# Tensor form of linear Hooke's law with E and v

• Engineering
• miraboreasu
In summary: Start with $$\sigma_{xx}=\frac{E}{(1+\nu)(1-2\nu)}[(1-\nu)\epsilon_{xx}+\nu\epsilon_{yy}+\nu\epsilon_{zz}]$$Rewrite this as :$$\sigma_{xx}=\frac{E}{(1+\nu)(1-2\nu)}[(1-2\nu)\epsilon_{xx}+\nu(\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})]$$$$=\frac{E}{(1+\nu)}\epsil miraboreasu Homework Statement Rewrite the linear Hooke's law with E and v Relevant Equations Linear Hooke's law Actually, this is not homework, but I think I need help like homework. It was raised from the notice that there is no tensor form of linear Hooke's law in terms of Young's modulus E, and Poission's ratio, v. For example, if we use lame parameters, we have G, \lambda, like The linear Hooke's law (vector-matrix form) is (https://physics.stackexchange.com/q...-materials-makes-stress-undefined-in-hookes-l) I tried to just use the relationship like: E= v = but, it ends up with an equation with 2 roots (the first eq for get G= f (E)), so I think I need help about write the notation form directly from the vector-matrix form of the linear Hooke's law Last edited: Start with$$\sigma_{xx}=\frac{E}{(1+\nu)(1-2\nu)}[(1-\nu)\epsilon_{xx}+\nu\epsilon_{yy}+\nu\epsilon_{zz}]$$Rewrite this as :$$\sigma_{xx}=\frac{E}{(1+\nu)(1-2\nu)}[(1-2\nu)\epsilon_{xx}+\nu(\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})]=\frac{E}{(1+\nu)}\epsilon _{xx}+\frac{E\nu}{(1+\nu)(1-2\nu)}(\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})=2G\epsilon_{xx}+\lambda (\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})$$Chestermiller said: Start with$$\sigma_{xx}=\frac{E}{(1+\nu)(1-2\nu)}[(1-\nu)\epsilon_{xx}+\nu\epsilon_{yy}+\nu\epsilon_{zz}]$$Rewrite this as :$$\sigma_{xx}=\frac{E}{(1+\nu)(1-2\nu)}[(1-2\nu)\epsilon_{xx}+\nu(\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})]=\frac{E}{(1+\nu)}\epsilon _{xx}+\frac{E\nu}{(1+\nu)(1-2\nu)}(\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})=2G\epsilon_{xx}+\lambda (\epsilon_{xx}+\epsilon_{yy}+\epsilon_{zz})$$Thank you, but sorry I didn't get it, how can I rewrite the vector-matrix form into the form like 2.9. I mean use tensor product, I, to represent the following miraboreasu said: Thank you, but sorry I didn't get it, how can I rewrite the vector-matrix form into the form like 2.9. I mean use tensor product, I, to represent the following View attachment 329468 Look at my equation again. It’s too easy. You have:$$G=\frac{E}{2(1+\nu)}$$and$$\lambda=\frac{E\nu}{(1+\nu)(1-2\nu)}

## What is the tensor form of Hooke's law?

The tensor form of Hooke's law expresses the relationship between stress and strain in a material using tensors. It is given by the equation: σ_ij = C_ijkl * ε_kl, where σ_ij is the stress tensor, ε_kl is the strain tensor, and C_ijkl is the fourth-rank stiffness tensor that characterizes the material's elastic properties.

## How are Young's modulus (E) and Poisson's ratio (ν) used in the tensor form of Hooke's law?

Young's modulus (E) and Poisson's ratio (ν) are used to derive the components of the stiffness tensor (C_ijkl) for isotropic materials. The stiffness tensor can be written in terms of E and ν as: C_ijkl = λ * δ_ij * δ_kl + μ * (δ_ik * δ_jl + δ_il * δ_jk), where λ and μ are the Lamé constants, which are functions of E and ν.

## What are the Lamé constants and how are they related to E and ν?

The Lamé constants, λ and μ, are material-specific constants used in the tensor form of Hooke's law. They are related to Young's modulus (E) and Poisson's ratio (ν) by the following equations: λ = (E * ν) / ((1 + ν) * (1 - 2ν)) and μ = E / (2 * (1 + ν)).

## How does the tensor form of Hooke's law simplify for isotropic materials?

For isotropic materials, the stiffness tensor C_ijkl has only two independent components due to the material's uniform properties in all directions. The tensor form of Hooke's law simplifies to: σ_ij = λ * δ_ij * ε_kk + 2 * μ * ε_ij, where δ_ij is the Kronecker delta, ε_kk is the trace of the strain tensor, and λ and μ are the Lamé constants.

## What is the significance of the Kronecker delta in the tensor form of Hooke's law?

The Kronecker delta, δ_ij, is a mathematical symbol used in the tensor form of Hooke's law to simplify expressions involving the identity matrix. It is defined as δ_ij = 1 if i = j and δ_ij = 0 if i ≠ j. In the context of Hooke's law, it helps in expressing the relations between different components of stress and strain tensors in a compact form.

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