Density driven damage mechanics (D3-M) model for concrete II: fully coupled chemo-mechanical damage

Mathematics and Mechanics of Solids

2021

In this short note, we develop a constitutive relation that is linear in both the Cauchy stress and the linearized strain, by linearizing implicit constitutive relations between the stress and the deformation gradient that have been put into place to describe the response of elastic bodies (Rajagopal, KR. On implicit constitutive theories. Applications of Mathematics 2003; 28: 279–319), by assuming that the displacement gradient is small. These implicit equations include the classical linearized elastic constitutive approximation as well as some classes of constitutive relations that imply limiting strain in tension, as special subclasses. Moreover, the constitutive relations that are developed allow the material moduli to depend on the density; thus, they can be used to describe the response of porous materials, such as porous metals, bone, rocks, and concrete undergoing small deformations.

Section: A Strain-limiting Constitutive Relation In Tensionmentioning

confidence: 95%

An implicit constitutive relation for describing the small strain response of porous elastic solids whose material moduli are dependent on the density

Mathematics and Mechanics of Solids

2021

“…It is worth noting that, if the modulus E 4 → 0, then the latter equation in (18) implies div(u) = 0, and together with (17) the limit relations describe the Stokes system for the incompressible solid:…”

Section: Analysis Of the Governing Equations Using The Three Fields ε T * And P As The Variablesmentioning

confidence: 99%

“…The variational equation ( 21) is obtained in a standard way after multiplication of the equilibrium equation ( 17) with v i , summing it over i = 1, 2, 3 and integrating by parts over with the help of boundary conditions (19) and (20). The variational equations in ( 22) are derived from the constitutive equations in (18) after taking the scalar product with the test functions S * and q.…”

Section: Analysis Of the Governing Equations Using The Three Fields ε T * And P As The Variablesmentioning

confidence: 99%

“…where I is the identity transformation, ε is the linearized strain, the β i , i = 1, 2, 3 are scalar valued functions that can at most depend linearly on ε, but arbitrarily on the invariants of T, while β i , i = 0, 4, 5 depends on the invariants of T. Since by virtue of the balance of mass: ρ R = ρ(detF), which when linearized leads to ρ R = ρ(1 + trε), trε can be replaced by ρ, which makes the constitutive relation useful in describing the response of porous materials as the porosity determines the density of the material (see [23] for discussion of the development and relevance of such constitutive relations). Gainfully exploiting the fact that such constitutive relations can accommodate the material moduli can be functions of the density, [18,19] studied the problem of initiation of damage in concrete.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On an Implicit Model Linear in Both Stress and Strain to Describe the Response of Porous Solids

Itou

Kovtunenko

2021

J Elast

We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder–Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.

“…The constitutive relations developed in [1] cannot be obtained by linearizing the Cauchy elastic bodies (see Cauchy [3], [4]) and hence Green elastic bodies (Green [5], [6]). Murru et al [7], [8] used a model that is a sub-class of the models developed in [1] to study damage in concrete, and Murru and Rajagopal [9] used a constitutive relation established in [1] to study the problem of stress concentration due to the presence of a hole in a slab that is subject to uniaxial extension. This study is a continuation of the previous analysis of stress concentration; we study the stress concentration due to the presence of a hole in a slab that is subject to biaxial extension and compression.…”

Section: Introductionmentioning

confidence: 99%

Stress concentration due to the bi‐axial deformation of a plate of a porous elastic body with a hole

Murru

2021

Z Angew Math Mech

Self Cite

Most solid bodies are porous and in such bodies we expect the material properties to vary with porosity and hence with the density. Such bodies whose properties depend on the density cannot be described by the classical linearized elastic constitutive relation when they are undergoing small deformations, as the material moduli cannot depend on density. A constitutive relation which is valid in the small displacement gradient range wherein the material moduli are dependent on the density can be developed by linearizing implicit constitutive relations for describing elastic bodies introduced recently by Rajagopal [1]. In this paper, we determine the stress concentration due to a hole in a slab that is subject to biaxial loading in a body described by such a constitutive relation. We find that the stress concentration can be much as 147% higher than that for classical linearized elasticity for the range of loadings considered in the study.