Hajime Matsuoka’s Profile

Step 1：

Deformation and failure of granular materials like sand are governed by interparticle friction (τ/σ, the ratio of shear stress τ to normal stress σ).

The plane with the largest τ/σ is the point where a line from the origin touches Mohr’s stress circle, and is therefore a plane that is more perpendicular than 45° plane.

It is well known that the slip plane (failure plane) of soil is｛45° + φ/2｝plane (φ: angle of internal friction at failure).

Step 2：

How should we consider the above discussion under three-dimensional stress conditions?

When three different principal stresses act on soil elements, i.e., the maximum principal stress σ₁, the intermediate principal stress σ₂, and the minimum principal stress σ₃(σ₁ > σ₂ > σ₃), I considered not only a two-dimensional mobilized plane between the maximum principal stress σ₁ and the minimum principal stress σ₃ (the AC plane in the figure below), but also a two-dimensional mobilized plane between the maximum principal stress σ₁and the intermediate principal stress σ₂(the AB plane in the figure below) and a two-dimensional mobilized plane between the intermediate principal stress σ₂ and the minimum principal stress σ₃ (the BC plane in the figure below).

A mobilized plane is a plane along which soil particles slide, eventually leading to the sliding plane (failure plane).

In my doctoral thesis (Deformation Characteristics of Soil, August 1973, Kyoto University), I successfully expressed the stress-strain relationship under the three principal stresses as a superposition of strains generated on these three two-dimensional mobilized planes (collectively named Compositely Mobilized Planes, or CMP).

(See Matsuoka, “A New Approach to Geotechnical Engineering,” Kyoto University Press, 2003, pp. 7-15.)

（See Matsuoka & Sun,”The SMP Concept-based 3D Constitutive Models for Geomaterials”,Taylor & Francis,2006,pp.1-9 in English)

Step 3：

Looking at the three two-dimensional mobilized planes AB, BC, and AC in the figure below (Figure 1.6(a)), a three-dimensional plane ABC(Figure 1.6(b)) began to emerge.

This is the Spatially Mobilized Plane (SMP).

By organizing the ratio of shear stress τ_SMP to normal stress σ_SMP on this SMP (τ_SMP/σ_SMP), shear strain, and normal strain, I jumped for joy when the results of a triaxial compression test (σ₁ > σ₂ = σ₃), a triaxial extension test (σ₁ = σ₂ > σ₃), and three-principal stress tests(σ₁ > σ₂ > σ₃) for the same sand (same density) all fell on a single curve.

It was when my graduate student, Teruo Nakai, completed his master’s thesis.

The SMP (Matsuoka-Nakai) failure criterion states that a granular material like sand will fail when τ_SMP/σ_SMP = a certain value.

This is a three-dimensional version of the commonly used Mohr-Coulomb criterion.

（See Matsuoka & Sun,”The SMP Concept-based 3D Constitutive Models for Geomaterials”,Taylor & Francis,2006,pp.9-14 in English)

Step 4：

I was casually looking at the two-dimensional failure criterion for metals, the Tresca criterion (1868), and the three-dimensional failure criterion for metals, the Mises criterion (1913), and I noticed something interesting.

While metallic materials, which are connected by a crystalline structure, are expressed on the coordinate axis of tensile stress, while granular materials, which consist of discrete particles, are expressed on the coordinate axis of compressive stress, there was a beautiful correspondence between the form of the failure criterion equation, the stress points on Mohr’s stress circle, and the shape of the failure plane under three-dimensional stress.

The fundamental difference between the two materials is that metallic materials fail at the maximum shear stress τ_max, while granular materials fail at the maximum shear stress/normal stress ratio (τ/σ)_max.

For details on the beautiful interrelationship between the two-dimensional failure criterion for metallic materials (Tresca’s criterion (1868)) and the three-dimensional failure criterion (Mises’s criterion (1913)), and the two-dimensional failure criterion for granular materials (Mohr’s (1882) and Coulomb’s (1773) criterion) and the three-dimensional failure criterion (Matsuoka and Nakai’s (SMP) criterion (1974)), please see the above-mentioned references: “New Approaches in Geotechnical Engineering,” pp. 22-29, and “Soil Mechanics” by Matsuoka, Morikita Publishing Co., Ltd., 1999, pp. 143-149.”

（See Matsuoka & Sun,”The SMP Concept-based 3D Constitutive Models for Geomaterials”,Taylor & Francis,2006,pp.14-20 in English)

Step 5：

I realized that if we can understand the extremes of truly frictional materials like granular materials with no bonds between particles (materials with cohesion c = 0, φ>0) and truly cohesive materials like metals with strong cohesion due to their crystalline structure (materials with internal friction angle φ = 0, c>0), all materials lie somewhere in between and are easy for humans to handle.

Therefore, we have considered a failure criterion for intermediate materials between metals and granular materials.

For details, please see the above-mentioned paper: A New Approach in Geotechnical Engineering, pp.30-35.

（See Matsuoka & Sun,”The SMP Concept-based 3D Constitutive Models for Geomaterials”,Taylor & Francis,2006,pp.20-24 in English)