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The actress said she is bombarded graphic videos, nude photographs and sexual messages, including rape threats.
Speaking to the Sunday Times, Ms Atack said: “Some of these men are married with children. They’ve got daughters in their profile pictures and things like that.
“How can you sleep at night knowing you’ve just tucked your daughters into bed, probably helped them with their homework, and then you’re sneakily going onto your phone [and sending that sort of message]?”
She told the newspaper she thought about telling the men’s wives but resisted because “their children don’t deserve” to be upended in that way.
Ms Atack, who is 31, said she had been targeted by social media trolls for years and that it has got worse since she became single.
“I am a woman on my own and I’m in my thirties, and it’s a trigger point because you receive messages like that,” she said.
“I can’t help but wonder, if I was in a secure marriage or whatever, whether it would still happen to me.”
In one terrifying incident a man messaged to say he would rape and kill her, describing the cushions on the sofa at her home. There was later a break-in but police had to drop the case after finding no leads.
She said part of the reason officers were unable to help was because social networks would not reveal the person’s identity.
Ms Atack said she was unable to come off social media because she used it so much for work but added that she had stopped reading messages from people she didn’t know.
“We use social media now like we use streets: it should be a safer space,” she said.
Ms Atack said the technology giants should not allow people with anonymous social media accounts to message others privately. She is also calling for a ban on naked images sent by direct message.
Instagram said it had “strict rules” against harassment and had built “several tools to help protect people”, including making it more difficult for people a user has blocked to message them again. Twitter said its “top priority” was to keep everybody who uses the platform “safe and free from abuse”.