By Vitor Dias da Silva

Provides a transparent and thorough presentation of the elemental ideas of mechanics and energy of fabrics. offers either the speculation and functions of mechanics of fabrics on an intermediate theoretical point. worthy as a reference software by means of postgraduates and researchers within the fields of strong mechanics in addition to training engineers.

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**Example text**

Since the only body force is the gravity force, we have X=Z=0 and Y = −ρ . Substituting these values in Expressions 5, we immediately see that they are satisﬁed. 3. In a body under a plane stress state the body forces are zero and the stresses have been approximated by the expressions (η, ρ, H and λ are constants) x σx = σy = ηρ H − y − τxy = 0 . λ (a) Verify that these functions cannot represent the stress distribution in the body. (b) Determine the conditions which the body forces have to obey so that these expressions can represent a possible stress distribution.

2⎭ ⎩ 2 ⎩ 2 2 n σ + τ2 σ12 σ22 σ32 l σ + m2 σ 2 + n2 σ 2 = σ 2 + τ 2 1 2 3 (40) 4 Only the upper half is considered, since it is not possible to make a general distinction between positive and negative shearing stresses in an inclined facet in a three-dimensional space. 34 II The Stress Tensor τ σ1 +σ3 2 A τmax facets parallel to σ1 facets parallel to σ2 facets parallel to σ3 σ1 −σ3 2 σ2 −σ3 2 σ3 σ1 −σ2 2 σ2 σ2 +σ3 2 σ1 σ σ1 +σ2 2 Fig. 14. Mohr’s representation of the stress state in the three-dimensional case The solution of this system may be obtained by means of determinants, yielding 1 1 1 τ 2 + (σ − σ2 ) (σ − σ3 ) 1 σ σ2 σ3 2 = l = D σ 2 + τ 2 σ22 σ32 (σ1 − σ2 ) (σ1 − σ3 ) 1 1 σ1 m = D σ12 1 σ σ2 + τ 2 1 1 σ 1 n2 = D σ12 1 σ2 σ22 2 1 τ 2 + (σ − σ1 ) (σ − σ3 ) σ3 2 = σ3 (σ2 − σ1 ) (σ2 − σ3 ) 1 τ 2 + (σ − σ1 ) (σ − σ2 ) σ , 2 2 = σ +τ (σ3 − σ1 ) (σ3 − σ2 ) where D = (σ1 − σ2 ) (σ2 − σ3 ) (σ3 − σ1 ) is the system’s determinant (Expr.

Lam´e’s Ellipsoid or stress ellipsoid This ellipsoid is a complete representation of the magnitudes of the stress vectors in facets around point O. It allows an important conclusion about the stress state: the magnitude of the stress in any facet takes a value between the maximum principal stress σ1 and the minimum principal stress σ3 . It must be mentioned here that this conclusion is only valid for the absolute value of the stress, since in expression 26 only the squares of the stresses are considered.