Question

As the title says, I'm interested in explicitly calculating the value of the Cauchy stress tensor in a static (non-moving) cube of some material that has uniform density an is isotropic (e.g. concrete), subjected to gravity forces. I'm interested in different configurations, e.g. the cube is sitting on the ground, or is glued to a vertical surface by extremely strong glue via one of the cube faces, or is glued to a horizontal surface from below, etc.

As far as I understand, the stress tensor itself σ
is a 2nd order symmetric tensor that acts as a linear operator on the normal to an imaginary surface going through the material to give the force acting on the surface. Since it is symmetric σij=σji
, I only need to compute 6 of the tensor components. The main equation of motion for the stress tensor is ∑j∂σij∂xj+Fi=0 where Fi
are the body forces acting on the cube. In my case, the only force is gravity, so F=rhog where rho is the density.

This gives us 3 equations, which, as I understand it, must be augmented with 3 more equations from some properties of the material (to produce 6 equations in total for the 6 unknowns), and some boundary equations (which can be derived from the configuration of the cube, i.e. the "gluing" or "sitting on" external surfaces).

My questions are:

What extra equations can I use for the stress tensor, assuming my material is isotropic?

Answer 1

To compute the Cauchy stress tensor for an isotropic material like concrete under gravity, Hooke's law along with the material's Young's modulus and Poisson's ratio are needed. Boundary conditions must match the cube's specific setup, such as being glued or resting on a surface. Solving the equilibrium equations with these conditions yields the stress distribution.

Step-by-step explanation:

To calculate the Cauchy stress tensor in static configurations of a cube under gravity, additional material equations are needed. For an isotropic material, these are typically the constitutive equations that relate stress to strain. In linear, isotropic elasticity, these equations take the form of Hooke's law, expressed as σij = λεkkδij + 2μεij, where λ and μ are the Lamé parameters of the material, εij is the strain tensor, and δij is the Kronecker delta. The Lamé parameters can be related to the more commonly known Young's modulus (E) and Poisson's ratio (ν) by λ = Eν/((1+ν)(1-2ν)) and μ = E/(2(1+ν)). Moreover, the isotropic bulk modulus (K) and the shear modulus (G or μ) are essential in describing the elastic response of materials and relate to volumetric and shape change under stress.

In practical terms, the boundary conditions depending on the cube's specific configuration (resting on the ground, glued to a vertical or horizontal surface) will provide additional equations necessary for a full solution. For the cube on the ground, the bottom face would have zero normal stress, while for the cube glued to a surface, the stress on that face would need to match the stress provided by the glue. Contrarily, free surfaces will have zero traction boundary conditions.

By solving the equilibrium equations with the correct boundary conditions and material properties, the stress distribution can be calculated. Typically, under gravity alone, we would expect a linear stress distribution for the case of a cube resting on the ground, with the maximum stress at the bottom and zero at the top surface. The stress will also depend on where the cube is fixed or glued, as these points will affect how the weight is distributed through the cube.