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Multidirectional grading influence on static/dynamic deflection and stress responses of porous FG panel structure: a micromechanical approach

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Abstract

This is the first time the multidirectional-graded porous panel structure modeled numerically using an equivalent single-layer higher-order polynomial model considering the cubic variation of extensional displacement to maintain the necessary stress/strain. The effect of porosity (even and uneven distributions) and variable grading patterns also included achieving the generality. Further, the deflection and stress values, the proposed bidirectional functionally graded (2D-FG) structure, are predicted under the variable loadings, i.e. static and dynamic. Three different types of grading pattern, i.e. power-law, exponential and sigmoid are introduced by varying the material constituents along their principal material axes (longitudinal and transverse). The current numerical solutions (deflection and stress) are obtained through a customized computer code (prepared in MATLAB), under the influences of the static and time-dependent loadings utilizing the higher-order finite element formulations. The dynamic deflections are obtained through the constant acceleration type Newmark’s time-integration steps. The predicted result accuracy is checked by comparing the previously published values in literature and different simulation models (ANSYS and ABAQUS). Besides, the batch input technique is adopted for the simulation material models for both the ANSYS and ABAQUS. Moreover, the python scripting is adopted first time to modify ABAQUS input files for the present 2D graded structure. The influential structure input parameter (power-law exponents, thickness ratio, aspect ratio, end conditions, geometry and curvature ratio) is varied to compute a few final responses (deflection and stress data) of multidirectional FG structure via the derived mathematical model and the final understandings listed the details.

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Abbreviations

P(X 1, X 3):

An effective material property of FG structure

P c and P m :

Corresponding properties of ceramic and metal, respectively

X 1 and X 3 :

Random points in the length and thickness direction

n x and n z :

Power-law exponents in the length and thickness direction

λ :

Porosity index

c and m :

Ceramic and Metal constituents

\(V_{{f_{{\text{c}}} }}\) and \(V_{{f_{{\text{m}}} }}\) :

Volume fractions of ceramic and metal constituents, respectively

A and b :

Length and width of the FG panel, respectively

h :

Thickness of the FG panel

R x 1 and R x 2 :

Radius of curvature along X1 and X3 axes, respectively

\(X_{{11}} ,\,\,X_{{22}} ,\,\,X_{{33}}\) and \( X_{{11_{0} }} ,\,X_{{22_{0} }} ,X_{{33_{0} }} \) :

Global and mid-plane displacement field along X1, X2 and X3 axes, respectively

\(\psi _{x}\) and \(\psi _{y}\) :

Rotation of transverse normal about the X2, and X1 axes, respectively

\(X_{{11_{0} }}^{*} ,\,\,X_{{22_{0} }}^{*} ,\,\,\psi _{x}^{*} ,\,\,\psi _{y}^{*}\) :

Higher-order terms of Taylor’s series expansion

\(X_{3}^{2}\) and \(X_{3}^{3}\) :

Square and cubic thickness coordinates, respectively

\(\varepsilon _{l}\) :

Linear strain tensor

\(\left[ {T_{l} } \right]_{{5 \times 20}}\) :

Linear thickness coordinate matrix

\(\left\{ {\overline{{\varepsilon _{l} }} } \right\}_{{20 \times 1}}\) :

Mid-plane strain terms matrix

\(\{ \delta _{0} \}\) :

Global displacement field vector

[N]:

Nodal shape function

\( \left\{ {\delta _{{0_{i} }} } \right\} \) :

ith node mid-plane displacement field vector

\(\left\{ {\overline{{\varepsilon _{l} }} } \right\}\) :

Mid-plane strain term

[B]20 × 9 :

The product form of shape functions and the differential operators

\(\left\{ \sigma \right\}\),\(\left\{ \varepsilon \right\}\) and \(\left[ {\overline{Q} } \right]\) :

Stress, strain and reduced stiffness matrix, respectively

\(U\) :

Total strain energy

[D]:

Material property matrix

T e :

Kinetic energy of the FG structure

ρ :

Mass density

\(\left\{ {\dot{\delta }} \right\}\) :

Velocity vector

[m]:

Elemental inertia matrix

W :

Workdone

q :

Applied transverse load

[F]:

Force vector

[M]:

Mass matrix

[K]:

Global stiffness matrix

δ and п :

Variation symbol and total energy functional, respectively

\(\ddot{\delta }_{0}\) :

Acceleration vector

Δt :

Time-step

T :

Total time

α, φ and β 0 to β 7 :

Newmark's integration parameters

\(\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{k} } \right]\) :

Effective stiffness matrix

\(\left[ {\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{F} } \right]\) :

Effective load matrix

w :

Actual deflection

\(\overline{w}\) :

Non-dimensional deflection

E c and E m :

Modulus of elasticity of ceramic and metal, respectively

µ c and µ m :

Poisson’s ratio of ceramic and metal, respectively

ρ c and ρ m :

Density of ceramic and metal, respectively

\(\overline{\sigma }\) :

Non-dimensional stress

\(\sigma\) :

Actual stress

τ xy :

Actual shear stress

\(\bar{\tau }\) xy :

Non-dimensional shear stress

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Authors and Affiliations

  1. Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela, Odisha, 769008, India

    Prashik Malhari Ramteke, Jainarayan Choudhary, Priyanshu Hissaria & Subrata Kumar Panda

  2. School of Mechanical Engineering, KIIT Bhubaneswar (Deemed to be University), Bhubaneswar, Odisha, 751024, India

    Nitin Sharma

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  1. Prashik Malhari Ramteke
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Ramteke, P.M., Sharma, N., Choudhary, J. et al. Multidirectional grading influence on static/dynamic deflection and stress responses of porous FG panel structure: a micromechanical approach. Engineering with Computers 38 (Suppl 4), 3077–3097 (2022). https://doi.org/10.1007/s00366-021-01449-w

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