# Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Model

**H**is the deformation gradient, i.e., the derivative of the current map ${\mathbf{R}}_{A}$ with respect to the reference map ${\mathbf{R}}_{B}$,

#### Field Equations and Relevant PDE Problems

## 3. Analytical Solution in the Case of Circular Cross-Sections

## 4. Analytical Solution in Terms of Cross-Sectional Strain Flow

#### Application Example

## 5. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic of a non-prismatic slender elastic body, occupying a region of volume V in its reference undeformed state (left), and a sketch of its current deformed state (right); the orange dot identifies a point in the reference undeformed cross-section (left); the deformed (warped) configuration of this latter is represented in red (right).

**Figure 2.**Schematic of a generic cross-section with indication of the two-dimensional domain ${\mathsf{\Sigma}}_{q}$ and its boundary lines, $\partial {\mathsf{\Sigma}}_{i}$ and $\partial {\mathsf{\Sigma}}_{e}$ (left), and case of the rectangular cross-section (right).

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**MDPI and ACS Style**

Migliaccio, G.
Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua. *Mathematics* **2023**, *11*, 4723.
https://doi.org/10.3390/math11234723

**AMA Style**

Migliaccio G.
Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua. *Mathematics*. 2023; 11(23):4723.
https://doi.org/10.3390/math11234723

**Chicago/Turabian Style**

Migliaccio, Giovanni.
2023. "Analytical Solutions of Partial Differential Equations Modeling the Mechanical Behavior of Non-Prismatic Slender Continua" *Mathematics* 11, no. 23: 4723.
https://doi.org/10.3390/math11234723