Stress - force

Japanese: 応力 - おうりょく

When an external force is applied to an object, if we imagine an imaginary surface inside the object and divide the object into two, both sides of the surface will exert a force on each other. This force per unit area is called stress. When the direction of this force is perpendicular to the surface, it is called normal stress, and when it is parallel to the surface, it is called shear stress. When it is oblique, there will generally be both components. When the normal stresses are in the direction of pushing against each other, it is called pressure, and when they are in the direction of pulling against each other, it is called tension.

To define stress, first, the direction of the surface (the normal direction of the surface) must be specified, and then the magnitude and direction of the forces acting through that surface must be given. In other words, stress is given by a vector indicating the direction of the surface and a vector indicating the force, and therefore stress is expressed as a second-order tensor. If stress is expressed as T , consider a surface perpendicular to the x- axis, and express the x , y , and z components of the forces acting through this surface as T _xx , T _xy , and T _xz . Similarly, T _yx , T _yy , T _yz , T _zx , T _zy , and T _zz can be considered, but the condition for the force moments to be balanced must be T _yz = T _zy , T _yx = T _xy , and T _xz = T _zx . Therefore, a stress tensor generally has six independent components, T _xx , T _yy , T _zz , T _yz , T _zx , and T _xy , and overall stress T is

Among these, T _xx , T _yy , and T _zz are the components of normal stress (positive is tension, negative is pressure), and T _yz , T _zx , and T _xy are the components of shear stress. By choosing the xyz axes appropriately, it is possible to find the xyz axes where all the shear stress components are zero. In this case, the stress tensor is

In this case, the xyz axes are called _the principal stress axes, _and Txx , Tyy , and Tzz are called _the principal stresses. In the case of _{hydrostatic pressure in a fluid, Txx = Tyy} ₌ Tzz = -P ( P is hydrostatic pressure) _.

[Yasushi Wada and Toshio Nishi]

[Reference] | Shear stress

Stress (tension, pressure, shear stress)
The forces exerted by the lower and upper parts of an object on the upper and lower parts, respectively, are equal in magnitude to and opposite in direction .

Stress (tension, pressure, shear stress)

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

Japanese:

物体に外から力を加えたとき、物体内部に仮想的な面を考えて物体を二つに分けて考えると、その面の両側は互いに力を及ぼし合っている。単位面積当りのこの力を応力という。この力の方向が面に垂直のとき法線応力、面に平行のときずり応力という。一般に斜めのときは、両方の成分があることになる。法線応力が互いに押し合う向きのとき圧力、引き合う向きのとき張力という。

　応力を定義するには、まず、面の方向（面の法線方向）を指定し、次のその面を通して及ぼし合う力の大きさと方向を与えなくてはならない。すなわち、応力は面の方向を示すベクトルと、力を示すベクトルとで与えられ、したがって、応力は2階のテンソルで表される。応力をTで表すと、x軸に垂直な面を考え、この面を通して及ぼし合う力のx、y、z成分をT_xx、T_xy、T_xzで表す。同様にT_yx、T_yy、T_yz、T_zx、T_zy、T_zzが考えられるが、力のモーメントがつり合っている条件としてT_yz＝T_zy、T_yx＝T_xy、T_xz＝T_zxでなくてはならない。したがって応力テンソルには一般にT_xx、T_yy、T_zz、T_yz、T_zx、T_xyの六つの独立な成分があり、全体として応力Tは

と書ける。このうちT_xx、T_yy、T_zzが法線応力（正のとき張力、負のとき圧力）の成分であり、T_yz、T_zx、T_xyがずり応力の成分である。xyz軸を適当に選ぶと、ずり応力成分がすべてゼロとなるようなxyz軸を探し出すことができる。このとき応力テンソルは

となる。このときのxyz軸を応力の主軸、T_xx、T_yy、T_zzを主応力という。流体中の静水圧の場合はT_xx＝T_yy＝T_zz＝－P（Pは静水圧）である。