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Mohr’s Circle for 3D Stress

1882-01-01

Christian Otto Mohr

For a general three-dimensional state of stress, the analysis is represented by three Mohr’s circles. These circles are drawn in the [latex]\sigma_n – \tau_n[/latex] plane using the three principal stresses ([latex]\sigma_1, \sigma_2, \sigma_3[/latex]) as diameters. The largest circle, defined by [latex]\sigma_1[/latex] and [latex]\sigma_3[/latex], encloses the other two and determines the absolute maximum shear stress, [latex]\tau_{abs max} = (\sigma_1 – \sigma_3)/2[/latex].

While the 2D Mohr’s circle is common, real-world 强调 states are three-dimensional. To analyze a 3D stress state, one first determines the three principal stresses, [latex]\sigma_1 \ge \sigma_2 \ge \sigma_3[/latex], which are the eigenvalues of the 3×3 Cauchy stress tensor. These three values are then used to construct three separate Mohr’s circles. The first circle is drawn between [latex]\sigma_1[/latex] and [latex]\sigma_2[/latex], the second between [latex]\sigma_2[/latex] and [latex]\sigma_3[/latex], and the third, largest circle between [latex]\sigma_1[/latex] and [latex]\sigma_3[/latex].

The stress state ([latex]\sigma_n, \tau_n[/latex]) for any arbitrarily oriented plane at the point will lie within the shaded area bounded by these three circles. A crucial insight from this 3D representation is the determination of the absolute maximum shear stress. Unlike the 2D case where the maximum in-plane shear is the radius, the absolute maximum shear stress for a 3D state is always the radius of the largest circle, given by [latex]\tau_{abs max} = R_{max} = (\sigma_{max} – \sigma_{min})/2 = (\sigma_1 – \sigma_3)/2[/latex]. This value is fundamental for applying failure criteria like the Tresca yield criterion in a general 3D context, as it represents the true maximum shear stress experienced by the material at that point.

3D, 故障分析, 有限元法（FEM）, 岩土工程, 材料科学, 机械工业, 机械, 应力腐蚀, 结构工程

UNESCO Nomenclature: 2203

– Classical mechanics

类型

抽象系统

中断

递增

使用方法

广泛使用

前体

Cauchy’s 3D stress tensor formulation
Eigenvalue analysis for 3×3 matrices
Mohr’s original 2D circle concept
Lamé’s stress ellipsoid concept

应用

analysis of complex stress states in mechanical components
geomechanics for understanding rock 力学 under triaxial stress
design of thick-walled 压力 vessels
aerospace engineering for analyzing fuselage and wing stresses

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Related to: 3D stress, Mohr’s circle, principal stresses, absolute maximum shear stress, cauchy stress tensor, triaxial stress, geomechanics, solid mechanics, failure analysis, continuum mechanics.