This model is able to represent the entire phenomenon observed during an impact on a slab. In this range of loading, the behavior of concrete is generally described by means of the plasticity theory where the spherical and the deviatoric responses are considered separately.
In multiaxial plasticity, we write yield functions in terms of invariants of so called "deviatoric" and "hydrostatic(spherical)" stress components.
looking at stress tensor, we can decompose it to above mentioned components as: sigma=s+pI. in which, s is deviatoric part and p=trace(sigma) and I is second order unit tensor.
the invariants for these tensors help us determine state of material at some specific stress level. three invariants are used. the first is I_1, which is first invariant of "stress" tensor. J_2 is second invariant of "deviatoric" tensor and J_3 is third invariant of deviatoric tensor.
we usually define the behavior of materials as combination of these invariants (linear or nonlinear) and this is the basis for constitutive equations for behavior of concrete materials.
they can be very easy (vonMises is one invariant sigma_y=sqrt(3J_2) for metals) or more complicated (Drucker-Prager, Willam Warnke for concrete). you can refer to attached links to find more about invariants and yield functions.
https://en.wikipedia.org/wiki/Stress_(mechanics)#Stress_deviator_tensor
https://en.wikipedia.org/wiki/Category:Yield_criteria
http://civil.colorado.edu/~willam/matl01.pdf
http://www.continuummechanics.org/cm/hydrodeviatoricstress.html
In multiaxial plasticity, we write yield functions in terms of invariants of so called "deviatoric" and "hydrostatic(spherical)" stress components.
looking at stress tensor, we can decompose it to above mentioned components as: sigma=s+pI. in which, s is deviatoric part and p=trace(sigma) and I is second order unit tensor.
the invariants for these tensors help us determine state of material at some specific stress level. three invariants are used. the first is I_1, which is first invariant of "stress" tensor. J_2 is second invariant of "deviatoric" tensor and J_3 is third invariant of deviatoric tensor.
we usually define the behavior of materials as combination of these invariants (linear or nonlinear) and this is the basis for constitutive equations for behavior of concrete materials.
they can be very easy (vonMises is one invariant sigma_y=sqrt(3J_2) for metals) or more complicated (Drucker-Prager, Willam Warnke for concrete). you can refer to attached links to find more about invariants and yield functions.
https://en.wikipedia.org/wiki/Stress_(mechanics)#Stress_deviator_tensor
https://en.wikipedia.org/wiki/Category:Yield_criteria
http://civil.colorado.edu/~willam/matl01.pdf
http://www.continuummechanics.org/cm/hydrodeviatoricstress.html
If you are talking about stresses, the hydrostatic or spherical part is the average normal stress in three perpendicular directions (i.e.: the pressure). The deviatoric part is everything else (i.e.: what is left of the stress state after subtracting the pressure).
If you are talking about strains, the hydrostatic or spherical part is the average normal strain in three perpendicular directions (i.e.: the change in volume). The deviatoric part is everything else (i.e.: the change in shape).
The spherical response will start with some elastic deformations up to cracking in tension and crushing in compression. The deviatoric response will also include an initial elastic response and eventually yielding in shear.
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