Problem 30 - Concepts and Applications of Finite Element Analysis

Question

\(17.30\) Starting with the basic definition of the rate of internal work for an element \(e\) as $$ \dot{W}_{e}^{\mathrm{in}}=\int_{v_{f}}\\{\dot{\epsilon}\\}^{T}\\{\boldsymbol{\sigma}\\} d V $$ show that the rate of internal work for the entire structure is \(\dot{W}^{\text {int }}=\\{\mathbf{D}\\}^{T}\left\\{\mathbb{R}^{\text {int }}\right\\}\). behavior, \(W_{n}^{\text {int }}=\frac{1}{2}\\{\mathbf{D}\\}_{n}^{T}[\mathbf{K}]\\{\mathbf{D}\\}_{n}\). Note: \(\frac{d}{d t}\left(\\{\mathbf{D}\\}^{T}[\mathbf{K}]\\{\mathbf{D}\\}\right)=2(\mathbf{D}\\}^{T}[\mathrm{~K}]\\{\mathbf{D}\\}\) if \([\mathbf{K}]\) is symmetric.

Step-by-Step Solution

Answer

The exercise involves the manipulation of the integral form of the rate of internal work to yield the expression desired. The final solution is \(\dot{W}_{n}^{\text {int }}=\frac{1}{2}\{\mathbf{D}\}_{n}^{T}[\mathbf{K}]\{\mathbf{D}\}_{n}\).

Step 1: Recall the provided definitions

The provided definitions are the basic formula for the rate of internal work for an element \(e\) given as: \(\dot{W}_{e}^{\mathrm{in}}=\int_{v_{f}}\{\dot{\epsilon}\}^{T}\{\boldsymbol{\sigma}\}d V\), and the equation for the rate of internal work for the entire structure is given as: \(\dot{W}^{\mathrm{int}}=\{\mathbf{D}\}^{T}\{\mathbb{R}^{\mathrm{int}}\}\). The goal is to simplify the first equation, in order to show that it equals the second.

Step 2: Integral simplification

The integral of a sum is equal to the sum of the integrals. Apply this theorem, altering the integral representation of \(\dot{W}_{e}^{\mathrm{in}}\) to that of the overall structure \(\dot{W}^{\mathrm{int}}\) such that \(\dot{W}^{\text {int }}=\{\mathbf{D}\}^{T}\{\mathbb{R}^{\text {int }}\}\), where \(\mathbb{R}^{\text {int }}\) equals the integral of the product of rate of strain and stress over the volume, \(\int_{v_{f}}\{\dot{\epsilon}\}^{T}\{\boldsymbol{\sigma}\}d V\).

Step 3: Make use of the given note

Using the note provided: \(\frac{d}{d t}\left(\{\mathbf{D}\}^{T}[\mathbf{K}]\{\mathbf{D}\}\right)=2(\mathbf{D}\)^{T}[\mathbf{K}]\{\mathbf{D}\}\), exploit this relationship to show: \(\dot{W}_{n}^{\text {int }}=\frac{1}{2}\{\mathbf{D}\}_{n}^{T}[\mathbf{K}]\{\mathbf{D}\}_{n}\), which leads to the final expression for the rate of internal work for the whole structure if \([\mathbf{K}]\) is symmetric.

Key Concepts

Internal Work Rate

The internal work rate is a fundamental concept in finite element analysis that represents the energy rate being absorbed or released within the structure due to internal forces. It's derived from the basic principles of mechanics where work can be stated as the integration of stress (\boldsymbol{\sigma}) times the strain rate (\boldsymbol{\dot{\epsilon}}) over the volume of the material (\boldsymbol{v_f}).

For an element of the structure, the formula given is \( \dot{W}_{e}^{\mathrm{in}}=\int_{v_{f}}\{\dot{\epsilon}\}^{T}\{\boldsymbol{\sigma}\}d V \) which signifies the rate at which strain energy is being stored or dissipated in a specific element. Integrating this quantity over the entire volume of the structure allows us to assess the structure’s overall capacity to do work internally, due to deformation at any given time.

To better understand this, consider that stress is a response to external loads, whereas strain rate measures how fast the deformation is happening. Their product, integrated across the structure, provides a dynamic view of the internal energy flow within the material, which is vital for analyzing the behavior under loading conditions.

Strain and Stress Relationship

The strain and stress relationship is a cornerstone of material mechanics, and it's essential in understanding how materials respond under load. Strain (\boldsymbol{\epsilon}) describes the deformation of material, while stress (\boldsymbol{\sigma}) represents the internal forces that resist this deformation.

Mathematically, this relationship for an isotropic, linear, and elastic material can be described by Hooke's Law which establishes a direct proportionality between stress and strain. Taking the rate of strain (\boldsymbol{\dot{\epsilon}}) into account, which is the change of strain over time, leads us to a dynamic scenario relevant for finite element analysis where materials deform under dynamic loading conditions.

In the exercise, the integral \( \int_{v_{f}}\{\dot{\epsilon}\}^{T}\{\boldsymbol{\sigma}\}d V \) effectively couples the rate of deformation with the stress to compute the internal work rate. Showing this interplay in a quantifiable manner allows engineers and researchers to predict failure, analyze stiffness, and ensure safety and reliability of the materials and structures under real-time loads and deformations.

Symmetric Stiffness Matrix

The stiffness matrix (\boldsymbol{[K]}) is a critical component in finite element analysis, embodying the structural characteristics that define how it responds to various loads. This matrix formulates the relationship between nodal displacements (\boldsymbol{\mathbf{D}}) and nodal forces and is derived from the elemental stiffness matrices which, in turn, are based on the materials' stress-strain relationship.

A crucial property of the stiffness matrix is its symmetry. This symmetry arises because of the physical principle that work done by a force during displacement is independent of the path taken – known as the work-energy principle. When the stiffness matrix is symmetric (\boldsymbol{[K]} = \boldsymbol{[K]}^T), it implies that the energy transferred is conserved, and consequently, the work done by the displacement due to a set of forces is equal to the work done by another set of displacements due to the same forces. This property simplifies the computational analysis by ensuring the matrix can be inverted and solved efficiently.

In the context of the exercise, the symmetric nature of the stiffness matrix allows the use of the given note \( \frac{d}{d t}(\{\mathbf{D}\}^{T}[\mathbf{K}]\{\mathbf{D}\})=2(\mathbf{D}\}^{T}[\mathrm{~K}]\{\mathbf{D}\}) \) to demonstrate the relationship between work rate and deformation behavior. This efficient simplification is only valid when the stiffness matrix is symmetric, highlighting its importance for both theoretical and practical applications in finite element analysis.