the components.
- is the transpose of .
- is the fourth-order 3D compliance tensor of layer i with the minor and major symmetries: , and it is positive definite. Its inverse is the 3D elasticity stiffness tensor and is denoted by for layer i. The tensor possesses the same symmetries as , and it is also positive definite.
- is monoclinic in direction z:
- are the in-plane stress components, are the transverse shear stresses and is the normal stress.
- are the in-plane strain components, are the transverse strains and is the normal strain.
- are the in-plane 3D displacement components, and is the normal 3D displacement component.
2.2. The 3D Model Equations
2.3. The Static of the SCLS1 Model
- is in-plane stress resultants tensor, related to the local stress in each layer i by:
- is the moment resultants tensor expressed in terms of the stress field in each layer i as follows:
- is the out-of-plane shear stress resultant vector, defined from the stress field in each layer i as follows:
- is the interlaminar shear stress at the interface between layer j, and layer for given by:
- is the normal stress at the interface between j and , for , given by:
- is the divergence of the interlaminar shear stress vector defined on the interface between layer j and for .
2.4. The Equilibrium Equations
2.5. Generalized Displacements
2.6. Generalized Strains
2.7. The Constitutive Equations of the SCLS1 Model
- Membrane constitutive equation of layer i:
- Bending constitutive equations of layer i:
- Transverse shear constitutive equation of layer i:
- Shear constitutive equation of interface :
- Normal constitutive equation of interface :
- Constitutive equation for the generalized stress at interface :
2.8. Finite-Element Displacement-Based Implementation
3. Hybridized Mixed Methods for 3D Continua
3.1. Continuous Variational Formulation
3.2. Finite-Element Discretization
4. Hybridization of a Mixed Method for the SCLS1 Model
4.1. Continuous Formulation
4.2. Finite-Element Implementation
- discontinuous Lagrange interpolation of degree p for ;
- discontinuous Lagrange interpolation of degree for U;
- discontinuous Lagrange interpolation of degree p on edges for V;
5. Illustrative Applications
5.1. Homogeneous Laminate
5.2. Laminate with a Circular Hole
5.3. Bending of a Laminate with Multi-Cracking
6. Conclusions and Perspectives
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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Discretization | U | V | Total | Condensed | |
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Mixed () | |||||
Mixed () | |||||
Displacement | – | – | – |
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Salha, L.; Bleyer, J.; Sab, K.; Bodgi, J. A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model. Mathematics 2022, 10, 1711. https://doi.org/10.3390/math10101711
Salha L, Bleyer J, Sab K, Bodgi J. A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model. Mathematics. 2022; 10(10):1711. https://doi.org/10.3390/math10101711
Chicago/Turabian StyleSalha, Lucille, Jeremy Bleyer, Karam Sab, and Joanna Bodgi. 2022. "A Hybridized Mixed Approach for Efficient Stress Prediction in a Layerwise Plate Model" Mathematics 10, no. 10: 1711. https://doi.org/10.3390/math10101711