I guess the trivial answer is that the stiffness tensor gives a linear relationship between two second rank tensors, the stress and the strain, and hence is a fourth rank tensor. Similarly, dielectric permittivity relates to vectors (first rank tensors) and hence is a second rank tensor, piezoelectricity relates strain (2nd) to electric field (1st) and is a 3rd rank tensor. One last example is thermal expansion, a second rank tensor, relating strain (2nd) to the scalar temperature (0th). Does this answer your question?
left hand side of the hook's law is the stress tensor (2 rank)is equal to product of stiffness tensor(4th rank) and strain tensor(2 rank).in the tensor multiplication , the stiffness tensor and the strain tensor the resulting stress tensor rank should br 4+2 that means 6 rank. how can you explain i went wrong?
The tensor operation you are referring to does not "expand" the rank of the resulting tensor, but rather "contracts" the rank. This is analogous to standard matrix operations in linear algebra, a 3x1 vector = a 3x3 matrix dotted with another 3x1 vector. This is similar to the dot product of two vectors being equal to a scalar.
where i, j = 1,2,3. We can make the expressions more compact by writing A_{ijM}, M=1 to 9 as A_{ijkl}, k, l = 1 to 3. So we have a relation of the form
sigma_{ij} = A_{ijkl} eps_{kl}
The quantity A looks like a fourth order tensor. The question then is: how do we know that A actually is a tensor?
Since a tensor is a physical quantity, the quantity itself should not change if the coordinate system is changed. However, the components of the tensor in a particular coordinate system may change. Strict transformation rules apply for changes from one coordinate system to another.
In this case, the rule for a fourth-order tensor is (for a Cartesian coordinate system with summation implied over repeated indices)
T'_ijkl = Q_{im} Q_{jn} Q_{pk} Q_{ql} T_{mnpq}
Doing the algebra shows that the tensor A is indeed a fourth-order tensor.
Mr chad landis ,thank you for your responding ,but if you refer this http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM20022…
in this page 10 it is given
"The rank of a new tensor formed by the product of two other tensors is the sum of their
individual ranks"
what could you say , i posted keeping this in mind.
The "product" that they are referring to here is some type of dyad product. Hooke's law involves an inner product. In component form a dyad product of two vectors looks like:
A_ij = B_i C_j
while the inner product looks like
A = summation over i of B_i C_i
In the first case, a second rank tensor was created by the dyad product of two first rank tensors, while in the second case a scalar (rank zero) was obtained from the inner product (dot product) of two vectors.
hook law
some one know
In reply to hook law by suman (not verified)
Hooke's law
I guess the trivial answer is that the stiffness tensor gives a linear relationship between two second rank tensors, the stress and the strain, and hence is a fourth rank tensor. Similarly, dielectric permittivity relates to vectors (first rank tensors) and hence is a second rank tensor, piezoelectricity relates strain (2nd) to electric field (1st) and is a 3rd rank tensor. One last example is thermal expansion, a second rank tensor, relating strain (2nd) to the scalar temperature (0th). Does this answer your question?
Chad
In reply to Hooke's law by Chad Landis
hook'sl aw
left hand side of the hook's law is the stress tensor (2 rank)is equal to product of stiffness tensor(4th rank) and strain tensor(2 rank).in the tensor multiplication , the stiffness tensor and the strain tensor the resulting stress tensor rank should br 4+2 that means 6 rank. how can you explain i went wrong?
In reply to hook'sl aw by suman (not verified)
Contraction, not expansion.
In reply to Contraction, not expansion. by Chad Landis
A bit more on Hooke's law
For a brief explanation of the thermodynamic basis of Hooke's law see
http://en.wikipedia.org/wiki/Hooke's_law
To expand upon Chad's explanation, a "linear" relationship between stress and strain means a 9 relations of the form:
sigma_{ij} = A_{ij1} eps_11 + A_{ij2} eps_12 + A_{ij3} eps_13 +
A_{ij4} eps_21 + A_{ij5} eps_22 + A_{ij6} eps_23 +
A_{ij7} eps_31 + A_{ij8} eps_32 + A_{ij9} eps_33
where i, j = 1,2,3. We can make the expressions more compact by writing A_{ijM}, M=1 to 9 as A_{ijkl}, k, l = 1 to 3. So we have a relation of the form
sigma_{ij} = A_{ijkl} eps_{kl}
The quantity A looks like a fourth order tensor. The question then is: how do we know that A actually is a tensor?
Since a tensor is a physical quantity, the quantity itself should not change if the coordinate system is changed. However, the components of the tensor in a particular coordinate system may change. Strict transformation rules apply for changes from one coordinate system to another.
In this case, the rule for a fourth-order tensor is (for a Cartesian coordinate system with summation implied over repeated indices)
T'_ijkl = Q_{im} Q_{jn} Q_{pk} Q_{ql} T_{mnpq}
Doing the algebra shows that the tensor A is indeed a fourth-order tensor.
-- Biswajit
hook's law
Mr chad landis ,thank you for your responding ,but if you refer this
http://www.grc.nasa.gov/WWW/K-12/Numbers/Math/documents/Tensors_TM20022…
in this page 10 it is given
"The rank of a new tensor formed by the product of two other tensors is the sum of their
individual ranks"
what could you say , i posted keeping this in mind.
In reply to hook's law by suman (not verified)
Hooke's law
The "product" that they are referring to here is some type of dyad product. Hooke's law involves an inner product. In component form a dyad product of two vectors looks like:
A_ij = B_i C_j
while the inner product looks like
A = summation over i of B_i C_i
In the first case, a second rank tensor was created by the dyad product of two first rank tensors, while in the second case a scalar (rank zero) was obtained from the inner product (dot product) of two vectors.
thank
thank sir , it is my mistake in unterstanding,sir.always masters are really masters,i am a newbie