3: Development of Constitutive Equations of Continuum, Beams and Plates
- Page ID
- 21489
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 3.1: Prologue to Development of Constitutive Equations for Continuum, Beams and Plates
- This page discusses constitutive equations that relate stress and strain in elastic materials, focusing on Hooke's law \( \sigma = E\epsilon \) where \(E\) represents Young's modulus. It notes the values of \(E\) for common materials like steels (approximately 2 GPa) and aluminum alloys (about 0.80 GPa). Additionally, it emphasizes the complexity of stress and strain interactions, indicating that a one-dimensional stress state is linked to a three-dimensional strain state.
- 3.2: Elasticity Law in 3-D Continuum
- This page elaborates on the 3-D elasticity law, establishing a connection between stress and strain using tensors and Lame' constants, assuming isotropy. It covers the definitions of variables, relationships of engineering constants like Young's modulus and Poisson's ratio to Lame' constants, and provides equations for stress-strain relations.
- 3.3: Specification to the 2-D Continuum
- This page discusses the concepts of plane stress, plane strain, and uniaxial strain, which are critical for analyzing thin plates and shells. Plane stress involves no stress in the z-direction with relevant strain relations, while plane strain has zero strain in one direction despite present stresses. Uniaxial strain occurs with restricted displacements and is common in material testing. Each condition possesses unique tensor representations that affect how materials behave under different loads.
- 3.4: Hook’s Law in Generalized Quantities for Beams
- This page explores the relationship between generalized forces (moment, axial force, shear) and generalized kinematic quantities (strain, curvature) in beam theory, noting inconsistencies from the lack of shear work displacement. It presents the Euler-Bernoulli hypothesis and derives formulas for axial force and bending moment, demonstrating their uncoupled responses.
- 3.5: Inconsistencies in the Elementary Beam Theory
- This page examines beam behavior under bending, focusing on shear stresses and warping, pointing out deficiencies in simplified models like the Euler-Bernoulli assumption. It highlights the negligible impact of small cross-sectional changes (up to 0.1%) on overall bending behavior and discusses how internal inconsistencies lead to minor errors in engineering applications.
- 3.6: Derivation of Constitutive Equations for Plates (Advanced)
- This page covers key concepts and equations related to the elasticity law for plates, focusing on bending moments, bending energy density, and strain-stress interactions. It introduces Hook's law for plane stress, the moment-curvature relationship, and the derivation of total bending and membrane energy densities, highlighting their mathematical formulations and relevance to mechanical behavior under loads.
- 3.7: Stress Formula for Plates
- This page explains the determination of axial stress in plates by drawing parallels with beams. It describes how bending moments and axial forces influence stress, and it transforms the stress-strain curve into relevant strain and curvature tensor expressions. It defines bending and axial rigidity relationships, leading to the derivation of stress equations.