Modeling of plates with multiple anisotropic layers and residual stress

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    Abstract

    Usually the analytical approach for modeling of plates uses the single layer plate equation to obtain the deflection and does not take anisotropy and residual stress into account. Based on the stress–strain relation of each layer and balancing stress resultants and bending moments, a general multilayered anisotropic plate equation is developed for plates with an arbitrary number of layers. The exact deflection profile is calculated for a circular clamped plate of anisotropic materials with residual bi-axial stress.From the deflection shape the critical stress for buckling is calculated and by using the Rayleigh–Ritzmethod the natural frequency is estimated. Using the Galerkin method, an approximate deflection shape is calculated for a rectangular plate, and for a square plate the expression can be simplified drastically. To support the results, the model has been compared to a FEM model, and an excellent agreement between the two models is seen with a relative difference of less than 2% for all calculations. The model was also used to extract the cell capacitance, the parasitic capacitance and the residual stress of a pressure sensor composed of a multilayered plate of silicon and silicon oxide. The extracted values were in good agreement with the expected and it showed that the behavior of devices with a plate could easily be predicted with a low uncertainty.
    Original languageEnglish
    JournalSensors and Actuators A: Physical
    Volume240
    Pages (from-to)70-79
    Number of pages10
    ISSN0924-4247
    DOIs
    Publication statusPublished - 2016

    Keywords

    • Anisotropic plate theory
    • Micromechanics
    • Stress
    • Multilayers

    Cite this

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    title = "Modeling of plates with multiple anisotropic layers and residual stress",
    abstract = "Usually the analytical approach for modeling of plates uses the single layer plate equation to obtain the deflection and does not take anisotropy and residual stress into account. Based on the stress–strain relation of each layer and balancing stress resultants and bending moments, a general multilayered anisotropic plate equation is developed for plates with an arbitrary number of layers. The exact deflection profile is calculated for a circular clamped plate of anisotropic materials with residual bi-axial stress.From the deflection shape the critical stress for buckling is calculated and by using the Rayleigh–Ritzmethod the natural frequency is estimated. Using the Galerkin method, an approximate deflection shape is calculated for a rectangular plate, and for a square plate the expression can be simplified drastically. To support the results, the model has been compared to a FEM model, and an excellent agreement between the two models is seen with a relative difference of less than 2{\%} for all calculations. The model was also used to extract the cell capacitance, the parasitic capacitance and the residual stress of a pressure sensor composed of a multilayered plate of silicon and silicon oxide. The extracted values were in good agreement with the expected and it showed that the behavior of devices with a plate could easily be predicted with a low uncertainty.",
    keywords = "Anisotropic plate theory, Micromechanics, Stress, Multilayers",
    author = "Mathias Engholm and Thomas Pedersen and Thomsen, {Erik Vilain}",
    year = "2016",
    doi = "10.1016/j.sna.2016.01.054",
    language = "English",
    volume = "240",
    pages = "70--79",
    journal = "Sensors and Actuators A: Physical",
    issn = "0924-4247",
    publisher = "Elsevier",

    }

    Modeling of plates with multiple anisotropic layers and residual stress. / Engholm, Mathias; Pedersen, Thomas; Thomsen, Erik Vilain.

    In: Sensors and Actuators A: Physical, Vol. 240, 2016, p. 70-79.

    Research output: Contribution to journalJournal articleResearchpeer-review

    TY - JOUR

    T1 - Modeling of plates with multiple anisotropic layers and residual stress

    AU - Engholm, Mathias

    AU - Pedersen, Thomas

    AU - Thomsen, Erik Vilain

    PY - 2016

    Y1 - 2016

    N2 - Usually the analytical approach for modeling of plates uses the single layer plate equation to obtain the deflection and does not take anisotropy and residual stress into account. Based on the stress–strain relation of each layer and balancing stress resultants and bending moments, a general multilayered anisotropic plate equation is developed for plates with an arbitrary number of layers. The exact deflection profile is calculated for a circular clamped plate of anisotropic materials with residual bi-axial stress.From the deflection shape the critical stress for buckling is calculated and by using the Rayleigh–Ritzmethod the natural frequency is estimated. Using the Galerkin method, an approximate deflection shape is calculated for a rectangular plate, and for a square plate the expression can be simplified drastically. To support the results, the model has been compared to a FEM model, and an excellent agreement between the two models is seen with a relative difference of less than 2% for all calculations. The model was also used to extract the cell capacitance, the parasitic capacitance and the residual stress of a pressure sensor composed of a multilayered plate of silicon and silicon oxide. The extracted values were in good agreement with the expected and it showed that the behavior of devices with a plate could easily be predicted with a low uncertainty.

    AB - Usually the analytical approach for modeling of plates uses the single layer plate equation to obtain the deflection and does not take anisotropy and residual stress into account. Based on the stress–strain relation of each layer and balancing stress resultants and bending moments, a general multilayered anisotropic plate equation is developed for plates with an arbitrary number of layers. The exact deflection profile is calculated for a circular clamped plate of anisotropic materials with residual bi-axial stress.From the deflection shape the critical stress for buckling is calculated and by using the Rayleigh–Ritzmethod the natural frequency is estimated. Using the Galerkin method, an approximate deflection shape is calculated for a rectangular plate, and for a square plate the expression can be simplified drastically. To support the results, the model has been compared to a FEM model, and an excellent agreement between the two models is seen with a relative difference of less than 2% for all calculations. The model was also used to extract the cell capacitance, the parasitic capacitance and the residual stress of a pressure sensor composed of a multilayered plate of silicon and silicon oxide. The extracted values were in good agreement with the expected and it showed that the behavior of devices with a plate could easily be predicted with a low uncertainty.

    KW - Anisotropic plate theory

    KW - Micromechanics

    KW - Stress

    KW - Multilayers

    U2 - 10.1016/j.sna.2016.01.054

    DO - 10.1016/j.sna.2016.01.054

    M3 - Journal article

    VL - 240

    SP - 70

    EP - 79

    JO - Sensors and Actuators A: Physical

    JF - Sensors and Actuators A: Physical

    SN - 0924-4247

    ER -