Lecture 6: Elastic Deformation of laminates

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Presentation transcript:

Lecture 6: Elastic Deformation of laminates At the end of this lecture you will have an understanding of: Basic knowledge of tensors and their link to the mechanical properties of materials Concepts behind the elastic deformation of a lamina Concepts behind the elastic deformation of laminates 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Scalar: physical quantity not related to any directions (temperature for example) Vector: physical quantity related to 1 or more directions and is expressed in relation to a set of axis Tensor of second rank: physical quantity which relates 2 vectors via and array of 9 components, each of which is associated with a pair of axis of a given coordinate system 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Consider vectors: 𝑃 = 𝑃 1 , 𝑃 2 , 𝑃 3 𝑄 = 𝑄 1 , 𝑄 2 , 𝑄 3 We can define a tensor T which will relate 𝑃 and 𝑄 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Or in matrix form With Tij a second rank tensor 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of vector components X3 X’3 P P’ X’1 X1 X2 X’2 𝑃 = 𝑃 1 , 𝑃 2 , 𝑃 3 𝑃′ = 𝑃′ 1 , 𝑃′ 2 , 𝑃′ 3 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of components of the second rank tensor We want to find the relation between the new and old components of tensor Tij, after changing the coordinate set of axis X3 X’3 aij X’1 X1 X2 X’2 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of components of the second rank tensor 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Transformation of components of the second rank tensor 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Stress tensor 𝜎 𝑖𝑗 𝜎 11 𝜎 12 𝜎 13 𝜎 21 𝜎 22 𝜎 23 𝜎 31 𝜎 32 𝜎 33 Principal/shear 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Strain tensor 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Elasticity a 4th rank tensor 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Hooke’s law 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Compliance/stiffness tensors 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Compliance/stiffness tensors: effect of symmetry 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Compliance/stiffness tensors: effect of symmetry With Cij=Cji 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Form of the compliance/stiffness tensors 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Form of the compliance/stiffness tensors 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Isotropic material 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Isotropic material 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Isotropic material 19/09/2018

Lecture 6: Elastic Deformation of laminates Basics of tensors and mechanical properties materials Isotropic material 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Isotropic material 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Single lamina loaded off axis 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Engineering constant of a lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Engineering constant of a lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Engineering constant of a lamina 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Examples 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Examples 19/09/2018

Lecture 6: Elastic Deformation of laminates Elastic deformation of a lamina Examples 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Loading of a stack of plies 19/09/2018

Lecture 6: Elastic Deformation of laminates Balanced laminates 19/09/2018

Lecture 6: Elastic Deformation of laminates Balanced laminates 19/09/2018

Lecture 6: Elastic Deformation of laminates Balanced laminates 19/09/2018

Lecture 6: Elastic Deformation of laminates Balanced laminates and internal stresses 19/09/2018

Lecture 6: Elastic Deformation of laminates Balanced laminates 19/09/2018